SOFTbank E-Book Center Tehran, Phone: , For Educational Use.

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1 SOFTbank E-Book Center Tehran, Phone: , For Educational Use.

2 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Predicting Outdoor Sound

3 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Also available from Taylor & Francis Urban Sound Environment J Kang Hb: Acoustics H Kuttruff Hb: Pb: Engineering Noise Control 3rd edition D Bies and C Hansen Hb: Pb: Fundamentals of Noise & Vibration F Fahy and J Walker Hb: Pb: Advanced Applications in Acoustics, Noise and Vibration F Fahy and J Walker Hb: Information and ordering details For price, availability and ordering visit our website Alternatively our books are available from all good bookshops.

4 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Predicting Outdoor Sound Keith Attenborough, Kai Ming Li and Kirill Horoshenkov LONDON AND NEW YORK

5 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. First published 2007 by Taylor & Francis 2 Park Square, Milton Park, Abingdon, Oxon OX 14 4RN Simultaneously published in the USA and Canada by Taylor & Francis 270 Madison Ave, New York, NY Taylor & Francis is an imprint of the Taylor & Francis Group, an informa business This edition published in the Taylor & Francis e-library, To purchase your own copy of this or any of Taylor & Francis or Routledge s collection of thousands of ebooks please go to Keith Attenborough, Kai Ming Li and Kirill Horoshenkov All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any efforts or omissions that may be made. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data Attenborough, K. (Keith). Predicting outdoor sound/keith Attenborough, Kai Ming Li and Kirill Horoshenkov. p. cm. Includes bibliographical references and index. I. Outdoor sounds Measurement. 2. Sound Recording and reproducing. I. Li, Kai Ming. II. Horoshenkov, Kirill. III. Title. QC246.A dc ISBN Master e-book ISBN ISBN 10: (Print Edition) ISBN 13: (hbk) ISBN 10: (Print Edition) ISBN 13: (ebk)

6 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Contents Preface List of symbols ix x 1 Introduction Early observations A brief survey of outdoor sound attenuation mechanisms Data including combined effects of ground and meteorology Classification of meteorological conditions for outdoor sound prediction Typical sound speed profiles Air absorption 21 2 The propagation of sound near ground surfaces in a homogeneous medium Introduction Mathematical formulation for a point source near to the ground The sound field above a locally reacting ground The sound field above a layered extended-reaction ground The propagation of surface waves above a porous ground Experimental data and numerical predictions The sound field due to a line source near the ground 60 3 Predicting the acoustical properties of outdoor ground surfaces Introduction Models for ground impedance Effects of surface roughness Effects of ground elasticity 97

7 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. 4 Measurements of the acoustical properties of ground surfaces and 110 comparisons with models 4.1 Impedance measurement methods Comparisons of impedance data with model predictions Measured and predicted roughness effects Measured and predicted effects of ground elasticity Comparisons between template fits and direct impedance fits for ground impedance 4.6 Measured flow resistivities and porosities Effective flow resistivities and other fitted parameters Predicting effects of source characteristics on outdoor sound Introduction The sound field due to a dipole source The sound field due to an arbitrarily orientated quadrupole Source directivity and railway noise prediction Predictions, approximations and empirical results for ground effect 177 excluding meteorological effects 6.1 Approximations for frequency and range dependency Approximations and data for A-weighted levels over continuous ground Predictions of the variation of A-weighted noise over discontinuous surfaces Influence of source motion on ground effect and diffraction Introduction A monopole source moving at constant speed and height above a ground surface 7.3 The sound field of a source moving with arbitrary velocity Comparison with heuristic calculations Diffraction of sound due to a point source moving at constant speed and height parallel to a rigid wedge 7.6 Source moving parallel to a ground discontinuity Source moving along a rigid barrier above the ground

8 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. 8 Predicting effects of mixed impedance ground Introduction Single discontinuity Propagation over impedance strips Effects of refraction above mixed impedance ground Predicted effects of porous sleepers and slab-track on railway noise Predicting the performance of outdoor noise barriers Introduction Analytical solutions for the diffraction of sound by a barrier Empirical formulations for studying the shielding effect of barriers The sound attenuation by a thin barrier on an impedance ground Noise reduction by a finite length barrier Adverse effect of gaps in barriers The acoustic performance of an absorptive screen Other factors in barrier performance Predicted effects of spectral variations in train noise during pass-by Predicting effects of vegetation, trees and turbulence Effects of vegetation and crops on excess attenuation spectra Propagation through trees and tall vegetation Meteorological effects on sound transmission through trees Combined effects of vegetation, barriers and meteorology Turbulence and its effects Analytical approximations including ground effect, refraction and 342 turbulence 11.1 Ray tracing Linear sound speed gradients and weak refraction Approximations for A-weighted levels and ground effect optimization in the presence of weak refraction and turbulence 11.4 A semi-empirical model for A-weighted sound levels at long range

9 SOFTbank E-Book Center Tehran, Phone: , For Educational Use Prediction schemes Introduction ISO CONCAWE Calculation of road traffic noise (CRTN) Calculation of railway noise (CRN) NORD HARMONOISE Performance of railway noise prediction schemes in high-rise cities Predicting sound in an urban environment Introduction Improved corrections for the reflection of road traffic noise from a building façade 13.3 Improved correction for the multiple reflections between parallel building façades Traffic noise attenuation along a city street Noise in tunnels Prediction of the acoustic effect of a single building façade with balconies Sound propagation between two parallel high-rise building façades Modelling of 3-D sound propagation between two high-rise building façades Sound propagation in city streets 453 Index 466

10 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Preface The subject of outdoor sound propagation is of wide-ranging interest not only for noise prediction but also in studies of animal bio-acoustics and in military contexts. The purpose of this book is to provide a comprehensive reference about aspects of outdoor sound and its prediction that should be useful to practitioners, and yet is respectable from the academic point of view. It is based on a joint experience of more than 50 years of research and consultancy. Many current prediction schemes for outdoor sound are empirical. To some extent this is understandable in view of the complicated source characteristics and complex propagation paths that are often of interest. Yet there has been significant progress in theories and computational methods for the various phenomena that are involved. These theories have been validated extensively by comparisons with data and help with our understanding of the important effects. No current text is devoted to bringing the leading theories and data together. Neither is the practitioner provided with the basis for deciding which model or scheme to use in a given situation. This text is a step towards a remedy for both of these deficiencies. The book covers recent advances in theory, new and old empirical schemes, available data and comparisons between theory and data. Where possible, examples of results of the application of prediction schemes have been included. Enough of the background theoretical detail is available to make the reader/user aware of the inherent approximations, restrictions and/or difficulties of any of the prediction methods being discussed. The book has had a long gestation since it was started in In 2001, Computational Atmospheric Acoustics by Erik Salomons was published. There have been three consequences of this publication for our endeavours in Predicting Outdoor Sound. The first is that we have not attempted to duplicate the fine, thorough and approachable treatments of the computational methods to be found in Salomons book. Second, given that Salomons book does not include any data, we have emphasized data where it is available. Finally, we have concentrated on those aspects of our research that complement Salomons work. Probably, students, researchers and consultants interested in outdoor sound prediction would do best if they are in possession of both texts. Keith Attenborough Kai Ming Li Kirill Horoshenkov November 2005

11 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Symbols A g A-weighted ground attenuation (ISO9613 2) c(z) Sound speed as a function of height c 0 Ambient (adiabatic) sound speed C(ω) Complex compressibility C Elastic constant for Biot model d Layer thickness, distance from carriageway edge D(z) Doppler factor f Frequency; hourly traffic flow F(w) Boundary loss factor Viscosity correction function g 0 3-D Green s function for sound propagation above rigid boundary g β 3-D Green s function for sound propagation above impedance boundary G 0 2-D Green s function for sound propagation above rigid boundary G β 2-D Green s function for sound propagation above impedance boundary k Von Karman constant k Propagation constant (=ω/c) h Mean propagation path height H Percentage molar concentration of water vapour; Elastic constant for Biot model Hankel function of the first kind and order 0 <H> Root mean square roughness height I Acoustic intensity; integral term Bessel function of the first kind and order 0 L Obukhov length; number of layers; outer (inertial) scale of turbulence; sound pressure level L Change in sound pressure level M Elastic constant for Biot model; Mach number n 1 Refractive index N Fresnel number N c Fractional cloud cover N PR Prandtl number P Pasquill class q 2 Tortuosity Q Spherical wave reflection coefficient r Horizontal range R 1 Source-receiver distance R 2 Length of specularly-reflected path R Shortest source-edge-receiver path R p Plane wave reflection coefficient R s Flow resistivity Effective flow resisitivity R eff

12 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. r h Relative humidity s 0 Steady flow shape factor S 0 Source strength s A, s B Dynamic pore shape parameters t e Emission time T Tortuosity T * Scaling temperature K T 0 Temperature C at zero height T av Average temperature C T(z) Temperature as a function of height u * Friction velocity (m/s) u 10 Wind speed at reference height of 10 m u(z) Wind speed as a function of height V j Plane wave reflection coefficient of layer j w Numerical distance W Ratio of minimum to mean roughness element spacing Z Specific normalized impedance z M Momentum roughness length z H Heat roughness length z 0 Roughness length α Air absorption coefficient; air absorption parameter α e Rate of porosity change β Specific normalized admittance β R Energy reflection coefficient γ Specific heat ratio γ g Ground parameter (Makarewicz) Γ Adiabatic correction factor Velocity potential, log2 base pore dimension λ Dimensionless parameter for complex density; wavelength Λ, Λ Viscous and thermal characteristic lengths φ Velocity potential µ Dynamic viscosity; polar angle ν Kinematic viscosity Mean square refractive index θ 0 Angle of incidence θ Polar angle θ s Scattering angle; angle of view ρ, ρ(ω) Density, complex density; transverse correlation σ Standard deviation of log-normal pore size distribution σ e Effective flow resistivity σ s Scattering cross section Variance of wind velocity fluctuations Variance of temperature fluctuations τ e, τ v Thermal and viscous relaxation times ψ Azimuthal angle

13 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. ψ M ψ H χ M χ H ω Ω Diabatic momentum profile correction (mixing) function Diabatic heat profile correction (mixing) function Inverse diabatic influence or function for momentum Inverse diabatic influence function for momentum Angular frequency Porosity

14 SOFTbank E-Book Center Tehran, Phone: , For Educational Use.

15 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Chapter 1 Introduction 1.1 Early observations The way in which sound travels outdoors has been of interest for several centuries. Initial experiments were concerned with the speed of sound [1]. The Francisan (Minimite) friar, Marin Mersenne ( ), suggested timing the interval between seeing the flash and hearing the report of guns fired at a known distance. William Derham ( ), the rector of a small church near London, was first to observe the influence of wind and temperature on sound speed and the difference in the sound of church bells at the same distance over newly fallen snow and over a hard frozen surface. Many records of the strange effects of the atmosphere on the propagation of sound waves have been associated with war [2, 3]. In June 1666, Samuel Pepys wrote that the sounds of a naval engagement between the British and Dutch fleets were heard clearly at some spots but not at others a similar distance away or closer. Pepys spoke to the captain of a yacht that had been positioned between the battle and the English coast. The captain said that he had seen the fleets and run from them, but from that hour to this hath not heard one gun. The effects of the atmosphere on battle sounds were not studied in a scientific way until after the First World War. During that war, acoustic shadow zones, similar to those observed by Pepys, were observed during the battle of Antwerp. Observers also noted that battle sounds from France only reached England during the summer months and were best heard in Germany during the winter. After the war there was great interest in these observations among the scientific community. Large amounts of ammunition were detonated throughout England and the public was asked to listen for sounds of explosions. Although there was considerable interest in atmospheric acoustics after the First World War, the advent of the submarine encouraged the greatest efforts in underwater acoustics research during and after the Second World War. The theoretical and numerical methods widely deployed in predicting sound propagation in the oceans have been adapted subsequently for use in atmospheric acoustics. A meeting organized by the University of Mississippi and held on the Mississippi Gulf Coast in 1981 was the first in which researchers in underwater acoustics met with scientists interested in atmospheric acoustics and stimulated the adoption and adaptation of the numerical methods, used for underwater acoustics, in the atmosphere [4]. 1.2 A brief survey of outdoor sound attenuation mechanisms Outdoor sound is attenuated by distance, by topography (including natural or artificial barriers), by interaction with the ground and with ground cover and by atmospheric effects including upward refraction and absorption. When the source is downwind of the

16 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Predicting outdoor sound 2 receiver, the sound has to propagate upwind. As height increases, the wind speed increases and the amount being subtracted from the speed of sound increases, leading to a negative sound speed gradient. In terms of rays, a negative sound gradient means that rays bend upwards. This is called upward refraction. Consequently, there is a ray that leaves the source, grazes the ground at some point and does not reach a receiver positioned beyond this point. Ray tracing ceases to be valid beyond this ground-grazing ray. The ray tracing model for outdoor sound is considered in more detail in Chapter 11. Upward refraction leads to the creation of a sound shadow at a distance from the source that depends on the gradient. The shadow zone is penetrated by sound scattered by turbulence and this sets a limit to the noise reduction within the sound shadow. Turbulence effects are considered further in Chapter 10. A negative sound speed gradient also results when the temperature decreases with height. This is called a temperature lapse condition and is the normal condition on a dry sunny day with little wind. A combination of slightly negative temperature gradient, strong upwind propagation and air absorption has been observed, in carefully monitored experiments, to reduce sound levels, 640 m from a 6 m high source over relatively hard ground, by up to 20 db more than expected from spherical spreading [5]. Atmospheric absorption acts as a low pass filter at long range. It results from heat conduction losses, shear viscosity losses and molecular relaxation losses. The total attenuation of a sound outdoors can be expressed as the sum of the reduction due to geometric spreading, atmospheric absorption and extra attenuation including, for example, ground effects, vegetation effects, refraction in the atmosphere and diffraction by barriers. Ground effects (for elevated source and receiver) are the result of interference between sound travelling directly from source to receiver and sound reflected from the ground. Since it involves interference, there can be enhancement as well as attenuation. Enhancement tends to occur at low frequencies. The presence of vegetation tends to make the surface layer of ground including the root zone more porous. The layer of partially decayed matter on the floor of a forest is highly porous. In addition, propagation through trees involves reverberant scattering by tree trunks and viscous scattering by foliage. These ground and scattering effects are explored in detail in Chapters 3 and 10 respectively. 1.3 Data including combined effects of ground and meteorology Pioneering studies of the combined influences of the ground surface and meteorological conditions were carried out by Parkin and Scholes [6 10] using a fixed Rolls Royce Avon jet engine as a source at two airfields (Hatfield and Radlett). In his 1970 Rayleigh Medal Lecture, one of the investigators, the late Peter Parkin, remarked [6] These horizontal propagation trials showed up the ground effect, which at first we did not believe, thinking there was something wrong with the measurements. But by listening to the jet noise at a distance, one could clearly hear the gap in the spectrum.

17 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Introduction 3 Parkin was among the first people to note and quantify the change in ground effect with the type of surface. Parkin and Scholes data showed a noticeable difference between the ground effects due to two grass covers. The ground attenuation at Hatfield, although still a major propagation factor, was less than at Radlett and its maximum value occurred at a higher frequency. The weather at the time of their measurements also enabled them to observe the different effect of snow. measurements [were] made at Site 2 [Radlett] with 6 to 9 in. of snow on the ground. The snow had fallen within the previous 24 hours and had not been disturbed. The attenuations with snow on the ground were very different from those measured under comparable wind and temperature conditions without snow. The maximum of the ground attenuation appears to have moved down the frequency scale by approximately 2 octaves. Examples of the Parkin and Scholes data are shown in Figure 1.1. They quoted their data as the difference in sound pressure levels at 19 m (reference location) and more distant locations corrected for the decrease expected from spherical spreading and air absorption. Clearly ground effect is sensitive to the acoustical properties of the surface. These depend on the substance of which the surface is composed. Different ground surfaces have different porosities. Soils have volume porosities of between 10 and 40%. Snow which has a porosity of around 60%, and many fibrous materials which have porosities of about 97%, have fairly low flow resistivities whereas a wet compacted soil surface will have a rather high flow resistivity. The thickness of the surface porous layer also is important and whether or not it has an acoustically hard substrate. The Parkin and Scholes data revealed the large effect at low frequencies (63 and 125 Hz octave bands) in the presence of thick snow. It should be noted moreover that even without snow there are significant differences between summer and winter excess attenuation in the Parkin and Scholes Radlett data. The classical experiments by Parkin and Scholes involved relatively little meteorological monitoring. In particular, the fine-scale fluctuations in wind speed and hence the turbulence were not monitored; perhaps since the important role of turbulence was not appreciated at the time. Recently there has been a similar experiment to that carried out by Parkin and Scholes. Simultaneous acoustic and meteorological measurements have been made using a jet engine source at a disused airfield operated as test facility by Rolls Royce at Hucknall [11]. In addition to wind and temperature gradient measurements, the fluctuation in wind velocity measurements was recorded and used as a measure of turbulence. Some of the data obtained under low wind and low turbulence conditions over continuous grassland are shown in Figure 1.2. Also shown is the third octave power spectrum of the Avon engine source between 100 and 4000 Hz deduced from the measured spectrum at m corrected for spherical spreading and ground effect. The data obtained at the longest range is noise-limited above 3 khz. The significant dips in the received spectra between 100 and 500 Hz are clear evidence of ground effect. The ground effect at Hucknall is different from that at either at Radlett or Hatfield.

18 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Predicting outdoor sound 4 Figure 1.1 Parkin and Scholes data for the level difference between 1.5 m high microphones at 19 and 347 m from a fixed jet engine source (nozzle-centre height 1.82 m) corrected for wavefront spreading and air absorption. The symbols and represent data over airfields (grass-covered) at Radlett and Hatfield respectively with a positive vector wind between source and receiver of 1.27 m s 1 (5 ft s 1 ). Crosses ( ) represent data over approximately 0.15 m thick (6 9 in.) snow at Hatfield with a positive vector wind of 1.52 m s 1 (6 ft s 1 ). The influence of small changes in the wind speed and turbulence strength on the measured spectra at the longest range is demonstrated in Figure 1.3. The associated meteorological conditions are detailed in Table 1.1. The ground effect between 100 and 400 Hz is fairly stable and is significantly greater for the low microphones and shifted in frequency compared with that for the high microphones. The data for both microphone heights show considerable variability between 400 and 2 khz as a result of changes in wind velocity and turbulence.

19 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Introduction 5 Figure 1.2 Data recorded at 1.2 m high receivers at horizontal ranges of m (solid line), 457 m (dotted line), 762 m (dashed line) and m (dash-dot line) from a fixed Rolls Royce jet engine source with the nozzle centre 2.16 m above an airfield at Hucknall, Notts. These data represent simultaneous recordings averaged over 26 s during zero wind and low turbulence conditions (block 20 of run 454, see Figure 1.3). Also shown (connected circles) is the deduced third octave power spectrum of the Avon jet engine source after subtracting 50 db. Figures 1.4 and 1.5 show A-weighted levels deduced from consecutive 26 s average spectra measured at 1.2 m height and ranges of m, m, m and m over grasslands at Hucknall. Figure 1.4 shows data for low wind speed (less than 2 m s 1 from source to receiver) and low turbulence conditions. Figure 1.5 shows data for moderate downwind conditions (approximately 6 m s 1 from source to receiver) and for higher turbulence intensities. The details of the meteorological conditions are listed in Tables 1.2 and 1.3.

20 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Predicting outdoor sound 6 Figure 1.3 Simultaneously measured narrow band (25 Hz interval) spectra at low (1.2 m) and high (6.4 m) microphones between 50 and 10 khz at m from a fixed Avon jet engine source averaged over 26 s intervals during low wind, low turbulence conditions at Hucknall (Notts, UK). The conditions are specified in Table 1.1 and the key.

21 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Run 454 Block No. Wind speed at ground (0.025 m) (m s 1 ) Table 1.1 Meteorological conditions corresponding to data in Figure 1.3 Wind speed at 6.4 m height (m s 1 ) Introduction 7 Direction relative to line of mics. ( ) Temperature at ground ( C) Temperature at 6.4 m ( C) Turbulence variable Figure 1.4 Comparison of A-weighted sound levels (26 s averages) deduced from low wind, low turbulence octave band measurements at 1.2 m height over grassland at Hucknall (Run 454 blocks 11, 12, 13( ); 14, 15, 16(+); 17, 18, 19, 20( )).

22 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Predicting outdoor sound 8 Note that there is a considerable spread in the measured levels at the longer ranges in Figure 1.4 as a result of the variation in wind speed and direction (up to approximately 2 m s 1 downwind at 6.4 m height) and turbulence levels. The data for stronger downwind conditions (up to approximately 6.5 m s 1 at 6.4 m height) in Figure 1.5 exhibit consistently higher levels than the relatively low wind speed data and a smaller spread. Although only four averages are shown in Figure 1.5, their spread is smaller than for any four averages exhibited in Figure 1.4. This is consistent with the assertion in ISO [12] that the variation in sound levels is less under moderate downwind conditions. The average downwind level measured at Hucknall is about 10 db higher than the levels for the lowest wind speed and turbulence conditions at 1.1 km from the source. Figure 1.5 Comparison of A-weighted sound levels deduced from consecutive 26 s average downwind octave band measurements at 1.2 m height above grassland at Hucknall (Run 453 blocks 3( ); 4(+); 5( ) and 6( )).

23 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Run 454 Block No. Wind speed at ground (0.025 m) (m s 1 ) Table 1.2 Meteorological data corresponding to sound level data shown in Figure 1.4 Wind speed at 6.4 m height (m s 1 ) Direction relative to line of mics. ( ) Temperature at ground ( C) Temperature at 6.4 m ( C) Turbulence variable a a Note a Also listed in Table 1.1. Run 453 Block No. Wind speed at ground (m s 1 ) Table 1.3 Meteorological data corresponding to sound level data in Figure 1.5 Wind speed at 6.4 m height (m s 1 ) Introduction 9 Direction relative to line of mics. ( ) Temperature at ground ( C) Temperature at 6.4 m ( C) Turbulence variable Classification of meteorological conditions for outdoor sound prediction The atmosphere is constantly in motion as a consequence of wind shear and uneven heating of the earth s surface (see Figure 1.6). Any turbulent flow of a fluid across a rough solid surface generates a boundary layer. Most interest from the point of view of outdoor noise prediction focuses on the lower part of the meteorological boundary layer called the surface layer. In the surface layer, turbulent fluxes vary by less than 10% of their magnitude but the wind speed and temperature gradients are largest. In typical daytime conditions the surface layer extends over m. Usually it is thinner at night.

24 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Predicting outdoor sound 10 In most common daytime conditions, the net radiative energy at the surface is converted into sensible heat. This warms up the atmosphere thereby producing negative temperature gradients as indicated in Figure 1.6. If the radiation is strong (high sun, little cloud cover), the ground is dry, and the surface wind speed is low, the temperature gradient is large. The atmosphere exhibits strong thermal stratification. If the ground is wet, most of the radiative energy is converted into latent heat of evaporation and the temperature gradients are correspondingly lower. In unstable daytime conditions, the wind speed is affected by the temperature gradient and exhibits slightly less variation with height than for the isothermal case. On the other hand, stable conditions prevail at night. The radiative losses from the surface cause positive temperature gradients. There is a considerable body of knowledge about meteorological influences on air quality in general and the dispersion of plumes from stacks in particular. Plume behaviour depends on vertical temperature gradients and hence on the degree of mixing in the atmosphere. Vertical temperature gradients decrease with increasing wind. The stability of the atmosphere in respect of plume dispersion is described in terms of Pasquill classes. This classification is based on incoming solar radiation, time of day and wind speed. There are six Pasquill classes (A F) defined in Table 1.4. Data are recorded in this form by meteorological stations and so, at first sight, it is a convenient classification system for noise prediction. Figure 1.6 Schematic of the daytime atmospheric boundary layer and eddy structures. The sketch graph on the left shows the mean wind speed (U) and the potential temperature profiles (θ=t+γ dz, where γ d =0.098 C km 1 is the dry adiabatic lapse rate, T is the temperature and z is the height).

25 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Introduction 11 Class A represents a very unstable atmosphere with strong vertical air transport, that is, mixing. Class F represents a very stable atmosphere with weak vertical transport. Class D represents a meteorologically neutral atmosphere. Such an atmosphere has a logarithmic wind speed profile and a temperature gradient corresponding to the normal decrease with height (adiabatic lapse rate). A meteorologically neutral atmosphere occurs for high wind speeds and large values of cloud cover. This means that a meteorologically neutral atmosphere may be far from acoustically neutral. Typically, the atmosphere is unstable by day and stable by night. This means that classes A D might be appropriate classes by day and D F by night. With practice, it is possible to estimate Pasquill Stability Categories in the field, for a particular time and season, from a visual estimate of the degree of cloud cover. The Pasquill classification of meteorological conditions has been adopted widely as the basis of a meteorological classification system for noise prediction schemes [e.g. 13]. However, it is clear from Table 1.2, that the meteorologically neutral category (C), while being fairly common in a temperate climate, includes a wide range of wind speeds and is therefore not very suitable as a category for noise prediction. In the CONCAWE scheme [13], this problem is addressed by defining six noise prediction categories based on Pasquill categories (representing the temperature gradient) and wind speed. There are 18 sub-categories depending on wind speed. These are defined in Table 1.5. CONCAWE category 4 is specified as one in which there is zero meteorological influence. So CONCAWE category 4 is equivalent to acoustically neutral conditions. Table 1.4 Pasquill (meteorological) stability categories Wind speed a Daytime incoming solar One hour before sunset Night-time cloud (m s 1 ) radiation mw cm 2 or after sunrise cover (octas) > <30 Overcast A A B B C D F or G b F D A B B C C D F E D B B C C C D E D D C C D D D D D D D >6.0 D D D D D D D D Notes a Measured to the nearest 0.5 m s 1 at 11 m height. b Category G is an additional category restricted to the night-time with less than 1 octa of cloud and a wind speed of less than 0.5 m s 1. Meteorological category Table 1.5 CONCAWE meteorological classes for noise prediction Pasquill stability category and wind speed (m s 1 ) (positive is towards receiver) A, B C, D, E F, G 1 v< <v< 0.5 v< <v< <v< 0.5 v< a +0.5<v< <v< <v< 0.5

26 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Predicting outdoor sound 12 5 v> <v< <v< v> <v<+3.0 Note a Category with assumed zero meteorological influence. The CONCAWE scheme requires octave band analysis. Meteorological corrections in this scheme are based primarily on analysis of the Parkin and Scholes data together with measurements made at several industrial sites. The excess attenuation in each octave band for each category tends to approach asymptotic limits with increasing distance. Values at 2 km for CONCAWE categories 1 (strong wind from receiver to source, hence upward refraction) and 6 (strong downward refraction) are listed in Table 1.6. Table 1.6 Values of the meteorological corrections for CONCAWE categories 1 and 6 Octave band centre frequency (Hz) Category Category Table 1.7 Estimated probability of occurrence of various combinations of wind and temperature gradient Zero wind Strong wind Very strong wind Very large negative temperature gradient Frequent Occasional Rare or never Large negative temperature gradient Frequent Occasional Occasional Zero temperature gradient Occasional Frequent Frequent Large positive temperature gradient Frequent Occasional Occasional Very large positive temperature gradient Frequent Occasional Rare or never Table 1.8 Meteorological classes for noise prediction based on qualitative descriptions W1 Strong wind (>3 5 m s 1 ) from receiver to source W2 Moderate wind ( 1 3 m s 1 ) from receiver to source, or strong wind at 45 W3 No wind, or any cross wind W4 Moderate wind ( 1 3 m s 1 ) from source to receiver, or strong wind at 45 W5 Strong wind (>3 5 m s 1 ) from source to receiver TG1 Strong negative: daytime with strong radiation (high sun, little cloud cover), dry surface and little wind TG2 Moderate negative: as T1 but one condition missing TG3 Near isothermal: early morning or late afternoon (e.g. one hour after sunrise or before sunset) TG5 Moderate positive: night-time with overcast sky or substantial wind TG6 Strong positive: night-time with clear sky and little or no wind Wind speed and temperature gradients are not independent. For example, very large temperature and wind speed gradients cannot coexist. Strong turbulence associated with high wind speeds does not allow the development of marked thermal stratification.

27 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Introduction 13 Table 1.7 shows a rough estimate of the probability of occurrence of various combinations of wind and temperature gradients [5]. With regard to sound propagation, the component of the wind vector in the direction between source and receiver is most important. So the wind categories (W) must take this into account. Moreover, it is possible to give more detailed but qualitative descriptions of each of the meteorological categories (W and TG, where W refers to wind and TG refers to temperature gradient see Table 1.8). Table 1.9 Qualitative estimates of impact of meteorological condition on noise levels W1 W2 W3 W4 W5 TG1 Large attenuation Small attenuation Small attenuation TG2 Large Small attenuation Small attenuation Zero meteorological Small attenuation influence enhancement TG3 Small Small attenuation Zero meteorological Small enhancement Small attenuation influence enhancement TG4 Small Zero meteorological Small enhancement Small enhancement Large attenuation influence enhancement TG5 Small enhancement Small enhancement Large enhancement In Table 1.9, the revised categories are identified with qualitative predictions of their effects on noise levels [5]. The classes are not symmetrical around zero meteorological influence. Typically there are more meteorological condition combinations that lead to attenuation than those that lead to enhancement. Moreover, the increases in noise level (say 1 5 db) are smaller than the decreases (say 5 20 db). It has been suggested [14] that noise calculation procedures should predict average levels, such as would occur under neutral conditions, and that following or opposing winds or temperature inversions will cause variations of ±10 db about the average values. Using the values at 500 Hz as a rough guide to the likely corrections on overall A- weighted broadband levels, it is noticeable that the CONCAWE meteorological corrections are not symmetrical around zero. The CONCAWE scheme suggests meteorological variations of between 10 db less than the acoustically neutral level for strong upward refraction between source and receiver and 7 db more than the acoustically neutral level for strong downward refraction between source and receiver. Zouboff et al. [5] have carried out a series of measurements using a loudspeaker source broadcasting broadband noise with maximum energy in the 500 and 1000 Hz octave bands over a flat homogeneous area, in the South of France, covered with pebbles and sparse vegetation. Acoustical data were collected at a series of microphones positioned between 20 and 640 m from the source. Meteorological parameters (mean air temperature and wind speed at three heights, together with wind direction, solar radiation and hygrometry) were monitored on a 22 m high tower located approximately at the centre of the measurement line. A hundred and ninety five 10-minute long samples were collected over a range of meteorological conditions and were expressed in terms of L Aeq. Since the ground condition changed very little, most of the variation may be attributed

28 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Predicting outdoor sound 14 Figure 1.7 Maximum, mean and minimum total attenuation deduced from Zouboff et al. [5] data for 10-minute L Aeq. to meteorological effects. Figure 1.7 shows the maximum, minimum and mean total attenuation from 80 to 640 m, deduced from levels measured at 1.5 m high microphones normalized to a level of 100 db at 20 m. These data offer further evidence for asymmetry of meteorological effects about the mean noise level. The difference between the minimum and mean levels is considerably less than the difference between the maximum and mean levels. Smaller differences were obtained with longer averaging times. For example, the 38 db range in 10-minute L Aeq at 640 m is reduced to a range of only 19 db when comparing 8-hour L Aeq during days differing in wind direction and cloud cover. Longer-term values of L Aeq will be dominated by the highest levels, even though they are relatively infrequent. Moreover, levels observed under downward refraction conditions exhibit less variability than those measured under upward refraction conditions. For these reasons, the ISO Scheme [12], predicts for moderate downwind conditions and distinguishes long-term (say seasonal or monthly) L Aeq from short-term (say daily) L Aeq. 1.5 Typical sound speed profiles Outdoor sound prediction requires information on wind speed, direction, temperature, relative humidity and barometric pressure as a function of height near to the propagation path. These are what determine the sound speed profile. Ideally, the heights at which the

29 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Introduction 15 meteorological data are collected should reflect the application. If this information is not available, then there are alternative procedures. It is possible, for example, to generate an approximate sound speed profile from temperature and wind speed at a given height using the similarity theory [15] and to input this directly. According to this theory, the wind speed component (m s 1 ) in the source-receiver direction and temperature ( C) at height z are calculated from the values at ground level and other parameters as follows: (1.1) (1.2) where u * Friction velocity (m s 1 ) (depends on surface roughness) z M Momentum roughness (depends on surface roughness) length z H Heat roughness length (depends on surface roughness) T * Scaling temperature K The precise value of this is not important for sound propagation. A convenient value is 283 K k Von Karman constant (=0.41) T 0 Temperature C at zero Again it is convenient to use 283 K height Γ Adiabatic correction factor = 0.01 C m 1 for dry air Moisture affects this value but the difference is small L Obukhov length (m) >0 stable, <0 unstable the thickness of the surface or boundary layer is given by 2L m T av Average temperature C It is convenient to use T av =10 so that (T av )=θ 0 ψ M Diabatic momentum profile correction (mixing) function ψ H Diabatic heat profile correction (mixing) function χ M Inverse diabatic influence or function for momentum =[1 16z/L] 0.25

30 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Predicting outdoor sound 16 χ H Inverse diabatic influence function for momentum =[1 16z/L] 0.5 For a neutral atmosphere, 1/L=0 and ψ M =ψ 11 =0. The associated sound speed profile, c(z), is calculated from (1.3) Note that the resulting profiles are valid in the surface or boundary layer only but not at zero height. In fact, the profiles given by these equations, sometimes called Businger- Dyer profiles [16], have been found to give good agreement with measured profiles up to 100 m. This height range is relevant to sound propagation over distances up to 10 km [17]. However, improved profiles are available that are valid to greater heights. For example [18], (1.4) Often z M and z H are taken to be equal. The roughness length varies, for example, between (still water) and 0.1 (grass). More generally, the roughness length can be estimated from the Davenport classification [19]. Figure 1.8 shows examples of sound speed (difference) profiles, (c(z) c(0)), generated from (1.2) (1.4) using (a) z M =z H =0.02, u * =0.34, T * =0.0212, T av =10, T 0 =6 (giving L= ) (b) z M =z H =0.02, u * =0.15, T * =0.1371, T av =10, T 0 =6 (giving L= 11.76) and Γ= These parameters are intended to correspond to a cloudy windy night and a calm clear night respectively [20]. Salomons et al. [21] have suggested a method for obtaining the remaining unknown parameters, u *, T * and L from the relationship (1.5) and the Pasquill Category (P).

31 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Introduction 17 Figure 1.8 Downward refracting sound speed profiles relative to the sound speed at the ground obtained from similarity theory. The continuous curve is approximately logarithmic corresponding to a large Obukhov length and to a cloudy, windy night. The broken curve corresponds to a small Obukhov length as on a calm clear night and is predominantly linear away from the ground. From empirical meteorological tables, approximate relationships between the Pasquill class P the wind speed u 10 at a reference height of 10 m and the fractional cloud cover N c have been obtained. The latter determines the incoming solar radiation and therefore the heating of the ground. The former is a guide to the degree of mixing. The approximate relationship is P(u 10, N c )=1+3 [1+exp( u N c )] 1 during the day =6 2 [1+exp (12 2u 10 2N c )] 1 during the night. (1.6)

32 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Predicting outdoor sound 18 A relationship between the Obukhov length L m as a function of P and roughness length z 0 <0.5 m, is (1.7) where B 1 (P)= P P 2 (1.8) and B 2 (P)=min(0, 0.045P 0.125) for 1 P 4 max (0, 0.025P 0.125) for 4 P 6. (1.9) Alternatively, values of B 1 and B 2 may be obtained from Table Equations (1.6) and (1.7) give L=L(u 10, N c, z 0 ). (1.10) Also u 10 is given by equation (1) with z=10 m, that is, (1.11) Equations (1.5), (1.10) and (1.11) may be solved for u *, T * and L. Hence it is possible to calculate ψ M, ψ H, u(z) and T(z). Figure 1.9 shows the results of this procedure for a ground with a roughness length of 0.1 m and two upwind and downwind daytime classes defined by the parameters listed in the caption. As a consequence of atmospheric turbulence, instantaneous profiles of temperature and wind speed show considerable variations with both time and position. These variations are eliminated considerably by averaging over a period of the order of 10 minutes. Models based on similarity theory give good descriptions of the averaged profiles.

33 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Introduction 19 Table 1.10 Values of the constants B 1 and B 2 in (1.7) for the six Pasquill classes Pasquill class A B C D E F B B Figure 1.9 Two daytime sound speed profiles (upwind dashed and dotted; downwind solid and dash-dot) determined from the parameters listed in Table The Pasquill Category C profiles shown in Figure 1.9 are approximated closely by logarithmic curves of the form where the parameter b (>0 for downward (1.12)

34 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Predicting outdoor sound 20 refraction and <0 for upward refraction) is a measure of the strength of the atmospheric refraction. Such logarithmic sound speed profiles are realistic for open ground areas without obstacles particularly in the daytime. A better fit to night-time profiles is obtained with power laws of the form [22] (1.13) where a=0.4(p 4) 1/4. The temperature term in effective sound speed profile given by equation (1.3) can be approximated by truncating a Taylor expansion after the first term to give (1.14) When combined with (1.1), this leads to a linear dependence on temperature and a logarithmic dependence on wind speed with height. By comparing with 12 months of meteorological data obtained at a 50 m high meteorological tower in Germany, Heimann and Salomons [22] have found that (1.14) is a reasonably accurate approximation to vertical profiles of effective sound speed even in unstable conditions and in situations where the similarity theory is not valid. By making a series of sound level predictions for different meteorological conditions, it was found that a minimum of 25 meteorological classes is necessary to ensure 2 db or less deviation in the estimated annual average sound level from the reference case with 121 categories. There are simpler, linear-segment profiles deduced from a wide range of meteorological data that may be used to represent worst case noise conditions, that is, best conditions for propagation [23]. The first of these profiles may be calculated from a temperature gradient of +15 C km 1 from the surface to 300 m and 8 C km 1 above that assuming a surface temperature of 20 C. This type of profile can occur during the daytime or at night downwind due to wind shear in the atmosphere or a very high temperature inversion (where the temperature gradient reverses at a great height above the ground). If this is considered too extreme, or too rare a condition for routine predictions, then a second possibility is a shallow inversion. A shallow inversion occurs regularly at night. A typical depth is 200 m. The profile may be calculated from a temperature gradient of +20 C km 1 from the surface to 200 m and 8 C km 1 above that assuming a surface temperature of 20 C [23]. The prediction of outdoor sound propagation also requires information about turbulence. Specifically it requires values of the mean-square refractive index, the outer length scale of the turbulence and a parameter representing the transverse separation between adjacent rays. The mean-squared refractive index may be calculated from the

35 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Introduction 21 measured instantaneous variation of wind speed and temperature with time at the receiver. (1.15) where is the variance of the wind velocity, is the variance of the temperature fluctuations, a is the wind vector direction, and C 0 and T 0 are the ambient sound speed and temperature respectively. In the absence of turbulence data, typical values of mean-squared refractive index are between 10 6 for calm conditions and 10 4 for strong turbulence. The outer length scale may be approximated by the height of the receiver, as long as source and receiver are close to the ground. The greatest effect of turbulence is predicted when the path separation is set to zero. A more detailed discussion of turbulence is in Chapter Air absorption During its passage through the air, sound is subject to two forms of dissipation of energy: classical and relaxation [24]. The classical absorption is associated with transfer of the energy of the coherent molecular motion to equivalent heat energy or random kinetic energy of translation of air molecules. The relaxation absorption mechanism is associated with redistribution of the translational or internal energy of the molecules. The relaxation mechanism may be divided into rotational and vibrational parts, the former being more significant at high values of frequency or pressure. All of these effects may be combined. For a plane wave, pressure P at distance x from a position where the pressure is P 0 is given by P=P 0 e αx/2 The frequency, humidity and temperature dependent attenuation coefficient a for air absorption may be calculated using equation (1.16) [24 28]. (1.16) where f r,n and f r,o are given by:

36 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Predicting outdoor sound 22 (1.17) (1.18) where f is the frequency, T is the absolute temperature of the atmosphere in degrees Kelvin, T 0 = K is the reference value of T(20 C), H is the percentage molar concentration of water vapour in the atmosphere=ρ sat r h p 0 /p s, r h is the relative humidity (%), p s is local atmospheric pressure and p 0 is the reference atmospheric pressure (1 atm= Pa). where C sat = (T 0 /T) These formulae give estimates of the absorption of pure tones to an accuracy of ±10% for 0.05<H<5,253<T<323, p 0 <200 kpa. Outdoor air absorption varies through the day and the year [29, 30]. Absolute humidity, H, is an important factor in the diurnal variation and usually peaks in the afternoon. Usually the diurnal variations are greatest in the summer. It should be noted that use of (arithmetic) mean values of atmospheric absorption may lead to overestimates of attenuation when attempting to establish worst case exposures for the purposes of environmental noise impact assessment. Investigations of local climate statistics, say hourly means over one year, should lead to more accurate estimates of the lowest absorption values. Figure 1.10 Variation of attenuation coefficient for air absorption with relative humidity.

37 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Introduction 23 Table 1.11 Total sound absorption in db km 1 versus relative humidity as a function of frequency at 20 C (68 F) Frequency (khz) Relative humidity (%) Figure 1.10 shows values of a versus relative humidity for air at 20 C and normal atmospheric pressure for frequencies between 2 and 12.5 khz [28]. Table 1.11 shows numerical values for db km 1 at specific frequencies. References 1 F.V.Hunt, Origins in Acoustics, publ. Acoustical Society of America, AIP (1992). 2 C.D.Ross, Outdoor sound propagation in the US Civil War, Appl. Acoust., 59: (2000). 3 M.V.Naramoto, A concise history of acoustics in warfare, Appl. Acoust., 59: (2000). 4 Proc. 1st International Symposium on Long Range Sound Propagation, Gulfport, MS, University of Mississippi, V.Zouboff, Y.Brunet, M.Berengier and E.Sechet, A qualitative approach of atmospherical effects on long range sound propagation, Proc. 6th International Symposium on Long Range Sound Propagation, ed. D.I.Havelock and M.Stinson, NRCC, Ottawa, publ. National Research Council of Canada, (1994). 6 P.Parkin, Acoustical reminiscences: the Rayleigh lecture, Proc. Inst. Acoust., 1:7 31 (1978). 7 P.H.Parkin and W.E.Scholes, The horizontal propagation of sound from a jet engine close to the ground at Radlett, J. Sound Vib., 1:1 13 (1965). 8 P.H.Parkin and W.E.Scholes, The horizontal propagation of sound from a jet engine close to the ground at Hatfield, J. Sound Vib., 2: (1965). 9 The results of measurements of the horizontal propagation of sound at Radlett, DSIR, Building Research Station, Internal Notes B215 and B249 (unpubl.). 10 The results of measurements of the horizontal propagation of sound at Hatfield, DSIR, Building Research Station, Internal Notes IN99 (1964) (unpubl.). 11 K.Attenborough, K.M.Li and S.Taherzadeh, Propagation from a broadband source over grassland: comparison of models and data, Proc. Inter-Noise 95, 1, 319 (1995). 12 ISO (1996), Acoustics Attenuation of Sound during Propagation Outdoors Part 2: General Method of Calculation. International Standard ISO :1996 (E). 13 K.J.Marsh, The CONCAWE model for calculating the propagation of noise from open-air industrial plants, Appl. Acoust., 15: (1982). 14 I.H.Flindell, Evidence to Heathrow T5 Enquiry (unpubl.). 15 A.S.Monin and A.M.Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence, Vol. 1, MIT press, Cambridge, MA (1979). 16 R.B.Stull, An Introduction to Boundary Layer Meteorology, Kluwer, Dordrecht, (1991).

38 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Predicting outdoor sound A.A.M.Holtslag, Estimates of diabatic wind speed profiles from near surface weather observations, Boundary-Layer Meteorology, 29: (1984). 18 E.M.Salomons, Downwind propagation of sound in an atmosphere with a realistic sound speed profile: a semi-analytical ray model, J. Acoust. Soc. Am., 95: (1994). 19 A.G.Davenport, Rationale for determining design wind velocities, J. Am. Soc. Civ. Eng., ST-86:39 68 (1960). 20 W.H.T.Huisman, Sound Propagation over Vegetation-covered Ground, PhD Thesis, University of Nijmegen, The Netherlands (1990). 21 E.M.Salomons, F.H.van den Berg and H.E.A.Brackenhoff, Long-term average sound transfer through the atmosphere based on meteorological statistics and numerical computations of sound propagation, Proc. 6th International Symposium on Long range Sound propagation, ed. D.I.Havelock and M.Stinson, NRCC, Ottawa, (1994). 22 D.Heimann and E.Salomons, Testing meteorological classifications for the prediction of longterm average sound levels, Appl. Acoust., 65: (2004). 23 J.M.Noble, U.S.Army research Laboratory, private communication (1993). 24 L.C.Sutherland and H.E.Bass, Atmospheric absorption in the atmosphere at high altitudes, Proc. 7th International Symposium on Long Range Sound Propagation, Lyon (1996). 25 ANSI SI , American National Standard Method for Calculation of the Absorption of Sound by the Atmosphere, Acoustical Society of America, New York (1995). 26 ISO (1993), Acoustics Attenuation of Sound during Propagation Outdoors Part 1: Calculation of the Absorption of Sound by the Atmosphere, International Organization for Standardization, Geneva, Switzerland (1993). 27 D.T.Blackstock, Fundamentals of Physical Acoustics, University of Texas, Austin, TX, John Wiley & Sons, Inc (2000). 28 C.M.Harris, Absorption of sound in air versus humidity and temperature, J. Acoust. Soc. Am., 40: (1966). 29 C.Larsson, Atmospheric absorption conditions for horizontal sound propagation, Appl Acoust., 50: (1997). 30 C.Larsson, Weather effects on outdoor sound propagation, Int. J. Acoust. Vib., 5:33 36 (2000).

39 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Chapter 2 The propagation of sound near ground surfaces in a homogeneous medium 2.1 Introduction Despite the fact that there are many important phenomena that affect the propagation of outdoor sound, the study of sound propagation in the homogeneous atmosphere above a flat ground, offers, perhaps, the simplest case that is amenable to theoretical analyses. In particular such a study has the distinct advantage that a closed-form analytic expression can be derived to represent the sound field. The analytical method, used for the general problem of point-to-point propagation above a boundary between two media, has been adapted from the study of the propagation of electromagnetic waves initiated by Sommerfeld [1] in 1909 and continued in Banos et al. [2]. With the benefit of these early studies in electromagnetic wave propagation, the propagation of sound above a porous ground surface was studied by Rudnick [3] in the 1940s, by Ingard [4] and Paul [5] in 1950s, by Wenzel [6], Donato [7], Chien and Soroka [8, 9] in the 1970s and by Attenborough et al. [10] and Kawai et al. [11] in the 1980s. In 1998, Li et al. [12] generalized the problem to allow for arbitrary layers in the lower medium. For situations in which multiple ground layers are important, they introduced the concept of effective impedance that greatly simplifies the computational requirements. The analytical treatment in this chapter is based on these studies [8, 12]. However, in anticipation of the subsequent discussions of more computationally intensive numerical schemes, the use of Fourier transformation method will be explored in considerable detail. 2.2 Mathematical formulation for a point source near to the ground A useful idealization of sound propagating over flat acoustically soft ground is that of the two-media problem shown in Figure 2.1 with the plane interface lying at z=0 in a rectangular coordinate system with x and y as the horizontal axes and z as the vertical

40 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Predicting outdoor sound 26 Figure 2.1 Ray-paths from a point source above an interface between two media. axis. The lower medium represents the porous ground which is treated as an effective fluid with complex density ρ 1 and complex propagation constant k 1. The upper medium represents the atmosphere. The air density and the speed of sound are denoted by ρ and c, respectively, and they are constant if the atmosphere is homogeneous. The problem to be considered is the radiation from a point source in the homogeneous atmosphere. We are primarily interested in the sound field in the upper medium in which the inhomogeneous Helmholtz equation for the acoustic pressure, p is (2.1) where k( ω/c) is the wave number, ω is the angular frequency of the source, time dependence factor e iωt is understood but suppressed throughout and S(x s ) is the source term at the point x s. For convenience, but without loss of generality, the source position is assumed to be located at (0, 0, z s ). In the lower medium, the governing equation for the acoustic pressure is (2.2) Here and subsequently, the properties of the lower medium are designated by the subscript 1. At the plane interface, the acoustic pressure and normal particle velocity should be continuous. This implies that (2.3) on z=0.

41 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. The propagation of sound near ground 27 We wish to derive the Green s function, G m (x x s ) for the sound field at the location x (x, y, z) due to a point monopole source situated at x s where S(x s )= δ(x)δ(y)δ(z z s ). Thus G m (x x s ) satisfies the inhomogeneous equation (2.4) where the second argument in the Green s function is suppressed for brevity, that is, G m (x) G m (x x s ). To find the solution for the inhomogeneous equation, it is convenient to introduce a Fourier transform pair for the Green s function as follows: (2.5a) and (2.5b) Then (2.4) becomes (2.6a) where and (2.6b) (2.6c)

42 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Predicting outdoor sound 28 By means of the mathematical technique we have used, the physical space (x, y) has been transformed into an imaginary κ-space (k x, k y ) to ease subsequent analyses. Using a similar Fourier transform pair for the lower medium, (2.2) can be simplified to (2.7a) where (2.7b) In (2.6b) and (2.7b), positive roots are chosen to ensure a finite and bounded solution for the inhomogeneous Helmoltz equation. Furthermore, the boundary conditions given in (2.3) do not change their form after the transformation. Hence, the boundary conditions in the transformed space, on z=0, are (2.8) From (2.6a) and (2.7a), the acoustic field due to a point monopole source at (0, 0, z s ) is (2.9a) (2.9b) (2.9c) where and U 1 are constants to be determined from the boundary conditions. The solutions given in (2.9a 2.9c) contain both outgoing ( for for z s >z 0 and for z 0) and incoming ( for z>z s, for z s >z 0 and for z 0) waves. We may interpret as waves reflected from the top of upper medium, as that reflected from the air/ground interface and as that reflected from the bottom of the lower medium. However, the Sommerfeld radiation condition requires that the sound field contains no incoming waves when z ±, therefore and U 1 must

43 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. The propagation of sound near ground 29 vanish. By using (2.9c) with U 1 =0, we can eliminate from the boundary condition (2.8) to give (2.10) on z=0, where ς 1 is the density ratio and n 1 is the index of refraction in the ground. They are given respectively by ς 1 =ρ/ρ 1 (2.11a) and N 1 =k 1 /k=c/c 1. (2.11b) In addition to the boundary condition (2.10) on z=0, we require the continuity of pressure and discontinuity of pressure gradient at the plane z=z s, that is, (2.12) The constants A u, A d and B 1 can be determined according to the conditions specified by (2.10) and (2.12), and hence the Green s function in the transformed space can be expressed as (2.13) where U 0 (which may be regarded as the corresponding reflection coefficient of the air/ground interface in the transformed space) is given by

44 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Predicting outdoor sound 30 (2.14) Substitution of (2.13) and (2.14) into (2.5b) leads to an integral expression for the Green s function for the sound field as follows: (2.15) The first two terms of the Green s function can be identified immediately as Sommerfeld integrals (see for example, Ref. 2.2, Ch. 2). These can be evaluated exactly as (2.16a) and (2.16b)

45 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. The propagation of sound near ground 31 where is the distance from the source located at (0, 0, z s ) to the receiver located at (x, y, z) and R 2 is the distance from the image source located at (0, 0, z s ) to the same receiving point, that is, The evaluation of the integral in the third term of (2.15) requires considerable effort. An exact solution is generally not possible but the integral can be estimated asymptotically by the method of steepest descents. Taking advantage of the fact that the solution is axisymmetric about the z=0 axis, the problem may be simplified by employing a polar coordinate system instead of the rectangular Cartesian coordinates. Making use of the transformation where κ and ε are, respectively, the magnitude and phase of the wavenumber in κ-space, we can write k x =κ cos ε (2.17a) k y =κ sin ε (2.17b) dk x dk y =κdκdε (2.11c) Similarly (x, y) transforms to (r, ψ) for the field points in the horizontal plane of constant z, namely,

46 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Predicting outdoor sound 32 x=r cos ψ (2.18a) and Y=r sin ψ, (2.18b) where r is the horizontal range from source to receiver. To cover the whole integration range with k x and k y varying from to +, the azimuth angle ψ is required to vary from 0 to 2π and κ must vary from 0 to. The unknown integral, which is denoted by I, can now be rewritten in the polar form as (2.19) The integral over ε can be evaluated by means of the integral expression for the Bessel function of zero order (Ref. 13; Eq ): Hence, (2.20) Before we proceed to evaluate this integral I, we consider two limiting cases in order to shed light on the influence of the ground, that is the lower medium, on the total sound field. The first special case is when the ground is acoustically hard. For example, the source might be located above a large thick metallic sheet or over sealed concrete such that air is unable to penetrate the surface. In this particular case, the density of the ground is much higher than the density of air. In other word, and the contribution of the integral I to the total sound field is negligibly small. So the Green s function is simply the sum of the two terms given in (2.16a) and (2.16b),

47 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. The propagation of sound near ground 33 (2.21) This solution is straightforward to interpret. The first term corresponds to the direct wave: the sound field caused by a source at (0, 0, z). The second term is the reflected wave, that is the sound field of a mirror image source located at (0, 0, z). However, if the ground is naturally porous (such as grassland or cultivated soil) or artificially porous (such as a pervious asphalt road surface), air is able to penetrate the ground. The assumption of is no longer valid as the density of air may not be negligible in comparison with the air within the ground. A more detailed consideration of the interaction of the sound wave with ground surfaces is required. In normal circumstances, c>c 1 so the index of refraction of the ground, n 1, is greater than 1. This is understandable since the propagation of sound in the lower medium will be somewhat impeded if it contains a combination of air gaps and solid particles. From Snell s law and with n 1 >1, we can conclude that the sound ray is refracted towards the normal as it propagates from air and penetrates the ground. For most outdoor ground surfaces, the speed of sound in the ground may be considered to be much smaller than that in the air, that is This means that further approximations may be made. In particular we can approximate impedance, Z as by n 1. We define the specific normalized (2.22a) Sometimes, it is convenient to use the specific normalized admittance, β, where Z=1/β. Consequently, according to (2.22a), we have β=ς 1 n 1. (2.22b) With this level of approximation, the boundary condition (2.10) becomes (2.23) The form given in (2.23) is called the impedance condition for the boundary surface. If sound waves at any angle of incidence will be strongly refracted downwards and travel normal to the surface. This type of ground surface is sometimes called a locally reacting ground because the air/ground interaction is independent of the angle of incidence of the incoming waves.

48 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Predicting outdoor sound The sound field above a locally reacting ground This section will give a derivation of the classical expression of the sound field due to a monopole source above a locally reacting ground. The fields above more complicated ground surfaces will be discussed in section 2.5. Using the approximation the integral I in (2.19) can be simplified to (2.24) To facilitate the analysis, we introduce a further transformation such that κ=k sin µ. (2.25a) It is straightforward to show dκ=k cos µdµ (2.25b) and (2.25c)

49 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. The propagation of sound near ground 35 Figure 2.2 Integration paths for the integral I in spherical polar co-ordinates. The integration limit for µ cannot be restricted to real values of this angle because κ is required to vary from 0 to. Consequently, k z varies from k to i according to (2.25c). The path of integration, Γ 1, for the new variable starts from µ=0, moves along the real axis to µ=π/2, then makes a right-angle turn and continues parallel to the imaginary axis from µ=π/2+i0 to µ=π/2 i, as shown in Figure 2.2. Taking the image source location as the centre, we write the separation between the source and receiver in a spherical polar coordinate (R 2, θ, ψ): (2.26a, b) It is also useful to note a property of the Hankel functions and their connection with the Bessel function: (2.27a, b)

50 SOFTbank E-Book Center Tehran, Phone: , For Educational Use. Predicting outdoor sound 36 Substitution of ( ) into (2.24) leads to After replacing µ with µ the second integral, noting the different integration limits and combining these two integrals, we obtain (2.28) with path of integration, Γ 2 shown also in Figure 2.2. The integral (2.28) can be approximated by a uniform asymptotic expansion [14] that combines the steepest descent approach and the pole subtraction method [8]. Only the solution and its interpretation are outlined here. The integral (2.28) can be evaluated asymptotically to yield where (2.29a) (2.29b)

51 The propagation of sound near ground 37 (2.29c) and erfc( ) is the complementary error function that can be computed readily [Ref. 13, Ch. 7]. represents a small correction term resulting from subtraction of the pole. The variable x 0, which fixes the location of the pole, is determined by (2.30) The significance of the pole location, and hence of x 0, will be discussed at a later stage when we address the intriguing physical phenomenon of the associated surface wave. In deriving (2.29), we have assumed that the following relationships hold: and These approximations mean that the resulting expressions will be valid only for long range and high frequency and with both the source and receiver located close to the ground surface. The Green s function for the sound field due to a monopole source radiating sound above a locally reacting ground can be determined by summing (2.16a), (2.16b) and (2.29a) to yield (2.31) where Φ p and are given in (2.29b) and (2.29c) respectively. However, the exact Green s function involves a considerable number of terms and may not be convenient for routine use. Additional assumptions are needed in order to simplify the expression given in (2.29b) and (2.29c). There are approximations corresponding to the relatively soft boundary case where the horizontal range where and other limiting cases. But it is found that the condition for a relatively hard boundary leads to the most versatile approximation. In this case, we assume that and r R 2. Then the factor in the curly bracket of (2.30) can be simplified to

52 Predicting outdoor sound 38 (2.32a) The Hankel function can be replaced by the first term of the asymptotic expansion, (2.32b) Substituting (2.32a) and (2.32b) into (2.29c), we obtain an approximation for Φ p as (2.33a) where by is sometimes called the numerical distance and is approximated (2.33b) Furthermore, it is possible to show that the correction term, that is, small when compared with Фp. Hence it can be ignored in (2.31) and the total sound field above a locally reacting ground can be computed by substituting (2.33a) into (2.31) to yield an approximate Green s function: (2.34) The computation of the error function can be implemented easily by using the following formulae for a large range of w. Note that the numerical distance is complex, w=w r +iw x. According to Abramowitz and Stegun [Ref. 13, p. 328], if w r >3.9 or w i >3,

53 The propagation of sound near ground 39 (2.35a) with an absolute error of less than If w r >6 or w i >6, (2.35b) with an absolute error less than For smaller values of w r and w i, the Matta and Reichel formula [15] may be used, in which (2.36a) (2.36b) (2.36c) where (2.36d)

54 Predicting outdoor sound 40 with (2.36e) The error bound E(h) in (2.36b) and (2.36c) can be estimated from (2.36f) taking h as 1 and E(h) Only three to four terms are needed for the infinite sum of H(w x, w r ) and K(w x, w r ) in order to meet the requirement [16]. If one reduces h to 0.8, then the magnitude of the error term becomes less than In this case, it is found that summing the series up to the fifth term will be sufficient to guarantee the required accuracy [13]. Note also that a typographical error in Ref. [13] has been corrected in (2.36b). The above numerical solutions have proved to be accurate for predicting outdoor sound when compared with other accurate numerical schemes. Although (2.31) is a more accurate asymptotic solution, the approximation (2.34) is found to be sufficiently accurate for most practical purposes. Moreover, (2.34) is preferred because it can be rewritten in a form that leads to a useful interpretation of each term. The solution enhances the physical understanding of the problem. To interpret each term in (2.34), let s consider a simpler but related problem: a plane wave impinges on an impedance boundary at an oblique angle, θ. From physical considerations and associated mathematical analysis [17], we would expect the total sound field, p t to consist of a direct wave, p d and a specularly reflected wave, p r multiplied by the plane wave reflection coefficient, (2.37) Hence p t =p d +R p p r. In an analogous way, we expect that the sound field due to a point source consists of:

55 The propagation of sound near ground 41 (a) direct contribution from the source, and (b) a specularly reflected contribution from the image source in the boundary modified by the reflection coefficient, Q for spherical waves. With this in mind, we find it useful to rewrite the solution for the integral (2.28) in its approximate form as (2.38) So the sound field due to a point monopole source above a locally reacting ground becomes (2.39) Regrouping the second and third terms of (2.39), we can write the sound field in a physically interpretable form as (2.40a) where F(w), sometimes called the boundary loss factor, is given by (2.40b) and the term in the square bracket of (2.40a) may be interpreted as the spherical wave reflection coefficient Q=R p +(1 R p ) F(w). (2.40c) The sound field consists of two terms: a direct wave contribution and a specularly reflected wave from the image source. At grazing incidence θ=π/2, so that in (2.37) R p = 1 which simplifies (2.40a) considerably leading to

56 Predicting outdoor sound 42 (2.41a) with the numerical distance, w given by (2.41b) Note that the use of the plane wave reflection coefficient (2.37) instead of the spherical wave reflection coefficient (2.40c), for grazing incidence would have led to the prediction of zero sound field for both source and receiver on the ground. The contribution of the second term in Q to the total field acts as a correction for the fact that the wavefronts are spherical rather than plane. This contribution has been called the ground wave, in analogy with the corresponding term in the theory of AM radio reception [18]. The function F(w) describes the interaction of a curved wavefront with a ground of finite impedance; if the wave front is plane (R 2 ) then w and F 0 and if the surface is acoustically hard, then β 0 which implies w 0 and F 1. There are many other accurate asymptotic and numerical solutions available and many numerical comparisons have been carried out but no significant numerical differences between various predictions have been revealed for practical geometries and typical outdoor ground surfaces. The formula (2.40a), which is known as the Weyl-Van der Pol formula in electromagnetic propagation theory [19], is the most widely used analytical solution for predicting sound field above a locally reacting ground in a homogeneous atmosphere. One of the interesting features of outdoor sound propagation is the existence of surface wave under certain ground conditions and geometries. The details of the surface wave will be discussed in section The sound field above a layered extended-reaction ground The use of (2.34) in predicting outdoor sound is satisfactory in many outdoor situations with different types of ground. However, there are some cases where it is unable to predict the ground effect accurately by modelling the ground surface as an impedance plane. Outdoor examples are porous road surfaces and a layer of snow. It is the case also with some porous materials used for sound absorption in buildings, industry or transport systems. The deficiency can be readily traced back to the fact that the index of refraction of the ground, n 1 is not sufficiently high to warrant the assumed condition of In this case, the refraction of sound wave depends on the angle of incidence as sound enters into the porous medium. This means that the apparent impedance depends not only on the physical properties of the ground surface but also, critically, on the angle of incidence. An improved formulation for the integral I in (2.20) is to use instead of β in the subsequent evaluation of the integral. Using the same transformation ((2.25) (2.27)), we can rewrite (2.20) in its full form as:

57 The propagation of sound near ground 43 (2.42) There are attempts to evaluate the above integral asymptotically with no further approximation [10]. Asymptotic expressions have been developed but the form is rather complicated and inconvenient to compute. A useful heuristic approximation is to replace with If one introduces an effective admittance, β e defined by then (2.42) can be recast as (2.43) (2.44) In view of our earlier analysis in section 2.3 and the analogous form of (2.28) and (2.44), we can see that both sound fields are identical [cf. (2.28) and (2.34)] except that the effective admittance (2.43) should be used instead of β for the sound field above a porous ground. We can even take a step backward and consider the boundary condition (2.10). It can be recast in a form of an effective admittance, cf. (2.43), as (2.45) In this analysis, there is an underlying assumption that the porous ground is a homogeneous and semi-infinite medium. In real life, there are many situations where there is a highly porous surface layer above a relatively non-porous substrate. This is the case, for example, with forest floors, freshly fallen snow on a hard ground or porous asphalt. Obviously, the assumption of a semi-infinite porous ground may not be adequate. It is important for us to derive the corresponding expression for the effective admittance of the surface of ground consisting of a hardback layer ground. To allow its derivation, let us consider the reflection of plane waves on such a ground: see Figure 2.3 and denote the layer thickness by d 1.

58 Predicting outdoor sound 44 Figure 2.3 Sound propagation from a point source over a hard-backed layer. We can use the technique described in section 2.2, although supplementary boundary conditions are required in order to satisfy the requirement for a hard-backed layer. The boundary conditions (2.8), and (2.9a) and (2.9b) remain unchanged but (2.9c) is replaced by (2.46) Combining (2.8), (2.9b) and (2.46), simplifying the expressions and making U 0 as the subject of the equation, we obtain (2.47) The quantity U 1 is determined by noting the condition for a hard backing, that is ( p 1 / z)=0 at z= d 1 where d 1 (>0) is the thickness of the porous medium. Hence, according to (2.46), Then, use of the expression in (2.47) yields (2.48) (2.49)

59 The propagation of sound near ground 45 Now we can extend the concept of effective admittance from the semi-infinite porous ground to that of a layer of finite thickness lying on a hard and impervious ground. In this case, the effective admittance can be deduced straightforwardly from (2.49) to give (2.50) At first sight, one might expect U 1 to be the plane wave reflection coefficient similar to U 0. If this were to be the case, then U 1 =1 for a rigid ground. However, this is not case as indicated in (2.48). A close examination reveals that the boundary planes are located at different levels. For U 0, the reflecting plane is situated at z=0 and, for U 1 it is located at z= d 1. As a result, a phase factor of is introduced in U 1. To have a consistent interpretation of the interaction at each interface, we introduce a plane wave reflection coefficient at each interface such that and (2.51a) Then (2.47) becomes (2.51b) (2.52) We can see from (2.52) that the plane wave coefficient of the air/ground interface, V 0 is a function of V 1 which, in turn, depends on the acoustical properties of subsequent layers. Typically, the reflection coefficient V 1 is 1 for a hard backing, 1 for a pressure-release backing and 0 for an anechoic backing (which absorbs all incoming sound energy). Between these limiting situations, there is a host of cases for different types of grounds. An interesting point worth noting is that the layer thickness d 1 tends to infinity if the ground is semi-infinite. The exponent factor, is negligibly small for a large value of d 1 because the imaginary part of K 1 is positive and non-zero. Hence (2.52) can be approximated by which has the same form as found previously for semi-infinite porous ground; see (2.14).

60 Predicting outdoor sound 46 (2.53) We can also estimate the minimum depth of the layer for which the ground can be treated as semi-infinite ground. A simple numerical calculation reveals that if Im(K 1 d 1 ) is greater than 6, say, then This suggests a simple condition for which the ground can be treated as a semi-infinite externally reacting ground, that is The minimum depth, d m to satisfy this condition depends on the acoustical property of the ground and the angle of incidence but we can consider two limiting cases. If we write k 1 =k r +ik x, then for normal incidence where κ=0 (or θ=0), the required condition is simply (2.54a) and for grazing incidence where κ=1 (or θ=π/2), the required condition is (2.54b) Often, it is found inadequate to model the ground as a single hard-backed layer. This may be the case, for example, with predicting propagation through forests. This motivates extension of the earlier analysis to allow for a multi-layered ground. Assume that the ground is composed of many layers (a total of L layers, say) of materials with different acoustical properties (see Figure 2.4).

61 The propagation of sound near ground 47 Each layer has a different but constant depth, d 1, d 2, d 3,, and so on. To facilitate the analysis, the location of each layer is denoted by z 0, z 1, z 2, z 3,, etc. with z 0 representing the height of the air/ground interface. In other words, Figure 2.4 Sound propagation from a point source over a multi-layer (n-th layer) ground. Again, we require the continuity of pressure and particle velocity at each layer: (2.55) In each layer, the pressure can be computed by summing the relevant incoming and outgoing waves in a form similar to (2.46) as follows: where (2.56a)

62 Predicting outdoor sound 48 (2.56b) The following definitions for the density ratio and the index of refraction of each layer (see (2.11)) are also found useful: (2.56c) and (2.56d) Note that in (2.55) and (2.56), the variable j varies from 1, 2, to L. Let s consider the j-th layer interface. By using the continuity conditions at the interface, we obtain (2.57a) (2.57b) We introduce the plane wave reflection coefficients V j (see (2.51b)) for all interfaces, which relate to the corresponding U j, as (2.57c) Eliminating B j and B j+1 by combining (2.57a) and (2.57b) and using V j in favour of U j through the use of (2.57c), we can write (2.58)

63 The propagation of sound near ground 49 Now we are in a position to determine the sound field above a layered porous ground by using (2.52) and (2.58). Suppose that the ground is assumed to consist of L rigid-porous layers and that the L-th interface is with a rigid plane. So we have V L =1 and according (2.58), V L 1 can be found by substituting j=l 1 to give (2.59) Next we substitute (2.58) into (2.52), put j=1, 2, 3,, L 2 in turn, and use (2.59) for V L 1. Then, in principle, a closed-form analytic expression for the plane wave reflection coefficient V 0 can be determined. Its derivation is fairly easy if L is small. When L=1, that is a single porous medium resting on a hard ground, we can obtain the same expression as shown in (2.49). For a double layer with L=2, V 0 becomes (2.60a) where (2.60b) It is interesting to consider some limiting cases of (2.60). If d 2 becomes very small, then (2.60a) can be reduced to (2.49), which is the expression for a hardback layer. If d 1 is very large, then (2.60a) can be simplified to the expression given in (2.53), which is the expression for a semi-infinite porous medium. As before, the effective admittance of the double layer can be inferred from (2.60a) to give (2.61a) where g 1 can be expressed in terms of ς j, n j and θ

64 Predicting outdoor sound 50 (2.61b) where ς j and n j are defined by (2.56c) and (2.56d). It is possible to derive an expression for the effective admittance for arbitrary number of layers. Even though the expression becomes increasingly complex for large L, obtaining its numerical values should be simple with the help of digital computers. Nevertheless, since sound waves can seldom penetrate more than a few centimetres in most naturally occurring outdoor ground surfaces, ground models consisting of more than two layers are unlikely to be required. The sound energy can hardly go beyond the first two layers and any further variations in the admittance of the lower layers contribute little to the total sound field above the ground. Finally, we wish to point out that the use of effective admittance greatly simplifies the analysis and the subsequent interpretation of the theoretical predictions. Although its introduction is somewhat heuristic, the formula gives satisfactory predictions for most practical situations. In the following sections and subsequent chapters, we assume that the specific normalized admittance of the ground should be treated as an effective parameter. We shall use β and β e interchangeably unless otherwise stated. 2.5 The propagation of surface waves above a porous ground Use of the Weyl-Van der Pol formula allows one to accurately predict the sound propagation outdoors in neutral atmospheric conditions. Although, as discussed in Section 2.3, there are standard routines for computing the contribution of ground wave, it is instructive to consider two approximations that provide useful physical insight into its behaviour as a function of the relative size of w. It is sufficient to consider the boundary loss factor, F(w) when we consider the ground wave term (second term of (2.40c)). For w <1, which implies small source/receiver separations and large impedances, (2.62a) and, for w >1, which requires large source/receiver separations and small impedances, (2.62b) where w x is the imaginary part of the numerical distance, w and H( ) is the Heaviside Step function defined by

65 The propagation of sound near ground 51 The Heaviside function arises as a result of the definition of the complimentary error function, erfc(x) for positive and negative arguments, that is, erfc( x)= 2 erfc(x). To satisfy the condition of w >1 for different frequencies and ranges, we consider ranges greater than 50 m and frequencies less than 300 Hz. In this case, the Heaviside term in (2.62b) makes a significant contribution. By substituting (2.62b) into (2.40a), we see that it gives the usual form of a surface wave, decaying principally as the inverse root of horizontal range and exponentially with height above the ground. To allow a precise expression for the surface wave, the more accurate asymptotic expansion (2.29) for (2.28) is required. The surface wave contribution is given by [12] (2.63a) in which, it will be turned on when Im(x 0 )<0, that is (2.63b) If we use the approximations (2.32a) and (2.32b) in (2.63a), we obtain an approximate expression for the surface wave as This is identical to the expression implied in (2.62b), as indeed, it should be the case. By use of the condition (2.63b), we can demonstrate that the required condition can be simplified slightly to By writing β=β r iβ x (or Z=Z r +iz x ), we can represent a cut-off for the existence of the surface waves as

66 Predicting outdoor sound 52 (2.63c) At grazing incidence, the condition for the existence of the surface wave is For typical outdoor surface with large impedance, where β 0, the condition is simply that the imaginary part of the ground impedance (the reactance) is greater than the real part (the resistance). The analysis is much simpler if we approximate Im(x 0 ) by (see (2.33b)), in which case, the cut-off for the surface wave is approximated by The condition can be rearranged in a more recognizable form as (2.64) which is an equation for a circle with centre at ( 2/cos θ, 2/cos θ) and radius of The condition (2.62) for the existence of surface waves can be summarized in Figure 2.5. As discussed in the last section, the ground wave contribution (the term in square brackets in (2.40a)), represents a contribution from the vicinity of the image of the source in the ground plane effectively a diffusing of the image for spherical wave incidence compared with the plane wave incidence. However, the surface wave is essentially a separate contribution propagating close to and parallel to the porous ground surface, and is associated with elliptical motion of air particles as the result of combining motion parallel to the surface with that normal to the surface in and out of the pores. The relative importance of each term will be shown when we discuss various ground impedance models in Chapter 4. We also note that the surface wave is predicted to have a speed less than the speed of sound in air. This forms a basis for detecting the surface wave as, at sufficient range, it should reach the receiver as a second arrival separating from the main body waves in experiments using a pulse sound source.

67 The propagation of sound near ground 53 Figure 2.5 The condition for the existence of surface waves for sound propagation over a ground surface. Despite the clear theoretical importance of the surface wave in sound propagation close to ground [20], its existence was the subject of considerable controversy in the early 1980s both from the theoretical and experimental points of view. The theoretical controversy was resolved in the late 1980s by Raspet and Baird [21]. They have analysed the propagation of a spherical wave above a complex impedance plane. They show that the existence of the surface wave is independent of the body wave in air. By taking the limit where the upper half space (air) becomes incompressible, they prove that the surface waves can still exist. Strictly speaking, the existence of an acoustic surface wave above a porous boundary is not really in doubt. Attenborough and Richards [22] conducted a more sophisticated analysis of the problem where the elasticity and porosity of the soil were taken into consideration. They show that the acoustic surface wave is one of two possible surface waves, the other being an air-coupled Rayleigh wave that travels below and parallel to the surface. At low frequencies, and under certain circumstances of ground impedance and geometry, the acoustic surface wave above a porous boundary is predicted to produce total pressure levels in excess of those that would be found over an acoustically rigid boundary. Careful indoor measurements in the mid-seventies, using sources of continuous sound and artificial surfaces, have confirmed this phenomenon and have shown that the surface wave term accurately predicts the acoustic field near grazing

68 Predicting outdoor sound 54 incidence and its attenuation with height above the surface [23, 24]. In their experiments, the artificial surface was designed such that the reactive part of the impedance was contrived to be much greater than the resistive part, thus artificially increasing the contribution of surface waves at low frequencies. Indoor experiments [25], using a pulse sound source above lighting diffuser lattices mounted on flat rigid boards, have been successful in demonstrating the surface wave as a separate arrival from that of a body wave arriving earlier. More recent indoor experiments, on sound propagation over impedance discontinuities [26] and sound propagation over the convex impedance ground [27], show conclusive evidence of the existence of the surface waves. On the other hand, early outdoor pulse experiments [28] in neutral atmospheric conditions at long range failed to register separate arrivals of the surface wave and direct wave. This is largely due to the fact that the resistive impedance components of most outdoor ground surfaces give rise to appreciable exponential attenuation along the surface. This outweighs the advantage otherwise possessed by the surface pressure component in that it decays with the inverse square root of range rather than inversely with range, as is true for other components. The ground types that are most likely to produce measurable surface waves are those that may be modelled as thin hardbacked layer (see Chapter 4). One of the most favourable ground surfaces is a thin layer of snow over a frozen ground. By firing blank pistol shots as the sound pulse, Albert confirmed the surface wave propagation for his outdoor experiments over snow [29]. In fact, Piercy et al. [18] have demonstrated, as early as in 1977, the presence of surface waves in sound propagation outdoors. Nevertheless, the existence of the surface wave above an impedance plane in a homogeneous atmosphere is now beyond doubt [30]. 2.6 Experimental data and numerical predictions To enable the prediction of sound propagation outdoors, the impedance of ground surface is required. There are a number of models for the acoustical properties of outdoor ground surfaces in which two semi-empirical models, namely a single-parameter model and a two-parameter model, are most frequently used. In Chapter 3, we shall discuss the basis for these and other ground impedance models. For the moment, it is sufficient just to quote the formulations so that we can illustrate ground effects on outdoor sound through numerical computations and comparisons with experimental data. The single-parameter model, due to Delany and Bazley [31], describes the propagation constant, k and normalized surface impedance, Z by a single adjustable parameter known as the effective flow resistivity, σ e which has units of Pa s m 2. It is called effective since in modelling outdoor ground surfaces, it rarely takes a value equal to the actual flow resistivity, that is a measure of the drop in pressure per unit length in the direction of a flow of air at unit speed. The Delany and Bazley model is suitable for a locally reacting ground as well as for an extendedreaction surface. The propagation constant and normalized surface impedance are determined according to (2.65a)

69 The propagation of sound near ground 55 (2.65b) For locally reacting ground, we can simply use (2.65b) to calculate the impedance for the prediction of outdoor sound. There is a slight complication for the extended-reaction ground but we can combine (2.65) to determine the ratios of sound speed and density of the air and ground (cf. (2.11)) by (2.66a) and (2.66b) which allows the effective impedance to be computed. Nevertheless, use of the single-parameter model to calculate the sound field permits us to study the contribution of surface waves straightforwardly. For simplicity, it is adequate just to consider a locally reacting ground. The magnitude and phase of impedance are given, respectively, by (2.67a) (2.67b) A close inspection of (2.66b) reveals that the phase angle of impedance varies from π/2 for (f/σ e ) 0 to 0 for (f/σ e ). Also, the effective flow resistivity varies from about 15 kpa s m 2 (for a very soft ground such as snow covered ground) to about 25,000 kpa s m 2 for a road surface made with hot-rolled asphalt.

70 Predicting outdoor sound 56 Let us now examine the numerical distance, w. We can expand (2.33b) by writing to give (2.68a) Given the ranges of effective flow resistivity, frequencies (10 Hz 10 khz) and source/receiver geometries of interest, the magnitude of the numerical distance varies from 0.01 to over 100. The phase of the numerical distance is limited to ±π/2. As discussed in section 2.3, the boundary loss factor F(w) plays an important role in neargrazing propagation. Hence it is worth showing its variation with the magnitude of the numerical distance. For simplicity, we consider the case of grazing propagation when the source and receiver are located on the ground surface In this case the numerical distance (2.68a) simplifies to (2.68b) which is essentially the same as (2.41b) except that w is expressed in the polar form. Figure 2.6 shows a plot of 20 log F(w) versus w for different phase angles of impedance, It is of interest to point out that the boundary loss factor is greater than 1 (indicating enhancement of sound fields) for a certain range of w when This is due to the addition of the surface wave component in the propagation of the outdoor sound. In this case, as shown in (2.68b), the numerical distance has a negative imaginary part.

71 The propagation of sound near ground 57 Figure 2.6 Magnitude of F(w) versus the numerical distance w for various phase angles of surface impedance. Source and receiver are located on the ground. At this point, it is worth mentioning that there is a minor typographical error in Ref. [30] since in figure 6 of that reference, a similar result is shown but w 2 is plotted instead of w as stated. To illustrate the relative importance of the various contributions of the total sound field, it is instructive to present some of the classical Parkin and Scholes data obtained in a series of measurements at two disused airfields in the 1960s using a fixed jet engine source [32, 33]. Figures 2.7(a) and (b) [34] show data for the corrected difference in levels at 1.5 m height between a reference microphone and a remote microphone at different ranges ( m for Figure 2.7(a) and 35 m for Figure 2.7 (b)) [34]. These Figures show also theoretical predictions using a single-parameter model for the surface impedance (see (3.2)). The predicted contributions to the total field is divided into the direct (D) wave, the reflected (R) wave, the ground wave (G) and the surface (S) wave. We can see from the theoretical fits to the experimental data that the ground waves and surface waves are major carriers of environmental noise at low frequencies over long distances.

72 Predicting outdoor sound 58 Figure 2.7 (a) Parkin and Scholes data ( ), reproduced from [34], for the corrected difference in levels from a fixed (1.82 m high) jet engine source between receivers at ranges of 19.5 m and m and at 1.5 m height over an airfield. Predictions use Delany and Bazley s impedance model with σ e =300 kpa s m 2 and show the individual contributions from the direct wave (D), the plane-wave reflected component (R), the ground wave (G) and the surface wave (S). (b) Data for sound level re free field obtained above snow (20 cm thick) at a range of 35 m (reproduced from [35]). Predictions use Delany and Bazley s impedance model (3.1, 3.2) with σ e =12 kpa s m 2. Again the predicted individual components are shown. The single-parameter model has been used to predict the impedance of a layered ground surface, which is subsequently used in the prediction of sound propagation outdoors. Nicolas et al. [35] have reported an extensive series of measurements over layers of snow from 5 to 50 cm thick and at propagation distance up to 15 m. They have compared their experimental data with tolerable success, but they were unable to obtain reasonable fits of data at low frequencies by assuming either a semi-infinite ground or a hard-backed single

73 The propagation of sound near ground 59 layer ground structure. Li et al. [12] have improved the agreement between predictions and data by introducing a double layer model. Figure 2.8 shows the measured behaviour over snow layers, nominally 6, 8 and 10 cm thick, together with various predictions. Table 2.1 shows the values of the effective flow resistivities and layer thicknesses used for the predictions on Figure 2.8. The double layer predictions (broken lines) are in somewhat better agreement with data than those assuming a single hard-backed layer (dotted lines) or a semi-infinite layer (continuous lines). Figure 2.8 Data and predictions for sound propagation over snow. The excess attenuation data (reported in [35]) were obtained with source height 0.6 m, receiver height 0.3 m and range 7.5 m. (a) 6-cm thick layer of snow over ice. (b) 8-cm thick layer of new snow over asphalt, (c) 10-cm thick layer of freshly fallen snow above 50 cm of old hardened snow covered previously by a crust of ice.

74 Predicting outdoor sound 60 Table 2.1 The layer thickness and flow resistivities used for the predictions shown in Figure 2.8 Ground type (a) (b) (c) Hard-backed layer (dotted line) σ e =15 kpa m s 2 σ e =15 kpa m s 2 σ e =10 kpa m s 2 d=6 cm d=8 cm d=10 cm Semi-infinite ground (solid line) σ e =15 kpa m s 2 σ e =15 kpa m s 2 σ e =10 kpa m s 2 Double layer (broken line) σ 1 =10 kpa m s 2 d 1 =8 cm σ 2 =15 kpa m s 2 d 2 =3 cm σ 1 =10 kpa m s 2 d 1 =10 cm σ 2 =55 kpa m s 2 d 2 =1 cm σ 1 =8 kpa m s 2 d 1 =12 cm σ 2 =15 kpa m s 2 d 2 =3 cm Another popular ground impedance model is the two-parameter model proposed by Attenborough [36] and revised by Raspet and Sabatier [37] see (3.13). Although this model is suitable only for a locally reacting ground, it allows better agreement with data obtained over many outdoor ground surfaces than that obtained with the single-parameter model. Two adjustable parameters, the effective flow resistivity (σ e ) and the effective rate of change of porosity with depth (α e ), are used. The impedance of the ground surface is predicted by (2.69) The effective rate of change of porosity with depth has units m 1. Of course, some improvement of the agreement between predictions and data is to be expected with two adjustable parameters instead of one. Impedance models using even more parameters will be discussed in Chapter 3. Finally in this section, we illustrate in Figure 2.9, the effects on the excess attenuation of the changes in the values of σ e and α e. It can be seen that the effect of increasing σ e is to increase the frequency of the ground effect dip, whereas the effect increasing α e is to increase the magnitude of the dip. 2.7 The sound field due to a line source near the ground We end this chapter by considering a closely related problem: the sound field due to a line source above a ground surface. This is a 2-D problem which is often used to model the traffic noise in highways. The solution can be obtained readily by using a similar approach as discussed earlier for the 3-D case. Let us use a rectangular coordinate system, x (x, z) with a line source located at x s (0, z s ). Without loss of generality, we assume x>0 as the field is symmetry for the regions of x>0 and x<0. The problem of finding the required Green s function, g m (x) can be stated as

75 The propagation of sound near ground 61 (2.70a) subject to the impedance boundary condition of (2.70b) at the ground surface z=0. Again, the admittance, β should be treated as an effective parameter where it can be used to model either a locally reacting or Figure 2.9 Predicted sensitivity of excess attenuation spectra to changes in effective flow resistivity σ e and the rate of change of porosity with depth α e in the two-parameter model (2.68) for the effective impedance of the ground surface.

76 Predicting outdoor sound 62 an extended-reaction ground surface. Application of a one-fold Fourier transform on both sides of (2.70a) and (2.70b) leads to (2.71a) (2.71b) where and the 1-D Fourier transform pair is defined by (2.71c) (2.71d) and (2.71e) With the boundary condition (2.71b), we can solve (2.71a) readily to yield (2.72) Substituting (2.72) into (2.7 1e), we can express the Green s function in an integral expression that can further be spilt into three terms as follows:

77 The propagation of sound near ground 63 (2.73) The evaluation of the integral can be simplified considerably if we use a polar coordinate centring on the source for the first integral and on the image source for the other two integrals. In the polar coordinate system, we have where R 1 and R 2 are the respective path lengths for the direct and reflected waves, respectively; and θ, are their corresponding polar angles. The required Green s function can be transformed to (2.74) The first and second integral can be identified as the sound field due to the source and its image source. These two integrals can be expressed in terms of Hankel functions as (2.75a) (2.75b) The third integral of (2.74), I β say, can not be evaluated exactly, but an asymptotic method may be applied to evaluate the integral and yields

78 Predicting outdoor sound 64 (2.76) Applying the same approximations as used in the 3-D case, that is, and we can express g m (x) by substituting (2.75a), (2.75b) and (2.76) into (2.74) to give The formula can be rearranged further to give the more recognizable form (2.77) (2.78) Hence the 2-D Green s function can be expressed in the classical form corresponding to the Weyl-Van der Pol formula for a point source. References 1 A.N.Sommerfeld, Propagation of waves in wireless telegraphy, Ann. Phys. (France) 333(4): (1909); 386 (25): (1926). 2 A.Banos, Jr, Dipole Radiation in the Presence of Conducting Half-space (Pergamon, New York) Ch. 2 4 (1966). 3 I.Rudnick, Propagation of an acoustic waves along an impedance boundary, J. Acoust. Soc. Am., 19: (1947). 4 K.U.Ingard, On the reflection of a spherical wave from an infinite plane, J. Acoust. Soc. Am., 23: (1951). 5 D.I.Paul, Acoustical radiation from a point source in the presence of two media, J. Acoust. Soc. Am., 29: (1959). 6 A.R.Wenzel, Propagation of waves along an impedance boundary, J. Acoust. Soc. Am., 55: (1974). 7 R.J.Donate, Spherical-wave reflection from a boundary of reactive impedance using a modification of Cagniard s method, J. Acoust. Soc. Am., 60: (1976). 8 C.F.Chien and W.W.Soroka, Sound proagation along an impedance plane, J. Sound Vib., 43:9 20 (1976). 9 C.F.Chien and W.W.Soroka, A note on the calculation of sound propagation along an impedance plane, J. Sound Vib., 69: (1980). 10 K.Attenborough, S.I.Hayek and J.M.Lawther, Propagation of sound above a porous half-space, J. Acoust. Soc. Am., 68: (1980). 11 T.Kawai, T.Hidaka and T.Nakajima, Sound propagation above an impedance boundary, J. Sound Vib., 83: (1982).

79 The propagation of sound near ground K.M.Li, T.Waters-Fuller and K.Attenborough, Sound propagation from a point source over extended-reaction ground, J. Acoust. Soc. Am., 104: (1998). 13 M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York (1972). 14 R.Wong, Asymptotic Approximations of Integrals, Academic Press, Inc., London (1989). 15 F.Matta and A.Reichel, Uniform computation of the error function and other related functions, Math. Comput., 25: (1971). 16 R.K.Pirinchieva, Model study of sound propagation over ground of finite impedance, J. Acoust. Soc. Am., 90: (1991). 17 A.P.Dowling and J.E.Ffowcs Williams, Sound and Sources of Sound, Ellis Horwood Ltd., Chichester Ch. 4 (1983). 18 T.F.W.Embleton, J.E.Piercy and N.Olson, Outdoor sound propagation over ground of finite impedance, J. Acoust. Soc. Am., 59: (1976). 19 L.M.Brekhovskikh, Waves in Layered Media, Academic Press, New York (1980). 20 K.Attenborough, Predicted ground effect for highway noise, J. Sound Vib., 81: (1982). 21 R.Raspet and G.E.Baird, The acoustic surface wave above a complex impedance ground surface, J. Acoust. Soc. Am., 85: (1989). 22 K.Attenborough and T.L.Richards, Transmission of sound at a porous interface, Proc. eleventh ICA, Paris, p. 178 (1 July 1983). 23 S.I.Thomasson, Sound propagation above a layer with large refraction index, J. Acoust. Soc. Am., 61: (1977). 24 R.J.Donate, Model experiments on surface waves, J. Acoust. Soc. Am., 63: (1978). 25 C.H.Howorth and K.Attenborough, Model experiments on air-coupled surface waves, J. Acoust. Soc. Am., 92:2431 (A) (1992). 26 G.A.Daigle, M.R.Stinson and D.I.Havelock, Experiments on surface waves over a model impedanve using acoustical pulses, J. Acoust. Soc. Am., 99: (1996). 27 Q.Wang and K.M.Li, Surface waves over a convex impedance surface, J. Acoust. Soc. Am., 106: (1999). 28 C.G.Don and A.J.Cramond, Impulse propagation in a neutral atmosphere, J. Acoust. Soc. Am., 81: (1987). 29 D.G.Albert, Observation of acoustic surface waves propagating above a snow cover, Proc. fifth Symposium on Long range Sound Propagation, (1992). 30 L.C.Sutherland and G.A.Daigle, Atmospheric sound propagation, in Handbook of Acoustics, edited by M.J.Crocker, JohnWiley & Sons, New York (1998). 31 M.E.Delany and E.N.Bazley, Acoustical properties of fibrous absorbent materials, Appl Acoust., 3: (1970). 32 P.H.Parkin and W.E.Scholes, The horizontal propagation of sound from a jet close to the ground at Radlett, J. Sound Vib., 1:1 13 (1965). 33 P.H.Parkin and W.E.Scholes, The horizontal propagation of sound from a jet close to the ground at Hatfield, J. Sound Vib., 2: (1965). 34 K.Attenborough, Review of ground effects on outdoor sound propagation from continuous broadband sources, Appl. Acoust., 24: (1988). 35 J.Nicolas, J.L.Berry and G.A.Daigle, Propagation of sound above a finite layer of snow, J. Acoust. Soc. Am., 77:67 73 (1985). 36 K.Attenborough, Ground parameter information for propagation modeling, J. Acoust. Soc. Am., 92: (1992); see also R.Raspet and K.Attenborough, Erratum: Ground parameter information for propagation modeling, J. Acoust. Soc. Am., 92: 3007 (1992). 37 R.Raspet and J.M.Sabatier, The surface impedance of grounds with exponential porosity profiles, J. Acoust. Soc. Am., 99: (1996).

80 Chapter 3 Predicting the acoustical properties of outdoor ground surfaces 3.1 Introduction In many outdoor noise prediction schemes, ground surfaces are considered as either acoustically hard, which means that they are perfectly reflecting, or acoustically soft, which implies that they are perfectly absorbing. According to ISO [1], any ground surface of low porosity is to be considered acoustically hard and any grass-, tree-, or potentially vegetation-covered ground is to be considered acoustically soft. Although this might be an adequate representation in some circumstances, it oversimplifies a considerable range of properties and resulting effects. Even the category of ground known as grassland involves a wide range of ground effects. Where the main objective is to predict outdoor sound at long range including effects of discontinuous ground, meteorological effects, diffraction by natural and artificial barriers and topography, it is sensible to use the simplest possible descriptions of the acoustical properties of ground surfaces. However, in relatively simple propagation conditions, for example over flat grassland around airports, better accuracy should result from attempts to characterize the ground more completely. The interaction of sound with the ground includes several phenomena known collectively as ground effect. A convenient indicator of this interaction is the spectrum of the ratio of the total sound level at a receiver to the direct sound that would be present in the absence of the ground surface. This has been called the excess attenuation due to the ground surface and has been described in Chapter 2. As long as the ground may be considered as locally reacting, that is the surface impedance is independent of the angle of incident sound, the excess attenuation at a given receiver may be calculated from knowledge of the surface impedance and the source-receiver geometry. Most naturally occurring outdoor surfaces are porous. As a result of being able to penetrate the porous surface, ground-reflected sound is subject to a change in phase as well as having some of its energy converted into heat. The acoustical properties of porous ground are affected by the ease with which air can move in and out of the ground. This is indicated by the flow resistivity which represents the ratio of the applied pressure gradient to the induced volume flow rate per unit thickness of material. If the ground surface has a high flow resistivity, it means that it is difficult for air to flow through the surface. Typically this would result from very low surface porosity. Hotrolled asphalt has a very high flow resistivity whereas freshly fallen snow has a low flow resistivity. Section 3.2 reviews the various models and parameters that have been used to calculate the impedance of smooth ground surfaces. As well as being porous, most naturally occurring outdoor ground surfaces are rough at some scale. The effects of roughness scales much smaller than the incident sound wavelengths are considered

81 Predicting the acoustical properties 67 separately in section 3.3. At low frequencies such as those encountered in blast noise, the elasticity of the ground may have an influence on outdoor sound and this possibility is explored in section 3.4. Methods of measuring ground impedance are described in Chapter Models for ground impedance Empirical and phenomenological models A widely used, semi-empirical single-parameter model, due to Delany and Bazley, for propagation constant and relative normal surface impedance ([2, 3], also introduced in Chapter 2) suggests that (3.1) Z c = X i0.087x 0.732, (3.2) where X=ρ 0 f/r s, ρ 0 being the density of air, R s the flow resistivity and f the frequency. These equations are based on best fits to a large number of (impedance tube) measurements made with many fibrous materials having porosities close to 1. Delany and Bazley suggest that the range of validity of their relationships is 0.01< X<1.0. Even for high porosity materials, many authors have found that (3.2) predicts the wrong frequency dependence for normal surface impedance at low frequencies and have suggested various modifications to these formulae [4, 5, 6]. Soils have porosities much less than 1 and rather higher flow resistivities than fibrous materials. Consider a soil of flow resisitivity 120 kpa s m 2. At 500 Hz, X=0.0005, which is well below the suggested range of validity for the semi-empirical relationships (3.1) and (3.2). Nevertheless, (3.1) and (3.2) have been used with tolerable success to characterize outdoor surface impedances as long as X=f/R eff ; where R eff, the effective flow resistivity, is treated as an adjustable parameter. For the high flow resistivities typical of soils, (3.2) predicts a reactance that exceeds the resistance over an appreciable frequency range. This is the kind of impedance behaviour that can be predicted for a rigid-porous layer (see section 3.2.3) or a porous semi-infinite medium with a rough surface (see section 3.3). Grass-covered surfaces often have a layered structure in which the root zone near the surface has a lower flow resistivity than the substrate ([7], also see section 3.4). Even though the wave penetrating the pores is predicted to have an attenuation that increases with the square root of frequency, near-surf ace layers may influence ground surface impedance. Rasmussen [7] found it possible to improve agreement with short-range propagation data over a grass-covered surface by assuming a hard-backed layer structure and by using

82 Predicting outdoor sound 68 equations (3.1) and (3.2) with the formula for the impedance of a hard-backed layer of thickness d, Z(d)=Z c coth( ikd). (3.3) Morse and Ingard [8] have suggested a phenomenological model for the acoustic behaviour of porous materials. This introduces three parameters: flow resistivity (R s ), porosity (Ω) and a structure factor (K). Their model may be expressed as (3.4) where k 0 =ω/c 0. Thomasson [9] has obtained good agreement with indoor and outdoor data by using a four-parameter version of the phenomenological theory, that is using (3.4) in (3.3), to predict the surface impedance of a hard-backed layer. Hamet [10] has used a modified form of this model, which allows for the frequency dependence of viscous and thermal effects (see section 3.2.2), to predict the acoustical properties of porous asphalt. His model may be written as follows: (3.5) where F µ =1+iω µ /ω, F 0 =1+iω 0 /ω, ω µ =(R s /ρ 0 )(Ω/K), ω 0 =ω µ (K/N PR ) and N PR is the Prandtl number for air Pore-based microstructural models The acoustical properties of the ground may be modelled as those of a rigid-porous material and characterized by a complex density, containing the influence of viscous effects, and a complex compressibility, containing the influence of thermal effects. Thermal effects are much greater in air-filled materials than in water-filled materials. Precise forms of these quantities may be obtained by considering a micro structure of narrow pores or tubes. This offers a more rigorous scientific basis for ground impedance prediction than phenomenological [8] or semi-empirical [2] approaches. According to Stinson [11], if the complex density in a (single) uniform pore of arbitrary shape is written as (3.6)

83 Predicting the acoustical properties 69 where λ is a dimensionless parameter, then the complex compressibility is given by (3.7) where is the adiabatic compressibility of air. H(λ) has been calculated for many ideal pore shapes [11 14] including circular capillary, infinite parallel sided slit, equilateral triangle and rectangle. The results are listed in Table 3.1. The dimensionless parameter λ may be related to the (steady) flow resistivity (R s ) of the bulk material through use of the Kozeny-Carman formula [15], (3.8) where the hydraulic radius, r h = wetted area/perimeter, s 0 is a steady flow shape factor and q 2 is tortuosity, defined as the square of the increase in path length per unit thickness of material due to deviations of the steady-flow path from a straight line. This means that it is possible to write λ 2 =2s 0 ωρ 0 q 2 /ΩR s where, for the pore shapes mentioned previously, r h and s 0 are defined in Table 3.2. Values of porosity and tortuosity for some granular materials are listed in Table 3.3 [16 20]. Once the complex density (ρ(ω)) and complex compressibility (C(ω)) of the individual pores are known, then the bulk propagation constant (k(ω)) and Table 3.1 Complex density functions for various pore shapes (where ν=µ/ρ 0 ) Pore shape λ H(λ) Slit (width 2b) b(ω/ν) 0.5 Cylinder (radius a) α(ω/ν) 0.5 Equilateral triangle (side d) Rectangle (sides 2a, 2b) Table 3.2 Hydraulic radius and steady flow shape factors Pore shape r h s 0 Slit (width 2b) b 1.5 Cylinder (radius a) a/2 1 Equilateral triangle (side d) 5/6 Square (side 2a) a/2 0.89

84 Predicting outdoor sound 70 Table 3.3 Some measured and calculated values of porosity and tortuosity Material Porosity Source/method of Tortuosity Source/method of (Ω) determination (q 2 ) determination Lead shot, 3.8 mm Measured by 1.6 Estimate from 1/Ω 0.5 (also fits diameter particles weighing acoustic data) Cell model predictions [16] Gravel, 10.5 mm grain size 0.45 Measured by weighing 1.55 Deduced by fitting surface admittance data Gravel, 5 10 mm grain size 0.4 Measured by weighing 1.46 Deduced by fitting surface admittance data 0.68 mm diameter glass beads Unspecified [17] Measured [18] Cell model predictions [16] Coustone 0.4 Unspecified [17] Measured [18] Clay granulate, later lite, 1 3 mm grain size [19] Measured [20] 1.25 Deduced by fitting surface admittance data (assuming porosity=0.52) Olivine sand Measured [20] Cell model predictions [16] relative characteristic impedance (Z c (ω)) of the bulk porous material may be calculated from (3.9a) (3.9b) Equation (3.9b) implies that the bulk compressibility C b (ω)=ωc(ω). There are various published methods that allow for arbitrarily shaped pores. Attenborough [21] scales the complex density function directly between pore shapes and introduces an adjustable dynamic pore shape parameter (s A ). For example, the bulk complex density function for parallel-sided slit pores is given by [14] (3.10) where

85 Predicting the acoustical properties 71 The pore shape parameter s A =1 for slit-like pores and 0.745<s A <1 for pore shapes varying between equilateral triangles and slits. If circular cylindrical pore shapes are used as the basis, then the dimensionless parameter may be written as This formulation retains explicit dependence on s A even in the low-frequency limit. Champoux and Stinson [12] have pointed out that to satisfy the correct limiting behaviour for the complex density, in particular that iωρ b R s as ω tends to zero, this method requires s A to be frequency-dependent. Their method of allowing for arbitrary pore shape is closer to that originally suggested by Biot [22] and the resulting pore shape parameter (s B ) has the advantage of being frequency-independent. Using this approach, with reference to the functions for slit-like pores, we write the complex density for the bulk material as (3.11) where (3.12) and 1<s B <1.342 for pore shapes varying from slits to equilateral triangles. This approach scales the viscosity correction or dynamic viscosity function F(λ) instead of the complex density function. Since F(λ) tends to unity in the low-frequency limit, this method of defining ρ b (λ) does not have any explicit dependence on a dynamic pore shape factor. In the low-frequency limit, the dependence on pore shape is only that implicit in the flow resistivity through (3.8).

86 Predicting outdoor sound 72 For a given bulk flow resistivity, porosity and tortuosity, although the complex density and complex compressibility, calculated from (3.6), (3.7) and Tables 3.1 and 3.2 are found to depend on pore shape, the complex bulk propagation constant, the complex characteristic impedance (given by (3.9)) and corresponding surface impedance of a hardbacked layer (given by (3.3)), are predicted to be relatively insensitive to pore shape [21]. This is true particularly for low flow resistivities and at frequencies less than a few thousand Hz. Other micro structural factors of significance in pore-based modelling are the variation of pore cross-sections along their lengths and the associated distributions of pore sizes and shapes. To allow simultaneously for arbitrary pore shapes and for pore cross-sections that change along their lengths, Johnson et al. [23] interpret 1/H(λ) as a dynamic tortuosity and introduce two characteristic lengths (Λ and Λ ), which represent two modifications of the dimensionless parameter λ used in (3.11) and (3.12). The simplest formulations for bulk material complex density and complex compressibility resulting from this approach, are based upon limiting forms for small and large characteristic lengths and may be written as [13], (3.13) where and (3.14) where In effect, the characteristic lengths introduce two dynamic pore shape factors s ρ (equivalent to s B ) and s C into complex density and complex compressibility respectively. Note that the formulations in equations (3.13) and (3.14) omit factors of 1/Ω and Ω

87 Predicting the acoustical properties 73 respectively from the definitions of bulk complex density and compressibility. While this gives rise to similar expressions for propagation constant to that given by (3.9a), the resulting definition of characteristic impedance differs from that given by (3.9b), which assumes that the wave impedance is identical to the impedance of a semi-infinite homogeneous rigid-porous material. Allard et al. [13] introduce a factor of (1/Ω) into their calculation of surface impedance. The need for two shape factors may be argued from the fact that the wider parts of each pore tend to be more important for the complex compressibility while the narrower pore cross-sections dominate the complex density. Theoretically it is expected that s C >s ρ, that is Λ >Λ, and typically this has been found to be the case. However, for arbitrary media, the viscous and thermal shape factors must be treated as independently adjustable parameters. In fitting acoustic data for glass beads, porous asphalt and a granular rubber compound values of s C <s ρ have been found to be necessary [14, 24]. A version of this model, based on the viscosity correction function for slit-like pores, which is likely to be more accurate at intermediate values of the dimensionless parameter, may be written as follows: (3.15) and (3.16) where It has been proved possible to calculate the shape factors (or characteristic lengths) for certain well-defined geometries (e.g. stacked cylinders [13] and stacked spheres [16]). Methods of measuring Λ by non-acoustic means and Λ by ultrasonic means have been published [25, 26]. By fitting data for fibrous materials [13] it has been found that s C 2s ρ (Λ 2Λ), whereas s C 5s ρ has been found appropriate for an ideal or model material having cylindrical pores with two diameters. Some values for the characteristic lengths in some granular materials are listed in Table 3.4 [27 30]. These show that the thermal characteristic length may exceed the viscous characteristic length by large factors. A theory for the acoustical characteristics of a medium consisting of identical stacked rigid spheres [16, 31] in terms of the radius of the spheres and packing

88 Predicting outdoor sound 74 Material Glass beads 0.68 ±0.12 mm 1.11±0.15 mm 1.64±0.15 mm Bulk density (kg m 3 ) Table 3.4 Values of characteristic dimensions and other parameters for some granular materials Viscous Λ(µ m) 93 a 16.8 a 18.8 a Thermal Λ (µ m) Porosity Tortuosity Flow resistivity R s (kpa s m 2 ) a 1.87 a 1.88 a Ref a [27] a 9.86 a Sand b [28] Sand b [29] Snow N/A [30] Notes a Calculated from cell model [16, 31] using measured Ω, q 2 and Λ. b Calculated assuming a density of 2600 kgm 3 for the grains. fraction has been developed. This model allows prediction of acoustical properties from knowledge only of (identical) grain size and porosity. For stacked identical spheres of radius R forming a granular medium with porosity h, the thermal characteristic length may be expressed as [31] (3.17) and the relationship between thermal and viscous characteristic lengths is (3.18) where (3.19) The grain shape in soils is usually far from spherical. Nevertheless the model for stacked spheres offers a method for estimating the likely values of the characteristic lengths in a granular medium from knowledge of the mean grain size and this makes it more feasible

89 Predicting the acoustical properties 75 to use multi-parameter models for predicting acoustical properties of arbitrary granular media. Hence (3.16) is replaced by (3.20) A method of allowing for a log-normal pore-size distribution, while assuming pores of identical (slit-like) shape has been developed [14] based on the work of Yamamoto and Turgut [32]. The viscosity correction function for a log-normal slit-pore distribution is given by (3.21) where the distribution in units. The bulk complex density is calculated from and σ is the standard deviation of (3.22) so that, using Stinson s relationship (3.7), (3.23) An advantage of this way of modelling the acoustical properties of air-filled granular materials is that as long as the influence of pore shape is ignored (by assuming identical pores of a certain geometrical shape) and the flow resistivity, porosity, tortuosity and pore-size distribution are available from non-acoustical measurements, it does not require any adjustable parameters. The numerical integration introduced in (3.21) can be avoided by means of Padé approximations [33, 34]. For a medium consisting of identical pores of arbitrary shape (3.4c) may be written as follows: (3.24a)

90 where and F(ε) is the viscosity correction function. Low- and high-frequency asymptotes for the viscosity correction function may be expressed as F(ε)=1+θ 1 ε 2 +O(ε 4 ),+O(ε 4 ), ε 0 (3.24b) and Predicting outdoor sound 76 F(ε)=θ 2 ε+o(1), ε. Hence, a Padé approximation for the viscosity correction function in a medium with a log-normal pore-size distribution has been proposed in the form (3.25) where a 1 =θ 1 /θ 2, a 2 =θ 1 and b 1 =a 1. Values of these coefficients can be determined analytically for certain pore shapes (see Table 3.5). However, as will be discussed in section 3.2.5, the influence of pore shape per se is relatively small Approximate models for high flow resistivities Equations (3.8)-(3.11) lead to simple approximations for characteristic impedance and propagation constant in the limit of small λ (corresponding to low frequencies and high flow resistivities). These approximations (based on approximation of Table 3.5 Expressions for coefficients in (3.23) where ξ=(σ In 2) 2 and σ is the standard deviation of the log-normal distribution in (such that pore dimension in ) Coefficients in (3.20) Slit-like pores Equilateral triangles Circular pores θ 1 (6/5)e 4ξ 1 (7/10)e 4ξ 1 (4/3)e 4ξ 1 θ 2 cylindrical pore functions) may be written as follows: (3.26)

91 Predicting the acoustical properties 77 (3.27) where T replaces q 2 used previously and represents an effective flow resistivity. Here s p is equivalent to s A used earlier but determined using circular cylinder solutions as a basis rather than those used for slits. The equivalent expressions using slitpore functions as the basis are (3.28) (3.29) Further approximation for high flow resistivity and low frequency produces (3.30) (3.31) where that is R e =R eff /Ω 2. It is useful to note the equivalent results to (3.28) and (3.29), obtained by approximating the Biot/Stinson/Champoux model (equations (3.8) (3.10) and (3.12)), which are (3.32)

92 Predicting outdoor sound 78 (3.33) These are independent of pore shape (except through R s ) and give identical results to (3.26) and (3.27) if s p =0.5. Moreover (3.30) and (3.31) are equivalent to the phenomenological model (3.4) with structure factor equivalent to tortuosity but with the isothermal value of sound speed in the pores. After further approximation for high flow resistivity, (3.31) becomes (3.34) Equations (3.29) and (3.31), which are identical if s p is set equal to 0.5, may be regarded as single-parameter models for the surface impedance of a homogeneous semi-infinite rigid-porous medium with high flow resistivity, or at low frequency, where the single parameter is effective flow resistivity. They predict surface impedances in which real and imaginary parts are equal and decrease as the square root of frequency. Equations (3.28), (3.29) and (3.3) may be used together with the first two terms of the power series expansion of the hyperbolic cotangent to produce a simple approximation for the surface impedance of a hard-backed thin high-flow-resistivity layer [35]: (3.35) where d e =Ωd is an effective depth. In forest floors, the flow resistivity of the surface porous (litter) layer is low, and it lies above a porous substrate with finite impedance, so it is necessary to implement a suitable multi-layer model. This may be deduced, from transmission line analysis, by means of the following equation: (3.36) where Z 1 and Z 2 are the relative characteristic impedances of the upper porous layer and porous substrate, respectively, k is the propagation constant in the upper porous layer and d is the layer thickness.

93 Predicting the acoustical properties 79 An approximation for a thin layer with an impedance backing such that the backing impedance is much higher than that of the layer is given by [35] (3.37) where d e =Ωd and d is the layer thickness. Several authors have considered the surface impedance of a high flow resistivity, rigid-porous medium in which the porosity decreases exponentially with depth [35 38]. The most rigorous approximation for the surface of such a medium is [37] (3.38) where α e =(n +2)α/Ω, where n is a grain shape factor such that the tortuosity is given by Both (3.33) and (3.34) imply X>R (as long as α e is positive), and may be written genetically in the form (3.39) where and b=c 0 /8πγ. For a non-hard backed thin layer, α e =4/d e. An approximation for the surface impedance of a high flow resistivity porous medium with the porosity increasing exponentially with depth is given by (3.32) with negative α e. If α e is negative then (3.34) predicts a resistance that exceeds the reactance at all frequencies. An improved approximation at high frequencies uses either (3.22) and (3.23) or (3.29) instead of (3.27) or (3.30) and may be written [39] as either (3.40a) or

94 Predicting outdoor sound 80 (3.40b) where Note that R e and are identical if s p =0.5. These introduce a third parameter T e =T/Ω 2 which depends on the tortuosity and porosity and influences the high frequency values of the impedance. Taraldsen [40] has suggested that a model connecting permeability and porosity, of a similar form to that used in geophysics, transforms the (three parameter) phenomenological model (3.4) into a one-parameter form that gives rather similar results to the Delany-Bazley model (3.2). Taraldesen s model for relative admittance, β(=1/z), may be expressed as follows: (3.41a) (3.41b) where (3.41c) (3.41d) Y=10 log(r eff ) (3.41e)

95 Predicting the acoustical properties 81 Figure 3.1 Comparison between predictions of Taraldsen s one-parameter model (3.41) and the Delany-Bazley model (3.2) for R eff =12,500 Pa s m 2. Figure 3.1 shows an example comparison between Taraldsen s one-parameter model and the Delany-Bazley model for a relatively low effective flow resistivity R eff =12,500 Pa s m 2.

96 Predicting outdoor sound Relaxation models By viewing the viscous and thermal diffusion in porous materials as relaxation processes, Wilson [41 43] has obtained models for the acoustical properties of porous materials in simple forms that, nevertheless, enable accurate predictions over wide frequency ranges. His results may be expressed as (3.42a) (3.42b) where, for identical uniform pores, τ e and τ v, the thermodynamic and aerodynamic characteristic times respectively, are given by (3.43a) and τ e =N PR s B 2 τ v. (3.43b) Essentially this represents a single-parameter model (the parameter being either τ e or τ v ) for a given pore shape, flow resistivity and porosity. The relaxation model is very similar in form to Hamet s modification of the phenomenological model [10]. By matching to the Delany and Bazley relationships for their range of validity, Wilson [42] has obtained particularly simple approximate relationships: (3.43c) and

97 Predicting the acoustical properties 83 (3.43d) When substituted into (3.42), these relationships provide a single-parameter model that mimics (3.1) and (3.2) for 0.01<X<1.0 but is valid outside this range. For materials in which the pore cross-sections vary, Wilson has suggested a slightly more complicated form, again based on (3.42) but and are replaced by where τ 1 and τ h are appropriate low- and high-frequency versions of τ e and τ v. Wilson has shown that results equivalent to (3.15) and (3.16) may be obtained with τ le =τ he =N PR s 2 K τ v and In its most general form, this represents a four-or five-parameter model. Relaxation models have been found to be a particularly useful basis for numerical time domain calculations [43] Relative importance of microstructural parameters It is interesting to consider the relative importance of various microstructural features for the acoustical properties of ground surfaces. Figure 3.2 shows predictions of the surface impedance of hard-backed layers corresponding to a snow-like that is low Figure 3.2 Predicted surface impedance of a 0.05 m thick layer of snow-like material, with flow resistivity 10 kpa s m 2, porosity 0.6 and tortuosity 4/3, composed of stacked identical grains or identical pores with specific shapes. flow resistivity medium with identical pores but assuming various different pore shapes. The key shows the dimensions of the corresponding spheres and pores. Clearly the predictions are not very dependent on the pore shape for a given flow resistivity and

98 Predicting outdoor sound 84 porosity. The only significant difference is between the prediction of the cell model for stacked spheres and the uniform pore models for the real part of the surface impedance at low frequencies. This is consistent with the finding of Pride et al. [44] who considered pores with varying cross sections. Figure 3.3 shows the predicted effects of varying the standard deviation of the size distribution between 0 and 1 on the acoustical characteristics of a hard-backed layer rigid-framed porous media having values of flow resistivity and porosity intended to represent a grass-like medium. Broadening the pore-size distribution tends to increase the real part of the impedance while leaving the imaginary part relatively unaffected. The corresponding short-range excess attenuation spectra are shown in Figure 3.4. The effects are marginal and therefore unlikely to be distinguishable in practice. It will be shown in the next section that the predicted effects of small-scale surface roughness are much more significant. 3.3 Effects of surface roughness Introduction As well as being porous, many outdoor surfaces are rough. Surface roughness scatters the sound both coherently and incoherently. The relative strengths of the coherent and incoherent parts of the scattered energy depend on the mean Figure 3.3 Predicted effect on surface impedance of varying the width of the poresize distribution in a 0.05 m thick layer of a grass-like slit-pore rigid-framed porous medium with flow resistivity 200 kpa s m 2, porosity 0.4 and tortuosity 5/2.

99 Predicting the acoustical properties 85 Figure 3.4 Predicted influence of the impedance variation shown in Figure 3.2 on the excess attenuation spectrum for source height 0.3 m, receiver height 0.3 m and range 1 m. size of the roughness compared with the incident wavelength. On disked or vegetationcovered soil, the roughness is small compared to the wavelengths of sound in the frequency range of interest ( Hz). Such roughness may be described as smallscale and, mainly, it gives rise to forward (coherent) scattering in the direction of specular reflection. However once the roughness size approaches the wavelengths of interest, nonspecular (incoherent) scattering dominates and ground interference effects are destroyed. The propagation of underwater sound over (or under) rough boundaries has been a topic of continuing interest for at least 30 years. Less attention has been paid to the equivalent situations in atmospheric acoustics. An important conclusion of previous work [45, 46], with regard to the coherent part of the scattering from the surface is that the impedance or admittance of the boundary is modified by the presence of small-scale roughness. Clearly this can be considered to have an influence on the (spherical wave) reflection coefficient and hence the ground effect, even above a ground medium that would have been considered acoustically hard if smooth. Tolstoy [47] has distinguished between two theoretical approaches, for predicting the coherent field resulting from boundary roughness where the typical roughness height is small compared to a wavelength. These are the stochastic and the boss models. Both of these reduce the rough surface scattering problem to the use of a suitable boundary condition at a smoothed boundary. According to Tolstoy [47], the boss method, originally derived by Biot [47] and Twersky [48], has

100 Predicting outdoor sound 86 the advantages that (i) it is more accurate to first order, (ii) it may be used even in conditions where the roughness shapes introduce steep slopes and (iii) it is reasonably accurate even when the roughness size approaches a wavelength. An advantage of the boss approach to modelling roughness effects in outdoor sound calculations is that it may be used to predict propagation over artificially embossed surfaces both for experimental validation of theory and for designing useful ground effect over hard surfaces [50, 54]. Analytical approximations for the effective impedance of rough surfaces deduced from boss theory are described in the next subsection and experimental data for roughness effects are presented in Chapter Impedance models including rough surface effects Hard rough surfaces Twersky has developed a boss model [53 55] to describe coherent reflection from a hard surface containing semi-cylindrical roughnesses in which the contributions of the scatterers are summed to obtained the total scattered field. Sparse and closely packed distributions of bosses have been considered and interaction between neighbour scatterers has been included. His results lead to a real part of the effective admittance of the rough hard surface which may be attributed to incoherent scattering. Consider a plane wave incident on an array of semi-cylinders of radius a and mean centre-to-centre spacing b on an otherwise plane hard boundary (Figure 3.5). Figure D representation of a plane wave incident in the direction of vector k on a surface containing a regularly spaced grating of semi-cylinders of radius a. If the angle of incidence with respect to the normal is denoted by a and the azimuthal angle between the wave vector and the roughness axes is denoted by φ, Twersky s results for the effective relative admittance β of a rough hard surface containing non-periodically spaced 2-D circular semi-cylinders are

101 Predicting the acoustical properties 87 β=η iξ (3.44a) with (3.44b) (3.44c) V=nπa 2 /2 is the raised cross sectional area per unit length, n is the number of semicylinders per unit length (=1/b), δ=2/1+i is a measure of the dipole coupling between the semi-cylinders, I=(a 2 /b 2 )I 2 where (1 W) 2 is a packing factor introduced for random distributions, W=nb*=b*/b, b* is the minimum (centre to centre) separation between two cylinders and k is the wave number. W=1 for periodic spacing and W=0 for random spacing. The real part η of the admittance (which represents the incoherent or non-specular scattering loss term) is zero for periodic distributions of bosses. For grazing incidence normal to the cylinder axes, a=π/2 and azimuthal angle we obtain (3.45a) With randomly distributed semi-cylindrical roughness, this simplifies further to

102 Predicting outdoor sound 88 (3.45b) According to Lucas and Twersky [50], for semielliptical cylinders with eccentricity K, so that V=nπa 2 K/2, (3.45c) Twersky s cylindrical boss theory may be generalized to scatterers of arbitrary shape by comparison with equivalent results from Tolstoy s work. Equations (3.44) (3.46) may be contrasted with the equivalent results from Tolstoy s boss theory [49] for the effective admittance of a surface containing 2-D roughnesses of arbitrary shape, after correcting his expression for a missing coefficient σ. According to Tolstoy, β= ikε(cos 2 φ σ cos 2 α), (3.46) where (3.47) s 2 =(1/2)(1+K) is a shape factor, K is a hydrodynamic factor depending on steady flow around a scatterer, ν 2 =1+(2πVs 2 /3b) is a scatterer interaction factor and V is the cross-sectional scatterer area above the plane per unit length. Values of K are known for various shapes [49]. For semielliptical cylinders K=a /b, where a is the height and b is the semibase. For a scatterer having an isosceles triangular section with side u and height h, K=1.05(h/u)+0.14(h/u) 2. For semicylinders the expressions for ε and σ can be simplified to obtain (3.48) and

103 Predicting the acoustical properties 89 (3.49) Equation (3.45b) can be rewritten as (3.50) Comparison of equations (3.47) and (3.51) suggests that Twersky s and Tolstoy s expressions for the imaginary part of the effective admittance are equivalent for circular semicylinders if δ is replaced by (ε/v)+1=2s 2 /ν 2. Moreover Twersky s result (3.46) for elliptical semicylinders shows the same dependence on K, the eccentricity, as can be obtained from Tolstoy [49], that is δ=(1+k)/[1+ik(1+k)/2]. This more general expression for δ gives a dependence on the hydrodynamic factor K (through ε) and thus on the shape of the scatterer. This suggests that generalized forms for the real and imaginary parts of effective admittance may be obtained from (3.45), and the following form (3.51) where V represents the scatterer volume per unit area (raised area per unit length in 2-D). These results allow predictions of propagation over bosses of the various shapes for which K is known. In addition to the known values for semicylinders, semi-ellipsoids and triangular wedges, K for thin slats may be deduced by assuming that each slat affects the fluid flow as if it were a lamina [49]. The expression for the virtual mass of a lamina of width 2a is identical to the one for a cylinder of radius a, that is K=1. Table 3.6 summarizes the known values of the hydrodynamic factor for the various roughness shapes Rough finite impedance surfaces Tolstoy has considered sound propagation at the rough interface between two fluids [49] and his results have been used to predict the effective impedance of a rough porous boundary [51]. Consider the general case involving a planar distribution

104 Predicting outdoor sound 90 Table 3.6 Hydrodynamic factors for use in effective admittance models Shape of 2-D roughness K Semi-cylindrical 1 Semi-elliptical, height a, semi-base b a /b Isosceles triangle, side u, height h 1.05(h/u)+0.14(h/u) 2 Thin rectangle (slat) 1 of small fluid scatterers, N per unit area, embedded in a fluid half space with density and sound speed ρ 3 and c 3 beneath a fluid half space characterized by ρ 1 and c 1. The scatterers have density ρ 2 and c 2, height h, centre-to-centre spacing l and occupy a total volume σ V above the plane. The effective admittance β* of surfaces with 3-D scatterers, is given by β*= ik 0 ε +β s (3.52) where β s represents the impedance of the imbedding plane, and if and c 1 <c 3, or ρ 2 =ρ 3 and c 2 =c 3 (3.50) reduces to (3.53)

105 Predicting the acoustical properties 91 For 2-D scatterers, the corresponding effective admittance is given by (3.54) where (3.55) and θ(=π/2 φ) is the angle between the source-receiver axis and the normal to the scatterer axis. If the embedding material and scatterers are rigid and porous, then they can be treated acoustically as if they are fluids but with complex densities and sound speeds. These complex quantities may be calculated from any of the models described in the previous section. If (3.54) and (3.55) may be approximated by and by (3.56) (3.57) respectively, where k s is the complex wave number within the lower half space (i.e. the imbedding material). Moreover for high-flow-resistivity surfaces and low frequencies, β s k s γω. Consequently (3.58) implies that only the imaginary part of the smooth surface admittance is changed by the presence of roughness. This might be a basis for the acoustic determination of roughness in some circumstances.

106 Predicting outdoor sound 92 However, as remarked earlier, Tolstoy s results for hard rough surfaces ignore incoherent or non-specular scatter. It is a straightforward heuristic extension to write the effective surface admittance of a porous surface containing 2-D roughnesses as (3.58) where η is given by (3.49), δ is calculated from (2s 2 /ν 2 ), s 2 and ν 2 being calculated from (3.56). For φ=0, a π/2 (normal to scatterer axes, but non-grazing incidence) For a=π/2, φ 0 (grazing incidence but not normal to scatterer axes) (3.59) (3.60) If the roughness is periodic, then W=1 and only the imaginary part of admittance is changed. If the packing is random, then W=0 and the real part of the admittance changes by a factor that increases with f 3. For non-grazing incidence, (3.60), with W=0, predicts the maximum roughness effect for a given value of Ω, V and β s. Roughness is predicted to be particularly significant for high flow resistivity or hard ground surfaces near grazing incidence. For hard rough surfaces, (3.61) The model for the effective admittance of the rough interface between two fluids was developed for semi-infinite media. However, for a high-flow-resistivity medium, the difference between the values of δi and δih calculated from 2s 2D /ν 2D and 2s 2 /ν 2 respectively is small. This is illustrated in Figure 3.6. Consequently, (3.60) and (3.61) may be evaluated by using any of the existing models for (smooth) surface admittance β s (see for example [14, 23, 26]) and by adding the effective admittance for hard rough ground (see section ). Figure 3.7 shows predictions of the impedance of various randomly rough finite impedance surfaces at a grazing angle of 5. At low frequencies, and for the listed parameters, it is predicted that there is a reduction in the real part of impedance compared with the imaginary part. Such behaviour at low frequencies is predicted also in the impedance of smooth ground surfaces if there is layering [14]. For larger roughness and layered ground, more complicated relationships between real and imaginary parts of

107 Predicting the acoustical properties 93 impedance may result. Note that the assumptions, kh<kb 1, are violated above 1000 Hz for the largest roughness case considered in Figure 3.7. The predicted high-frequency limit of the impedance of a smooth porous surface is determined by the ratio of tortuosity to porosity and, therefore, is always greater Figure 3.6 Comparison of δ1 =2s 2D /ν 2D and δ1h=2s 2 /ν 2 for parameter values: flow resistivity 500 kpa s m 2, porosity 0.4, tortuosity 2.5 and semi-cylindrical roughness radius m, mean spacing 0.03 m.

108 Predicting outdoor sound 94 Figure 3.7 Predicted impedance of smooth and rough finite impedance surfaces. The porous medium is assumed semi-infinite with uniform triangular pores, flow resistivity 500 kpa s m 2, porosity 0.4 and tortuosity=2.5. The solid lines represent the predicted relative impedance of the smooth porous surface. The broken lines correspond to random semi-cylindrical roughness m high with mean spacing 0.03 m. The dotted lines correspond to random semi-elliptical roughness, height m, eccentricity (K) 1.5 and mean spacing 0.06 m. The rough surface impedance is predicted for a grazing angle of 5. than 1. A common feature of predictions for the real part of impedance of rough finite impedance surfaces, is that, at high frequencies, they are less than would be expected for smooth surfaces with the same porosity and flow resistivity. The incoherent component of scatter ensures that the effective impedance of a randomly rough finite impedance surface tends to zero at high frequencies. It is clear from Figure 3.7, that the presence of even relatively sparse slight 2-D roughness, less than m in height and with mean spacing of 0.05 m, is predicted to cause a significant change in the smooth surface impedance.

109 Predicting the acoustical properties 95 Figure 3.8 shows that the predicted normalized surface impedance of a rough porous surface may be approximated by formula (3.2) with an effective flow resistivity given by 0.8 the actual flow resistivity. Clearly this simple single-parameter prediction is similar to that due to the more sophisticated rough finite impedance model after parameter adjustment. Predictions such as those shown in Figure 3.8 may explain why (3.2) has been so successful in modelling the impedance of outdoor ground surfaces. Figure 3.8 Predicted surface impedance of rough ground. Solid lines represent the predicted normalised surface impedance of a porous surface with 20/m semi-cylindrical roughness as specified in the caption for Figure 3.7. The dotted lines represent the result predicted by (3.2) with R se =0.8 the actual flow resistivity. The dashed lines are predictions of equation (3.63). Also shown in Figure 3.8 are predictions of the formula (3.62) where and is the rms roughness height. There are noticeable differences between the predictions above 2 khz. However such frequencies are seldom of importance in predicting outdoor sound. Figure 3.7 shows that the predicted effect of roughness (small compared with wavelength) is to reduce the real part of the effective surface impedance while leaving the imaginary part relatively unaffected. Equation (3.63) and Figure 3.8 indicate that a tolerably successful prediction of the effective normalized impedance of a rough surface

110 Predicting outdoor sound 96 is that of the smooth surface but with a reduced real part. There is an important limitation on the domain of (3.63) which avoids unphysical predictions, that is a negative real part of impedance, when the flow resistivity and roughness height are large. Comparisons between data and impedance models are presented in Chapter 4. Figure 3.9 shows the predicted effect of various roughness specifications on the impedance of a grass-like layer. Figure 3.10 shows the corresponding excess attenuation predictions. The predictions in Figure 3.10 show that the effective impedance changes resulting from increase in mean roughness height from 1 to 4 cm causes the main ground effect to be shifted to lower frequencies. The predicted shifts are significant for excess attenuation and may be important in short-range ground characterization. Figure 3.9 Predicted effect of increasing roughness parameter values on the normalized surface impedance of a grass-like 5 cm thick layer: flow resistivity 200 kpa s m 2, porosity 0.4, tortuosity=1/porosity, triangular pores. Continuous lines correspond to a smooth boundary. Dashed lines correspond to 2D roughness consisting of 0.01 m radius randomly spaced semi-cylinders, mean centreto-centre spacing 0.04 m with axes normal to source-receiver axis. Dotted lines correspond to 2D roughness with 0.04 m radius, mean centre-to-centre spacing 0.16 m.

111 Predicting the acoustical properties 97 Figure 3.10 Predicted excess attenuation spectra corresponding to the impedance spectra shown in Figure Effects of ground elasticity Noise sources such as blasts and heavy weapons shooting create low-frequency impulsive sound waves which propagate over long distances, and can create a major environmental problem for military training fields. Such impulsive sound sources tend to disturb neighbours more through the vibration and rattling induced in buildings, than by the audible airborne sound itself [57, 58]. Human perception of whole body vibration includes frequencies down to 1 Hz [59] and the fundamental natural frequencies of buildings are in the range 1 10 Hz. Planning tools to predict and control such activities must therefore be able to handle sound propagation down to these low frequencies. Despite their valuable role in many sound propagation predictions, impedance models that assume rigid-porous ground have an intrinsic theoretical shortcoming that they fail to account for air-to-ground coupling through interaction between the atmospheric sound wave and surface waves in the ground. Air-ground coupling has been of considerable interest in geophysics, both because of the measured ground-roll caused by intense acoustic sources and the possible use of air sources in ground layering studies. Theoretical analyses have been carried out for spherical wave incidence on a ground consisting either of a fluid or solid layer above a fluid or solid half space [60]. However, to describe the phenomenon of acoustic-toseismic coupling accurately it has been found that the ground must be modelled as an elastic porous material [61 63]. The resulting theory is relevant not only to predicting the ground vibration induced by low-frequency acoustic sources but also, as we shall see, in predicting the propagation of low-frequency sound above the ground.

112 Predicting outdoor sound 98 The classical theory for a porous and elastic medium predicts the existence of three wave types in the porous medium: two dilatational waves and one rotational wave. In a material consisting of a dense solid frame with a low density fluid saturating the pores, the first kind of dilatational wave has a velocity very similar to the dilatational wave (or geophysical P wave) travelling in the drained frame. The attenuation of the first dilatational wave type is however, higher than that of the P wave in the drained frame. The extra attenuation comes from the viscous forces in the pore fluid acting on the pore walls. This wave has negligible dispersion and the attenuation is proportional to the square of the frequency, as is the case for the rotational wave. The viscous coupling leads to some of the energy in this propagating wave being carried in the pore fluid as the second type of dilatational wave. In air-saturated soils, the second dilatational wave has a much lower velocity than the first and is often called the slow wave, the dilatational wave of the first kind being called the fast wave. The attenuation of the slow wave stems from viscous forces acting on the pore walls and from thermal exchange with the pore walls. Its rigid-frame limit is very similar to the wave which travels through the fluid in the pores of a rigidporous solid [64]. It should be remarked that the slow wave is the only wave type considered in the previous discussions of ground effect. When the slow wave is excited, most of the energy in this wave type is carried in the pore fluid. However, the viscous coupling at the pore walls leads to some propagation within the solid frame. At low frequencies, it has the nature of a diffusion process rather than a wave, being analogous to heat conduction. The attenuation for the slow wave is higher than that of the first in most materials and, at low frequencies, the real and imaginary parts of the propagation constant are nearly equal. The rotational wave has a very similar velocity to the rotational wave carried in the drained frame (or the S wave of geophysics). Again there is some extra attenuation due to the viscous forces associated with differential movement between solid and pore fluid. The fluid is unable to support rotational waves, but is driven by the solid. The propagation of each wave type is determined by many parameters relating to the elastic properties of the solid and fluid components. Considerable efforts have been made to identify these parameters and determine appropriate forms for specific materials. In the formulation described here, only equations describing the two dilatational waves are introduced. The coupled equations governing the propagation of dilatational waves can be written as [65] (3.63) (3.64)

113 Predicting the acoustical properties 99 where is the dilatation or volumetric strain vector of the skeletal frame: is the relative dilatation vector between the frame and the fluid; u is the displacement of the frame, U is the displacement of the fluid, F(λ) is the viscosity correction function, ρ is the total density of the medium, ρf is the fluid density, µ=dynamic fluid viscosity and k is the permeability. The second term on the RHS of (3.65), (µ/k)( ξ/ t)f(λ), allows for the damping through viscous drag as the fluid and matrix move relative to one another. m is a dimensionless parameter that accounts for the fact that not all the fluid moves in the direction of macroscopic pressure gradient as not all the pores run normal to the surface and is given by m=τρ 0 /Ω where τ is the tortuosity and Ω is the porosity. H, C and M are elastic constants that can be expressed in terms of the bulk moduli K s, K f and K b of the grains, fluid and frame respectively and the shear modulus, µ, of the frame. Assuming that e and ξ vary as e iωt, / t can be replaced by iω and equation (4.40) can be written as (3.65) where ρ(ω)=(τρ 0 /Ω)+(iµ/ωk) F(λ) is the dynamic fluid density. The original formulation of F(λ) (and hence of ρ(ω)) was a generalization from the specific forms corresponding to slit-like and cylindrical forms but assuming pores with identical shape. Expressions are available also for triangular and rectangular pore shapes and for more arbitrary microstructures (see section 3.2). If plane waves of the form e=a exp(i(lx ωt)) and ξ=b exp(i(lx ωt)) are assumed, then the dispersion equations for the propagation constants may be derived. These are: A(l 2 H ω 2 ρ)+b(ω 2 ρ 0 l 2 C)=0 (3.66) and A(l 2 C ω 2 ρ 0 )+B(mω 2 l 2 M+iωF(λ) η/k)=0. (3.67) A non-trivial solution of (3.67) and (3.68) exists only if the determinant of the coefficient vanishes, giving (3.68) There are two complex roots of this equation from which both the attenuation and phase velocities of the two dilatational waves are calculated.

114 Predicting outdoor sound 100 At the interface between different porous elastic media, there are six boundary conditions that may be applied. These are as follows: 1 continuity of total normal stress, 2 continuity of normal displacement, 3 continuity of fluid pressure, 4 continuity of tangential stress, 5 continuity of normal fluid displacement, 6 continuity of tangential frame displacement. At an interface between a fluid and the poroelastic layer (such as the surface of the ground) the first four boundary conditions apply. The resulting equations and those for a greater number of porous and elastic layers are solved numerically [13]. The spectrum of the ratio between the normal component of the soil particle velocity at the surface of the ground and the incident sound pressure, the acoustic-to-seismic coupling ratio or transfer function, is influenced strongly by discontinuities in the elastic wave properties within the ground. At frequencies corresponding to peaks in the transfer function, there are local maxima in the transfer of sound energy into the soil [61]. These are associated with constructive interference between down- and up-going waves within each soil layer. Consequently there is a relationship between near-surface layering in soil and the peaks or layer resonances that appear in the measured acoustic-to-seismic transfer function spectrum: the lower the frequency of the peak in the spectrum, the deeper the associated layer. Figure 3.11 Measured and predicted acousticto-seismic coupling ratio for a layered soil (range 3.0 m, source height 0.45 m).

115 Predicting the acoustical properties 101 Figure 3.11 shows example measurements and predictions of the acoustic-to-seismic transfer function spectrum at the soil surface [63]. The measurements were made using a loudspeaker sound source and a microphone positioned close to the surface, vertically above a geophone buried just below the surface of a soil which had a loose surface layer. Seismic refraction survey measurements at the same site were used to determine the wave speeds. The predictions have been made by using a computer code FFLAGS (Fast Field Program for Air-Ground Systems) that models sound propagation in a system of fluid layers and porous elastic layers [62]. The numerical theory (FFLAGS) may be used also to predict the ground impedance at low frequencies [66]. In Figure 3.12, the predictions for the surface impedance at a grazing angle of are shown as a function of frequency for the layered porous and elastic system described by Table 3.7 and compared with those for a rigid porous ground with the same surface flow resistivity and porosity. The influence of ground elasticity is to reduce the magnitude of the impedance considerably below 50 Hz. Potentially, this is very significant for predictions of lowfrequency noise, for example blast noise, at long range. Figure 3.13 shows that the surface impedance of this four-layer poroelastic system varies between grazing angles of 5.7 and 0.57 but remains more or less constant for smaller grazing angles. The predictions show two resonances. Figure 3.12 Predicted surface impedance at a grazing angle of for poro-elastic and rigid porous ground (4-layer system, Table 3.7).

116 Predicting outdoor sound 102 Table 3.7 Ground profile and parameters used in the calculations for Figures 3.11 and 3.12 Layer Flow resistivity (kpa Porosity Thickness P-wave speed S-wave speed Damping s m 2 ) (m) (m s 1 ) (m s 1 ) ,740, ,740, ,740, Halfspace The lowest frequency resonance is the most angle-dependent. The peak in the real part changes from 2 Hz at 5.7 to 8 Hz at On the other hand the higher frequency resonance peak near 25 Hz remains relatively unchanged with range. The peak at the lower frequency may be associated with the predicted coincidence between the Rayleigh wave speed in the ground and the sound speed in air (Figure 3.14). Compared with the near pressure doubling predicted by classical Figure 3.13 Normalized surface impedance predicted for the 4-layer structure, sound speed in air=329 ms 1 (corresponding to an air temperature of 4 C) for grazing angles between and 5.7.

117 Predicting the acoustical properties 103 ground impedance models, the predicted reduction of ground impedance at low frequencies above layered elastic ground can be interpreted as the result of coupling of a significant fraction of the incident sound energy into ground-borne Rayleigh waves. Numerical theory has been used also to explore the consequences of this coupling for the excess attenuation of low frequency sound above ground [66]. Figure 3.15 shows the excess attenuation predictions for 6.3, 7.2 and 12.6 km ranges. There are significant dips in the predicted excess attenuation spectrum. According to the convention used here, these represent maxima in the excess attenuation. The depths of the dips increase with range. The depth of the dip predicted at 3.5 Hz is 9 db and shows a small frequency dependence with changing range. The enhanced (above +6 db) excess attenuation spectrum immediately before the 3.5 Hz dip in the prediction for 12.6 km range is symptomatic of a surface wave. However as with the A/S impedance predictions, the Excess attenuation spectra predicted by FFLAGS for the four-layer poroelastic structure are rather dependent on the Figure 3.14 Rayleigh wave dispersion curve predicted for the system described by Table 3.7 [66]. Reprinted with permission from Elsevier.

118 Predicting outdoor sound 104 Figure 3.15 Excess attenuation spectra predicted for source and receiver heights of 2 m and 0.1 m and three ranges assuming a sound speed in air of 329 m s 1. Reprinted with permission from Elsevier. Figure 3.16 Excess attenuation spectra below 50 Hz predicted at ranges of 6300 m, 7200 m and 12,600 m, source height 2.0 m, receiver height 1 m, with an assumed sound speed in air of 332 m s 1.

119 Predicting the acoustical properties 105 assumed sound speed in air. Figure 3.16 shows corresponding predictions with an assumed sound speed of 332 m s 1. The lowest frequency dip is predicted to move to 2 Hz from 3.5 Hz and to become shallower. The dip between 20 and 30 Hz remains at the same frequency but becomes more pronounced. It is important to consider whether it is possible to propose a model for the surface impedance of the ground that will be more accurate than classical impedance model predictions at low frequencies. Figure 3.17 shows the excess attenuation spectra predicted for source height 2 m, receiver height 0.1 m and horizontal range of 6.3 km over a layered ground profile corresponding to Table 3.5 (assumed sound speed in air of 329 ms 1 ) and by classical theory for a point source above an impedance (locally reacting) plane (2.40) using the impedance calculated for degrees grazing angle (Zeff, broken lines in Figure 3.14). The predictions show a significant extra attenuation between 2 and 10 Hz. The predictions indicate also that, for an assumed sound speed in air of 329 ms 1, and, apart from an enhancement near 2 Hz, the excess attenuation spectrum might be predicted tolerably well by using modified classical theory instead of a full poroelastic layer calculation. However Figure 3.18 shows the same information as Figure 3.17 but for an assumed sound speed of 332 m s 1. In this case the low frequency dip in the excess attenuation spectrum is not predicted as well by assuming an effective impedance. Figure 3.17 Excess attenuation spectra predicted for source height 2 m, receiver height 0.1 m and horizontal range of 6.3 km by FFLAGS (assumed sound speed in air of 329 ms 1 ) and by classical theory using impedance calculated for grazing angle. Reprinted with permission from Elsevier.

120 Predicting outdoor sound 106 Figure 3.18 As for Figure 3.17 but with an assumed sound speed in air of 332 m s 1. It is difficult to measure the surface impedance of the ground at low frequencies [67, 68]. Consequently, as yet, the predictions of significant ground elasticity effects have been validated only by data for acoustic-to-seismic coupling, that is by measurements of the ratio of ground surface particle velocity relative to the incident sound pressure [66]. The comparisons between predictions and data for acoustic-to-seismic coupling are discussed in Chapter 4. References 1 ISO :1996 Acoustics attenuation of sound propagation outdoors part 2: general method of calculation (International Standards Organisation 1996). 2 M.E.Delany and E.N.Bazley, Acoustical properties of fibrous absorbent materials, Appl. Acoust., 3: (1970). 3 M.E.Delany and E.N.Bazley, Acoustical properties of fibrous absorbent materials, National Physical Laboratory Aero Report AC71 (1971). 4 F.P.Mechel, Ausweitung der Absorberformel von Delany and Bazley zu teifel Frequenzen, Acustica, 35: (1976). 5 I.P.Dunn and W.A.Davern, Calculation of acoustic impedance of multilayer absorbers, Appl. Acoust., 19: (1986). 6 Y.Miki, Acoustical properties of porous materials modifications of Delany and Bazley models J. Acoust. Soc. Jpn., 11:19 24 (1990). 7 K.B.Rasmussen, Sound propagation over grass covered ground, J. Sound Vib., 78(2): (1981). 8 P.M.Morse and K.U.Ingard, Theoretical Acoustics, Princeton University Press, Princeton, NJ (1986). 9 S.I.Thomasson, Sound propagation above a layer with a large refractive index, J. Acoust. Soc. Am., 61: (1977).

121 Predicting the acoustical properties J.F.Hamet and M.Bérengier, Acoustical Characteristics of Porous Pavements: A New Phenomenological Model, Internoise 93, Leuven, Belgium (1993); see also M.Bérengier, M.Stinson, G.Daigle and J.F.Hamet, Porous road pavements: acoustical characterization and propagation effects, J. Acoust. Soc. Am., 101: (1997). 11 M.R.Stinson, The propagation of plane sound waves in narrow and wide circular tubes and generalisation to uniform tubes of arbitrary cross-sectional shape, J. Acoust. Soc. Am., 89(1): (1991). 12 Y.Champoux and M.R.Stinson, On acoustical models for sound propagation in rigid frame porous materials and the influence of shape factors, J. Acoust. Soc. Am., 92(2): (1992). 13 J.F.Allard, Propagation of Sound in Porous Media, Elsevier Applied Science, London (1993). 14 K.Attenborough, Models for the acoustical properties of air-saturated granular media, Acta Acust., 1: (1993). 15 P.C.Carman, Flow of Gases through Porous Media, Butterworths, London (1953). 16 O.Umnova, K.Attenborough and K.M.Li, Cell model calculations of dynamic drag parameters in packings of spheres. J. Acoust. Soc. Am., 107(6): (2000). 17 K.V.Horoshenkov and M.J.Swift, The acoustic properties of granular materials with pore size distribution close to log-normal, J. Acoust. Soc. Am., 110: (2001). 18 J.F.Allard, B.Castagnede, M.Henry and W.Lauriks, Evaluation of tortuosity in acoustic porous materials saturated by air, Rev. Sci. Instrum., 65: (1994). 19 M.J.Swift, The Physical Properties of Porous Recycled Materials, PhD Thesis, University of Bradford, UK, (2000). 20 P.Leclaire, O.Umnova, K.V.Horoshenkov and L.Maillet, Porosity measurement by comparison of air volumes, Rev. Sci. Instrum., 74: (2003). 21 K.Attenborough, On the acoustic slow wave in rigid framed porous media, J. Acoust. Soc. Am., 81: (1987). 22 M.A.Biot, Theory of elastic waves in a fluid-saturated porous solid. II High-frequency range, J. Acoust. Soc. Am., 28: (1956). 23 D.L.Johnson, J.Koplik and R.Dashen, Theory of dynamic permeability and tortuosity in fluidsaturated porous materials, J. Fluid Mech., 176: (1987). 24 P.Allemon and R.Hazelbrouck, Influence of pore size distribution on sound absorption of rubber granulates, Proc. Inter Noise 96, Publ. IOA, Book 2, (1996). 25 P.Leclaire, M.J.Swift and K.V.Horoshenkov, Determining the specific area of porous acoustic materials from water extraction data, J. Appl. Phys., 84: (1998). 26 P.Leclaire, L.Kelders, W.Lauriks, C.Glorieux and J.Thoen, Determination of the viscous characteristic length in air-filled porous materials by ultrasonic attenuation measurements, J. Acoust. Soc. Am., 99(4): (1996). 27 Z.Fellah, C.Depollier and M.Fellah, Application of fractional calculus to the sound waves propagation in rigid porous materials: validation via ultrasonic measurements, Acust. Acta Acust., 88:34 39 (2002). 28 J-F.Allard, M.Henry and J.Tizianel, Pole contribution to the field reflected by sand layers J. Acoust. Soc. Am., 111(2): (2002). 29 M.Henry, P.Lemarinier, J.F.Allard, J.L.Bonardet and A.Gedeon, Evaluation of the characteristic dimensions for porous sound-absorbing materials J. Appl. Phys., 77:17 (1995). 30 M.Boeckx, G.Jansens, W.Lauriks and D.Albert, Modelling acoustic surface waves above a snow layer, Acust. Acta Acust., 90(2): (2004). 31 O.Umnova, K.Attenborough and K.M.Li, A cell model for the acoustical properties of packings of spheres, Acust. Acta Acust., 87: (2001). 32 T.Yamamoto and A.Turgut, Acoustic wave propagation through porous media with arbitrary pore size distributions, J. Acoust. Soc. Am., 83(5): (1988).

122 Predicting outdoor sound K.V.Horoshenkov, K.Attenborough and S.N.Chandler-Wilde, Pade approximants for the acoustical properties of rigid frame porous media with pore size distribution, J. Acoust. Soc. Am., 104: (1998). 34 K.V.Horoshenkov and M.J.Swift, The acoustic properties of granular materials with pore size distribution close to log-normal, J. Acoust. Soc. Am., 110(5): (2001). 35 K.Attenborough, Ground parameter information for propagation modeling, J. Acoust. Soc. Am., 92: (1992); see also R.Raspet and K.Attenborough, Erratum: Ground parameter information for propagation modeling, J. Acoust. Soc. Am., 92: 3007 (1992). 36 R.Raspet and J.M.Sabatier, The surface impedance of grounds with exponential porosity profiles, J. Acoust. Soc. Am., 99(1): (1996). 37 R.J.Donate, Impedance models for grass covered ground, J. Acoust. Soc. Am., 61: (1977). 38 K.Attenborough, Acoustical impedance models for outdoor ground surfaces, J. Sound Vib., 99: (1985). 39 S.Taherzadeh, Private Communication (1997). 40 G.Taraldsen, The Delany-Bazley impedance model and Darcy s law, Acust. Acta Acust., 91:41 50 (2005). 41 D.K.Wilson, Relaxation-matched modeling of propagation through porous media including fractal pore structure, J. Acoust. Soc. Am., 94: (1993). 42 D.K.Wilson, Simple relaxation models for the acoustical properties of porous media, Appl. Acoust., 50: (1997). 43 D.K.Wilson, V.E.Ostashev, S.L.Collier, N.P.Symons, D.F.Aldridge and D.H.Marlin, Time-domain calculations of sound interactions with outdoor ground surfaces, Appl. Acoust. (in press). 44 S.R.Pride, F.D.Morgan and A.F.Gangi, Drag forces of porous medium acoustics, Phys. Rev. B., 47(9): (1993). 45 M.S.Howe, On the long range propagation of sound over irregular terrain, J. Sound Vib., 98(1):83 94 (1985). 46 I.Tolstoy, Smoothed boundary conditions, coherent low frequency scatter, and boundary modes, J. Acoust. Soc. Am., 75(1):1 22 (1984). 47 M.A.Biot, Reflection on a rough surface from an acoustic point source, J. Acoust. Soc. Am., 29: (1957). 48 V.F.Twersky, On scattering and reflection of sound by rough surfaces, J. Acoust. Soc. Am., 29: (1957). 49 I.Tolstoy, Coherent sound scatter from a rough interface between arbitrary fluids with particular reference to roughness element shapes and corrugated surfaces, J. Acoust. Soc. Am., 72(3): (1982). 50 R.J.Lucas and V.Twersky, Coherent response to a point source irradiating a rough plane J. Acoust. Soc. Am., 76: (1984). 51 K.Attenborough and S.Taherzadeh, Propagation from a point source over a rough finite impedance boundary, J. Acoust. Soc. Am., 98(3): (1995). 52 P.M.Boulanger, K.Attenborough, T.Waters-Fuller and K.M.Li, Rough hard boundary ground effects, J. Acoust. Soc. Am., 104(1): (1998). 53 V.Twersky, Scattering and reflection by elliptically striated surfaces, J. Acoust. Soc. Am., 40: (1966). 54 V.Twersky, Multiple scattering of sound by correlated monolayers, J. Acoust. Soc. Am., 73:68 84 (1983). 55 V.Twersky, Reflection and scattering of sound by correlated rough surfaces, J. Acoust. Soc. Am., 73:85 94 (1983). 56 H.Lamb, Hydrodynamics, Dover, New York (1945) p C.Madshus and N.I.Nilsen, Low frequency vibration and noise from military blast activity prediction and evaluation of annoyance, Proc. Inter Noise 2000, Nice, France (2000).

123 Predicting the acoustical properties E.Øhrstrøm, Community reaction to noise and vibrations from railway traffic. Proc. Internoise 1996, Liverpool, UK. 59 ISO :1998, Mechanical vibration and shock evaluation of human exposure to whole body vibration part 2: vibration in buildings. International Organization for Standardization. 60 W.M.Ewing, W S.Jardetzky and F.Press, Elastic Waves in Layered Media, McGraw-Hill Book Company, New York (1957). 61 J.M.Sabatier, H.E.Bass, L.M.Bolen and K.Attenborough, Acoustically induced seismic waves, J. Acoust. Soc. Am., 80: (1986). 62 S.Tooms, S.Taherzadeh and K.Attenborough, Sound propagation in a refracting fluid above a layered porous and elastic medium, J. Acoust. Soc. Am., 93(1): (1993). 63 N.D.Harrop, The Exploitation of Acoustic-to-Seismic Coupling for the Determination of Soil Properties, PhD Thesis, The Open University (2000). 64 K.Attenborough, On the acoustic slow wave in air filled granular media, J. Acoust. Soc. Am., 81: (1987). 65 R.D.Stoll, Theoretical aspects of sound transmission in sediments, J. Acoust. Soc. Am., 68(5): (1980). 66 C.Madshus, F.Lovholt, A.Kaynia, L.R.Hole, K.Attenborough and S.Taherzadeh, Air-ground interaction in long range propagation of low frequency sound and vibration field tests and model verification, Appl. Acoust., 66(5): (2005). 67 G.A.Daigle and M.R.Stinson, Impedance of grass covered ground at low frequencies using a phase difference technique, J. Acoust. Soc. Am., 81:62 68 (1987). 68 M.W.Sprague, R.Raspet, H.E.Bass and J.M.Sabatier, Low frequency acoustic ground impedance measurement techniques, Appl. Acoust., 39: (1993).

124 Chapter 4 Measurements of the acoustical properties of ground surfaces and comparisons with models 4.1 Impedance measurement methods Impedance tube Early measurements of reflection coefficients of outdoor ground surfaces at normal incidence used adaptations of the standing wave or impedance tube technique, employed primarily to obtain the absorbing properties of building materials [1]. The method assumes plane wave incidence and requires the measurement of the ratio of the maximum sound pressure to the level of the first (or subsequent) pressure minimum (as counted from the surface of interest) as well as the distance from the surface to the first minimum at each frequency. The rather different acoustical, physical and biological properties of outdoor grounds result in several problems not encountered often with architectural absorbents. These include the following: (i) the crucial importance of low background noise, probe end and tube absorption corrections [2] in measuring the relatively high impedances associated with many ground surfaces, (ii) the difficulties of establishing the surface plane location (particularly important in determining the phase (imaginary part) of the impedance), (iii) the changing micro-meteorological conditions and the formation of worm casts during lengthy measurements [3], (iv) the unrepresentative nature of the (necessarily) small tube sample in view of the lateral inhomogeneity typical of many grounds and (v) the possibility of destruction of the local ground structure during insertion of the lower end into the ground. If a probe microphone is used in the tube, there is the additional problem of devising a stable system for vertical probe traversal. Nevertheless several useful data sets for impedance as a function of frequency have been obtained with a vertical impedance tube and probe microphone [4, 5]. The impedance tube method has proved particularly successful as a laboratory technique for measuring the acoustical properties of snow [6]. However, this involved use of a horizontal tube with automatic tracking of the probe microphone and measurements both of hard-backed and quarter wavelength (air)-backed finite length samples. By this means it was possible to evaluate the normal surface impedance and the propagation constant; both quantities are important when describing propagation over such a low flow resistivity, potentially externally reacting surface as snow. A modification of the impedance tube method that has been adopted for use in building acoustics [7 8] measures the complex transfer function between two microphones positioned fairly near to the sample surface. The method has the advantage

125 Measurements of the acoustical properties 111 that it is possible, by use of a broadband signal, to obtain impedance data over a wide frequency range simultaneously rather than at one frequency at a time as required by the probe microphone technique. The technique however is not as accurate as the probe method [9 10] and gives rise to particular problems when the microphone spacing corresponds to integer numbers of quarter or half wavelengths. The microphone spacing (s) determines the frequency range of the measurement s validity according to and n is an integer. This technique has been used to determine the normal surface impedance of forest floors [11] Impedance meter The normal surface impedance of the ground has been obtained from direct measurements of pressure and volume velocity at the ground surface by means of a mechanically driven cylindrical Helmholtz resonator chamber, which was pounded into the ground [12]. The sound pressure was measured by a microphone, mounted flush in the wall of the chamber, and an optical monitor of the piston driver displacement permitted measurement of the phase angle between the volume velocity and the pressure. This enabled deduction of the real and imaginary parts of impedance over the frequency range 300 to 1000 Hz. While this method overcomes several of the problems associated with outdoor use of the impedance tube, it shares the difficulties associated with small sample size and the destructiveness of its insertion Non-invasive measurements Direct measurement of reflection coefficient Dickinson and Doak [3] used a free field version of the impedance tube technique at normal incidence. Standing waves were set up along an axis between a loudspeaker source and the ground. The loudspeaker was mounted on an A-frame about 4 m high. The sound field was explored by a microphone traversing the axis and the normal surface impedance deduced in a similar manner to that used on impedance tube data. However, allowance must be made for spherical divergence of the various wavefronts. As long as the measurements are made at normal incidence and the sum of the source and receiver heights is a sufficient number of wavelengths above the ground, then the total field is given with adequate accuracy by the assumption of plane rather than spherical reflection coefficients in the analysis. Embleton et al. [4] adapted this free field technique to oblique incidence by measuring the interference field with a probe microphone along the track of the specularly reflected ray. Van der Heijden [13] avoided the need for a probe microphone in the free field

126 Predicting outdoor sound 112 techniques both at normal and oblique incidence by use of up to 16 microphone positions. A free field adaptation of the two-microphone impedance tube technique, developed initially for use in building acoustics [14], has been used successfully to measure the impedance of outdoor ground surfaces at normal and oblique incidence [15]. The typical geometry used included a source height of 2.1 m, an upper microphone height of 135 mm, microphone spacing of 75 mm and an angle of incidence of 45. With this, geometry data for the complex transfer function between the two microphones and hence for impedance were obtained over the frequency range 200 to 2000 Hz. The method was found to be very sensitive to meteorological conditions and to the assumed ground surface location. Background noise problems were alleviated by use of narrow-band signals. At low frequencies it is difficult to achieve the source height required for the plane wave reflection coefficient assumption and to achieve the low background noise important to the free field standing wave technique. Daigle and Stinson [16] and Sprague et al. [17] have measured the phase gradient between a closely spaced, phase-matched pair of microphones in the normal incidence interference pattern instead. This technique requires accurate measurements only in regions of the peaks and valleys in the phase gradient. However it is extremely difficult to carry out, being extremely sensitive to unwanted reflections and to meteorological conditions. Using a loudspeaker source at a height of about 7 m and four different microphone spacings (between 10 and 80 cm), data have been obtained between 25 and 300 Hz. Tone bursts or more general impulses may be used as an alternative to continuous sound sources for impedance measurements. If the incident pulse is short enough, then unwanted reflections may be isolated and their effects may be eliminated. Van der Heijden et al. used tone bursts on ground surfaces brought into the laboratory [18]. Cramond and Don [19 20] have developed an outdoor technique in which a blank-firing rifle is used as a source and two microphones are deployed so that the ground-reflected signal at one arrives at the same time as the direct signal at the other. With appropriate choice of source and receiver heights (approximately 2 m) this enables direct deduction of reflection coefficient as a function of frequency within the limitations of the source spectrum (400 to 5 khz). The method has been shown capable of distinguishing the impedance of a tyre track from that of the surrounding uncompacted soil. Also it has been used to show the existence of high-frequency (>1 khz) peaks in the surface impedance of a soil wetted so that a thin (1 to 2 cm thick) saturated layer was formed at the surface. On the other hand the frequency range of the method is limited by lack of source energy at low frequencies and by turbulent scattering at high frequencies Impedance deduction from short-range measurements If a point source is located on a locally reacting surface and emits broadband sound, then for a reception point also on the surface R p = 1 and (2.40) is simplified considerably [4]. Specifically, (4.1)

127 Measurements of the acoustical properties 113 where for (4.2) and (4.3) This means that the excess attenuation (sound pressure level with respect to free field) is given by (4.4) Equation (4.4) enables deduction of the magnitude of the normal surface impedance as a function of frequency from measurements of the excess attenuation spectrum (or of excess attenuation as a function of range at each frequency of interest). This method has been used by Embleton et al. [4] on grassland. Results were found to be in reasonable agreement with both impedance tube and free field data obtained over the same surface. An elaboration of this technique has been used by Habault and Corsain [21]. The excess attenuation at grazing incidence was obtained as a function of range at several frequencies and the impedance was deduced from a minimization algorithm using a grazing-incidence approximation [22]. The measurement technique was found to be restricted practically to frequencies above 500 Hz since the required range of measurement increases as the frequency of interest is decreased. Nevertheless the resulting data shows remarkably little scatter. More generally, measurements of the magnitude of the excess attenuation from a point source at non-grazing incidence may be deduced, by means of least-squares or template fitting to the theories described in Chapter 2, and yield impedance as a function of frequency [23]. Such a procedure may be regarded as a simple example of the matched field algorithms employed in underwater acoustics. It should be noted that excess attenuation is independent of the source spectrum, and consequently, in principle, any source may be used as long as it is broadband and the measurements are taken at sufficient distance that the source may be regarded as a point source. On the other hand the measurement range should be sufficiently small (<5 m) that meteorological influences are kept to a minimum [24]. Bolen and Bass [25] obtained impedances of grassland, soils and sands as a function of frequency from 40 to 300 Hz using a large loudspeaker as a source. However their data (even for the homogeneous sand) shows considerable scatter between successive frequencies. Probably, this is the result of the relatively large ranges

128 Predicting outdoor sound 114 (>15 m) used for their measurements and the fact that no constraint was imposed on the behaviour of the real and imaginary parts of impedance as separate functions of frequency. Such constraints may be imposed by the adoption of an appropriate model for the acoustical properties of the ground [23]. This may involve either short-range measurements of excess attenuation or the level difference between vertically separated microphones [23, 26 29]. The fitting of excess attenuation or level difference spectra, obtained near grazing incidence, is achieved by means of the classical theory for propagation from a point source over an impedance plane. As well as not requiring the assumption of plane waves, which is valid only at sufficient source height, higher frequencies and near normal incidence, the short-range propagation method also includes effects of small-scale surface roughness which may be considered to alter the surface impedance compared with a smooth surface with the same pore structure near grazing incidence (as discussed in section 3.3). Wempen [26] has measured and recorded a broadband signal in an anechoic chamber and then used this as a reference signal, transmitting it across several ground surfaces. In addition, the direct and ground-reflected waves were synchronized by feeding them into an FFT analyser and adjusting the time delay between them. This enabled high signal-to-noise ratios to be obtained with relatively short average times. An alternative to an anechoic chamber measurement to establish the free field spectrum level at the relevant range is to obtain a reference measurement after raising the source and receiver sufficiently far above the surface that ground effects may be ignored. This is possible in situ particularly when using a Maximum Length Sequence Signal Analysis (MLSSA) system. In this system the source is controlled to give a pseudo-random sequence of impulses. The received signal is analysed for this particular sequence. The resulting analysis is robust to background noise and gives the time domain or impulse response, which may be used to eliminate unwanted portions of the time series. Examples of excess attenuation spectra obtained in this way outdoors and at short range are shown in Figure 4.1. It should be noted, however, that MLSSA is not reliable in a time-varying environment. Deduction of excess attenuation requires knowledge of the free field spectrum of the source. However it may be more convenient to use the difference in spectra between two separately located reception points than to use the excess attenuation spectrum at a single point for the deduction of ground impedance. The points may be separated either horizontally or vertically. In particular the level difference between two vertically separated microphones at horizontal ranges of less than 3 m from a point source has been used extensively [23, 26 29]. The level difference spectrum may be calculated as the difference between the excess attenuation spectra at the two receivers. Ideally with the lower microphone on the surface the level difference spectrum corresponds closely in form to the excess attenuation spectrum at the upper microphone. In practice the lower microphone may be at a non-zero height but must be sufficiently low so that the first interference minimum in the corresponding excess attenuation spectrum lies above the frequency range of interest. This serves to avoid the effect of the very large sound velocity gradients that may occur close to the ground. On the other hand at very short range (<2 m) refraction effects are likely to be small.

129 Measurements of the acoustical properties 115 Figure 4.1 Excess attenuation spectra obtained with source and receiver at 0.1 m height and separated by 1 m over (a) grass 1, (b) soil, (c) grass 2, (d) bean plants and (e) wheat. The wheat was 0.55 m high. The different lines in each graph represent results of successive measurements.

130 Predicting outdoor sound Model parameter deduction A large number of measurements of level difference spectra and excess attenuation spectra have been obtained at horizontal source-receiver ranges of 2 m or less, both in the laboratory and outdoors. These data have been fitted using some of the impedance models discussed in Chapter 3 [26 29]. Example laboratory measurements of excess attenuation spectrum over a sand surface, theoretical predictions and fits are shown in Figure 4.2. Wempen [26] has fitted his measured excess attenuation spectra data comparatively with several impedance models. A comprehensive set of outdoor excess attenuation measurements, at short range (between 7 and 15 m), have been made by Embleton et al. [30]. These data have been fitted by the single-parameter Delany-Bazley model (3.2) giving a wide range of effective flow resistivities for several surfaces (see section 4.6). Figure 4.2 Measured excess attenuation spectrum with point source height=receiver height=0.1 m, horizontal separation 1 m, above 0.3 m thick sand (dashed line), compared with predictions of a two parameter model (3.38) (solid line) and a prediction for a distribution of triangular pore sizes (dotted line) using measured flow resistivity (473 kpa s m 2 ), assumed porosity (0.4), assumed tortuosity (2.5) and best-fit standard deviation of lognormal pore size distribution Bestfit values for two parameter model are σ e =499 kpa s m 2 and α e = 185 m 1. By making excess attenuation measurements at short range (4 m) and by making use of a phenomenological model ((3.3) and (3.4)) for the acoustical properties of a hard-backed

131 Measurements of the acoustical properties 117 porous layer, Thomasson [31] was able to deduce best-fit parameter values for grassland, hay and newly planted soil. If the impedance model requires fitting of more than two parameters then problems of non-uniqueness will arise [23, 32, 33]. These can be overcome if one of the parameters can be fixed beforehand. For example, the porosity of the surface may be known to within ±20%. Reference 68 in Chapter 3 discusses automation of the parameter fitting procedure. The deduction of parameters through the impedance-model-based approach could be of particular interest for acoustical monitoring of soils. If the ground surface is sloping, the source and receiver heights in any ground characterization should be measured perpendicular to the surface. As discussed in section 3.3, the qualitative effect of irregular or random roughness on porous surfaces is to change the effective admittance. The effects may be predicted quantitatively as long as the irregularity or roughness is small compared with the wavelengths of interest and either has identifiable simple shapes or known mean height and spacing. This means that, if an independent measurement of flow resistivity is available, it may be possible to deduce ground roughness from acoustical measurements The template method for model parameter deduction There is an ANSI standard method based on fitting measurements to templates of theoretical excess attenuation or level difference spectra [34]. These templates are based on the likely range of ground surface parameters and the resulting variation in excess attenuation or level difference spectra predicted according to various impedance models. The best-fit parameters for the impedance model are deduced by eye from the closest template excess attenuation prediction to the data. An example of the use of the template method based on predictions of the two-parameter model (3.38) is shown in Figure 4.3. It is noted in the standard that, in the context of predicting outdoor sound, it would be preferable to deduce impedance directly since the result would be independent of the adequacy of any particular impedance model Direct impedance deduction A method of direct impedance deduction has been proposed based on 2-D minimization of the difference between data and theory at each frequency [35]. This requires considerable computation and is relatively inefficient. An alternative numerical method is based on root-finding [36]. This technique takes advantage of the fact that the classical approximation to the spherical wave reflection coefficient is an analytic function of the impedance. The resulting saving in computation time can be up to a hundred-fold without any loss in accuracy. Examples of results obtained with this method are given in sections 4.2 and 4.4.

132 Predicting outdoor sound 118 Figure 4.3 Excess attenuation spectrum measured (continuous line) at a microphone 0.1 m high and at a horizontal distance of 1 m from a point source loudspeaker 0.1 m above a sports field and predictions (dot-dash, broken, dotted lines) using various values of the parameters (see Legend) in a two-parameter impedance model. Clearly the prediction corresponding to an effective flow resistivity of 100 kpa s m 2 and effective alpha 10/m gives the best fit. 4.2 Comparisons of impedance data with model predictions Figures 4.4 and 4.5 compare data obtained by Cramond and Don using a pulse method [19, 20] with predictions of the semi-emprical Delany and Bazley model (3.2), and porebased models ((3.11), (3.12) and (3.30)). The data are for compacted earth and the same soil with the top 0.02 m loosened. There are significant differences between the measured impedance of compacted and loose soil. Although it is possible to obtain reasonable fits with the semi-empirical model in both cases, improved agreement is possible with models based on an assumed slit-pore microstructure. Figure 4.6 shows impedance data for a hard-backed sample of snow obtained in an impedance tube by Buser [6]. The measured flow resistivity and bulk density of this

133 Measurements of the acoustical properties 119 Figure 4.4 Impedance of compacted soil measured by Cramond and Don [20] and predictions using (3.2) with effective flow resistivity 450 kpa s m 2 (dotted lines), (3.11) and (3.12) and (3.28) (solid lines) with flow resistivity 275 kpa s m 2, porosity 0.4 and tortuosity 2. sample were 9.6 kpa s m 2 and 208 kg m 3 respectively. Assuming an ice density of 913 kg m 3, the latter value corresponds to a porosity of The continuous lines in Figure 4.6 correspond to predictions of an identical slit-pore model assuming tortuosity is equal to inverse porosity. The dotted lines represent predictions of the relaxation model with s B =1. The dashed lines represent predictions of the Hamet phenomenological model (3.5) and the dash-dot lines represent predictions of the Delany-Bazley model (3.1) (3.3) with effective flow resistivity given by the product of measured flow resistivity and porosity. The predictions of the relaxation model can be improved by adjusting the value of s B (1.4 gives better agreement with data). However it should be noted that this removes one of its advantages as a simple model. The slit-pore predictions are obtained using the measured flow resistivity and porosity, without any adjustment, and assuming that tortuosity is given by the inverse of porosity. It is noticeable that use of an effective flow resistivity given by the product of flow resistivity and porosity, in the Delany-Bazley model, gives relatively poor predictions for these data. Figure 4.7 shows example impedance values obtained by the direct deduction method (see section ) over a smoothed sand surface [37] together with fits based on (3.11) (3.13).

134 Predicting outdoor sound 120 Figure 4.5 Impedance of loosened soil (0.02m thick) above compacted soil measured by Cramond and Don [20] and predictions using (3.2) with effective flow resistivity 450 kpa s m 2 (dotted lines); (3.1), (3.2) and (3.3) (dashdot lines) with layer thickness 0.02m; (3.11), (3.12) and (3.30) (solid lines), with upper layer flow resistivity 100 kpa s m 2, porosity 0.4, tortuosity 1.5 and thickness 0.02 m; substrate flow resistivity 800 kpa s m 2, porosity 0.2 and tortuosity 3 and (3.31) (broken lines) in which R e =50 kpa s m 2 and α e =133 m Measured and predicted roughness effects Near-grazing propagation measurements at short-range have been made over a corrugated polymer foam surface in the laboratory and have demonstrated the roughness effects predicted by (3.54) [38]. However the assumed location of the effective admittance plane had to be adjusted to improve agreement with data at higher frequencies. Laboratory measurements of excess attenuation from an elevated continuous point source of white noise have been made in the frequency range 500 to 5 khz over various hard and finite impedance boundaries containing 2-D strip roughnesses [39, 40]. The resulting data has been used to confirm the predicted effects of roughness size, shape,

135 Measurements of the acoustical properties 121 concentration and the acoustical properties of their constituent material, on ground effect. As noted previously, relative SPL spectra measured over acoustically hard rough surfaces show a shift in ground effect to lower frequency than would be expected for a smooth acoustically hard surface. Figure 4.6 Measured and predicted impedance of a 0.05 m thick layer of snow with measured flow resistivity 9.6 kpa s m 2 and porosity Predictions use identical slit pore model ((3.11) and (3.12)) with measured flow resistivity and porosity and tortuosity=1/porosity (solid lines), Hamet- Berengier (dashed), Wilson-relaxation (dotted) and Delany-Bazley (dash-dot) with R e =Ω R s. Measurements [40] have shown that there are considerable differences between the ground effects caused by periodically and randomly spaced roughnesses with the same packing density. Periodically spaced roughnesses yield additional diffraction grating effects and give greater relative SPL minima. Predictions of ground effect due to regularly spaced roughnesses based on an extended effective admittance model are sensitive to small deviations from exactly periodic spacing. Incoherent scattering has been shown to play an important role for the source-receiver geometries and roughness

136 Predicting outdoor sound 122 sizes studied. The effective admittance model has been generalized for arbitrary scatterer shape. Diffraction grating effects have been predicted both by the boundary element model and by a heuristic modification of the classical analytical approximation for propagation from a point source near to an impedance boundary. The resulting approximation gives some good predictions of propagation over wooden slats and triangular wooden rods on a flat hard surface. However, for all of the larger scatterers considered, the boundary element code gives better predictions. Figure 4.8 shows results obtained over an artificially rough hard surface made from inverted ceramic tiles. Figure 4.7 Impedance deduced from three complex excess attenuation measurements with a point source over smoothed sand (source height=receiver height=0.1 m (dotted and dot-dash lines), source height=receiver height=0.05 m (broken and dotted lines), range=1 m). The predictions (solid lines) use a semi-infinite slit-pore model [37], measured flow resistivity 420 kpa s m 2, assumed porosity=0.4 and tortuosity=1/porosity. Figure 4.9 shows impedance data obtained by direct deduction from complex excess attenuation measurements (see section ) over a rough sand surface [37]. Figure 4.10 indicates that the boss model, including randomness and incoherent scatter, may be used to explain certain grassland impedance spectra [41]. The data were obtained over an established area of grass at Silsoe, UK, using the direct deduction technique described in section Note that in Figure 4.10 the measured and predicted impedance tends to zero above 3 khz.

137 Measurements of the acoustical properties 123 There is relatively little published in situ data specifically demonstrating the effects of roughness rather than the combined effects of changing roughness and flow resistivity outdoors. Data obtained by Aylor [42] demonstrates a considerable change in excess attenuation over approximately 50 m range after disking a soil and without any significant change in meteorological conditions. Figure 4.11 shows excess attenuation measured at short range over ground with various surface treatments [43]. Figures 4.12 and 4.13 show spectra of the difference in sound pressure levels received by vertically separated microphones over ploughed and subsoiled heavy (boulder) clay. The microphones were at 1 m and 0.1 m heights and located 30 m from the Electro-Voice loudspeaker source (centre at 1.65 m height) on a sunny Figure 4.8 Measured excess attenuation spectrum (solid line) with source and receiver at heights of 0.25 m and 0.3 m respectively and 2.5 m apart over a surface consisting of 12 equally spaced glazed ceramic tile halfcylinders of radius m. Also shown are predictions without and with scatterer interaction (dash-dot and dashed lines respectively) and of a boundary element model (dotted line).

138 Predicting outdoor sound 124 Figure 4.9 Comparison of measured (solid lines) and predicted (dotted and broken lines) impedance of a sand surface containing (approximately) semicylindrical roughness, height m and centre-to-centre spacing 0.04 m. Predictions use (3.45) and assume semicylindrical roughness with given dimension and spacing and either the predicted (dotted line) or the measured (broken line) smooth surface impedance (see Figure 4.6).

139 Measurements of the acoustical properties 125 Figure 4.10 Normalized impedance data (open circles) obtained from complex excess attenuation measurements over established grassland at Silsoe, Beds, UK [41]. The theoretical predictions (solid lines) are for the impedance of a semi-infinite slit pore medium, with flow resistivity 400 kpa s m 2, porosity 0.4, tortuosity=1/porosity. A partially random (randomness parameter W=0.5) close packed semicylindrical roughness with 0.02 m radius is assumed. Predictions using these parameters but excluding roughness are represented by dotted lines.

140 Predicting outdoor sound 126 Figure 4.11 Measured excess attenuation (15) at short range (source height 0.35 m, receiver height 0.38 m and horizontal separation 1.5 m) above soil subject to the indicated surface treatments [43].

141 Measurements of the acoustical properties 127 Figure 4.12 Average level difference spectrum obtained with a loudspeaker source at 1.65 m height and vertically separated microphones at 1 and 0.1 m height at a range of 30 m over sub-soiled ground (solid line). Predictions assume either the smooth surface of a hardbacked slit pore medium with flow resistivity 80 kpa s m 2, porosity 0.4, tortuosity 2.5 and thickness 0.08m (broken line) or a surface with

142 Predicting outdoor sound 128 these parameters plus 0.05 m high closepacked but partially random (randomness factor 0.75) semicylindrical roughness (dotted line). Predictions are (a) without and (b) with turbulence characterized by L 0 =1 m (see Chapter 9). Figure 4.13 Average level difference spectrum obtained at a range of 30 m over ploughed ground (solid line). Predictions assume either the smooth surface of a hard-backed slit pore medium with flow resistivity 10 kpa s m 2, porosity 0.4, tortuosity 2.5 and thickness 0.175

143 Measurements of the acoustical properties 129 m (broken line) or a rough surface with these parameters plus 0.08 m high, close-packed, semi-elliptical cylinder (eccentricity 1.6) roughness (dotted line) (a) without and (b) with turbulence. The turbulence parameters assumed are the same as for Figure day with light wind. Vertical level differences are surrogates for excess attenuation since they are independent of the source spectrum. There are clear differences in the ground effects. Figures 4.12 and 4.13 also show theoretical predictions for the level difference spectra at 30 m. Predictions are shown both for rough and smooth surfaces in each case. Although some of the change in ground effect between the two conditions will result from differences in flow resistivity, improved agreement with data is obtained in both cases by including roughness effects and by attributing different surface roughness to each ground. Predictions are shown with and without allowance for turbulence. This suggests that, for the conditions and source-receiver geometry obtaining during these measurements, turbulence affects only the higher frequencies and is not responsible for the behaviour of the data around 1 khz which seem consistent with the predicted effects of surface roughness. The roughness size assumed to fit the data over ploughed ground is greater than that assumed for the sub-soiled ground. This is consistent with the fact that sub-soiling produces less surface roughness than ploughing. The fitted thickness (0.175 m) for the surface layer of the ploughed ground is consistent with the expected depth of the plough pan. Figure 4.14 shows the predicted impedance corresponding to the parameters used to fit the level difference data in Figures 4.12 and The impedance predicted for the ploughed field is affected by layer resonance, whereas that predicted for the sub-soiled ground has the rough finite impedance form in which the impedance tends to zero (see section 3.6).

144 Predicting outdoor sound 130 Figure 4.14 Surface impedance of subsoiled (continuous lines) and ploughed clay soil (broken lines) predicted using the parameters identified by fitting the data shown in Figures 4.7 and Measured and predicted effects of ground elasticity As discussed in section 3.4, ground may be described as a layered poroelastic solid [44 49]. Surficial soils have sufficiently low stiffness and rigidity to have seismic wave speeds that may be rather less than the speed of sound in air. Ground elasticity is predicted to have its greatest influence on the surface impedance at low frequencies [49]. However, as remarked already, the measurement of surface impedance in situ at low frequencies is rather difficult [16, 17]. Moreover the influence of ground elasticity depends strongly on the attenuation of the elastic waves in the surface layer for which there is a paucity of data. Consequently the available measurements of impedance are rather inconclusive about the effects of elasticity. Figure 4.15 shows predictions of rigid- and elastic-framed models for the surface impedance of a soil at Bondville, IL, USA characterized as a porous layer over a nonporous elastic intermediate layer and a non-porous elastic substrate and by the parameters listed in Table 4.1 [17]. The impedance is predicted to be slightly less at low frequencies if the soil elasticity is taken into account. Data for the ratio of air pressure 0.1 m above the ground to the vertical soil particle velocity measured at a collocated geophone buried just beneath the ground surface measured in response to C4 explosions have been reported [49]. These data were

145 Measurements of the acoustical properties 131 Figure 4.15 Predicted surface impedance at low frequencies for a soil at Bondville, IL, USA [17]. The continuous lines represent predictions including effects of elasticity whereas the broken lines represent predictions that ignore these effects. A double layer structure was found by a shallow seismic survey and parameters (listed in Table 4.1) were found by direct measurement and by fitting data for the acoustic-to-seismic coupled spectrum [17]. Reprinted with permission from Elsevier. Table 4.1 Seismic parameters measured and deduced for Bondville site Layer Thickness (m) P-wave speed (ms 1 ) S-wave speed (m s 1 ) Density (kg m 3 ) Porosity

146 Predicting outdoor sound 132 Table 4.2 Seismic layering deduced by Spectral Analysis of Surface Waves (SASW) at Finnskogen in Norway Layer Thickness (m) P-wave speed (m s 1 ) S-wave speed (m s 1 ) Damping Halfspace obtained at ranges between 2 and 17 km at Finnskogen in Norway. One objective was to derive an improved ground impedance model for low frequencies that takes ground elasticity into account. Calculations of the impedance of a multi-layered poroelastic system at an angle of incidence of 90 (grazing incidence) were obtained by means of a plane wave model (Multipor) [33]. This model was used also to compute the acousticseismic coupling ratio (A/S), that is the ratio of incident pressure to the component of solid particle velocity normal to the surface. This ratio has the same units as impedance (Pa s m 1 ) and may be termed an acousto-seismic (A/S) impedance. It was shown that the computed quantities are sensitive to ground elasticity. Moreover it was found possible, using the plane wave model, to match the computed acoustic-seismic coupling ratio to data measured at ranges of 2 km and above during blast propagation trials in Norway. The seismic profile deduced on site (see Table 4.2) using the Spectral Analysis of Surface Waves (SASW) method is altered to enable such data fitting (see Table 4.3). The plane wave model predicts results that are independent of range. On the other hand it is noticeable that the A/S impedance data have a significant spread below 7 Hz. Data have been obtained at horizontal ranges between 2 and 17 km. It is noticeable also, that with one possible exception, the data [43] do not show as sharp a minimum as predicted by Multipor. Minima in the A/S impedance data appear between 1.5 and 3 Hz. Table 4.3 Parameters used for fitting acoustic-to-seismic data at Finnskogen, Norway Layer Thickness Flow resistivity (kpa Porosity P-wave speed S-wave speed Damping (m) s m 2 ) (m s 1 ) (m s 1 ) , Halfspace 1,74, Plane wave predictions show a significant minimum in the acoustic-to-seismic coupling ratio at 2.5 Hz which is the frequency at which the (non-porous elastic) Rayleigh wave speed predicted from the dispersion curve for the profile given in Table 4.3, coincides with the sound speed in air. Calculations for propagation from a point source that take ground elasticity, porosity and layering into account may be made using FFLAGS (Fast Field program for Layered

147 Measurements of the acoustical properties 133 Air-Ground Systems). FFLAGS enables atmospheric refraction to be included also but for the calculations presented here the atmosphere is assumed to be homogeneous. Details of FFLAGS are given elsewhere [45]. The flow resistivities and porosities that have been used in the calculations reported here are shown in Table 4.3. It should be noted that the assumed porosities are not consistent with the assumed densities but this inconsistency has little effect on the predictions. Example comparisons between predicted and measured acoustic-seismic impedance (i.e. the ratio of sound pressures at microphones divided by vertical soil particle velocities measured at collocated geophones), are shown in Figure 4.16(a). Points (circles, boxes and crosses) in Figure 4.16(a) represent data. Predictions by FFLAGS at ranges of 6.3 km, 7.2 km and 12.6 km for source height 2 m, receiver heights 0.1 m (microphone) and 0.05 m (geophone) and the four-layer ground profile (Table 4.3) are shown in Figure 4.16(b) and (c). Clearly FFLAGS predictions are in tolerable agreement with data, similar to the plane wave predictions [49] but, unlike these, show range dependence. The predictions are consistent with the differences between data at different ranges. The main minimum in the acoustic-seismic coupling spectrum below 5 Hz is predicted to be relatively shallow and to depend on range. Comparison of predictions in Figure 4.16(b) and (c) indicates that predictions have a significant dependence on the assumed sound speed in air. Figure 4.17 shows predictions obtained at 6300 m range with four different air sound speeds. Sound speeds in air of 329 ms 1, 332 ms 1, 335 ms 1 and 343 ms 1 correspond to temperatures of 4 C, 1 C, 6 C and 20 C respectively at sea level. In particular the predicted A/S impedance below 5 Hz is much altered by different air sound speed values. This is consistent with the interpretation that an air-coupled Rayleigh wave is propagating below 5 Hz since the slope of the predicted Rayleigh wave dispersion curve is small near 329 m s 1 [33, 49].

148 Predicting outdoor sound 134 Figure 4.16 (a) Acoustic-seismic coupling ratios deduced from measurements [33] at ranges of 6.3 km (circles), 7.2 km (boxes) and 12.6 km (crosses); (b) predictions of FFLAGS for sound speed in air of 329 ms 1 ; (c) predictions of FFLAGS for sound speed in air

149 Measurements of the acoustical properties 135 of 332 ms 1. The predictions for 6.3 km are represented by solid lines; for 7.2 km by broken lines and for 12.6 km by dotted lines. Figure 4.17 Predicted acoustic-seismic coupling ratio at 6300 m assuming the layered ground system given by Table 4.3 and sound speeds in air of 329 ms 1 (thick solid line), 332 ms 1 (broken line), 335 ms 1 (dash-dot line) and 343 ms 1 (thin solid line) respectively. The data at a given range [49] display some variability. However, there will have been some sources of variability, for example meteorology, topography and changing conditions with range, which are not taken into account by FFLAGS. Predictions of the surface impedance of layered poroelastic ground were shown and discussed in Chapter 3 (Figures ). As well as A/S coupling and surface impedance, FFLAGS may be used to predict excess attenuation spectra (i.e. the sound pressure spectra relative to free field). Such predictions were discussed in Chapter 3 also. 4.5 Comparisons between template fits and direct impedance fits for ground impedance As part of a US Army field trial [50], surface impedance measurements have been made at various positions in a field of grass-covered sandy soil, on two gravel pits, over adjacent ploughed ground. Measurements of complex sound pressure level were made using a source consisting of mid-range Hi-Fi driver in a cylindrical cabinet and Maximum Length Sequence Signal Analysis (MLSSA-Half Blackman Harris window) for signal generation, acquisition and analysis (Paper 1, figures 8 and 9). Relatively shortrange measurements (maximum 3 m horizontal separation) with low source and receiver

150 Predicting outdoor sound 136 heights (no greater than 0.6 m) were made. Similar measurements were made over the ploughed ground and the gravel pits. An MLSSA generated signal consists of a pseudorandom sequence of pulses which sounds continuous and has a broad spectrum. Use of an MLSSA system for generation and acquisition gives good signal-to-noise ratio since it can be expected that background noise does not contain the same sequence as the source signal. However there may be problems with the use of MLS methods in time-varying environments. To enable calculation of the sound levels relative to free field (excess attenuation), free field data were obtained at the relevant ranges by raising the source and receiver heights to over 1 m. The analysis uses both a method similar to the Template Method for Measuring Ground Impedance [34], and the method of Direct Impedance Fitting [36]. The template method uses only the magnitude of the measured excess attenuation spectra whereas the direct impedance fitting requires the complex excess attenuation data, that is magnitude and phase, to calculate the real and imaginary parts of surface impedance. Figures show example results. In each case the left-hand graph shows the data and results of impedance model fitting using the template method. For the template method, three impedance models have been used: a single-parameter semi-empirical model (Delany and Bazley (3.2)), a two-parameter (variable porosity) model (3.12), in which the parameters are effective flow resistivity and rate of change of porosity with depth and a sixparameter rough porous layer model ((3.3) and (3.60)), assuming identical triangular pores and with two of the parameters, namely porosity and tortuosity, fixed at 0.4 and 2.5 respectively. The continuous lines represent the loudspeaker-gathered data, broken lines represent predictions using the two-parameter model, and dotted lines represent predictions using the six-parameter model. The dash-dot lines represent predictions using the single-parameter Delany and Bazley model (3.2). The right-hand graph in each of Figures 4.18 through 4.21 shows the directly deduced impedance spectra as black continuous lines. The other lines represent the various impedance model predictions using the best fit parameter values deduced from the excess attenuation spectra. Although it is not the only model that gives tolerable agreement with the measured excess attenuation spectra, the six-parameter model consistently gives best agreement with the directly deduced impedance spectra, particularly for the plowed ground and gravel pits. Gravel Pit 1 contained three layers of gravel within a total depth of 1.5 m. For the purposes of the template fitting for ground impedance, Pit 1 has been treated as a single layer and the effective depth adjusted for best fit. With this simplification, the apparent flow resistivity is smaller than the measured flow resistivity for Pit 2 which is consistent with the larger stone size in the layer nearest the surface of Pit 1. In the case of gravel Pit 2, template fitting (after fixing the flow resistivity at the measured value for compacted 8 mm gravel) has been applied assuming either hard-backing or sandy soil backing. The latter slightly reduces the oscillations in the predictions observed at low frequencies. The directly deduced (linear) impedance spectra for both gravel pits suggest more or less pure resistance. Table 4.4 summarizes the best-fit values of the four adjustable parameter values for all of the surfaces according to the six-parameter model.

151 Measurements of the acoustical properties 137 Figure 4.18 Parts (a) and (c) show data for the sound level with respect to free field and deduced complex impedance of a sandy soil ground surface. The sound level differences were obtained from free field measurements with raised source and receiver and from measurements nearer the ground. For (a), the microphone and sound source were 0.54 m above the ground and the horizontal separation was 2 m. For (c), microphone and sound source were 0.23 m above the ground and the horizontal separation was 2 m. Parts (b) and (d) show real and imaginary parts of the impedance calculated from the measured and predicted complex sound pressure level differences as solid lines. Three impedance models have been used to obtain best fits with the magnitude of the sound level differences and predictions of the surface impedance. The dash-dot lines represent predictions of the single-parameter model (3.2); broken lines represent predictions of the two-parameter model (3.12); dotted lines represent predictions of the six-parameter model (3.3 and 3.60). For the single parameter model, the effective flow resistivity was 200 kpa s m 2. For the two-

152 Predicting outdoor sound 138 parameter model, effective flow resistivity was 80 kpa s m 2 and the porosity change rate was 0/m. For the six-parameter model, flow resistivity was 150 kpa s m 2, porosity was 0.4, tortuosity was 1/porosity or 2.5, roughness height was m and the roughness spacing was 0.05 m. In this case the rough layer is assumed to be semi-infinite. Figure 4.19 Parts (a) and (c) show data for the sound level with respect to free field and deduced complex impedance of the sandy soil surface obtained after ploughing. The sound level differences were obtained from free field measurements with raised source and receiver and from measurements near the ground across the furrows. For (a) microphone and sound source were 0.54 m above the ground and the horizontal separation was 3 m. For (c), microphone and sound source were 0.54 m above the ground and the horizontal separation was 2 m. Parts (b) and (d) show real and imaginary parts of the impedance calculated

153 Measurements of the acoustical properties 139 from the measured and predicted complex sound pressure level differences. The various lines represent predictions of impedance models as in Figure For the single parameter model, the effective flow resistivity was 50 kpa s m 2. For the two-parameter model, effective flow resistivity was 30 kpa s m 2 and the porosity change rate was 100/m. For the six-parameter model, flow resistivity was 10 kpa s m 2, porosity was 0.4, tortuosity was 1/porosity or 2.5, roughness height was 0.04 m, roughness spacing was 0.1 m and layer thickness was 0.1 m. Figure 4.20 Parts (a) and (c) show data for the sound level with respect to free field and deduced complex impedance of the ground surface obtained for a gravel pit containing three layers with different stone sizes; the largest stones being near the surface. The sound level differences were obtained from free field measurements with raised source and receiver and from measurements near the pit

154 Predicting outdoor sound 140 surface. For (a), microphone and sound source were 0.54 m above the ground and the horizontal separation was 2 m. For (c), microphone and sound source were 0.54 m above the ground and the horizontal separation was 1.5 m. Parts (b) and (d) show real and imaginary parts of the impedance calculated from the measured and predicted complex sound pressure level differences. The various lines represent predictions of impedance models as in Figure For the single parameter model, the effective flow resistivity was 80 kpa s m 2. For the two-parameter model, effective flow resistivity was 20 kpa s m 2 and the porosity change rate was 250/m. For the six-parameter model, flow resistivity was 0.5 kpa s m 2, porosity was 0.4, tortuosity was 1/porosity or 2.5, roughness height was 0.01 m, roughness spacing was 0.05 m and layer thickness was 4.5 m. Figure 4.21 Parts (a) and (c) show data for the sound level with respect to free field and deduced complex impedance of the surface of a pit containing medium sized gravel. The sound level differences were obtained from

155 Location Measurements of the acoustical properties 141 free field measurements with raised source and receiver and from measurements near the pit surface. For (a), microphone and sound source were 0.18 m above the ground and the horizontal separation was 2 m. For (c), microphone and sound source were 0.39 m above the ground and the horizontal separation was 1.5 m. Parts (b) and (d) show real and imaginary parts of the impedance calculated from the measured and predicted complex sound pressure level differences. The various lines represent predictions of impedance models as in Figure For the single parameter model, the effective flow resistivity was 20 kpa s m 2. For the two-parameter model, effective flow resistivity was 25 kpa s m 2 and the porosity change rate was 200/m. For the six-parameter model, flow resistivity was 1.65 kpa s m 2 (the independently measured value of the compacted pea gravel), porosity was 0.4, tortuosity was 1/porosity or 2.5, roughness height was m, roughness spacing was 0.02 m and layer thickness was 1.5 m. Table 4.4 Summary of the four adjustable best-fit parameters for ground impedance (Figures ) using the six-parameter model Effective flow resistivity (kpa s m 2 ) Layer depth (m) Roughness height (m) Unploughed ground near m site Ploughed ground (across furrows) Pit 1 (triple layer) Pit 2 (medium gravel with 1.65 (measured 1.5 (actual sandy soil backing) compacted value) depth) Table 4.5 Measured flow resistivities and porosities of grassland Roughness spacing (m) Ground type Flow resistivity (kpa Porosity (air-filled/ waterfilled) s m 2 ) or total Loamy sand beneath lawn (no roots) 677± /0.137 Grass covered compact sandy soil 463± /0.052 Grass-covered field /0.160

156 Predicting outdoor sound 142 Loamy sand beneath lawn (0.06 m 237± thick with roots) Grass 220 Grass root-filled layer 189±91 Loamy sand with roots (mixed grass overgrowth) 114± / Measured flow resistivities and porosities Tables 4.5 to 4.7 show the range of flow resistivities and porosities that have been measured for outdoor ground surfaces [6, 13, 51 53]. The range of flow resistivities is particularly wide. According to Table 4.5, the flow resistivity of a ground described simply as grassland could vary by a factor of five. According to Table 4.6, the flow resistivity of granular surfaces including soils, sands and gravel could vary by a factor of more than 200. Table 4.6 Measured flow resistivities and porosities of granular surfaces Ground type Flow resistivity (kpa s m 2 ) Porosity (air-filled/ waterfilled) or total Wet sandy loam Compacted silt Mineral layer beneath mixed 540± /0.15 deciduous forest Sand (moistened) Loamy sand on plain 422± /0.112 Hard clay field 400 Sand (dry) Bare sandy plain 366± /0.093 Dry sandy loam Humus on pine forest floor 233± /0.161 Sand (dry) Sand (grain dia mm) Sand (dry) Sand (grain dia mm) Gravel (mean max. grain dimension 1.81 mm) Table 4.7 Measured values for low flow resistivity outdoor surfaces Ground type Flow resistivity (kpa s m 2 ) Porosity (air-filled/ waterfilled) or total Litter layer on mixed deciduous forest 30±31 floor ( m thick) Wet peat mul 24±5 0.55/0.29

157 Measurements of the acoustical properties 143 Beech forest litter layer ( m 22± thick) Snow (old) Pine forest litter ( m thick) 9± /0.286 Snow (new) Gravel (mean max. grain dimension mm) Railway ballast [53] Table 4.8 indicates that the presence of acoustically soft organic root layer (the root zone) above rootless mineral soil reduces the flow resistivity near the surface by a factor of between 2 and 5 compared with that in the substrate. The measured effects of roots on porosity are less dramatic. Table 4.8 Influence of root zones on flow resistivity and porosity Ground type Flow resistivity (kpa s m 2 ) Porosity (volume %) Bare loamy sand 422± ±1.7 Grass root layer in loamy sand 153± ±4.4 Grass root layer in loamy sand 237± ±9.3 Loamy sand beneath root-zone 677± ±1.7 Loamy sand with roots 114± ±4.5 Table 4.9 Ranges of effective flow resistivity (when used in (3.2)) that give the best agreement between predicted and measured excess attenuation or level difference spectra over various types of ground surface Description of surface Flow resistivity/cgs rayls (1 cgs rayl=1000 Pa s m 2 ) Dry snow, new fallen 0.1 m over about 0.4 m older snow Sugar snow In forest, pine or hemlock Grass: rough pasture, airport, public buildings, etc Roadside dirt, ill-defined small rocks up to 0.1 m mesh Sandy silt, hard packed by vehicles Asphalt, sealed by dust and light use ~30,000 Upper limit set by thermal-conduction and viscous to boundary layer 4.7 Effective flow resistivities and other fitted parameters A set of excess attenuation measurements at short range (between 7 and 15 m) made by Embleton et al. [30] over various surfaces have been fitted subsequently by the single-

158 Predicting outdoor sound 144 parameter Delany-Bazley model (3.2) giving a wide range of effective flow resistivities for several surfaces (see Table 4.9). There are impedance measurements that are not fitted well by using (3.2) [54, 55]. In some cases the ground impedance can be modelled adequately by a single parameter according to (3.30). In the many cases, where the soil is not homogeneous, (3.31) or (3.35) may turn out to give better fits. Using these equations and best fits to the experimental data, Table 4.10 may be constructed. Also listed in Table 4.10 [13, 19, 20, 56, 57], where available, are the non-acoustically determined values of flow resistivity and (air-filled) porosity. Table 4.10 Ground parameter data and corresponding best-fit impedance models Data source Soil type Measured parameters Best-fit parameters Frequency range (khz) R s (kpa s Ω R e (kpa s α e d e /m m 2 ) (air) m 2 ) Homogeneous van der Heijden Sand (15 cm) [13] van der Heijden [13] Sandy soil Don and Grassland Cramond [20] Don and Grass-covered field Cramond [20] Cramond and Grassland Don [19] Variable porosity van der Heijden Lawn [13] Don and Cramond [20] Hay-covered field van der Heijden Meadow with grass [13] 8 10 cm high Hard-backed layer lchida [56] Snow van der Heijden [13] Floor of a mixed forest containing firs van der Heijden [13] Meadow covered in low vegetation Huisman [57] Floor of pine forest It is noticeable that in many cases the acoustically deduced or effective flow resistivities are smaller than the measured ones.

159 Measurements of the acoustical properties 145 References 1 American Society for the Testing of Materials, ASTM, Standard test method for impedance and absorption of acoustical materials by the tube method, (1977). 2 Y.Ando, The directivity and acoustic centre of a probe microphone, J. Acoust. Soc. Jpn., 24: (1968). 3 P.J.Dickinson and P.E.Doak, Measurement of the acoustical impedance of ground surfaces J. Sound Vib., 13(3): (1970). 4 T.F.W.Embleton, J.E.Piercy and N.Olson, Outdoor sound propagation over a ground of finite impedance, J. Acoust. Soc. Am., 59(2): (1976). 5 R.Talaske, The Acoustic Impedance of a Layered Forest Floor, MS Thesis, Pennsylvania State University. 6 O.Buser, A rigid frame model of porous media for the acoustic impedance of snow, J. Sound Vib., 111:71 92 (1986). 7 American Society for the Testing of Materials ASTM E1 050, Standard testing method for impedance and absorption of acoustical materials using a tube, two microphones and a digital frequency analyzer (1986). 8 J.Y.Chung and D.A.Blaser, Transfer function method of measuring a duct acoustic properties: experiment, J. Acoust. Soc. Am., 68: (1980). 9 F.J.Fahy, Rapid method for the measurement of sample acoustic impedance in a standing wave tube, J. Sound Vib., 97: (1984). 10 W.T.Chu, Further experimental studies on the transfer technique for impedance tube measurements, J. Acoust. Soc. Am., 83: (1988). 11 G.M.Heissler, O.McDaniel and M.Dahl, Measurements of normal impedance of six forest floors by the tube method, Proc. 2nd Symposium on Long Range Sound Propagation and Acoustic/Seismic Coupling, Vol.2, , University of Mississippi PARGUM (1985). 12 A.J.Zuckerwar, Acoustic ground impedance meter, J. Acoust. Soc. Am., 73(6): (1983). 13 L.A.M.van der Heijden, The Influence of Vegetation on Acoustic Properties of Soils, PhD Thesis, University of Nijmegen, The Netherlands (1984). 14 J.F.Allard and B.Sieben, Measurements of acoustic impedance in a free field with two microphones and a spectrum analyzer, 77: (1985). 15 D.Waddington, Acoustical Impedance Measurement using a Two-Microphone Transfer Function Technique, PhD Thesis, University of Salford (1990). 16 G.A.Daigle and M.R.Stinson, Impedance of grass covered ground at low frequencies using a phase difference technique, J. Acoust. Soc. Am., 81:62 68 (1987). 17 M.W.Sprague, R.Raspet, H.E.Bass and J.M.Sabatier, Low frequency acoustic ground impedance measurement techniques, Appl. Acoust., 39: (1993). 18 L.A.M.van der Heijden, J.G.E.M. de Bie and J.Groenewoud, A pulse method to measure the impedance of semi-natural soils, Acustica, 51(3): (1982). 19 A.J.Cramond and C.G.Don, Reflection of impulses as a method of determining acoustic impedance, J. Acoust. Soc. Am., 77: (1984). 20 C.G.Don and A.J.Cramond, Soil impedance measurements by an acoustic pulse technique, J. Acoust. Soc. Am., 77(4):1601 (1985). 21 D.Habault and G.Corsain, Identification of the acoustical properties of a ground surface, J. Sound Vib., 100(2): (1985). 22 D.Habault and P.Filippi, Ground effect analysis: surface wave and layer potential representations, J. Sound Vib., 79: (1981). 23 C.Hutchinson-Howorth, K.Attenborough and N.W.Heap, Indirect in-situ and free-field measurement of impedance model parameters or surface impedance of porous layers, Appl Acoust., 39: (1993).

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161 Measurements of the acoustical properties 147 model verification, Appl. Acoust., 66(5): (2005). Reprinted with permission from Elsevier. 50 P.Schomer and K.Attenborough, Basic results of tests at Fort Drum, Noise Control Eng. J., 53: (2005). 51 M.J.M.Martens, L.A.M.van der Heijden, H.H.J.Walthaus and W.J.J.M.van Rens, Classification of soils based on acoustic impedance, air flow resistivity and other physical soil parameters, J. Acoust. Soc. Am., 78: (1985). 52 O.Umnova, K.Attenborough, H.-C.Shin and A.Cummings, Response of multiple rigid porous layers to high levels of continuous acoustic excitation, J. Acoust. Soc. Am., 116(2): (2004). 53 K.Attenborough, P.Boulanger, Q.Qin and R.Jones, Predicted influence of ballast and porous concrete on rail noise, Proc. Inter Noise 2005, Rio de Janeiro, Paper D.L.Johnson, J.Koplik and R.Dashen, Theory of dynamic permeability and tortuosity in fluidsaturated porous materials, J. Fluid Mech., 176: (1987). 55 P.Leclaire, L.Kelders, W.Lauriks, C.Glorieux and J.Thoen, Determination of the viscous characteristic length in air-filled porous materials by ultrasonic attenuation measurements, J. Acoust. Soc. Am., 99(4): (1996). 56 T.Ishida, Acoustic properties of snow, Contrib. Intensity Low Temp. Sci. Series A, 20: (1965). 57 W.H.T.Huisman, Sound Propagation over Vegetation-covered Ground, PhD Thesis, University of Nijmegen, The Netherlands (1990).

162 Chapter 5 Predicting effects of source characteristics on outdoor sound 5.1 Introduction The sound field due to a monopole source in a homogeneous medium above an absorbing ground is cylindrically symmetric so that there is no azimuthal variation. But many outdoor noise sources are directional and do not behave as simple monopole sources. Source descriptions in terms of multipoles may be more useful. Although multipole sources radiate a much weaker sound field than the corresponding monopole source, there are many situations when the multi-pole source strength is very large and there is no significant contribution from monopole radiation. For instance, sound radiation by a vibrating sphere or an unenclosed loudspeaker can be represented by a dipole. Lighthill s aeroacoustic analogy establishes that the sources of a quadrupole nature may be used to model jet noise. 5.2 The sound field due to a dipole source In Chapter 2, we have derived a general expression for the sound field above a ground surface due to a point monopole source. We have seen that the interaction of the sound waves with the ground can be modelled using the concept of the effective impedance. For an extended-reaction ground the plane wave reflection, R p depends on the angle of incident waves. As the ability for the sound waves to penetrate the ground diminishes, the ground becomes locally reacting so that the plane wave reflection coefficient is essentially independent of the angle of incidence. This is an useful interpretation of the ground impedance as it greatly simplifies our analysis for sources other than a monopole. Apart from the monopole, the simplest source of practical importance is a dipole. It can be imagined as two closely spaced point monopole sources with the same amplitude but opposite in sign, that is to say, they are 180 out of phase. The sound field can be calculated by summing the individual contribution due to these two monopoles and taking the limit of a vanishingly small separation. Typical examples are the sound generated by an oscillating sphere, by an enclosed loud speaker and the component of train noise resulting from wheel/rail interaction. These noise sources can be modelled successfully by using a dipole. In view of this, we continue this section with the study of the sound field due to a dipole source above a ground surface. The ground is assumed to have normalized effective admittance, β, throughout. Previously, sound radiation from a dipole has been studied only in an unbounded medium [1, 2].

163 Predicting effects of source characteristics on outdoor sound A horizontal dipole Let us assume that two out-of-phase monopoles are placed at the same height, z s above the ground and at a horizontal distance of 2 apart. The line joining the two monopole sources (or the so-called dipole axis) can be aligned so that it is parallel to the x-axis. If it is not so aligned, we can always achieve this by rotating the x y plane about the z-axis appropriately. Without loss of generality, we take the centre of the dipole at the position (0, 0, z s ) and the source strength of each monopole to be S 0. Since the monopoles are 180 out of phase, we can refer to them as positive and negative monopoles. We adopt the convention that the dipole axis is drawn from the negative monopole to the positive as shown in Figure 5.1. The sound field due to such a system of sources can be found by solving the Helmholtz inhomogeneous equation. subject to the usual boundary condition at the ground, z=0 of (5.1a) (5.1b) Figure 5.1 The source/receiver geometry for a dipole source above an impedance ground: (a) 2D view and (b) 3D view. In the limit of small, the right side of (5.1a) can be simplified by taking the difference of the Taylor expansions of the two terms in the square bracket to give (5.2)

164 Predicting outdoor sound 150 where S 1 =2S 0 may be regarded as the source strength of the dipole. We look for the Green s function, G h (x) for the sound field due to a horizontal dipole locating at x s (0, 0, z s ) and receivers at x (x, y, z) with the source strength S 1 =1. Using the method of Fourier transformation (see (2.5)), the Helmholtz equation can be reduced to (5.3a) and the boundary condition at z=0 becomes (5.3b) The problem as stated in (5.3) can be solved in the same manner as shown in Chapter 2. The solution is (5.4) Hence the sound field can be cast in an integral form of (5.5) where A direct evaluation of the first integral of (5.5) is a challenging, if not impossible, task. A more subtle method for its solution is to reformulate the problem as stated in (5.2): it is equivalent to finding the direct wave field, p d in the absence of any boundary. Replacing the acoustic pressure by p d = φ/ x and reversing the order of differentiation, we obtain

165 Predicting effects of source characteristics on outdoor sound 151 (5.6) Obviously, φ is the solution for a monopole source located at (0, 0, z s ). The solution is well known and it can be evaluated exactly as Hence the direct wave term is simply, (5.7a) Sometimes, it will be more convenient to express the solution in the form of spherical polar coordinates centring on the source position. Then the solution becomes (5.7b) The typical characteristics of the solution for a dipole sound field are evident in the above equation. Essentially, the solution comprises of two components: a near field dominated by the term and a far field term given by ik/r 1. Similarly, the second integral of (5.5) can be identified as the sound field due to the image source situating at (0,0, z s ). By analogy, the solution is (5.8a) Expressing the solution in the form of spherical polar coordinates (R 2, θ, ψ) centred on the image source, we obtain (5.8b) In (5.5), we are left with the evaluation of the third integral (5.9)

166 Predicting outdoor sound 152 in order to specify fully the analytical solution for the Green s function of the horizontal dipole. Again, we find it fruitful to transform the integral expression from (k x, k y ) to (κ, ε) in the same way as we do for the monopole source in Chapter 2. Then (5.9) can be transformed to (5.10) and Using the general form of the integral expression for the Bessel function of n-th order [1, Eq. (9.1.21)], (5.11a) together with the identity of (5.11b) we see that I can be written in terms of the Bessel function of the first order as (5.12) With the use of the following identities, and the substitution of κ=k sin µ in (5.12), we arrive at a somewhat more familiar form as follows: (5.13)

167 Predicting effects of source characteristics on outdoor sound 153 Comparing with (2.28), we find that (5.13) has an extra sin µ term and the Hankel function of the first order is used instead of zero-th order owing to the fact that the horizontal dipole is considered here. At first sight, the seemingly different Hankel functions used for monopole and horizontal dipole appears to pose a problem for the asymptotic evaluation of (5.13). Nevertheless, we find that it is possible to generalize the method from that of the monopole to a source of higher-order pole by using the following integral expression for Hankel functions and their asymptotic forms: and (5.14a) (5.14b) The integral can be evaluated asymptotically in the same way as that for monopole and the details are described elsewhere [2]. The solution can be expressed as where (5.15a) (5.15b) With the use of (5.14b) with n=1 and the usual approximations, can simplify I to and θ π/2, we (5.16) Summing all contributions, (5.7) and (5.16), the required Green s function for a horizontal dipole is

168 Predicting outdoor sound 154 (5.17) To express the required Green s function in the form of Weyl-Van der Pol formula (see (2.38)), we can rewrite (5.16) as (5.18) where and θ π/2 are assumed. These assumptions are generally valid for most practical outdoor sound predictions. Hence the Green s function can be cast in the classical Weyl-Van der Pol form as: (5.19) where Q is the spherical reflection coefficient given in (2.40c). We note the interesting point that the Green s function can be derived also, by stating (5.20) Substituting (2.40a) into (5.20), differentiating each term with respect to x and assuming that (Q sin θ)/ x is negligibly small, we can arrive at the same expression as given in (5.19). Nevertheless, we show the fuller derivation here because it is instructive to extend the asymptotic analysis from that of the monopole source to the dipole case through this relatively simple control case. As our knowledge grows, we can handle cases for higher-order sources with confidence. We also wish to point out that the corresponding Green s function can be readily extended to an arbitrarily oriented horizontal dipole. Noting that the azimuthal angle is the angle measured from the dipole axis that is aligned along the x-axis, by a simple rotation of axis we can see that the sound field can be written as (5.21)

169 Predicting effects of source characteristics on outdoor sound 155 where ψ 1 is azimuthal angle of the axis of the arbitrary orientated dipole. A more rigorous analysis for the validity of this expression will be left to section when we study the sound field due to an arbitrarily oriented dipole. Before we proceed to study the sound field due to a vertical dipole, we show the predicted sound fields due to a horizontal dipole which is aligned along the x-axis, that is ψ 1 =0. Figure 5.2(a) compares the excess attenuation, defined as the total sound field relative to the direct field for monopole and horizontal dipole. The source and receiver heights are 1.0 and 0.5 m respectively, and the separation is 5.0 m. The two-parameter impedance model (3.32), is used throughout this chapter. Substituting values for the constant parameters of air, we arrive at the following equation: which is essentially the same as (3.38). For dipole sources, assumed default parameter values for σ e and α e are 38 kpa s m 2 and 15 m 1. Unless stated otherwise, the default parameters are used in all calculations for the dipole. With the assumed geometry, there is little difference between the predicted excess attenuation spectra for monopole and horizontal dipole sources because the angles for the direct and the reflected wave components, and θ respectively, are very close. Hence, the excess attenuation spectra are virtually indistinguishable, except at the interference minima, for monopole and dipole sources. Figure 5.2(b) shows the same comparison but at only 1 m range. For this geometry, the angle between and θ is large and so the difference between the excess attenuation spectra predicted for monopole and horizontal dipole sources is significant. Also shown in Figure 5.2(a)(b) are the prediction of sound fields due to a vertical dipole. Their interpretations will be discussed in the next section.

170 Predicting outdoor sound 156 Figure 5.2 The comparison of excess attenuation spectra due to a monopole (solid line), horizontal dipole (dotted line) and a vertical dipole (dashed line). The source/receiver geometry is z s =1.0 m and z=5.0 m: (a) range=5.0 m (b) range=1.0 m The sound field due to a vertical dipole Suppose that the dipole is aligned so that its axis is aligned along the vertical z-axis as shown in Figure 5.3. The governing inhomogeneous Helmholtz equation is (5.22) where a unit dipole source strength is assumed. The required Green s function after the Fourier transformation is (5.23) subject to the same boundary condition (5.3b).

171 Predicting effects of source characteristics on outdoor sound 157 We introduce a potential function such that potential function is modified to dφ/dz+ikβφ=0 at z=0. The boundary condition for the Equation (5.23) can then be simplified to Figure 5.3 A vertical dipole above an impedance ground. (5.24) Since the original Helmholtz equation (5.22) contains a source term proportional to δ(z z s )/ z, we expect continuity of pressure gradient but a pressure jump at the plane z=z s. This condition can be stated in terms of the potential function φ (see also (2.12)) as (5.25) Hence the solution for (5.24), which satisfies the boundary conditions (5.25), is (5.26) where the sign function

172 Predicting outdoor sound 158 (5.27) Replacing the potential function with and substituting it into the inverse Fourier integral, we can express the sound field for the vertical dipole as (5.28) We can identify immediately that the first integral corresponds to the direct field, p d while the second is the sound field due to the image source, p r. Again, the solution for a free field dipole can be used for a vertical dipole as we did for a horizontal dipole in the last section. The only requirement is to determine the angle between the dipole axis and the line radiating from the source to receiver, which is for the case of the direct wave. Hence, it is not difficult to see that (5.29a) Similarly, the angle between the dipole axis and the line radiating from the image source to receiver is θ. The corresponding sound field can be represented by (5.29b) Note, in the direct wave, that the sign function has been incorporated in the cosine term by a precise definition of the direction of polar axes (dipole axis and the spherical polar coordinate axis. We choose that all the polar angles are measured from the direction of the negative z-axis (see Figure 5.3). When the source is located above the receiver, the polar angle for the direct wave is less than π/2 which implies is positive. On the other hand, when the source is placed below the receiver, then which implies a negative value for the cosine term. An immediate consequence is that the principle of reciprocity does not hold for a vertical dipole. Exchanging the source and receiver

173 Predicting effects of source characteristics on outdoor sound 159 position will not lead to the same sound field. We confirm this later where we derive the full expression for the sound field due to the vertical dipole. In (5.28), we are left with the third term for further investigation. Just like the horizontal dipole, this term can be simplified to (5.30) The integral with respect to ε can be evaluated by using the identity (5.11 a) to yield which can be further reduced to (5.31a) We resolve the integrand by partial fractions leading to (5.31b) A close examination of the above integrals reveals that the first term is akin to the direct wave term of a monople and the second term is analogous to the boundary wave term (see (2.28)). Hence, no extra effort is required to obtain the solutions for these two integrals and the boundary wave term for the vertical dipole is simply (5.32) Summing (5.29) and (5.32), we get the total sound field due to a vertical dipole (or the Green s function) as

174 Predicting outdoor sound 160 (5.33) With the extra requirement that it is possible to write the Green s function in the classical form of the Weyl-Van der Pol formula as (5.34) where cos µ p (with µ p as a complex angle) is determined according to cos µ p = β. (5.35) The term cos µ p is used to facilitate the interpretation of the theoretical formula as it is linked to the propagation of ground wave. We can imagine it as the direction that characterizes such propagation. It is tempting, as for the horizontal dipole, to replace µ p with θ. However, numerical experiments suggest that it will not lead to a satisfactory approximation. Figure 5.4(a)(b) show predictions (solid line) of the sound field due to a vertical dipole as a function of horizontal range at 1000 and 100 Hz. The source and receiver heights are located at 2.0 and 1.0 m above the ground surface. The dotted line represents the predicted sound levels due to a monopole source at the same source/receiver geometry and frequencies as the case of vertical dipole. The source strength for the dipole is normalized to give the same sound level as the vertical dipole at 0 m range. In general, the interference pattern of the dipole is essentially the same as that of the monopole source but the level is somewhat lower. Predictions of the excess attenuation spectra due to a vertical dipole are shown also plotted in Figure 5.2(a)(b) in the last section. Compared with the corresponding spectra for a monopole or a horizontal dipole, they exhibit greater interference effects. Figure 5.5

175 Predicting effects of source characteristics on outdoor sound 161 Figure 5.4 The predicted sound pressure level (SPL) from a vertical dipole (solid line) above an impedance ground. The predicted sound level due to a monopole (dotted line) is shown also. The source and receiver heights are 2.0 m and 1.0 m respectively: (a) frequency, f=1000 Hz and (b) frequency, f=100 Hz. Figure 5.5 Predicted excess attenuation of sound due to a monopole (solid line) and vertical dipole (dashed line) above an

176 Predicting outdoor sound 162 impedance ground. The dotted line is the ground wave component of the sound field due to the vertical dipole. The horizontal dipole for this case is indistinguishable from that for a monopole. The geometrical configuration used in the plot is as follows: the source and receiver heights are 0.1 m and m respectively and range is 2.0 m. shows predictions of excess attenuation spectra with vertical dipole (solid line), monopole (dotted line) sources and receiver close to an impedance ground surface with heights of 0.1 m and 0.25 m respectively. Propagation of near-grazing sound, the ground wave contribution due to the vertical dipole is predicted to be particularly significant. In Figure 5.5, the predicted ground wave contribution due to a vertical dipole is shown by the dotted line. It is of interest to point out that the predicted sound fields are different by changing the position of source and receiver. This is illustrated in Figure 5.6 by excess attenuation predictions at a range of 1 m but with the source at m and the receiver at 0.1 m over the default impedance ground surface. The discrepancy can be explained by the fact that the polar angles joining the source directly with the receiver are different by π as shown in Figure 5.7(a)(b). The magnitude of the direct wave is the same for both cases but they are different by a factor of 1. Therefore, the sound fields for both situations are not identical. Consequently, a straightforward application of the reciprocity theorem is not applicable in the case for a vertical dipole source above the ground surface. Figure 5.6 Predicted excess attenuation of sound due to a monopole (solid line) and vertical dipole (dotted line) above an impedance ground with source and receiver heights of 0.25 m and 0.1 m respectively and

177 Predicting effects of source characteristics on outdoor sound 163 range of 1.0 m. The dashed line represents the prediction for a vertical dipole with source and receiver heights of 0.1 m and 0.25 m respectively and the same range. Figure 5.7 The differences in the polar angles when the position of source and receiver are interchanged The Green s function for an arbitrarily orientated dipole Having gone through the analyses for the sound field due to a horizontal and vertical dipole, it is natural to expect that the Green s function for an arbitrarily orientated dipole is a combination of these two fundamental building blocks. To fix ideas, let s think about the representation of an arbitrarily orientated dipole the separation of the two equal and opposite monopoles being 2. The process is much easier if one introduces the direction cosines l (l x, l y, l z ) of the dipole axis. Sometimes, these direction cosines are referred as the dipole-moment amplitude [1, 2, pp. 165]. As in our previous analysis, the spherical polar coordinates are frequently used. So it is convenient to define the dipole axis as (5.36) where γ 1 is the polar angle measured from the negative z-axis and ψ 1 is the azimuthal angle measured from the positive x-axis, see Figure 5.8. Take, for example, that the direction cosines for a horizontal dipole aligned in the x-direction is (1, 0, 0). Hence, the

178 Predicting outdoor sound 164 polar and azimuthal angles γ 1 and ψ 1 are π/2 and 0 respectively. On the other hand, the direction cosines for a vertical dipole aligned in the z-direction and its corresponding polar and azimuthal angles are (0, 0, 1), 0 and π/2 respectively. The source on the right side of (5.1a), Γ 1 (r s ) is simply Γ 1 (r s )= S 0 [δ(r [r s +1 ]) δ(r [r s 1 ])] (5.37a) Figure 5.8 An arbitrarily oriented dipole above an impedance ground. where r s and r are the source and receiver position. Expanding the terms in Taylor s series and grouping the resulting terms, we modify the source term to Γ 1 (r s )=S 1 [δ (x)δ(y)δ(z z s )+δ(x)δ (y)δ(z z s )+δ(x)δ(y)δ (z z s )], (5.37b) where the primes denote the derivatives of the delta functions with their respective arguments. We can now apply the method of Fourier transformation to (5.1a) with (5.37b) as the source term. Then the Green s function, G d (x, y, z) for a dipole of unit strength can be reduced to (5.38) subject to the same impedance boundary condition, dp/dz+ikβφ=0 as before.

179 Predicting effects of source characteristics on outdoor sound 165 Following the same procedure as detailed in section and 5.2.3, we obtain a solution for as (5.39a) where Λ =k x l x +k y l y +sgn(z z s )k z l z (5.39b) Λ + =k x l x +k y l y +k z l z (5.39c) and (5.39d) Since each of the terms, Λ and Λ +, can be decomposed into a horizontal and a vertical component, the horizontal component contains a further of two terms, k x l x and k y l y. There is only a single term, k z l z for the vertical component. It is obvious that the Green s function can be considered as a sum of two components that can be written as G d (x, y, z)=g h (x, y, z)+g v (x, y, z), (5.40a) where, (5.40b) and

180 Predicting outdoor sound 166 (5.40c) The inverse Fourier transforms for horizontal and vertical dipoles can also be written in the polar form as (5.41a) where the reflection coefficient V is transformed to (5.41b) (5.41c) The results in the preceding sections can be applied for the evaluation of (5.41a) and (5.41b) with some minor modifications. The solutions are (5.42a) and (5.42b)

181 Predicting effects of source characteristics on outdoor sound 167 For an arbitrary orientated dipole, the total sound field can be calculated by substituting (5.42) into (5.40a). To write the expression in a more compact form, we can introduce the following unit vectors: (5.43a) (5.43b) (5.43c) where and are the unit vectors pointing radially outward from the dipole centre towards the observation point for the direct and reflected wave respectively. In addition, we may regard as the unit vector that characterizes the direction of propagation of the ground wave as it involves the complex angle µ p. Noting that l is the direction cosine of the dipole axis, we can show that (5.44a) and (5.44b) (5.44c) For near-grazing propagation where we can substitute ( ) into (5.40a) to obtain a close form analytical solution for the sound field due to an arbitrarily orientated dipole as follows:

182 Predicting outdoor sound 168 (5.45) Finally, it is of interest to point out that although the sound field generated by a dipole can be obtained, in principle, by differentiating the monopole sound field with respect to the appropriate spatial coordinates [see for example, the asymptotic expression given in 3], in practice however, the asymptotic expression for the monopole sound field is not sufficiently precise to yield satisfactory results, especially when calculating the verticaldipole component of the sound field. 5.3 The sound field due to an arbitrarily orientated quadrupole In the last section, the asymptotic solution of the sound field due to a dipole has been derived. It is natural to ask whether a similar asymptotic solution can be developed for an arbitrarily orientated quadrupole because it can be imagined as two closely spaced dipoles with equal but opposite dipole-moment amplitude vectors. Also, we are interested to see whether the classical form of the Weyl-Van der Pol formula can be extended to the corresponding sound field due to the quadrupole. Study of the field due to a quadrupole source above a ground surface is relevant to modelling jet engine testing noise outdoors. In this section, we shall derive the corresponding asymptotic formula for the sound field due to the quadrupole. As the construction of a quadrupole is realizable by putting two closely spaced dipoles of equal but opposite dipole-moment amplitude vectors, the source term for the inhomogeneous Helmholtz equation is simply Γ 2 (r s )=Γ 1 (r s +m ) Γ 1 (r s m ) (5.46) where Γ 1 (r s ) is the source term for a dipole (see (5.37)), and m (m x, m y, m z ) are the direction cosines that characterize the orientation of the quadrupole axis. Also, it is convenient to introduce the corresponding spherical polar coordinates with the polar angle of γ m and azimuthal angle ψ m. Using the definition for the dipole source (5.37b) in (5.46), expanding the corresponding terms by Taylor Series and simplifying the resulting expression, we can show that the quadrupole source term can be written in a rather compact form as (5.47a)

183 Predicting effects of source characteristics on outdoor sound 169 where and S 2 =2S 1 is the quadrupole source strength. The term can be expanded further to yield Γ 2 (r s )=S 2 {l x m x δ (x x s )δ(y y s )δ(z z s ) +l y m y δ(x x s )δ (y y s )δ(z z s ) +l z m z δ(x x s )δ(y y s )δ (z z s ) +(l x m y +l y m x )δ (x x s )δ (y y s )δ(z z s ) +(l y m z +l z m y )δ(x x s )δ (y y s )δ (z z s ) +(l z m x +l x m z )δ (x x s )δ (y y s )δ (z z s )}. (5.47b) The method for deriving the required asymptotic solution for arbitrary orientated quadrupole is similar to that for the dipole, which has been described in section 5.2. The derivation involves some tedious algebraic manipulations, but the approach is straightforward. In this section, we outline the method and obtain the solution. Using the method of Fourier transformation, as before, with the source term specified by (5.47), the approximate solution for an arbitrarily orientated quadrupole of unit strength is [4] (5.48) It is evident from (5.48) that the amplitude of the sound field due to a quadrupole source is a function of frequency. This characteristic is quite different from that of a monopole source in which the monopole source is proportional to 1/R. This implies that the extra path length of the image wave (i.e. R 1 R 2 ) becomes significant at high frequencies and the excess attenuation spectrum tends to large values. This feature, which is more prominent at smaller incident angles, is therefore not related to the effective impedance of the ground surface, but rather to the fact that the amplitude of the quadrupole field is dependent on the frequency. Two types of quadrupole, longitudinal and lateral, are classified normally although it is possible to introduce the horizontal and vertical components just like we do for the dipole. A longitudinal quadrupole has the characteristic such that the direction cosines, l and m, are parallel while they are perpendicular for a lateral quadrupole (see Figure 5.9). In the following plots, we show the numerical results of our asymptotic solution (5.48) for these two types of quadrupole. In particular, we consider the case of a longitudinal quadrupole such that it is aligned perpendicular to the ground surface, γ m =γ 1 =0. We shall describe it as the vertical longitudinal quadrupole. A lateral quadrupole, with γ m =π/2 and

184 Predicting outdoor sound 170 Figure 5.9 Two basic types of quadrupole: a longitudinal quadrupole and a lateral quadrupole. The direction of arrows indicate the orientation of the dipole axis (l) and the quadrupole axis (m). γ 1 =0, is considered also. Ground parameters with σ e and α e of 100 kpa s m 2 and 50 m 1 are used for the predictions in this section. Figures 5.10 and 5.11 show the predicted excess attenuation spectra for a vertical longitudinal quadrupole and a lateral quadrupole. The corresponding spectra due to a vertical dipole and monopole are also shown in the figures for comparison. The horizontal quadrupole [γ m =γ 1 =π/2], which has a similar characteristic as the monopole and horizontal dipole source, will not be discussed here. In Figure 5.10, the source and receiver heights are assumed to be 2.5 and 1.2 m respectively and the separation is 50 m. As shown in section 5.2.2, the predicted excess attenuation for a vertical dipole shows greater interference effects as compared with that due to a monopole (the dotted and dashed lines in 5.10(a)). Furthermore, the frequencies of the subsequent interference dips are about the same for the monopole, vertical dipole and longitudinal quadrupole, but the interferences are much less significant for the longitudinal quadrupole. The

185 Predicting effects of source characteristics on outdoor sound 171 Figure 5.10 The excess attenuation spectra for a quadrupole (solid line), a vertical dipole (dotted line) and a monopole (dashed line). The source and receiver heights are 2.5 m and 1.2 m respectively and the horizontal range is 50 m. (a) A vertical longitudinal quadrupole, and (b) A lateral quadrupole with γ m =π/2, γ 1 =0, and ψ m =π/2, ψ 1 =0. The lateral quadrupole and the vertical dipole spectra are almost coincident within the thickness of the line.

186 Predicting outdoor sound 172 Figure 5.11 The excess attenuation spectra for a quadrupole (solid line) a vertical dipole (dotted line) and a monopole (dashed line). The source and receiver heights are 2.5 m and 1.2 m respectively and the range is 320 m. (a) A vertical longitudinal quadrupole, and (b) A lateral quadrupole with γ m =π/2, γ 1 =0, and ψ m = π/2, ψ 1 =0.The lateral quadrupole and the vertical dipole spectra are almost coincident within the thickness of the line. predicted spectra for a lateral quadrupole, a vertical dipole and a monopole are shown in 5.10(b). At this source/receiver geometry, there is little difference in the predicted spectra between a lateral quadrupole and a vertical dipole. Figure 5.11 (a) shows prediction of attenuation spectra with a vertical longitudinal quadrupole (solid line), a vertical dipole (dotted line) and a monopole (dashed line) source with the same source and receiver heights, but the range is 320 m. It can be seen, in this near-grazing propagation, that the predicted excess attenuation due to the longitudinal quadrupole is lower than that due to a vertical dipole, but it is still higher than that due to a monopole. In Figure 5.11(b), we show the predicted attenuation, for the same source/receiver geometry, for a lateral quadrupole (solid line), a vertical dipole (dotted line) and a monopole (dashed line). There is not much difference in the predicted excess attenuation between a lateral quadrupole and a vertical dipole, and these predictions are different from that due to a

187 Predicting effects of source characteristics on outdoor sound 173 monopole. It should be noted that typographical errors in reference [6] have been corrected in Figures 5.10(b) and 5.11(b). Finally, we remark that the resulting formula may be used to study the propagation of jet noise above a ground surface. In an idealized case, jet noise source is represented by either a randomly orientated longitudinal or a randomly orientated lateral quadrupole. Taherzadeh and Li [7] have applied the above formulation to investigate the directivity pattern of a simple quadrupole and have suggested that there is a significant influence of the impedance of ground surface on the directivity of jet noise. This is a consequence of near cancellation of the intensity field at the near-grazing angles for both longitudinal and lateral quadrupoles. Furthermore, the directivity pattern is explicitly dependent on the frequency and the source height rather than the product of kz s as suggested by Smith and Carpenter [8] when they consider turbulent jet noise above a hard ground. Such a modification is due to the frequency dependence of the reflected wave term. 5.4 Source directivity and railway noise prediction Accurate prediction of noise from railways has become increasingly important as a result of increased rail traffic and the emphasis on high-speed rail links. There are several forms of noise generation on modern trains: traction noise, rail/wheel interaction noise, auxiliary equipment noise and aerodynamic noise. In many studies of noise from trains travelling at speeds below km h 1, it is often assumed that traction and aerodynamic noise are unimportant and that the predominant noise mechanism is the rail/wheel interaction. In this respect, it has been suggested that railway noise at these speeds can be modelled by a set of sources which are located at some positions between 0 m and 1 m above either the centreline of the nearest track or the nearside rail. It has been known for many years that wheel/rail noise emitted by a train is directional and the approximate representation by a line of incoherent dipole sources provides good agreement with the measured data. In particular, most electrically hauled trains radiate sound with dipole source characteristics. As a result, the Austrian ÖAL model [9] uses a combination of dipole and monopole sources with the ratio of 15% monopole and 85% dipole type radiation. An alternative numerical model based on the boundary integral equation method has been considered by Morgan [10]. According to this model, the boundary integral form (8.7) is modified so that the standard 2-D Green s function for sound propagation from a monopole above an impedance ground, G(r, r s ), is replaced by (5.49) where u x, u y are the components of u=(u x, u y ), the unit vector along the axis of the dipole, and the derivatives G(r, r s )/ x s and G(r, r s )/ y s can be calculated using a very efficient technique [11]. Usually, in railway noise calculations, the dipoles are assumed to be orientated horizontally, that is u=(1, 0). Morgan used the boundary integral equation method to predict railway noise propagation in the presence of noise barriers and rigid train bodies [10]. The cross-section

188 Predicting outdoor sound 174 of the track considered in this study is reproduced in Figure Two railway tracks here are modelled separately with a pair of dipole sources at the railheads above a porous layer of ballast. The assumed spectral strength of these sources is that suggested by Hemsworth [12] for a passenger coach. The ground is flat and consists of grassland on the receiver side including the vicinity of a 2.0 m noise barrier (NB). The left side of the noise barrier is absorbing. The acoustic surface admittance of the porous ballast, grassland and absorbing barrier treatment can be modelled, for example, using the four-parameter Attenborough model [13] for which Morgan suggested the values of the non-acoustic parameters listed in Table 5.1. The receiver positions are located at distances of 20, 40 and 80 m on the right-hand side from the noise barrier and at 1.5 m above the ground. The insertion losses are calculated in 1/9-octave bands in the frequency range between 63 and 3150 Hz Figure 5.12 Assumed railway cross-section (adapted from [10]). Table 5.1 A summary of the non-acoustical parameters used in a four-parameter model for porous grassland, ballast and absorbing barrier treatment (adapted from [10]) Surface type Flow resistivity, R Porosity Tortuosity Layer depth Pore shape (kpa s m 2 ) (Ω) (q) d (m) factor (s p ) Grassland Ballast Absorbing barrier treatment

189 Predicting effects of source characteristics on outdoor sound 175 Table 5.2 Comparison of the broad band A-weighted insertion loss (db) for two types of sources of railway noise (adapted from [10]) Barrier height (m) Source type Nearside track (m) Farside track (m) Monopole Dipole Monopole Dipole using the boundary integral equation method for both monopole (equation (8.7) with Green s function (8.10)) and dipole sources (equation (8.7) with Green s function (5.49)). The following formula is then used to combine the 1/9-octave band results (5.50) so that the broadband insertion losses, LB, presented in Table 5.2 [10] can be obtained. Here L 0 (f n ) are the N 1/9-octave railway noise spectra predicted in the absence of the barrier and L B (f n )=L 0 (f n ) IL P (f n ), where IL P (f n ) is the predicted 1/9-octave barrier insertion loss. The results demonstrate that, for the assumed railway noise spectrum, the difference between the predictions based on the monopole and dipole models is marginal and vanishes at the receiver positions more distant from the noise barrier. This prediction that the difference is small may be largely attributed to the effect of acoustic scattering from the ballast, train body and from the noise barrier. A small discrepancy between these results can be observed also with an increased barrier height and this is explained by the actual difference in the emission characteristics. References 1 A.D.Pierce, Principles and Applications, second printing, Acoust. Soc. Am., New York (1991). 2 A.P.Dowling and J.E.Ffowcs Williams, Sound and Sources of Sound, Ellis Horwood Ltd., Chichester (1983). 3 M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York (1972). For other identities of Bessel functions, see also G.N.Watson, A Treatise on the Theory of Bessel Functions, Cambridge UP, Cambridge (1944). 4 K.M.Li, S.Taherzadeh and K.Attenborough, Sound propagation from a dipole source near an impedance plane, J. Acoust. Soc. Am., 101: (1997). 5 A.V.Generalov, Sound field of a multipole source of order N near a locally reacting surface, Sov. Phys. Acoust., 33: (1987). 6 K.M.Li and S.Taherzadeh, The sound field of an arbitrarily-oriented quadrupole above a ground surface, J. Acoust. Soc. Am., 101: (1997).

190 Predicting outdoor sound S.Taherzadeh and K.M.Li, On the turbulent jet noise near an impedance surface, J. Sound Vib., 208: (1997). 8 C.Smith and P.W.Carpenter, The effect of solid surfaces on turbulent jet noise, J. Sound Vib., 185: (1995). 9 ÖAL Richtlinie, Nr. 30, Calculation of Noise Emission from Rail Traffic (in German). 10 P.A.Morgan, Boundary element modelling and full scale measurement of the acoustic performance of outdoor noise barriers, PhD Thesis, University of Bradford, November (1999). 11 S.N.Chandler-Wilde and D.C.Hothersall, Efficient calculation of the Green s function for the acoustic propagation above a homogeneous impedance plane, J. Sound Vib., 180 (5): (1995). 12 B.Hemsworth, Prediction of train noise, in P.M.Nelson (Ed.), Transportation Noise Reference Book, Butterworth, London (1987). 13 K.Attenborough, Acoustical impedance models for outdoor ground surfaces, J. Sound Vib., 99(4): (1985).

191 Chapter 6 Predictions, approximations and empirical results for ground effect excluding meteorological effects 6.1 Approximations for frequency and range dependency Several analytical approximations for ground effect have been derived. Rasmussen [1] has obtained an approximation for frequency-dependent ground effect in the form of the direct and plane wave reflected fields plus a correction. This approximation is based on the Delany and Bazley single parameter model for ground impedance (3.2). His result for the sound level relative to free field may be expressed as follows: EA=10 log{l+( R(θ) r 1 /r 2 ) 2 +2( R(θ) r 1 /r 2 ) cos[k(r 1 r 2 )+a]+a 2 +2 exp( 5cos(θ))A cos ξ[1+( R(θ) r 1 /r 2 ) cos[k(r 1 r 2 )+α]] (6.1) where the complex plane wave reflection coefficient, R(θ)=RR+iRI, is calculated from Z=R+iX is the normalized surface impedance, α= tan 1 ( R I /R R ), A= (1 R R )/[1+k 2 (36.4r/R se )] 2 and ξ= 4 tan 1 [k(36.4r/r se )]. An example comparison between (6.1) and a calculation based on the Weyl-Van der Pol expression (2.40) is shown in Figure 6.1. Although it avoids computation of complementary error functions of complex argument, the approximation does not reproduce the main ground effect dip particularly well for the chosen parameters and it is restricted to a single-parameter ground impedance model (see Chapter 3 for a discussion of impedance models). Moreover, it is not difficult to compute the full solution for any impedance model or measured impedance (see Chapter 2 (2.34) et seq). Defiance and Gabillet [2, 3] have derived two approximations, also based on the Delany and Bazley impedance model. The two approximations are, respectively,

192 Predicting outdoor sound 178 Figure 6.1 Comparison between the approximation represented by (6.1) and a calculation using the Weyl-Van der Pol formula for source at 1.65 m, receiver at 1.2 m and separated horizontally by 10 m over a Delany-Bazley impedance given by an effective flow resistivity of 200 kpa s m 2. for shorter ranges over relatively hard surfaces and longer ranges over soft surfaces. This is a consequence of the intended application to predictions in the presence of barriers where the ground on the source side of the barrier (a road surface for example) tends to be acoustically harder than that on the receiver side. Their results may be expressed as EA 1 =10 log {[1 δ/r B+cos (kδ)] 2 +[B sin(kδ)] 2 } (6.2) where (6.3) where

193 Predictions, approximations and empirical results 179 Figures 6.2 and 6.3 show example comparisons between the predictions of these equations and those of (2.40) (the Weyl-Van der Pol formula), for a relatively short range (20 m). The approximations are reasonable for their intended combinations of range and Figure 6.2 Comparison between predictions of the Defrance approximation (6.2) (solid and dash-dot lines) for source height 0.3 m, receiver height 2 m and 20 m range over two relatively hard surfaces. Corresponding predictions, using the Weyl-Van der Pol formula (2.40), are shown as broken and dotted lines. The ground impedances are given by the Delany and Bazley model (3.2) with R se =5000 kpa s m 2 and 1000 kpa s m 2 respectively.

194 Predicting outdoor sound 180 Figure 6.3 Comparison between predictions of the Defrance approximation (6.3) (solid line) for source height 0.3 m, receiver height 2 m and 200 m range over softer ground (6.3). The corresponding Weyl-Van der Pol type predictions are represented by the dashed line. The ground impedance is computed from the Delany and Bazley formula (3.2) with R se = 100 kpa s m 2. flow resistivity. Nevertheless, again (6.2) and (6.3) are limited to a single impedance model of limited applicability for outdoor surfaces. Moreover (6.3) fails to be a good approximation at short range for low flow resistivity surfaces. As mentioned earlier, given the relative ease with which the more exact analytical approximation (2.40) can be calculated, there seems little reason for these frequencydependent approximations. There is a better computational case for such approximations if predictions of the variation of A-weighted levels with distance are required. 6.2 Approximations and data for A-weighted levels over continuous ground Makarewicz and Kokowski [4] have deduced a simple semi-empirical form for carrying out calculations of the variation in A-weighted levels from a stationary source for ranges up to 150 m taking into account wavefront spreading and ground effect only. Their result for propagation from a point source may be expressed as

195 Predictions, approximations and empirical results 181 (6.4) Here E is an adjustable parameter intended to include the effect of the presence of the ground on radiation of sound energy from the source (2 E 1), L WA is the A-weighted sound power level of the source and γ g is an adjustable ground parameter. The lower the impedance of the ground, the larger is the value of γ g. Figure 6.4 shows a comparison of predictions of (6.4) for the decrease in A-weighted level with range from a source with a sound power spectrum similar to that of an Avon jet engine but 30 db less. The circles represent predictions obtained from (2.40) using the Delany and Bazley single parameter ground impedance model (R se =200 kpa s m 2 ). The solid line represents predictions from (6.4) with E=2 (the value for hard ground) and γ g = The agreement between the predictions is rather good. Strictly (6.4) is intended for use at relatively short ranges (<150 m). Makarewicz and others [5 8] have suggested approximations that can be used for predicting for A- weighted levels for longer ranges. These depend on the assumed impedance model and take meteorological factors into account as well as ground effect and wavefront spreading. They will be discussed in Chapter 12. The values of the parameters to be used in (6.4) to predict the curve shown in Figure 6.4 have been derived from trial-and-error fitting. However, where the sound power level of the source is unknown, Makarewicz suggests the use of in situ sound level measurements (L A1 and L A2 ) at two locations (r 1, h 1 ) and (r 2, h 2 ). The ground parameter γ g may be deduced subsequently from (6.5) where value for γ g in (6.4). The value of E may then be obtained by substituting the

196 Predicting outdoor sound 182 Figure 6.4 Predictions of A-weighted levels up to 110 m from a source with a broad band sound power spectrum similar to that of an Avon jet engine (see Chapter 4) but 30 db less. The source and receiver heights are 1 m. Circles represent predictions obtained from (2.40) and (3.2) with R se = 200 kpa s m 2. The solid line represents values obtained from Makarewicz formula (6.4) with E=2 (the value for hard ground) and γ g = Figure 6.5 [4, 9] shows average data for the A-weighted sound level with distance up to 50 m over three surfaces. Two of the surfaces were of adjacent portions of a field. One part of the field was covered in wheat stubble and the other had been ploughed. The rate of sound attenuation over the ploughed portion is clearly greater than that over the wheat stubble. The third surface was covered with oil seed rape plants shortly after flowering. No quantitative meteorological data is available for the measurements over the wheat stubble field. However, in the oil seed rape field, there was with a slight wind component (between 0.9 and 1.9 m s 1 ) from source to receivers. The same source (an Electro-Voice loudspeaker) was used to obtain all of these data which have been normalized to the level measured at 1 m. Since this level was uncalibrated, the data show relative rather than absolute values of sound levels. The shape of the A-weighted sound power spectrum deduced from the measured sound level spectrum at 1m, assuming spherical spreading from the source, is shown in Figure 6.6. The open diamonds in Figure 6.5 represent data obtained with the source at 1.65 m height and receivers at 1.2 m height (except at 1 m range where the receiver was at 1.65

197 Predictions, approximations and empirical results 183 m height) above the oil seed rape crop which was about 1 m high. Also shown (as a dotted line), are predictions obtained from the simple formula for Figure 6.5 A-weighted levels at receivers 1.2 m high due to a broad-band sound signal generated by a loudspeaker source 1.6 m high over an oil seed rape field shortly after flowering (diamonds) and adjacent portions of a wheat stubble field unploughed (joined crosses) and ploughed (joined circles). Also shown are predictions of the Makarewicz and Kokowski approximation [4] (solid, broken and dash-dot lines respectively) and the ISO scheme [9] for broad-band A-weighted levels excluding air absorption (dotted line).

198 Predicting outdoor sound 184 Figure 6.6 A-weighted sound spectrum at 1 m from the loudspeaker (arbitrary reference) used to generate the levels shown in Figure 6.5. Table 6.1 Values of the ground parameter γ g (6.4) obtained from measurements at ranges up to 50 m over three surfaces (see Figure 6.5) Surface γ g Oil seed rape field (recently flowered) Undisturbed part of wheat stubble field Ploughed part of wheat stubble field A-weighted ground attenuation A g given in the ISO scheme [9], that is, A g =4.8 (2h/r)[17+(300/r)], A g 0. (6.6) The measured data indicate significantly larger ground attenuation up to 50 m than predicted by the ISO broadband formula. The differences in attenuation are much greater than the expected contribution of air absorption at the ranges concerned. Moreover they have been observed under slightly downwind conditions (wind component between source and receiver approximately 2 m s 1 ). The ISO formula is intended to predict for moderate downwind conditions. The solid lines in Figure 6.5 shows predictions from (6.4) with E=1 (the value for soft ground) and the value of γ g listed in Table 6.1. The agreement between the (6.4) predictions and data (open diamonds) obtained over the oil seed rape field is good. The procedure based on (6.5) and the measured levels at 10 and 50 m have been used to fit the data shown in Figure 6.5 (open circles and crosses) above subsoiled and undisturbed portions of the wheat stubble field respectively. This yields the values of γ g listed in Table 6.1. In each case, the resulting predictions of (6.4) at the three other locations (between

199 Predictions, approximations and empirical results and 40 m) are good, whereas the ISO broadband formula (6.6) underestimates the measured attenuation by up to 10 db. It should be noted that the values of γ g that fit these data are significantly larger than those used by Makarewicz and Kokowski [4] to fit data obtained over grassland. 6.3 Predictions of the variation of A-weighted noise over discontinuous surfaces Comparison between predictions and data for traffic noise The empirical formula in CRTN [10] for predicting ground effect from free-flowing traffic is L CRTN =5.2 log(3h/d), 1 h d/3 (6.7a) where h is the mean propagation path height and d is the distance from edge of the nearside carriageway. Hothersall and Chandler-Wilde [11] have used a boundary element method to make a series of predictions of excess attenuation for A-weighted traffic noise. They assumed a continuous line source with a standard traffic noise spectrum over a hard surface and 3.5 m from the edge of soft ground. The soft ground impedance was represented by the Delany and Bazley model (3.2) using an effective flow resistivity of 250 kpa s m 4. They calculated the excess attenuation compared to a reference level at 10 m from the source. A good fit to their calculations for continuous soft ground beyond the edge of the hard surface was found to be given by the formula L BEM =8 log(3.72h/d) 1 h d/3.72. (6.7b) A comparison of predictions using (6.6) and (6.7) assuming a receiver height of 1.5 m is shown in Figure 6.7. The predictions of the standard (CRTN) method, for a receiver height of 1.5 m, appear relatively conservative. On the other hand, the boundary element predictions are for an isothermal stationary atmosphere whereas the standard predictions are intended to apply to moderate downwind conditions and the data on which it is based will include turbulence effects. In particular, turbulence will tend to reduce the ground effect Measurements and predictions of noise from aircraft engine testing Noise measurements have been made to distances of 3 km during aircraft engine run-ups with the aim of defining noise contours in the vicinity of airports [12].

200 Predicting outdoor sound 186 Figure 6.7 Comparison between predictions of a formula deduced from boundary element calculations [11 (solid line) and the empirical relationship for excess attenuation used in CRTN [10] for free flowing traffic (broken line). Measurements were made for a range of power settings during several summer days under near to calm weather conditions (wind speed <5 ms 1, temperature between 20 and 25 C). Between 7 and 10 measurements were made at every measurement station (in accordance with ICAO Annex 16 requirements) and the results have been averaged. Example results are shown in Figure 6.8. It can be shown that these data are consistent with nearly acoustically neutral conditions [12]. Note that at 3 km, the measured levels are more than 30 db less than would be expected from wavefront spreading and air absorption only. Figure 6.8 shows predictions based on the known engine spectrum in the direction of the measurements (Figure 6.9), (2.40) and the Delany and Bazley one-parameter impedance model (3.2) with effective flow resistivities of 20,000, 300 and 2000 Pa s m 2 respectively for concrete, grass and soil. Up to distances of between 500 and 700 m from the engine, the data in Figure 6.8 suggest attenuation rates near to the concrete or spherical spreading plus air absorption predictions. Beyond 700 m, the measured attenuation rate is nearer to the soil or between the soil and grass predictions. These results are consistent with the fact that the run-ups took place over the concrete surface of an apron and further away (i.e. between 500 and 700 m in various directions) the ground surface was soil and/or grass. Indeed these data can be shown to be consistent

201 Predictions, approximations and empirical results 187 Figure 6.8 Measured differences (joined crosses) between the A-weighted sound level at 100 m and those measured at ranges up to 3 km during an II-86 aircraft s engine test in the direction of maximum jet noise generation (~40 from exhaust axis) and predictions for levels due to a point source at the engine centre height assuming spherical spreading plus air absorption and various types of ground [12]. Reprinted with permission from Elsevier. Figure 6.9 SPL Spectra, normalized to a distance of 1 m from an IL-86 engine test, as a function of angle (theta) (forward of the aircraft=0 ) [12]. Reprinted with permission from Elsevier.

202 Predicting outdoor sound 188 Figure 6.10 Measured differences (joined crosses) between the A-weighted sound level at 100 m and those measured at ranges up to 3 km from an AI-24 turbo-prop engine (on an An-24 aircraft) in the direction of maximum propeller noise generation (~80 from axis of engine inlet) [12]. Reprinted with permission from Elsevier. with predictions that assume an impedance discontinuity between 500 and 1000 m from the source [12]. Figure 6.10 shows equivalent data and predictions for the attenuation from a propeller aircraft. The averaged data in both the maximum jet noise and the maximum propeller noise directions indicate a change from one rate of attenuation to another at distances greater than 500 m.

203 Predictions, approximations and empirical results 189 Figure 6.11 Measured sound levels in the direction of maximum propeller noise from a turboprop aircraft and the fit given by (6.8). Figure 6.11 shows that a fit to the data beyond 100 m in the direction of maximum propeller noise is given by L A = log(d) 0.005d, (6.8) where d m is the horizontal distance. This suggests attenuation considerably in excess of the sum of spherical spreading together and air absorption. References 1 K.B.Rasmussen, Approximate formulae for short-distance outdoor sound propagation, Appl. Acoust., 29: (1990). 2 J.Defrance, Methode Analytique pour 1e calcul de propagation de bruit exterieur, PhD Thesis, L Université du Maine, France (1997). 3 J.Defrance and Y.Gabillet, A new analytical method for the calculation of outdoor noise propagation, Appl. Acoust., 57(2): (1999). 4 R.Makarewicz and P.Kokowski, Simplified model of ground effect, J. Acoust. Soc. Am., 101: (1997). 5 R.Makarewicz, Near-grazing propagation over soft ground, J. Acoust. Soc. Am., 82: (1987). 6 K.M.Li, K.Attenborough and N.W.Heap, Comment on Near-grazing propagation over soft ground, J. Acoust. Soc. Am., 88: (1992). 7 R.Makarewicz, Reply to comment on Near-grazing propagation over soft ground, J. Acoust. Soc. Am., 88: (1992).

204 Predicting outdoor sound K.Attenborough and K.M.Li, Ground effect for A-weighted noise in the presence of turbulence and refraction, J. Acoust. Soc. Am., 102: (1997). 9 ISO , Method for predicting outdoor sound Part 2 a general method of calculation, HMSO Calculation of Road Traffic Noise, Dept. of the Environment and the Welsh office, HMSO S.N.Chandler-Wilde and D.C.Hothersall, Propagation of road traffic noise over ground of mixed type, in Proc. Inst. Acoust., 7(2): (1985). 12 O.Zaporozhets, V.Tokarev and K.Attenborough, Predicting noise from aircraft operated on the ground, Appl. Acoust., 64: (2003). Reprinted with permission from Elsevier.

205 Chapter 7 Influence of source motion on ground effect and diffraction 7.1 Introduction Many of the previous chapters have been concerned with the sound field due to a stationary monopole source above an absorbing ground. For example, the asymptotic solution for a stationary monopole was considered in Chapter 2. However, there is an increasing concern about the noise from all forms of transportation. Such sources are neither stationary nor monopole in nature. We studied the sound field due to a directional source in Chapter 5. In this chapter, we study the effect of source motion on sound propagation outdoors. Previous analyses have simplified the problem of modelling a moving source either by treating it as a quasi-stationary point source or, in the case of road traffic noise, as a continuous line source. This assumption is valid for a vehicle travelling at a low speed but is unlikely to be adequate for a source travelling at a non-negligible Mach number such as a high-speed train, an advancing helicopter or aircraft approaching an airport during landing (or taking-off). A better physical understanding of the effects of source motion should lead to better designs of suitable noise control devices. There have been many theoretical and experimental efforts to study the effects of motion on aerodynamic noise. However, these were restricted to either an unbounded medium or propagation above a rigid half plane [1 3]. In this chapter we investigate the effect of motion of the source on propagation of sound above an absorbing ground by means of analytical approximation. The problem to be addressed can be considered as an extension to the problem of predicting the free field sound pressure due to a moving source. This type of problem has a long history and has been addressed by many authors. In his seminal paper Lowson [4] pointed out the importance of stating whether the theoretical derivation is based on the wave equation for the velocity potential or on the wave equation for the perturbation pressure. Similarly, Graham and Graham [5] have distinguished two different types of models for a moving source: a conventional moving source that injects fluid as it moves, and, a modified source that can be depicted as an expanding and contracting balloon in uniform motion. Despite the considerable previous theoretical work, there are relatively few in-depth studies addressing outdoor sound propagation from sources moving at high speeds. In this chapter, we shall provide the theoretical basis for such developments. Our study is based on the wave equation for the velocity potential as it is used to describe the sound field due to a moving monopole source [3]. The use of velocity potential is tied in closely with the fundamental theory of fluid mechanics whereas an approach based on the wave equation for the perturbation pressure has little physical significance.

206 Predicting outdoor sound A monopole source moving at constant speed and height above a ground surface Consider a point monopole source of unit strength moving at a constant speed of u( cm) along the x-axis where c is the speed of sound and M is the Mach number. The source is moving at a constant height, z s above an impedance ground with a specific admittance β. The intention is to determine an analytical expression for the sound field due to a moving source above an impedance ground in the homogeneous atmosphere. The governing equation is the space-time wave equation given by (7.1a) and the boundary condition, at z=0, is determined by (7.1b) Here, φ is the velocity potential relating to the acoustic pressure p by p= φ/ t, (7.2) ω 0 is the angular velocity of the source (x, y, z) are the rectangular coordinates of space and t is time. The analysis can be simplified considerably by using a Lorentz transformation [5] such that (7.3) where (7.4) This transformation reduces the governing wave equation and the boundary condition (at z L =0) to

207 Influence of source motion on ground effect 193 (7.5a) (7.5b) where h s =γz s and the subscript L denotes the corresponding parameters in the Lorentz coordinates. The resulting equation (see (7.5a)), is analogous to that of a stationary source except that the source strength and the time-dependent factor are modified accordingly. In addition, the boundary condition (7.5b), is somewhat different from that of a stationary source above an impedance ground. It is apparent from (7.5b) that use of a Lorentz transformation leads to a relatively simple equation compared to that resulting from use of a Galilean transformation [6, 7]. The interpretation of different terms in the analytic expression will be left until section 7.5 after we have derived the corresponding expression for a source moving perpendicularly to the ground surface in section 7.3. The time-dependent factor, can be factored out by separating the variables in the solution of (7.6) into spatial and temporal domains as (7.6) where G L (X L ) is the required Green s function for the Helmholtz equation with the source located at (0, 0, h s ). Then (7.5) becomes (7.7a) (7.7b) where k 0 ( ω 0 /c) is the wavenumber. We introduce a Fourier transformed pair for the Green s function in the Lorentz space (cf. (2.5a) and (2.5b) of Chapter 2), (7.8a) and

208 Predicting outdoor sound 194 (7.8b) using the same analysis as described in Chapter 2, we can show that (7.9) Substitution of (7.9) into (7.8b) gives an integral representation of the sound field which can be estimated by evaluating the integrals asymptotically. The solution for (7.9) is wellknown for the case of a stationary source and it is sometimes referred to as the Weyl-Van der Pol formula. The method of steepest descent is used to derive the asymptotic solution. The total field is given by a sum of three components: a direct wave term G 1, a contribution due to an image source G 2 and a boundary wave term G b. The boundary wave term is particularly important for the near-grazing propagation above a ground of finite impedance. The same analytical method can be applied to the corresponding result for a moving source and leads to an accurate asymptotic solution. It follows that φ 1 and φ 2 can be found straightforwardly as (7.10) (7.11) where d 1 and d 2 are the direct and reflect path lengths in Lorentz space: (7.12) The boundary wave term, G b, can be expressed as (7.13)

209 Influence of source motion on ground effect 195 The evaluation of the above integral can be simplified considerably by using spherical polar coordinates centred on the source and the concept of effective admittance such that β L =γβ(1+m sin θ L cos ψ L ) (7.14) where θ L and ψ L are, respectively, the polar and the azimuthal angles of the reflected wave in the Lorentz space. The polar angle θ L may be interpreted as the angle of incidence of the reflected wave also. Using the procedure detailed in Chapter 2, the boundary wave term can be evaluated to give (7.15a) where (7.15b) Noting the relationship (7.6) and summing the contributions due to the direct wave term, image source term and the boundary wave term, we have (7.16) where (7.17a) (7.17b) Equation (7.16) reduces to the classical Weyl-Van der Pol formula (2.40) in the case of a stationary source (M=0). It should be pointed out that the ground wave term in (7.16) is ignored in earlier publications [5 7]. However, it has been shown that the ground wave term is particularly important for a stationary source at low frequencies near the ground

210 Predicting outdoor sound 196 effect dip. Since it is expected that the same situation will apply to a moving source above an impedance plane, ignoring the ground wave term will lead to erroneous predictions especially at frequencies near the ground effect dip. Equation (7.16) is an asymptotic expression for the sound field due to a source moving at constant speed parallel to the ground. Next we aim to give the physical interpretation of each term in (7.16). Let us first consider the amplitude of the direct and reflected waves. It is sufficient to study the reflected wave term as they have similar characteristics. In general, sounds heard at different positions are heard at different times. It is apparent that the instantaneous relative positions between the moving source and a stationary receiver have little relevance in the determination of the sound field. It is convenient to use the emission time geometry. This is the geometry we have already used to describe the sound field due to a stationary source in the homogeneous atmosphere. Ignoring the constant factor, 1/4π, the virtual distance travelled is d 2 /γ 2 =r 2 (t) say. Replacing the Lorentz coordinates with the emission time coordinates, the virtual distance is simply (7.18a) It should be remarked that the instantaneous distance, R 1 (t) between the source and receiver is given by (7.18b) Suppose τ 2 is the emission time of sound heard at x at time t. An implicit relationship between t and τ 2 can be expressed as (7.19a) This can be solved to obtain τ 2 explicitly for any locations of observer as follows, (7.19b) The spurious roots have been introduced as a result of squaring both sides of (7.19a) in order to obtain a solution for Since must be real and less than t, there is only one root with the choice of negative sign that satisfies these conditions. To write the virtual distance r 2 in terms of R 2, it is useful to combine (7.18a) and (7.19b). With some simple algebraic manipulations, we obtain

211 Influence of source motion on ground effect 197 r 2 (t)=c(t τ 2 ) M(x cmτ 2 ). (7.20) The above equation can be simplified further if we introduce a Doppler factor, 1 M r2, such that M r2 =M(x cmτ 2 )/R 2 (τ 2 )=M sin θ cos ψ. (7.21) where θ and ψ are the polar angle and azimuthal angle between the source and observer s position at emission time. Hence, it follows from (7.20) and (7.21) that the virtual distance is given by (7.22) The virtual distance, r 2, may be interpreted as the distance between the image source and receiver s position at the emission time τ 2. This distance is further modified by the wellknown Doppler factor, 1 M r2. The emission time τ 2 may be regarded as the retarded time as a result of the time delay for the sound propagation from the image source to the receiver s position. We seek to represent the phase of the reflected wave in terms of the retarded time. The phase function of the reflected wave, can be transformed back from the Lorentz space to the emission time coordinates as follows: (7.23) Similarly, the plane wave reflection coefficient R p and the numerical distance w L can be written in the emission time coordinates as (7.24a)

212 Predicting outdoor sound 198 (7.24b) where w and R p are, respectively, the numerical distance and the plane wave reflection coefficient in the emission time geometry. The following identities, which are adapted from Morse and Ingard [3], are found useful in deriving (7.24). and (7.25a) (7.25b) The source motion has no effect on the plane wave reflection coefficient but the numerical distance has been modified by a factor (1 M r2 ) 1/2. The same transformation can be applied to the direct wave term, although R p and w L are not required in this situation. Denoting the Doppler factor for the direct wave by 1 M r1, we can express (7.16) in the emission time coordinates: (7.26) In principle the sound field can be obtained by differentiating (7.26) because it is related to the velocity potential, φ, through the use of (7.2). However, this approach will lead to a somewhat lengthy evaluation [8]. An alternative method for obtaining an asymptotic form for p is to differentiate both sides of the space-time equation. Noting (7.2) and expanding the right-hand side of (7.1a), we obtain (7.27a)

213 Influence of source motion on ground effect 199 where the prime denotes the derivative respect to its argument. Similarly, the boundary condition (cf. (7.1b)) can be re-stated in terms of the sound pressure as (7.27b) A close examination of (7.27a) reveals that there are two source terms consisting of a monopole and horizontal dipole in the space-time equation. Applying the principle of superposition, we can find the sound field corresponding to each individual component source and assume that the total sound field is a sum of these two contributions. From the analysis earlier in this section, the monopole sound field, p m can be written down immediately as (7.28) The asymptotic solution for the sound field due to the horizontal dipole is available also because we developed it in Chapter 5. The sound field due to a moving horizontal dipole, p d can be found by starting from the solution for a moving monopole. The asymptotic solution is obtained by differentiating the appropriate solution for the moving monopole to give (7.29) This can be transformed back to the emission time coordinates by using the same approach as we did for the sound field due to a monopole source. After some algebraic manipulation, it is straightforward but tedious to show that the total sound field from combining (7.28) and (7.29) yields the rather compact form

214 Predicting outdoor sound 200 (7.30) This is the asymptotic solution for the sound field due to a monopole source moving at constant speed parallel to a ground surface. It may be called the Doppler Weyl-Van der Pol formula. As expected, the solution can be reduced to the classical form (2.40) when the speed of the source vanishes. 7.3 The sound field of a source moving with arbitrary velocity In the last section, we saw that the use of a simple transformation serves to bring the source at rest relative to the ground surface in the frame of Lorentz space. This is possible only when the source is moving at a constant height above the ground surface. Generalization of the method to other situations faces considerable geometrical complexities even for one of the simplest case where the source just moves at constant speed vertically upwards. This is because the ground surface is now moving away from the source after the transformation. This renders subsequent analyses intractable. Consequently, it is rather difficult to extend the method of Lorentz transformation to other more general situations. So, if it is of interest to derive an asymptotic formula for a source moving at arbitrary velocity above a ground surface, the method of Lorentz transformation is not applicable even if we are prepared to take up the challenge of modifying the solution to meet the boundary condition of a moving impedance plane. A new approach is required to allow the derivation of the required asymptotic formulae. Before we move on to derive a generalized expression for a source moving at arbitrary speed above a ground surface, it is instructive to consider the classical prediction of the free field due to a moving source. The space-time wave equation is (7.31) where r (x, y, z) is the field point and r s (x s, y s, z s ) is the instantaneous position of the source. Differentiation of r s with respect to t gives the instantaneous velocity, u=cm (cm x, cm y, cm z ) of the source. Without resort to the use of the method of Fourier transformation, there is a subtle way in deriving an exact solution for the sound field due to a moving source. The method is as follows: the field generated by a source (or even an extended source) of strength q(r, t) is given by the well-known integral representation [9] of

215 Influence of source motion on ground effect 201 (7.32) where y is the source s location and τ is the retarded time which varies over the source. It is given by τ=t r y /c. By the use of generalized function, δ(t τ r y /c, (7.32) is identical to (7.33) Now, for a point moving harmonic source, the source strength is simply q(r, t)= exp( iω 0 t)δ(r r s (t)). Hence, the sound field given in (7.33) may be rewritten as (7.34) Changing the order of integration, the triple integral with respect to y can be evaluated straightforwardly and exactly because of the δ-function and its solution is (7.35) The integral with respect to τ can be evaluated also. Using a property of δ-functions where f and g are any arbitrary functions and τ* is a zero of g, that is, any solution for g(τ)=0, and the prime denotes the derivative with respect to τ, we can identify that f=exp( iω 0 τ)/4π r r s (τ) and g=t τ r r s (τ) /c. The differentiation of g with respect to τ gives

216 For a subsonic source speed, Predicting outdoor sound 202 g =1 M r where Comparing with (7.21), we can see M r is the component of the source Mach number in the direction of the observer. With the aid of this analysis, the integral with respect to τ in (7.35) can be evaluated exactly to yield (7.36) where R(τ*)= r r s (τ*) and τ* satisfies c(t τ*)= r r s (τ*). (7.37) Now we are in a position to investigate the sound field due to a source moving above a ground surface with an arbitrary velocity. Again, the sound field is governed by the space-time wave equation given in (7.31) subject to the impedance boundary condition of (7.1b). As expected, the sound field is composed of two terms: a direct wave φ 1 and a reflected wave φ 2. With this analysis, the solution for the direct wave can be written immediately as (7.38a) where τ 1 is a zero of (7.38b)

217 Influence of source motion on ground effect 203 c(t τ 1 )=R 1 (τ 1 ), (7.38c) and M r1 is the component of the source Mach number in the direction of the observer. The same method is readily adaptable to the reflected wave by noting the source strength q(r, t) in (7.32) is replaced heuristically by q(r, t)= exp( iω 0 t)qδ(r r s (t)), (7.39a) where Q is the spherical wave reflection coefficient given by Q=R p +(1 R p )F(w). (7.39b) Then, by using the same procedure, we can obtain an asymptotic formula for the reflected wave as (7.40a) where τ 2 is a zero of (7.40b) c(t τ 2 )=R 2 (τ 2 ), (7.40c) and M r2 is the component of the image source Mach number in the direction of the observer. The spherical wave reflection coefficient Q in (7.40a) is calculated at the emission time τ 2. If we consider the special case of a source moving at a constant speed in the x-direction, we discover that the reflection wave term derived in this section is the same as that derived in section 7.2, except that the numerical distance w differs by a correction term (1 M r ) 1/2. The correction term, which is introduced as a result of approximating the source strength in (7.39a), is small if the source traverses at low subsonic speed. To allow an uniform asymptotic solution for two derivations, we assume that the numerical distance should be modified by the same correction term as derived in

218 Predicting outdoor sound 204 section 7.2. Hence, summing the contribution of the direct and reflected wave terms, we obtain the velocity potential for a source moving at arbitrary velocity as follows: (7.41) To obtain an asymptotic expression for the acoustic pressure, we need to differentiate the velocity potential with respect to t. It is useful just to consider the direct wave, p 1 ; (7.42) Making use of the following identities, where is the component of the source acceleration in the direction of the observer, we can show that the direct wave is given by (7.43) Equation (7.43) is the free field sound pressure due to a source moving at arbitrary velocity. An extra term is introduced as a result of the acceleration of the source, which is significant when This furnishes a generalization of the result derived

219 Influence of source motion on ground effect 205 by Morse and Ingard [3] who have considered the sound field due to a source moving at a constant speed, apart from a minor typographical error in their equation ( ). For a distant slow source, variations of the spherical coefficient Q with emission time t are generally small. Hence, the reflected wave term can be derived in an analogous manner as that for the direct wave. The solution is (7.44) The total sound field is obtained by summing the contributions of the direct and reflected waves to give, (7.45) The ground attenuation for stationary sources is usually expressed by means of the excess attenuation (EA), that is the ratio in db of the total pressure field to the direct field (see Chapter 2). For sources in motion, we refer to the instantaneous excess attenuation which is that observed at the emission time t e for which the interfering direct and reflected fields are calculated by means of the retarded times t L and respectively. An example numerical prediction of the instantaneous excess attenuation [10], for a source moving above a porous ground at a height of 3 m at Mach number 0.3 in uniform circular motion with radius 2 m about a vertical axis, is shown in Figure 7.1. The impedance of the porous ground is modelled by using (3.13) with σ e =140 kpa s m 2 and α e =35 m 1. Equation (7.45) can be reduced to (7.30) for the special case of a monopole source moving parallel to the ground surface at constant speed. The solutions for a moving source ((7.45) and (7.30)) have a similar form to that for a stationary source. The only difference is that the magnitudes of the direct and reflected waves are modified due to the effect of source motion. The apparent impedance of the ground surface is unaffected by the convection of the source but there is a small correction to the numerical distance. This slightly alters the boundary loss factor and has a relatively small effect on the overall sound field especially for a relatively slow moving source. In the next section, we shall compare the accuracy of another approximation scheme with the exact formulation derived in this section.

220 Predicting outdoor sound 206 Figure 7.1 (a) Predicted instantaneous excess attenuation for a source in circular motion at Mach 0.3 about the vertical axis, (b) Excess attenuation for the corresponding stationary source. 7.4 Comparison with heuristic calculations As a first approximation for low source speed, the effects of motion are sometimes ignored. For example, a road or railway track may be modelled as a coherent line source [11]. This assumption produces a simple solution since the asymptotic solution for a line source (see section 2.7) can be used. An alternative heuristic approach is to model the problem as the sound field due to a quasi-stationary point source with the source frequency adjusted by the Doppler factor, (1 M r ) 1. The sound field can be computed by using the Weyl-Van der Pol formula (2.40) with the source frequency of f 1 f 0 (1 M r ) 1. Moreover, the ground admittance can be calculated using the Dopplerized frequency for the interfering reflection. Numerical calculations of the pressure field due to a source in uniform motion above an impedance plane have been carried out (M Buret thesis) for Mach numbers up to 0.3 (370 km h 1 ). The modified Weyl-Van der Pol calculation (7.16) is compared with the heuristic approach for the calculation of the sound field in Figure 7.2. The source is located at height z s =1 m above the ground, and is travelling along the x- axis. The receiver height is 1.2 m. The ground admittance is calculated by means of the variable porosity model (3.13) with parameters σ e =140 kpa s m 2 and α e =35 m 1. Predictions of the two models for a moving point source are compared with the calculation for stationary source in Figures 7.2 and 7.3. The varying time parameter used here and subsequently relates to the position of the

221 Influence of source motion on ground effect 207 Figure 7.2 Predicted instantaneous excess attenuation at a receiver 1.2 m high and 50 m from an approaching source moving parallel to a horizontal flat ground at a height of 1 m (a) using (7.16), (b) using the heuristic approach with Dopplerized frequency and (c) assuming a stationary source. source corresponding to the emission of the interfering reflection. For Figure 7.2 the separation distance between the source path and the receiver is 50 m. With the chosen geometry, there is near-grazing incidence for all source positions so the values taken by the emission times for interfering direct and reflected waves are indistinguishable. The observation time t varies non-linearly with the emission position. Figures 7.3 (a) and (b) show that the ground effect dip is shifted towards the low frequencies when the source approaches the receiver (the ground is seen as softer), and towards higher frequencies when the source is receding (the ground is seen as harder). However, the heuristic, that is Dopplerized frequency, calculation shows greater sensitivity to the motion than the modified Weyl-Van der Pol calculation (7.16). Although effects due to source motion are observable in the frequency domain, the influence on the intensity of the ground attenuation on the variation in magnitude of the ground effect dip is only a few db. For the chosen geometrical and ground parameters, at Mach number 0.3, there is a shift of about 100 Hz according to the modified Weyl-Van der Pol formula (solid lines) and 170 Hz with the heuristic calculation (dashed lines). Figure 7.4 shows the frequency shift of the ground effect dip due to source motion at Mach numbers 0.1 (white), 0.2 (grey) and 0.3 (black). The calculation of the frequency of the ground effect dip is carried out by locating the first minimum in the instantaneous excess attenuation. Because this calculation has been carried out numerically, the smoothness of the resulting curves is rather poor. For the chosen

222 Predicting outdoor sound 208 Figure 7.3 Instantaneous excess attenuation predicted (A) when the source approaches the receiver: and

223 Influence of source motion on ground effect 209 (B) when the source is receding from the receiver: (a) Solid lines represent predictions of the modified Weyl-Van der Pol formula (7.16); dashed lines represent predictions using the Heuristic approach (i.e. Dopplerized frequency). The dotted line represents predictions for a stationary source. Figure 7.4 Frequency shift of the ground effect dip due to source motion. The configuration is the same as that for Figure 7.3. Solid lines represent predictions using the modified Weyl- Van der Pol formula. Broken lines represent predictions using the heuristic approach. geometrical and ground parameters, at Mach number 0.3, the predicted shift is about 100 Hz according to the modified Weyl-Van der Pol formula (solid lines) whereas a shift of 170 Hz is predicted with the heuristic calculation (dashed lines). For higher source speeds, the shift of the ground effect dip is predicted to have a significant influence on the resulting sound level, particularly for a source whose spectrum peaks in the frequency range where the maximum attenuation is observed.

224 Predicting outdoor sound Diffraction of sound due to a point source moving at constant speed and height parallel to a rigid wedge Kinematics Consider a semi-infinite rigid wedge with arbitrary top angle T=2π T 1 T 2 and constant cross section in the y z plane. A sound source S is moving along the x-axis, parallel to the edge of the wedge and a cylindrical coordinate system [ξ, r, x] can be introduced as shown in Figure 7.5, with r S [ξ S, r S, x S ] representing the source coordinates. For uniform source motion at Mach number M, the only varying parameter is the source-receiver offset x x S. If the initial position of the source is at x=0, then, where c 0 denotes the speed of sound, Figure 7.5 Geometry describing a source in uniform motion along a semi-infinite wedge. x S (t)=c 0 Mt. (7.46) The projection of the geometry in the cross-sectional plane remains unchanged with source motion so there is an analogy with a stationary source in a 2-D situation. According to the geometrical theory of diffraction, the diffracted sound ray is a broken segment linking the source S to the edge of the wedge E to the receiver R. The diffraction angle ε is the same on both source and receiver side [12]. If the receiver is at x=0, (7.47) where t e is the time of emission of the diffracted wave reaching the receiver at time t and L the total length of the diffracted ray. Also, from the geometry,

225 Influence of source motion on ground effect 211 (7.48) and (7.49a) where (7.49b) M E =M cos ε being the component of the Mach number along the diffracted ray Diffracted pressure for a source in uniform motion The solution for wedge-diffracted wave from a stationary point source is well known. The result given by Pierce [12], based on the electromagnetic theory, is used here. Morse and Ingard [3] have derived the sound field due to a monopole in uniform motion in the free field by means of a Lorentz transformation. Buret et al. [13] have used this transformation to formulate the Doppler-Weyl-Van der Pol formula (7.30). The coordinates and time (x L, y L, z L, t L ) in the Lorentz space are given by (7.3). In the Lorentz space, it can be shown that the acoustic pressure field can be expressed as the sum of contributions from two stationary sources: a monopole and a dipole oriented in the direction of source motion [14]. Denoting these relative contributions by p (0) and p (1) respectively, the total pressure p due to a source of strength P 0 can be expressed as [13]: (7.50) With constant Mach number, the Lorentz transformation involves uniform expansion along the three dimensions, as well as a time-dependent translation along the direction x of source motion. As a result, the transformed geometry is analogous to that for a stationary source in the physical space. A cylindrical coordinate system [ξ L, r L, x L ] is introduced and is related to that in the physical plane by (7.51)

226 Predicting outdoor sound 212 The polar angle is unaffected by the transformation. As a result, the top angle of the wedge has the same value in both the transformed and the physical space. The expression for the monopole component of the diffracted wave is deduced straightforwardly from Pierce s formulation [12] as (7.52a) where L L is the diffracted path in the transformed coordinates and A D is the diffraction integral given by (7.52b) (7.32c) (7.52d) where ν=π/(2π T) is the wedge index and ρ S =γr S. Buret et al. [15] have derived the expression for the 3-D dipole field diffracted by a wedge and they have validated this expression through laboratory measurements. For a dipole lying along the x L -axis, (7.53) In (7.53), dipole orientation cosines missing from eq. (14a) of Ref. [15] have been reintroduced, that is (l x, l y, l z )=(cos ε L, 0, 0). For long diffraction paths and high frequencies, X L+ and X L take small values and the second term in (7.53) can be neglected [15]. This approximation is particularly sensible for source motion along the wedge, as the diffracted path length increases with the time-dependent source-receiver offset x x S. The dipole component of the pressure then takes a form similar to the monopole pressure given in (7.52), except for a directivity factor and a strength factor: (7.54)

227 Influence of source motion on ground effect 213 (7.55a) (7.55b) and (7.55c) where the total diffracted pressure is expressed by substituting (7.52) and (7.54) into (7.50). Hence the diffracted acoustic pressure wave due to a monopole in uniform motion in parallel to a rigid wedge can be formulated as follows: (7.56) where A D is defined as in (7.52b) with (7.57) For a receiver in the shadow zone of the obstacle, the total pressure field is given by (7.56). However, for arbitrary source or receiver position, it involves potential contributions from a direct pressure wave, as well as reflections on the sides of the wedge. The formulation of the total pressure in the transformed space is straightforward by means of (7.45) together with the expression for diffraction of sound from arbitrarily positioned monopoles [12] and its extension to dipoles by Buret et al. [15]. This is not developed here for brevity s sake. In Figure 7.6, predictions of the diffracted wave are plotted for discrete frequencies 100 Hz, 1 khz and 5 khz, for a point source moving at Mach number 0.3 in the vicinity of a semi-infinite rigid half plane (T=0; ν=1/2) with source, receiver and obstacle edge at

228 Predicting outdoor sound 214 heights 1 m, 1.2 m and 2 m respectively. The shortest horizontal separation from the obstacle to the source is 10 m and that to the receiver on the other side is 20 m. Solid lines show the calculation by means of (7.56). Calculations by means of the complete dipole component as in (7.53) are indistinguishable from the latter. The approximation used to derive (7.56) is valid along the whole source path. Broken lines show the calculation for the corresponding stationary omni-directional point source located at the position x S (t e ) of emission. Figure 7.6 Diffracted pressure predicted at discrete frequencies 100 Hz ( ), 1 khz ( ) and 5 khz ( ) for a source in uniform motion at Mach 0.3 along a half-plane. The source path is 1 m below the edge, at a horizontal separation of 10 m. The receiver is 0.8m below the edge at a distance of 20 m. Solid lines correspond to the approximate solution. Predictions without approximation (i.e. using the full dipole component) are undistinguishable from the approximate results. Dashed lines (with different thicknesses) correspond to predictions for the corresponding stationary source at the point of emission.

229 Influence of source motion on ground effect 215 Comparison of the predictions for a stationary source with those for a moving source enables assessment of the effects of source motion. As the moving source approaches the receiver the Sound Pressure Levels (SPLs) are predicted to be higher than for a stationary source. When the source is receding the SPLs are predicted to be lower than for a stationary source. In particular, the maximum in SPL is predicted to occur when the source is approaching the receiver, that is before the position of closest approach. These effects are related to the Doppler factor (1 M cos ε) 1 and the convection coefficient (1 M cos ε) 2 in (7.56). Both are larger than unity on approach and smaller than unity on recession. These observations are consistent with previous results for a source in the free field [3]. 7.6 Source moving parallel to a ground discontinuity Introduction A semi-empirical formula for the sound field due to a stationary source in the presence of a ground discontinuity has been derived by De Jong, considering the superposition of two half planes with different admittance [16]. This model was shown to be valid in 3-D situations for which the propagation path is not necessarily perpendicular to the discontinuity [16] and was later and extended to dipole sources [15]. Derivation of the acoustic pressure field for source motion in parallel to the discontinuity is then straightforward (see (7.45)). Consider that a point source is in uniform motion parallel to an impedance jump coinciding with the x-axis and at height z S (see Figure 7.7). Figure 7.7 Geometry describing a source in uniform motion parallel to a discontinuity in ground impedance. The ground on the source side is characterized by specific admittance β 1 and that on the receiver side by β 2. The direct wave reaching the receiver R at time t has been emitted at instant t e, following path R=c 0 (t t e ) with azimuthal angle ψ and elevation angle The reflected wave reaching R at instant t, and denoted subsequently by dashed parameters in

230 Predicting outdoor sound 216 the physical space, has been emitted at retarded time and has followed path with azimuthal and elevation angles and respectively. For brevity, the expressions for and R, ψ, θ are not detailed here since their derivation is straightforward [14, 15]. The diffracted wave follows path L defined by (7.48) and its time of emission is calculated from (7.48) and (7.49). During the source motion, the point of specular reflection remains on a line parallel to the admittance discontinuity and the source path. Consequently, the proportion of each type of ground covering on the source-receiver path also remains constant during the motion of the source. This is of particular interest if there are strips of different grounds, parallel to the source path. Some models for such a situation involve tedious calculations whereby the diffracted waves at each discontinuity are summed [18 20]. However Nyberg [17] has shown that for an infinite alternation of strips with respective admittance β 1 and β 2, the ground effect could be modelled using the area-averaged admittance (7.58a) with a and b the spatial periods for each strip. This theory is valid for short periods compared with the incident wavelength λ [18], that is for (7.58b) This condition might not be fulfilled for source motion parallel to the strips, since the effective width of the strips along the sound propagation path is increased by a factor 1/cos ψ due to the source-receiver offset. However, some tolerance in the accuracy of the model is acceptable for source positions at large offsets for which the distance attenuation is large. Moreover, Nyberg s theory has shown to give good agreement with measurements beyond its limit of approximation [18]. Calculation of the acoustic pressure field for a source moving along periodic strips of ground is straightforward, by substituting the area-averaged impedance into the Doppler-Weyl-Van der Pol formula (7.16) Uniform motion parallel to a single discontinuity After Lorentz-transforming De Jong s solution for a stationary source [16], the monopole component of the acoustic pressure near a single impedance discontinuity is

231 Influence of source motion on ground effect 217 (7.59) where L L is the diffraction path in the Lorentz space for a horizontal half plane whose edge coincides with the ground discontinuity and D L =A D (X L+ )+ A D (X L ), that is the sum of the two diffraction integrals given in (7.52). Q G is the spherical wave reflection coefficient at the point of specular reflection, with G 1 or 2 depending on the geometry. Q 1 and Q 2 are the spherical wave reflection coefficients for each type of ground. They are calculated by using the Doppler-Weyl-Van der Pol formula (7.16), that is (7.60a) where is the component of the Mach number in the direction of propagation of the reflected wave, and R p,i, the plane wave reflection coefficient, is given by (7.60b) (7.60c) The expression for the boundary loss factor F is (7.60d) where erfc denotes the complementary error function and wi the numerical distance is given by

232 Predicting outdoor sound 218 (7.60e) When the point of specular reflection coincides with the admittance step, and X L+ =0. Since A D (0)= (1 i)/2, the continuity of the monopole component of the acoustic pressure across the jump is ensured. The dipole component is also expressed as the sum of a direct, a reflected and a diffracted pressure wave to which are applied the appropriate reflection coefficients. The formulations for the direct and the reflected fields are given in Li et al. [21] and the form of the diffracted wave is given in Buret et al. [15]. Hence the total dipole contribution may be expressed as (7.61a) In (7.61a) (7.6 1b) represent the directivity factors where and θ L are the azimuthal angle and the elevation angles for the direct and reflected ray in the Lorentz space and θ E,L is the elevation angle of the ray linking the source to the admittance step. Corresponding angles in the physical space are shown in Figure 7.7. It should be noted that when the point of specular reflection lies on the impedance jump, and continuity of the acoustic pressure across the jump is ensured. Since variations of the spherical wave coefficient in the horizontal directions are known to be small [15], the last term in (7.61a) can be neglected and (7.61) can be simplified to an expression analogous to (7.59) for the monopole component. The total pressure field is derived by substituting (7.59) and (7.60) into (7.45). Transformation back into the physical space is straightforward by using (7.55) and noting that [13] (7.62a)

233 Influence of source motion on ground effect 219 (7.62b) and (7.62c) A double dash denotes the parameter in the physical space (see Figure 7.7). The pressure field for a source in uniform motion along an admittance discontinuity is then expressed as: Where (7.63a) (7.63b) (7.63c) (7.63d) Figure 7.8(a) shows the predictions of the SPL according to (7.63) as a function of the source position at emission of the reflected wave with a 33%-hard ground mix. The source is supposed to be moving at Mach number 0.3 with its track located 10 m away from the impedance jump at height 1 m above hard ground. The receiver is located 20 m away from the discontinuity, 1.2 m above soft ground characterized by the two-parameter impedance model (3.13) with flow resistivity σ e =140 kpa s m 2 and porosity rate α e =35 m 1. Figure 7.8(b) shows the corresponding SPL predictions for a stationary source at offset locations between 0 m and 200 m. Comparison of these Figures enables the effects of source motion to be assessed.

234 Predicting outdoor sound 220 Figure 7.9 shows the same calculations but for the source track and receiver 20 m and 10 m away from the impedance jump respectively (66% of hard ground). The corresponding predictions of instantaneous excess attenuation (10 log( p(t) / p dir (t) )) are shown in Figure 7.10, for the source at offset 100 m on approach thick solid line) and recession thin solid line) and for the corresponding stationary source (broken line). Thick and thin dotted lines represent the predictions at the closest position for moving and stationary source respectively and are undistinguishable from one another. Figure 7.8 (a) Sound Pressure Level (SPL, re. arbitrary reference) predicted for a source moving at Mach number 0.3 parallel to an admittance discontinuity, 10 m away from it, at height 1 m. The receiver is 20 m away from the step-line at height 1.2 m. The ground on the source side (33%) is hard, the ground on the receiver side is characterized by σ e =140 kpa s m 2 and α e =35 m 1. (b) SPL for a stationary source in the same geometry as in (a), but with offset from 0 to 200 m.

235 Influence of source motion on ground effect 221 Figure 7.9 Same as Figure 7.8, but for a discontinuity at 20 m from the parallel source path (66% hard ground). Figure 7.10 Instantaneous excess attenuation for the same configurations as in (a) Figure 7.8, (b) Figure 7.9. Approach Recession stationary source at offset x S =100 m,...moving source at closest separation the results for a stationary source at closest separation are undistinguishable from the latter. As observed for homogenous ground in section 7.4 (see also [13]), the predicted effects of motion are small. The predicted SPL contours are asymmetric along the source motion

236 Predicting outdoor sound 222 direction due to the Doppler shift and the convection factor that affect the location and magnitude of the ground effect dip. This results in lower SPL being predicted than for a stationary source when the source approaches the receiver up to the frequency of the ground effect dip and higher levels being predicted from the dip up to the higher frequency destructive interference. This trend in the predictions is reversed when the source recedes from the receiver. As the source approaches, the ground effect dip is predicted to be slightly sharper and deeper than that for a stationary source whereas, as the source recedes it is predicted to be slightly broader and shallower. Deformation of the SPL pattern and hence sensitivity to motion is predicted to be stronger on approach than on recession. On the other hand, change of the proportion in the ground mix covering the source-receiver path is predicted to have little effect. 7.7 Source moving along a rigid barrier above the ground Barrier over hard ground Sound propagation in the presence of a rigid barrier above the ground is a classical problem. In the case of hard ground on both sides of the barrier, the solution is straightforward, by means of mirror images. When there is no direct sound and no reflection on the barrier, the total sound field is the sum of four diffracted waves, Figure 7.11 Geometry describing a source in uniform motion parallel to a thin barrier above the ground. The diffraction path 12 with ground reflections on both sides of the barrier is represented by thick lines. corresponding to the paths linking the source S and its image with respect to the ground S to the receiver R and its image R via the edge of the barrier (Figure 7.11).

237 Influence of source motion on ground effect 223 The paths including a ground reflection on the source side are denoted with subscript 1 and subscript 2 is used to indicate the paths for which there is a ground reflection on the receiver side. The total field is given by p=p 0 (p 0 +p 2 +p 12 ) (7.64) where p 0 stands for the diffracted wave obtained for path 0, linking source to edge to receiver with no ground reflection. Using the same reference systems as in section 7.5, the coordinate vectors of the image source and the image receiver are and r respectively. In the Cartesian and the cylindrical coordinates (x, y, z) and [r, ξ, x] they are expressed as (7.65a) r (t)=(0, y, z) [r, φ, x] (7.65b) where φ S and φ are the elevation angles of the paths linking the projection of the image source and image receiver to the top of the barrier (see Figure 7.11). If z E denotes the barrier height, (7.66a, b) The four diffracted paths have different lengths and as a result, for each contribution in L K =c 0 (t t K ) (7.67a) where the L K s are the diffracted path lengths, (7.67b)

238 Predicting outdoor sound 224 The subscripts 0, 1, 2 and 12 have the same meanings as in (7.64). The components of the Mach number in the direction of the diffracted rays M K are determined from the diffraction angles ε K by (7.68) After solving the system of equations in (7.67), each pressure wave component is calculated from (7.56) by substituting the appropriate (image) source and (image) receiver coordinates together with the corresponding diffracted path and angle, L K and ε K. The total pressure is then calculated by substituting the component pressures into (7.64). Figure 7.12(a) shows predictions of the sound field for the same configuration as used for Figure 7.6, but above hard ground (at zero height). The SPL predicted for the corresponding stationary source locations is given for comparison in Figure 7.12(b). Figure 7.12 Predicted sound field due to a source moving at Mach number 0.3 along a thin barrier of height 2 m above hard ground. The source path is 10 m away from the barrier, at height 1 m. The receiver is in the shadow zone, 20 m away from the barrier and at height 1.2 m Again, the pressure field is predicted to be asymmetric along the source travel direction. Lower levels and hence stronger barrier attenuation are predicted as the source recedes from the receiver (x S (t 0 ) 0). On the other hand, sensitivity of the sound field to source motion is predicted to be more important on source approach (x S (t 0 ) 0), for which the deformation of the sound pressure level pattern is stronger. Strong interference between the four components of the acoustic pressure is observed in the high frequencies.

239 Influence of source motion on ground effect Motion parallel to a barrier over arbitrary impedance ground If the ground on either or both source and receiver side of the barrier has finite impedance, the corresponding reflection coefficients differ from 1 (the value for acoustically hard ground). Equation (7.64) must be amended to account for the ground reflections [22]. It is then necessary to consider separately the monopole and the dipole component of the pressure field given in (7.45). We use β 1 and β 2 to denote the admittance of the ground on the source and receiver side respectively. As in the previous section, subscript 1 denotes a ground reflection on the source side, whereas subscript 2 corresponds to a ground reflection on the receiver side. The monopole component is obtained by transforming to the Lorentz space the sound field due to a stationary monopole in the presence of a barrier over the ground [22]: (7.69) The primes on the reflection coefficients in the last term denote the fact that the emission time for p 12 is different from those for p 1 and p 2. The diffracted monopole pressures are calculated from (7.52). Using Huygens s principle and considering that the diffracted wave is emitted by a secondary source located at the edge of the obstacle, the spherical wave reflection coefficients are calculated for the paths joining the image source and the image receiver to the edge of the barrier [23]. Hence Using the Doppler-Weyl-Van der Pol formula (7.16), with i 1, 2 (7.70) (7.71a) (7.71b) (7.71c) (7.71d)

240 Predicting outdoor sound 226 cos θ 1 =(z S +z E )/ρ 1 ; cos θ 2 =(z+z E )/ρ 2 (7.71e) (7.71a e) can be used as appropriate also to calculate and To obtain the dipole component of the pressure field is expressed in the Lorentz space, we recall the solution for the stationary dipole [15]. For a dipole orientated along the x L - axis, (7.72) where the diffracted dipole pressures are calculated using (7.54). The last term in (7.72) can be neglected due to the small variations of the spherical wave reflection coefficient in the horizontal plane. Hence after summing the monopole and the dipole contributions and transforming back into the physical space, (7.73a) (7.3b) where K 0, 1, 2, 12. Figure 7.13 shows predictions of the pressure field for the same source and barrier geometries as were used for Figure 7.11 but with hard ground on the source side and soft ground on the receiver side. Figure 7.14 shows the corresponding predictions for soft ground on both sides. As observed in respect of Figure 7.11, lower levels are predicted when the source is receding (x S (t 0 ) 0). Two dips related to the reflections on each side of the barrier are identifiable, particularly at large offsets and for source recession. As before, sensitivity to motion is predicted to be stronger on source approach. Also, source motion is predicted to be more important for soft ground. This was not so obvious in the predictions involving a

241 Influence of source motion on ground effect 227 Figure 7.13 Predictions for the same situation as in Figure 7.12 but with hard ground on the source side and soft ground (σ e =140 kpa s m 2 and α e =35 m 1 ) on the receiver side. Figure 7.14 Predictions for the same situations as obtain in Figures 7.12 and 7.13 but with soft ground on both sides. ground discontinuity as the proportion of hard ground was varied. This suggests that the source motion effect is related to the fact that, in the presence of a barrier, the angles of incidence of the reflected waves take higher values.

242 Predicting outdoor sound 228 References 1 D.G.Crighton, Scattering and diffraction of sound by moving bodies, J. Fluid. Mech., 72: (1975). 2 A.P.Bowling, Convective amplification of real simple sources, J. Fluid. Mech., 74: (1976). 3 P.M.Morse and K.U.Ingard, Theoretical Acoustics, Princeton University Press, Princeton, NJ (1986), Chapter M.V.Lowson, Sound field for singularities in motion, Proc. Royal Soc. Series A, 286: 559 (1965). 5 E.W.Graham and B.B.Graham, Theoretical acoustical sources in motion, J. Fluid. Mech., 49:481 (1971). 6 T.D.Norum and C.H.Liu, Point source moving above a finite impedance reflecting plane experiment and theory, J. Acoust. Soc. Am., 63: (1978). 7 S.Oie and R.Takeuchi, Sound radiation from a point source moving in parallel to a plane surface of porous material, Acustica, 48: (1981). 8 S.Oie and R.Takeuchi, Sound radiation from a point source moving vertical to a plane surface of porous material, Acustica, 48: (1981). 9 G.Rosenhouse and N.Peled, Sound field of moving sources near impedance surfaces, International Conference on Theoretical and Computational Acoustics, Mystic, CT, USA, 5 9 July 1993, vol. I, (1993). 10 M.Buret, New Analytical Models For Outdoor Moving Sources of Sound, PhD Thesis, The Open University, UK (2002). 11 A.P.Dowling and J.E.Ffowcs Williams, Sound and Sources of Sound, Ellis Horwood Ltd., Chichester (1983), Chapter A.D.Pierce, Acoustics, an Introduction to its Physical Principles and Applications, Acoustical Society of America, New York (1989). 13 M.Buret, K.M.Li and K.Attenborough, Optimisation of ground attenuation for moving sound sources, Appl. Acoust., 67: (2006). 14 K.M.Li, M.Buret and K.Attenborough, The propagation of sound due to a source moving at high speed in a refracting medium, Proc. Euro-Noise, 98, Vol.2, pp (1998). 15 M.Buret, K.M.Li and K.Attenborough, Diffraction of sound from a dipole source near to a barrier or an impedance discontinuity, J. Acoust. Soc. Am., 113(5): (2003). 16 B.A.De Jong, A.Moerkerken and J.D.Van der Toorn, Propagation of sound over grassland and over an earth barrier, J. Sound Vib., 86(1):23 46 (1983). 17 D.C.Hothershall and J.N.B.Harriot, Approximate models for sound propagation above multiimpedance plane boundaries, J. Acoust. Soc. Am., 97(2): (1995). 18 P.Boulanger, T.Waters-Fuller, K.Attenborough and K.M.Li, Models and measurements of sound propagation from a point source over mixed impedance ground, J. Acoust. Soc. Am., 102(3): (1997). 19 M.R.Bassiouni, C.R.Minassian and B.Chang, Prediction and experimental verification of far field sound propagation over varying ground surface, Proc. Inter-Noise 83, Vol.1, pp (1983). 20 C.Nyberg, The sound field from a point source above a striped impedance boundary, Acta Acust., 3: (1995). 21 K.M.Li, S.Taherzadeh and K.Attenborough, Sound propagation from a dipole source near an impedance plane, J. Acoust. Soc. Am., 101(6): (1997). 22 H.G.Jonasson, Sound reduction by barriers on the ground, J. Sound Vib., 22(1): (1972). 23 P.Koers, Diffraction by an absorbing barrier or by an impedance transition, Proc. Inter-Noise 83, vol. 1, pp (1983).

243 Chapter 8 Predicting effects of mixed impedance ground 8.1 Introduction Many outdoor sound propagation situations involve at least one visible change in the ground impedance. Frequently there is a change from acoustically hard ground near the source to acoustically soft ground near the receiver. A typical example is sound propagation from a road where the road traffic source is over an acoustically hard surface such as hot-rolled asphalt but the receiver might be above grassland adjacent to the road. Other situations might involve propagation over a strip of acoustically hard ground such as a service road or acoustically soft ground such as occurs in a belt of trees. In addition, the impedance of naturally occurring acoustically soft ground varies from place to place along the propagation path. There might be several areas of differing impedance. Do these variations have a significant influence on propagation? Can the influence be predicted using an average or effective impedance and hence the models described in Chapters 2 and 3? In current prediction schemes for outdoor sound, allowance is made for mixtures of hard and soft ground. Either the computed soft ground effect is reduced linearly in proportion to the fraction of soft ground between source and receiver [1, 2], or it is computed only over the soft ground proportion of the path [3]. In this chapter, we review models for predicting the effects of impedance changes along propagation paths and the resulting understanding of these effects. We begin by discussing propagation over a single discontinuity. Subsequently we describe models and measurements over strips of different impedance. Also, we present some predictions of the resulting effects on A- weighted sound levels. Finally we consider the simultaneous effects of meteorological conditions and impedance change. 8.2 Single discontinuity De Jong s semi-empirical method Various analytical methods of solution for the two-impedance boundary have been proposed. These range from an exact solution in the form of a triple integral [4] to an approximate solution of the boundary integral [5]. Semi-empirical formulas have been proposed by Koers [6] and by De Jong et al. [7]. The former presented a model based on Kirchoff diffraction theory where the diffracted field at an impedance transition is calculated by taking it as a wedge with a top angle of 180 and having a different impedance on each side. De Jong considered Pierce s formulation of sound diffraction

244 Predicting outdoor sound 230 from a wedge [8] with different surface acoustic impedances at either side. He then considered the limiting case where the wedge opens flat and derived his expressions for the propagation of sound over an impedance discontinuity. The form of solution proposed by De Jong et al. [7] is particularly useful since it is an empirical combination of the formulas for diffraction of spherical waves at a rigid half plane and the field due to a point source over an impedance plane. This means that it includes the spherical wave reflection coefficient calculation described earlier (Chapter 2) and, hence, it offers a relatively straightforward computation. Consider the situation shown in Figure 8.1. According to the De Jong model, the excess attenuation over the single discontinuity between portions of ground with impedance Z a and Z b, the source being over Z a, is given by (8.1) where (8.2) Q a,b is replaced by Q a, the spherical wave reflection coefficient for the portion of the ground with impedance Z a and the positive sign in the curly brackets is used when the point of specular reflection falls on that portion of ground. Conversely, Q a,b is replaced by Q b, the spherical wave reflection coefficient for the portion of the ground with impedance Z b, and the negative sign in the curly brackets is used when the point of specular reflection falls on that portion of ground. and are Fresnel integrals of Figure 8.1 The geometry and path lengths for the De Jong formula (8.1).

245 Predicting effects of mixed impedance ground 231 the form and the path lengths are defined in Figure 8.1. R d is the source-discontinuity-receiver path. According to Daigle et al. [9] and Hothersall and Harriott [10], the De Jong model fails at grazing incidence. On the other hand it agrees remarkably well with data obtained when source and receiver are elevated (see section 8.2.5) Rasmussen s method Rasmussen [11] has suggested a numerical method for determining the sound field over a plane containing an impedance discontinuity. A hypothetical planar source 40 wavelengths wide and 20 wavelengths tall is placed above the discontinuity. The planar source is discretized into an array of point sources a fifth of a wavelength apart. The relative strength of each source is calculated using the usual point source theory for propagation over infinite impedance Z 1. The received field is calculated as the sum of the contributions from each of the constituent planar sources over an infinite Z 2. However, comparison between predictions of this relatively numerically intensive method, the De Jong semi-empirical formula, and data indicated [11] that the De Jong formula is adequate for engineering purposes. The method is known also as the substitute sources method and has been developed further recently [12] Fresnel zone methods Following a suggestion by Slutsky and Bertoni [13], Hothersall and Harriott [10] have developed an approach based on Fresnel diffraction theory. They observe that if the oscillations due to diffraction are ignored, the excess attenuation with range from the source essentially varies between the values appropriate to point-to-point propagation over each impedance alone. The transition is within a well-defined region around the point of specular reflection between source and receiver. This may be defined as a Fresnel zone (see Figures 8.2 and 8.3) at the boundary of which the path length is some fraction (F) of a wavelength (e.g. λ/3) greater than the specularly reflected path. Figure 8.2 Source-receiver geometry, specular reflection point P, equivalent specularly reflected path (S R) and path (S P R) via point P on the Fresnel zone boundary so that S P +RP R 2 =Fλ.

246 Predicting outdoor sound 232 Figure 8.3 The elliptical boundary of the Fresnel zone in the ground plane. Hothersall and Harriott [10] have shown that path length differences between λ/3 and λ/4 are acceptable. The locations P of points on the surface of the Fresnel volume define the boundary of an ellipsoid with foci S and R and its major axis along S R with semi-major axis a=(r 2 +Fλ)/2 and semi-minor axis the ground plane at an ellipse (see Figure 8.3) with equation The Fresnel ellipsoid cuts (8.3) where c=r 2 /2 SP and the origin of the coordinate system is the point of specular reflection. The area of this ellipse may be computed from the locations of the points x 1,2 and y 1,2 shown in Figure 8.3. These locations are given by (8.4a) (8.4b) where

247 Predicting effects of mixed impedance ground 233 and The expression for y 1,2 differs from that given by Hothersall and Harriott [10], since they interchange variables x and y without being consistent with their notation. Moreover their corresponding expression is not dimensionally correct. The area of the ellipse is πx 1 y 1. The frequency-dependent and geometry-dependent elliptical area defined in this way may be interpreted as the area of ground between point source and receiver which is important in determining the ground effect. The reduction in Fresnel zone size with increasing frequency is demonstrated in Figures 8.4 to 8.6 for source and receiver at 1 m height above the ground, separated horizontally by 10 m and F=1/3. Note that the major axis of the ellipse exceeds the source-receiver separation (10 m) at 100 Hz (see also the 63 Hz ellipse in Figure 8.6). This implies that the area of ground of interest at this frequency extends beyond that between source and receiver. The elliptical area becomes circular at normal incidence. If it is assumed that the excess attenuation is linearly dependent on the proportion of the different surfaces (µ) along the line between source and receiver representing the Fresnel zone, then (8.5) The line representing the Fresnel zone is defined by the intersection between the Fresnel zone ellipse ((8.4) and Figure 8.3)) and the vertical plane through source, receiver and specular reflection point.

248 Predicting outdoor sound 234 Figure 8.4 Semi-axes of elliptical Fresnel zone (8.3) as a function of frequency calculated for hs=hr=1 m, r=10 m and F=1/3. The solid line represents the semi-major axis and the broken line represents the semi-minor axis. Figure 8.5 Area of elliptical Fresnel zone (8.3) calculated for hs=hr=1 m, r=10 m and F=1/3.

249 Predicting effects of mixed impedance ground 235 Figure 8.6 Fresnel (F=1/3) zones centred at the specular reflection point (5 m from source or receiver along source-receiver axis) at three frequencies for source and receiver at 1 m height and separated by 10 m. The projections of the source and receiver positions on the (x, y) plane are at (0, 5) and (0, 5) respectively. Apart from the factors µ and (1 µ), the terms on the right hand side of (8.5) represent the excess attenuation over uniform boundaries with surface impedance Z 1 or Z 2 respectively. The linear combination of excess attenuation given by (8.5) is implicit in standard calculations of ground effect [1, 2]. Alternatively, it is possible to assume that the excess attenuation components are proportional to the areas of the Fresnel zone occupied by each impedance. Other possibilities are that either the pressure or the intensity is proportional to µ. For example, if a linear interpolation of pressure is used,

250 Predicting outdoor sound 236 (8.6) where µ is calculated from the areas of each impedance inside the Fresnel zone The Boundary Element Method The increasing availability of powerful computational resources is making numerical schemes based on Boundary Elements more attractive. The computational complexity of the Boundary Element approach is reduced considerably if it is used only for 2-D problems. Let a line source produce a time-harmonic sound field, in a medium, D, bounded by a locally reacting impedance surface, S. This impedance plane can have features such as impedance discontinuities, etc. The Boundary Integral form of the wave equation can be written as (8.7) where r s is the position vector of the boundary element ds, and n is the unit normal vector out of ds. The parameter, ε, is dependent on the position of the receiver [14]. It is equal to 1 for r in the medium, 1/2 for r on the flat boundary and equal to the Ω/2π at edges where Ω is the solid angle. The Green s function, G(r, r 0 ), is the solution of the wave equation in the domain without the effect of scatterers. The integral is then the contribution of the scatterer elements to the total sound field at a receiver position. This integral formulation (first derived by Kirchhoff in 1882) is called the Helmholtz- Kirchhoff wave equation. It is the mathematical formulation of Huygen s principle. For receiver points on the boundary, one obtains an integral equation for the field potential at the boundary. This Boundary Integral Equation is a Fredholm integral equation of the second kind. Once solved, the contribution of the scatterers can be determined by evaluating the integral in (8.7) and calculating the total field for any point in the entire domain, D. This is the main BIE equation for the acoustic field potential in the presence of non-uniform boundary. The BEM represents the acoustic propagation in a medium by the boundary integral equation and solves this integral equation numerically. The locally reacting impedance boundary condition is as follows: (8.8) where β, the admittance, can be a function of the position on the boundary. This applies to the surface of the plane boundary and to the surfaces of the scatterers. Thus the

251 Predicting effects of mixed impedance ground 237 derivative term of the unknown potential can be written in terms of the potential itself. Hence, (8.9) The integral equation (8.9) has to be solved numerically through discretizing the boundary [14 18]. When discretizing the boundary surface, it is assumed that the unknown potential is constant in each element, thereby reducing the integral equation to a set of linear equations. One method involves using a quadrature technique (Simpson s rule or Gauss) which is similar to a finite difference formulation. To minimize the number of elements, the Green s function includes reflection from the flat impedance surface. Consider a 2-D problem with an infinitely long line source radiating cylindrical waves in the medium. In this case the boundary integral is a line integral and its evaluation by numerical means is a simple matter. In this case the Green s function takes the form [14] with (8.10) (8.11) In these expressions H (1) ( ) is the Hankel function of the first kind, r, r 0 and are the receiver, source and image source position vectors respectively. The wavenumber k and the complex admittance β are dependent on the frequency. The function P β represents the ground wave term. The derivatives of the Green s function in the x and z directions are also available [15] Data obtained over a single discontinuity There have been many laboratory measurements of propagation from a point source over a single impedance discontinuity [19 21]. Figure 8.7(a) and (b) illustrate the good

252 Predicting outdoor sound 238 Figure 8.7 Comparison of De Jong et al. model predictions with laboratory measurements over a single surface discontinuity between hardwood and sand. Source and receiver heights were 0.1 m and they were separated by 1 m. (a) Source over wood, discontinuity at 0.8 m from source (b) source over sand, discontinuity at 0.6 m from source. Figure 8.8 Path lengths for propagation over an impedance strip. agreement obtained between predictions of the De Jong formula and data obtained using a small-scale geometry in the laboratory [20]. A model discontinuous surface was composed of varnished hardwood board and 0.3 m deep sand contained within a box measuring 1.1 m 2 m 0.3 m. The box was placed in an anechoic chamber. The impedance of the continuous sand surface was derived from complex excess attenuation spectra by the root-finding method described in Chapter 4 and was shown in Figure 4.9. Figure 8.8 compares predictions of the De Jong model with data of the spectrum level due to a point source at 0.1 m height at a receiver at 0.1 m height and separated by 1 m. The predictions use the impedance for the sand deduced from the excess attenuation data over a continuous sand surface (see Figure 4.9) and assume that the hardwood board is acoustically hard. These laboratory data correspond to grazing angles of about 11. Rasmussen [11] has compared predictions of the De Jong model and the numerical model described in section

253 Predicting effects of mixed impedance ground with outdoor data taken under light wind conditions (<2 m s 1 ). The source height was 1 m, the receiver height was 0.1 m and their horizontal separation was 10 m. This corresponds to a grazing angle of about 5.7. The agreement between predictions and data was good. 8.3 Propagation over impedance strips Single absorbing strip Consider a single strip of absorbing material, impedance Z 2 lying perpendicular to the line between source and receiver in a plane of less absorbing material with impedance Z 1 (see Figure 8.8). The Boundary Integral Equation formulation developed in this way by Chandler- Wilde and Hothersall [14] may be written as (8.12) where is the pressure at the receiver for a surface with homogeneous impedance are the positions vectors of the source, the receiver and a point in the boundary respectively, is the Green s function associated with propagation over a boundary of impedance Z 2 (see for example (8.11)) and S is the surface of the strip. The integral can be calculated numerically by a standard boundary element technique once the pressure is known at point in the strip. If the source and receiver are restricted to a vertical plane perpendicular to the impedance discontinuity, then the surface integral is replaced by a line integral which saves computation time. Hothersall and Harriott [10] have shown that it is straight-forward to apply the empirical model developed by De Jong to a single strip. Figure 8.8 defines the various path lengths needed in the expression of the acoustic pressure over a ground of impedance Z 2 with a single strip of impedance Z Multiple discontinuities It is straightforward to extend the model to multiple strips by summing over all the diffraction terms, each term arising from each impedance discontinuity [20]. The pressure at the receiver relative to the free field in the case of multiple strips of impedance Z 1 imbedded in a plane of impedance Z 2 is given by

254 Predicting outdoor sound 240 (8.13) where j is the index characterizing the various strips and n is the total number of strips between source and receiver. In accordance with previous definitions, The spherical wave reflection coefficients Q 1 or Q 2 are used when the specular reflection point falls on the ground of impedance Z 1 or Z 2 respectively. The sequences of values (Q 1,2, γ j, δ j ) appear in (8.13) according to the following combinations: (Q 2, 1, 1) if the specular reflection point is on the source side of the j-th strip, (Q 2, 1, 1) if the specular reflection point is on the receiver side of the j-th strip, (Q 1, 1, 1) if the specular reflection point is inside the j-th strip. However, comparisons with laboratory data [20] have shown that extension of the De Jong formula to multiple strips is not very successful. Nyberg [22] has solved the Helmholtz equation with the boundary condition for a point source above an infinite plane surface consisting of strips with alternating impedance by using Cartesian coordinates and a Fourier transform technique. The ground effect due to a two-valued, infinitely periodic (i.e. striped) impedance (see Figure 8.9), may be determined, approximately, from that predicted for a point source (see Chapter 2). Hence, the expression for the field at height z, due to a source at height z 0, is (8.14) where a, b are the periods of the two types of strips, and L 0 is the usual integral for a point source (see Chapter 2) using the area-averaged admittance (β 0 b+β 1 a)/(a+b). Also, α is the horizontal wavenumber and This approximation requires

255 Predicting effects of mixed impedance ground 241 Figure 8.9 Magnitude of a surface admittance varying regularly between values of β 0 and β 1 with periods a and b respectively. Figure 8.10 Predictions of excess attenuation due to the ground from a point source over ground containing a single discontinuity (solid lines), continuous impedance 1 (dotted lines),

256 Predicting outdoor sound 242 continuous impedance 2 (broken lines) and the area-averaged impedance (dot-dash lines) for receiver height 1.2 m, separation 25 m, distance to discontinuity 3.5 m and two source heights above impedance 1 (a) 0.5 m, (b) 0.1 m. The impedances are those predicted by the two-parameter model (3.13) with effective flow resistivities of 10,000 and 100 kpa s m 2 respectively (α=100 m 1 ). Figure 8.10 illustrates that use of the area-averaged impedance does not lead to good predictions in the case of a single discontinuity. Predictions of excess attenuation due to the ground using the De Jong formula are compared with those obtained with either continuous impedance and with the continuous averaged impedance for two geometries Comparison between predictions and laboratory data for impedance strips Laboratory measurements have been made with a broadband point source over the edge of a box measuring 1 m 0.8 m 0.08 m containing alternate wood and sand strips [17]. Figures 8.11 and 8.12 show the good agreement obtained between the predictions of Nyberg s theory [22] and examples of these data. Figure 8.11 Sound level relative to free field (solid line) obtained with a point source and receiver at 0.1 m height and 1 m apart over alternating wood and sand strips [20] m wide blocks of wood were separated by m wide strips of sand. A sand strip was at the

257 Predicting effects of mixed impedance ground 243 specular reflection point. This corresponds to 80% hard surface cover. Predictions of the Nyberg theory (dash-dot line) have used the measured geometry and sand impedance (see Figure 4.9). Figure 8.12 Sound level relative to free field obtained with a point source and receiver at 0.1 m height and 1 m apart over alternating wood and sand strips [20] m wide blocks of wood were separated by m wide strips of sand with either wood (thick line) or sand (thin line) at the specular reflection point. These data correspond to 50% hard surface cover. Predictions of Nyberg s theory (dash-dot line) used the measured geometry and sand impedance (Figure 4.9). The regular oscillations in these data may be due to the edges of the box containing the sand and the strips below the source and receiver. However, they might also be the consequence of multiple interferences between the strips. Recently it has been shown that similar oscillations occur in the excess attenuation above an acoustically hard surface containing randomly positioned semicylinders and that these are predicted by an accurate multiple-scattering theory [23]. It should be noted that the Nyberg theory does not predict any difference between the excess attenuation due to the ground for a hard or soft strip at the point of specular reflection. However, the data show a difference between these conditions (see Figure 8.12). The BEM formulation is extended easily to enable calculation of propagation over multiple strips and has been found to predict a different attenuation according to whether

258 Predicting outdoor sound 244 a hard or soft strip occupies the specular reflection point [20]. An example is shown in Figure Figure 8.13 Spectra of the sound levels relative to free field for a 60% hard-strip boundary with (a) a strip of wood at the specular reflection point and (b) a strip of sand at the specular reflection point. The dotted lines represent the measured data and the solid lines represent the boundary element predictions. 8.4 Effects of refraction above mixed impedance ground The Parabolic Equation (PE) method is a useful technique for incorporating both a refracting medium and range-dependent impedance. For example, Robertson et al. [24]

259 Predicting effects of mixed impedance ground 245 have used a PE routine to investigate low-frequency propagation over an impedance mismatch, and You and Yoon [25] have used a polar PE formulation to investigate the effect of road-grass verge discontinuity on sound field in a refracting medium. Berengier et al. [26] have used the PE method successfully to model propagation involving impedance variation, topography and wind speed gradients. Here we describe an extension of De Jong s formulation to allow for a refraction and range-dependent impedance. To test the validity of the assumptions inherent in our heuristic extensions to the De Jong s model to allow for refraction, predictions of the modified De Jong formulation are compared with an alternative numerical procedure based on the Boundary Integral Equation [27]. The predictions from the two methods, in the absence of turbulence, are compared at high and low frequencies. Equation (8.2) can be extended to allow for atmospheric refraction using ideas of geometrical optics. In downwind or temperature inversion conditions, there may be multiple ray arrivals and rays may have multiple reflections from the ground. The second term in (8.2) becomes a sum over all rays with Q a,b replaced by (8.15) where l is the number of the bounces the ray makes at the region a and m is the number of bounces in region b. There will be more than one diffracted ray path from the source to the point of discontinuity and then to the receiver. In general, if the numbers of ray paths from source to the discontinuity and from the discontinuity to the receiver are m and l respectively, then the total number of diffracted ray paths is m l. Since there is more than one R 3, the term inside the brackets that includes the Fresnel function should also be replaced by a sum over all possible R 3 s. Furthermore, the term under the square root in the argument of the Fresnel function can now have negative value, but this is unrealistic. To avoid this we revert to the original expression by Pierce [9] and introduce a modulus function. The expression for the total field becomes (8.16) In this expression R n are sound ray trajectories from source to the receiver with n=1 taken as the direct ray path and M, the total number of possible ray paths, is equal to m l. The parameters, µ 1,j and µ 2,j are the polar angles of the incident and diffracted wave at the point of discontinuity for any R d respectively and sgn(x) is the sign function. Its value

260 Predicting outdoor sound 246 is 1 for a negative x and +1 otherwise. The evaluation of R d,j requires computation of all possible combinations of ray trajectories from the source to the point of discontinuity at the ground plus trajectories from the point of discontinuity to the receiver. This will involve multiple values of R d in the downwind case. In the presence of a positive gradient, caused by a temperature inversion or downwind, the rays that go through n reflections on the ground can be determined from the following quartic equation: (8.17) where (8.18) and x is the distance from the source to the first reflection from the ground. Only real roots of the equation are to be considered. This equation is to be solved for all n until no real solutions exist. A more useful parameter is the angle of incidence of any particular ray. This is given in terms of x by (8.19) where x is determined from (8.18). All other ray parameters can easily be determined from this angle. We define three terms: (8.20) (8.21) (8.22)

261 Predicting effects of mixed impedance ground 247 where (8.23) These are the horizontal distances from the source, receiver or the point of reflection at the ground, to the turning point of the ray path. The angle µ i is the Figure 8.14 Definitions of the terms used to describe a ray path for the extended De Jong model. polar angle (the angle that the tangent to the ray makes with the positive z-axis) of the ray at the source or receiver position. These distances and angles are shown in Figure Once the angle of incidence for a particular ray is known, the phase and amplitude of each ray can be evaluated. There are four cases that can be distinguished. 1 Range=T 1 +T 2 +T 3 (8.24) and (8.25) 2 Range=T 1 +T 2 T 3

262 Predicting outdoor sound 248 (8.26) and (8.27) 3 Range=T 1 T 2 +T 3 (8.28) and (8.29) 4 Range=T 1 T 2 T 3 (8.30) and (8.31) The geometrical path length, R g, of the ray is simply the arc length of the ray path and can be determined by the integral (8.32) where µ 1 and µ 2 are the polar angles of the beginning and the end points respectively and R c is the radius of curvature of the ray. R g determines the amplitude along the ray. The

263 Predicting effects of mixed impedance ground 249 phase of the ray is determined by its acoustic path length, R a. This takes into account the change of the speed of sound along the ray path and is given by (8.33) It is important to note that for n>0 more than one option may be valid. In fact for n=1 there may be one or three rays present, and for each n>1 there may be two or four rays up to a maximum value of n depending on the gradient and the range. Figure 8.15(a) and (b) compare the predictions of the extended De Jong model (8.16) and numerical hybrid BIE/FFP code [24] in a downward-refracting condition corresponding to a positive wind speed gradient of 0.25 m s 1 m 1 over an impedance discontinuity at frequencies of 200 Hz and 1 khz. Source and receiver are assumed at 5 m height. For Figure 8.14(a), the discontinuity is 200 m from the source and the ground is assumed acoustically hard from the source to the discontinuity and absorbing thereafter. At the higher frequency, the hard section extends only up to 100 m. While the two models agree well at 1 khz up to 300 m range, the performance of the extended De Jong model is not as good at the lower frequency. Figure 8.15 (a) Predictions, obtained by using the extended De Jong model (broken line) and the BIE/FFP (solid line) at a frequency of 200 Hz, of the sound field in a downwind condition over an impedance discontinuity at 200 m. The source and receiver heights are 5 m. (b) As for (a) but at a frequency of 1000 Hz and with the impedance jump at only 100 m from the source. 8.5 Predicted effects of porous sleepers and slab-track on railway noise

264 Predicting outdoor sound Predicted effects of porous sleepers and slab-track on railway noise As a consequence of noise mapping of major railways in accordance with the European Directive on Environmental Noise (Commission of the European Communities 2002), it is likely that additional noise mitigation will be required. Methods for controlling railway noise that have been explored include barriers, cuttings and tunnels. Less attention has been paid to options relating to track design. These options include optimizing the characteristics and deployment of railway ballast and the introduction of porous sound absorption into sleeper and track materials. Such options will be particularly important in respect of wheel/rail noise. Here we present the results of a brief study using a boundary element code to predict the likely effects on wheel/rail noise of introducing porous sleepers and using porous concrete on slab track [28]. Figure 8.16 shows the track profile assumed for considering the effect of sleepers. This profile includes two tracks, a Cess walkway and a cable trough. The walkway and cable trough are assumed to be acoustically hard. The edges on either side of the railway track are assumed to be flat and grass covered. The track consists of areas of ballast and sleepers, that is a mixed impedance situation in respect of ground effect. The receiver height is assumed to be 1.5 m above the grass surface and the horizontal source/receiver range is denoted by d i where i=1 to 4 corresponding to each of the four rails. The assumed values of d i are d 1 =28.3 m, d 2 =26.9 m, d 3 =24.9 m and d 4 =23.5 m. In the BEM used for this study, the rail/wheel excitation is modelled by a coherent line of monopole sources, that is monopole sources in the vertical plane perpendicular to Figure 8.16 Assumed railway track profile. the track. If Cartesian coordinates O xy are adopted, with the y-axis vertically upwards, the boundary (ground surface and track) lies entirely on or above the line y=0. For some of the calculations reported here, the boundary is assumed locally reacting and β(r s ) is used to denote the surface admittance relative to air at a point r s =(x s, y s ) on the boundary. It is also assumed that β(r s )=β c, some constant value, at points on the boundary sufficiently far from the tracks. The numerical procedure treats the above problem as a perturbation of the case in which the boundary is coincident with the line y=0 and β=β c. In this simple case the propagation problem can be solved exactly. Let G βc (r, r 0 ) denote the Green s function for this problem, that is G βc (r, r 0 ) denotes the acoustic pressure at r when a unit strength monopole source is located at r 0 above a flat homogeneous plane of admittance β c. The boundary element method is a numerical method applied to the reformulation of the Helmholtz equation as a boundary integral equation. In this study the integral equation employed is

265 Predicting effects of mixed impedance ground 251 (8.34) γ denotes those parts of the boundary on which β β c or which lie above the line y=0. Equation (8.34) holds for points r=(x, y) on or above the boundary, where k is the wavenumber, p(r) denotes the pressure at r and p 0 (r) the pressure that would be measured if the boundary were flat, coinciding with the line y=0, and had constant admittance β c. For points r=(x, y) on the boundary, ε(r)=ω(r)/(2π) if y>0 or Ω(r)/π if y=0, where Ω(r) is the corner angle at r(=π if r is not a corner). ε(r)=1 at points above the boundary. Equation (8.34) is solved by the piecewise constant collocation boundary element method [29, 30]. In this study, two monopole sources are assumed on each track, that is four sources altogether. The sources are located close to the wheel rail interfaces, that is 0.05 m above the top of the m high rails. A separate simulation is made for each source position and the resulting predicted noise levels are added logarithmically in pairs, assuming that the sources are incoherent. The sources on the track furthest from the receiver give rise to the highest predicted sound reductions. This is a consequence of the fact that, for the nearer track sources, the Fresnel zone for sound interaction with the ground falls largely on the (assumed) grass edge and the acoustical properties of the track area play a lesser role in the overall sound attenuation. BEM-predicted A-weighted SPL spectra in vertical planes through either acoustically hard sleepers or porous concrete sleepers (red and blue broken lines respectively) with simultaneous sources at positions (1) and (2) on the further track are compared in Figure The reference source spectrum used is that shown in Figure Sleepers made of porous concrete mix 1 are predicted to give greater sound reduction than the reference case (in which sleepers are assumed to be acoustically hard) below 1500 Hz. Also shown are predictions in a vertical plane through ballast only (black diamonds). The parameters assumed for predicting the acoustical properties of ballast and porous concrete are listed in Table 8.1. The acoustical properties have been predicted by means of the Johnson-Allard- Umnova model [31] (see also (3.15) and (3.16)). According to the latter model, the dynamic (complex) tortuosity α(ω) may be approximated by (8.35)

266 Predicting outdoor sound 252 Figure 8.17 BEM-predicted A-weighted SPL spectra for sources at positions (1) and (2) in the assumed railway profile (see Figure 8.16) for hard, soft and absent sleepers. Figure 8.18 Idealized slabtrack railway profile, source/receiver geometry and the discretizations used for BEM calculations.

267 Predicting effects of mixed impedance ground 253 Table 8.1 Parameter values assumed for railway ballast and porous concrete Material Flow resistivity (Nsm 4 ) Porosity Tortuosity Viscous characteristic length Ballast Porous concrete mix Porous concrete mix where ω is the angular frequency, A is the characteristic viscous length, η is the coefficient of dynamic viscosity and ρ 0 the density for air. The dynamic compressibility C(ω) is assumed to be that deduced for stacked spheres (see ( )). Hence (8.36) where Θ 0.675(1 Ω). The impedance is given by (8.37) It should be noted that the flow resistivity of railway ballast is sufficiently low that it is externally reacting and the acoustic surface impedance depends on the angle of incidence. Another consequence of the low flow resistivity is that the surface impedance depends on the depth of the ballast even when this is of the order of 0.5 m. A thickness l=0.3 m has been assumed here. The angle-dependence of the relative surface impedance of a hardbacked ballast layer has been computed from (8.38) where k 0 is the propagation constant in air, l is the layer depth, ρ 0, ρ 1 are the (real) density of air and (complex) density of the porous material respectively and θ is the incidence angle measured from the normal to the surface. The ballast impedance has been evaluated for each boundary element length of the ballast sections in the discretized version of the profiled track. The incidence angle has been defined as that between the line from the source to the middle of the boundary

268 Predicting outdoor sound 254 element segment and the normal to the segment. Where, as a consequence of the assumed track profile, the normal to the segment and the straight line to the source are separated by more than 90, the incidence angle has been set to 90. The grassland is assumed to have the impedance of an infinitely thick homogeneous porous layer predicted by a fourparameter model (see Chapter 3) using the values shown in Table 8.2. These values are consistent with those used by Morgan et al. [32]. The predictions for a 2-D cross-section through ballast only show the greatest attenuation at low frequencies. Predictions of the broadband SPL, obtained from a logarithmic summation of the narrow-band values in the SPL spectra, for simultaneous sources (1) and (2), show that in the vertical plane through the sleepers (assumed infinitely long), a 3 db reduction of overall A-weighted SPL is predicted if acoustically hard sleepers are replaced by porous concrete sleepers. Since the BEM allows only 2-D calculations, that is in a vertical plane, the 3-D effects of ballast between the sleepers have been ignored. One way of adjusting for the presence of ballast between the sleepers along the track axis is to calculate an energy equivalent continuous level over a ballast plus sleeper period assuming a constant velocity source. It is possible then to compute the energy equivalent level from the sum of the energies (i.e. squared pressures) weighted according to the width of the sleeper and the width of the ballast between the sleepers. The results of this time-average approach for sources on the further rail from the receiver are shown in Table 8.3 (to nearest 0.5 db). Figure 8.18 shows the track profile assumed for predicting the effect of using porous concrete instead of sealed concrete for slab track. The two rail beds Table 8.2 Parameter values used to predict the grass impedance Flow resistivity σ (Nsm 4 ) Porosity (Ω) Tortuosity q Pore shape factor s p 125, Table 8.3 BEM-predicted A-weighted broadband SPL for simultaneous sources (1) and (2) computed from timeaveraged levels Porous concrete sleepers Reference hard sleepers A-weighted L eq (db)

269 Predicting effects of mixed impedance ground 255 Figure 8.19 BEM-predicted A-weighted SPL spectra over continuous slab tracks with grass edges for sources at positions (1) and (2). are assumed to be either continuous porous concrete slab track, or continuous acoustically hard (non-porous) concrete. BEM-predicted A-weighted SPL spectra for vertical planes through porous concrete mixes 1 and 2 and for hard slab (joined squares, joined circles and joined stars respectively) with simultaneous sources at positions (1) and (2) on the further track are shown in Figure Also shown is the reference A-weighted spectrum (joined diamonds), that is the SPL measured at 1 m, for a train running at a speed of 145 km h 1. These results assume spherical spreading from point source rail/wheel contacts. If cylindrical spreading were to be assumed instead, then all of the predictions would be shifted to higher levels. The predicted effects of porous concrete are to reduce the A-weighted sound level above 200 Hz, compared with acoustically hard slab track. The predicted difference between hard and porous slab track is nearly 8 db at 2 khz. Porous concrete mix 1 is predicted to give greater sound reduction than mix 2 below 1000 Hz. The difference between the predicted A-weighted levels for the two mixes is nearly 5 db between 50 and 500 Hz for simultaneous sources at positions (1) and (2), that is on the further track.

270 Predicting outdoor sound 256 Table 8.4 Predicted overall A-weighted levels for source spectrum shown in Figure 8.19 and the track profile shown in Figure 8.18 Mix 1 Mix 2 Hard slab track Sources (1) and (2) Sources (3) and (4) Despite the superior performance predicted for mix 1 at low frequencies, the predicted reductions in the overall A-weighted level compared with an acoustically hard track for both porous concrete mixes are identical for the assumed reference source spectrum. The predicted reductions in the overall A-weighted levels are summarized in Table 8.4. Even for the nearest rail, the predicted reduction due to replacing non-porous by porous slab track is significant. References 1 ISO Acoustics attenuation of sound during propagation outdoors part 2: a general method of calculation (1996). 2 The calculation of road traffic noise, Department of Environment HMSO (1985). 3 K.J.Marsh, The CONCAWE model for calculating the propagation of noise from open-air industrial plants, Appl. Acoust., 15: (1982). 4 B.O.Enflo and P.H.Enflo, Sound wave propagation from a point source over a homogeneous surface and over a surface with an impedance discontinuity, J. Acoust. Soc. Am., 82: (1987). 5 J.Durnin and H.Bertoni, Acoustic propagation over ground having inhomogeneous surface impedance, J. Acoust. Soc. Am., 70: (1981). 6 P.Koers, Diffraction by an absorbing barrier or by an impedance transition, Proc. Internoise, (1983). 7 B.A.De Jong, A.Moerkerken and J.D.Van Der Toorn, Propagation of sound over grassland and over an earth barrier, J. Sound Vib., 86:23 46 (1983). 8 A.D.Pierce, Diffraction of sound around corners and over wide barriers, J. Acoust. Soc. Am., 55(5): (1974). 9 G.A.Daigle, J.Nicolas and J.-L.Berry, Propagation of noise above ground having an impedance discontinuity, J. Acoust. Soc. Am., 77: (1985). 10 D.C.Hothersall and J.N.B.Harriott, Approximate models for sound propagation above multiimpedance plane boundaries, J. Acoust. Soc. Am., 97: (1995). 11 K.B.Rasmussen, Propagation of road traffic noise over level terrain, J. Sound Vib., 82 (1):51 61 (1982). 12 J.Forssen, Calculation of noise barrier performance in a turbulent atmosphere by using substitute-sources above the barrier, Acust. Acta Acust., 86: (2000). 13 S.Slutsky and H.L.Bertoni, Analysis and programs for assessment of absorptive and tilted parallel barriers, Transportation Research Record 1176, Transportation Research Record, National Research Council, Washington, DC, (1987). 14 S.N.Chandler-Wilde and D.C.Hothersall, Sound propagation above an inhomogeneous impedance plane, J. Sound Vib., 98: (1985).

271 Predicting effects of mixed impedance ground S.N.Chandler-Wilde, Ground Effects in Environmental Sound Propagation, PhD Thesis, University of Bradford (1988). 16 R.P.Shaw, Boundary integral equation methods applied to wave problems, in Developments in Boundary Element Methods-1, Eds, P.K.Banerjee and R.Butterfield, Applied Science Publishers, London (1980). 17 D.F.Mayers, Quadrature methods for fredholm equations of the second kind, in Numerical Solution of Integral Equations, Eds, L.M.Delves and J.Walsh, Clarendon Press, Oxford (1974). 18 D.C.Hothersall and S.N.Chandler-Wilde, Prediction of L10 noise levels from road traffic: distance corrections for inhomogeneous ground cover, Acust. Acta Acust., 10(2): (1988). 19 C.Klein, Kritische studie van het geluidsveld, opgewekt door een monopool, in de nabijheid van een reflekterend oppervlak, PhD Dissertation, Catholic University of Leuven (1983). 20 P.Boulanger, T.Waters-Fuller, K.Attenborough and K.M.Li, Models and measurements of sound propagation from a point source over mixed impedance ground, J. Acoust. Soc. Am., 102(3): (1997). 21 M.R.Bassiouni, C.R.Minassian and B.Chang, Prediction and experimental verification of far field sound propagation over varying ground surfaces, Proc. Internoise, (1983). 22 C.Nyberg, The sound field from a point source above a striped impedance boundary, Acta Acust., 3: (1995). 23 P.Boulanger, K.Attenborough, Q.Qin and C.M.Linton, Reflection of sound from random distributions of semi-cylinders on a hard plane: models and data, J. Phys. D., 38: (2005). 24 J.S.Robertson, P.J.Schlatter and W.L.Siegmann, Sound propagation over impedance discontinuities with the parabolic approximation, J. Acoust. Soc. Am., 99: (1996). 25 C.S.You and S.W.Yoon, Predictions of traffic noise propagation over a curved road having an impedance discontinuity with refracting atmospheres, J. Korean Phys. Soc., 29(2): (1996). 26 M.C.Bérengier, B.Gauvreau, Ph. Blanc-Benon and D.Juvé, Outdoor sound propagation: a short review on analytical and numerical approaches, Acta Acust., 89: (2003). 27 S.Taherzadeh, K.M.Li and K.Attenborough, A hybrid BIE/FFP scheme for predicting barrier efficiency outdoors, J. Acoust. Soc. Am., 110(2): (2001). 28 K.Attenborough, P.Boulanger, Q.Qin and R.Jones, Predicted influence of ballast and porous concrete in railway noise, Proc. InterNoise 2005, Rio de Janeiro, Paper IN05_ O.Umnova, K.Attenborough, E.Standley and A.Cummings, Behavior of rigid-porous layers at high levels of continuous acoustic excitation: theory and experiment, J. Acoust. Soc. Am., 114(3): (2003). 30 S.N.Chandler-Wilde and D.C.Hothersall, Efficient calculation of the Green s function for acoustic propagation above a homogeneous impedance plane, J. Sound Vib., 180: (1995). 31 D.C.Hothersall, S.N.Chandler-Wilde and N.M.Hajmiraze, Efficiency of single noise barriers, J. Sound Vib., 146: (1991). 32 P.A.Morgan and G.R.Watts, Investigation of the screening performance of low novel railway noise barriers. Unpublished project report, TRL, PR SE/705/03 (2003).

272 Chapter 9 Predicting the performance of outdoor noise barriers 9.1 Introduction Noise reduction by barriers is a popular method for mitigating the effects of noise from transportation. Near a highway, railway or airport, the receiver can be shielded by a barrier, which intercepts the line-of-sight from the noise source. As long as the direct transmission of sound through the barrier is negligible, the acoustic field in the shadow region is mainly dominated by the sound diffracted around the barrier. The many different types of barriers include a normal straight-edge barrier, a top-bended or cranked barrier, an inclined barrier, a louvre barrier and a barrier with multiple edges. To design noise barriers with optimum acoustic performance, it is important to have accurate prediction schemes for calculating the sound reduction by barriers. The aims of this chapter are 1 to introduce the basic principles of the diffraction of sound by noise barriers and identify the important parameters, 2 to discuss both analytical and empirical models for studying the acoustic performance of noise barriers, 3 to determine the degradation of performance due to presence of gaps in barriers, 4 to explore the effectiveness of a barrier in screening a directional source and 5 to investigate the consequences of assuming a fixed source spectrum when calculating the performance of noise control elements such as barriers. 9.2 Analytical solutions for the diffraction of sound by a barrier Since the diffraction of spherical, cylindrical or plane waves by a thin half plane is of interest in optics as well as in acoustics, it has been studied, experimentally and theoretically, since the eighteenth century. In the optical context, it was suggested that the diffraction pattern behind a half plane is the result of the superposition of waves scattered by the edge and the unobstructed portion of the incident waves. The first mathematically rigorous solution of a half-plane diffraction problem was formulated in 1896 by Sommerfeld who considered the two-dimensional problem of a plane wave incident on a thin, perfectly reflecting, half plane and solved the corresponding partial differential equations [1, 2]. The resulting solutions are composed of two main terms. The first term is the direct wave and is expressed exactly according to the principles of geometrical acoustics. The second term, which is identified as the contribution of the diffracted wave, is expressed in terms of Fresnel integrals. Subsequently, Carslaw [3, 4] and MacDonald

273 Predicting the performace of outdoor noise barriers 259 [5] extended Sommerfeld s approach to the generalized problems of the diffraction of cylindrical and spherical incident waves by wedges. In a similar manner to Sommerfeld s solution, their solutions are expressed as the sum of two integrals, associated with the incident and scattered (diffracted) fields respectively. An alternative to seeking a formal solution for the governing partial differential equations that arise when solving diffraction problems directly is an integral formulation. Copson [6] and Levine and Schwinger [7a, b] have used this approach to diffraction problems and obtained exact solutions by applying the Wiener-Hopf method, which is a standard technique for solving certain types of linear partial differential equations subject to mixed boundary conditions and involving semi-infinite geometries [8, 9]. Tolstoy [10, 11] has obtained an alternative exact and explicit solution for sound waves diffracted by wedges. In his solution, the diffracted sound field can be represented by a sum of an infinite series. The coefficient of each term of the series is given by a set of linear equations which can be solved by a simple recursion scheme. The edge diffractions are taken into account exactly without the need for an asymptotic approximation of integrals. However, the practical application of this formulation is limited by the slow convergence of the series particularly at high frequencies Formulation of the problem As described earlier, there are many different models for calculating the diffraction of sound by a thin plane. For the moment, putting aside the mathematical complexity of the analytical solution, we consider simply the geometrical configuration of the problem. It is convenient to use a cylindrical polar coordinate system to describe the relative position of source and receiver from the edge of a thin plane. Let the origin be located at the edge and (r, θ, y) be the cylindrical polar coordinates. The y-axis coincides with the edge of the semi-infinite half plane and the initial line of the polar coordinates lies on the right-hand surface of the half plane. Hence, the thin plane is made of two surfaces at θ=0 (right surface) and θ=2π (left surface) with zero thickness and it has an exterior angle of 2π. With this coordinate system, all radial distances are measured from the edge and all angular positions are measured in the counter clockwise direction as shown in Figure 9.1.

274 Predicting outdoor sound 260 Figure 9.1 Diffraction of sound by a semiinfinite thin plane. Without loss of generality, a point source is placed in front of the left surface of the thin plane at (r 0, θ 0, y 0 ), that is 2π θ 0 π and the receiver is located at (r r, θ r, y r ) where 2π θ r 0. Using the principle of geometrical acoustics, the sound field consists of a diffracted field, p d and a geometrical solution that combines the incident and reflected waves, p i and p r. The dashed lines at θ=3π θ 0 and θ=θ 0 π in Figure 9.1, subdivide the field into three separate regions, I, II and III. The ray that strikes the top edge of barrier reflects back along the path represented by a broken line, θ=3π θ 0. According to the geometrical acoustics, none of the reflected rays can penetrate region II and III and they are confined to region I. The dividing line at θ= 3π θ 0 separates region I from region II and creates a shadow boundary B r for the reflected wave. Similarly, the direct wave cannot penetrate region III because of the presence of the thin plane preventing direct line-of-sight contact between source and receiver. Region III is called the shadow zone. The dividing line at θ=θ 0 π separates region II from region III and it is called the shadow boundary, B i, for the direct wave. If the source faces the right-hand side of the thin plane, then there is a wave reflected on the θ=0 face. The occurrence of each wave type can be summarized as follows: 1 a direct wave where π θ r θ 0 0; 2 a wave reflected from the θ=0 face when π (θ r +θ 0 ) 0; 3 a wave reflected from the θ=2π face when (θ r +θ 0 ) 3π 0 and 4 a diffracted wave from edge of plane when 0 θ 0, θ r 2π.

275 Predicting the performace of outdoor noise barriers 261 Items (2) and (3) mentioned earlier are mutually exclusive as the source cannot see both surfaces of the half plane simultaneously. The analytical solution for the total sound field in the vicinity of a semi-infinite half plane can be decomposed into three terms. The first term takes account of the contribution of a direct wave, p i, the second term accounts for the contribution of a reflected wave from the surface of the semi-infinite half plane, p r and the last term deals with the contribution of the diffracted wave at the edge of the half plane, p d. Except the diffracted wave term, the occurrence of the direct and reflected wave terms depends very much on the relative position of source, receiver and the thin plane. In each region, the total sound field, p T, is Region I: pt=p i +p r +p d ; Region II: p T =p i +p d ; and Region III: p T =p d. (9.1a) (9.1b) (9.1c) In the following sections, we shall discuss two popular solutions that have been frequently used to calculate the diffracted wave term. Nevertheless, it should be noted that the diffracted sound field may be treated as having two components which are associated with the direct and reflected fields, p di and pdr respectively. The geometrical solution becomes discontinuous at the shadow boundary of the reflected wave because the reflected wave term vanishes in region II. However, the associated component of the diffracted wave for the reflected fields p dr also becomes discontinuous on crossing the shadow boundary of the reflected wave. As a result, the total sound field remains continuous. Similarly, the geometrical solution at the shadow boundary of the direct wave becomes discontinuous due to the absence of the direct wave term but the diffraction solution associated with the direct field p di becomes discontinuous also and that leads to a continuous solution throughout The MacDonald solution Consider the sound field due to a point monopole source which is placed near to a rigid half plane. The source has unit strength and the time-dependent factor, e iωt, is omitted for simplicity. Using the cylindrical polar coordinate system described in the last section such that the distance from the source and its image (see Figure 9.2) to the receiver, R 1 and R 2, can be determined according to (9.2a)

276 Predicting outdoor sound 262 and (9.2b) Figure 9.2 The geometry of source, image source and receiver in the proximity of a semiinfinite plane. We can also determine the shortest distance from the source (or image source) to receiver after diffraction from the half plane, that is the shortest source-edge-receiver path as (9.3) Following Sommerfeld [1], MacDonald [5] developed a solution in a spherical polar coordinate system. This was subsequently recast in the cylindrical polar system by Bowman and Senior [12] to give the total field as a sum of two contour integrals as follows:

277 Predicting the performace of outdoor noise barriers 263 (9.4) where k is the wavenumber of the incident wave and is the Hankel function of the first kind. The limits of the contour integrals are determined according to and (9.5a) (9.5b) where sgn(u), the sign function, takes the value of +1 or 1 depending on the sign of the argument. Note that both terms in (9.4) contain integrals that are related to the integral representation of spherical wave where (9.6) The integrals solution of (9.4) can be spilt into two parts: a geometrical and diffraction contribution. The total sound field in regions I, II and III can be expressed in the form given in (9.1a c) with the direct wave, reflected wave and diffracted wave determined respectively by (9.7a) (9.7b) and

278 Predicting outdoor sound 264 (9.7c) A limiting case that should be checked is the high-frequency limit, for which k. Both ς 1 and ς 2 tend to infinity in this limit. Hence, the diffraction integrals in (9.7c) become negligibly small and the geometrical solution is recovered as it should. Another limiting case of interest is the situation where the source or the receiver (or both) are close to the edge of the half plane such that and where R is defined by (9.3c). We can expand the integrands by means of Taylor series and integrate the resulting expression to yield an approximate solution for the total pressure as (9.8) The last case is when both the source and receiver are far from the edge and the shadow boundaries, that is MacDonald [5] derived an asymptotic solution for this case in which the contribution from the diffracted wave can be written in terms of Fresnel integrals as follows: (9.9) The function G(u) is defined as (9.10a) where F r (u)=c(u)+is(u), (9.10b) and the Fresnel integrals C(u) and S(u) are defined [13] by (9.11a)

279 Predicting the performace of outdoor noise barriers 265 and (9.11b) The corresponding Fresnel numbers of the source and image source are denoted, respectively, by N 1 and N 2. They are defined as follows: (9.12a) and (9.12b) If the Fresnel numbers are sufficiently large the asymptotic expansions for the Fresnel integrals can be used in (9.9) to yield an approximate solution for the diffracted sound: (9.13) The approximation in (9.13) is invalid at the shadow boundaries. In this case, the exact integral representation of the total field (cf. (9.4)) should be used instead. Thus, the total sound field at the shadow boundary of the reflected wave is (9.14) When the receiver is located at the shadow boundary of the direct wave, the solution becomes (9.15)

280 Predicting outdoor sound The Hadden and Pierce solution for a wedge The last section considered the diffraction of sound by a thin half plane. Now we investigate the sound diffraction around wedges for arbitrary source and receiver locations. An accurate integral representation of the diffracted wave from a semi-infinite wedge has been given by Hadden and Pierce [13]. Consider the geometrical configuration of the problem in Figure 9.3. We are interested in studying the diffraction Figure 9.3 Diffraction of a spherical sound wave by a wedge. field of a wedge with an exterior angle of In a similar fashion to diffraction by a thin screen, the solution is composed of a geometrical solution of a direct wave term, a wave reflected from the right face of the wedge and a wave reflected from the left face. The direct wave is zero unless the receiver can see the source. There are no contributions from the reflected waves unless one can construct specularly reflected waves from either faces of the wedge. In addition to the geometrical solutions, the total field consists of a diffracted wave which can be decomposed into four terms:

281 Predicting the performace of outdoor noise barriers 267 (9.16) where ς 1 = θ r θ 0 (9.17a) (9.17b) ς 3 =θ r +θ 0 (9.17c) (9.17d) Each of the diffracted wave terms correspond to the sound paths between the source S 0, its image in the barrier the receiver R 0 and its image in the barrier that is paths and for the wedge shown in Figure 9.4. For each path the diffracted field V(ς i ) can be calculated from (9.18a) where (9.18b) (9.18c) (9.18d)

282 Predicting outdoor sound 268 Figure 9.4 Source, image source, receiver and image receiver for a wedge. (9.18e) (9.18f) and R, the shortest diffraction path from source to receiver, is given in (9.3c). In (9.18b), the function H(u) is the Heaviside step function which is 1 for positive argument and zero otherwise. The parameter υ is the wedge index with υ=1/2 for a thin screen and υ=2/3 for a right-angle wedge. The diffracted field can be computed readily, using (9.18a f), as the integral is dominated by the e y factor, the rest of the integrand is non-oscillatory and its magnitude is bounded by unity. The formulation allows one to evaluate the sound fields for arbitrary point source locations in the vicinity of a rigid wedge. This accurate representation of the sound field has been shown to agree well with experiment. Unlike the MacDonald solution, the formulation allows for reflections of the source and receiver on the faces of the barrier (wedge). As a result, it is possible to consider the effect of barrier surfaces with finite impedance by incorporating appropriate spherical wave reflection coefficient(s) for the paths from the image sources. However, instead of pursuing this idea here, we explore a special case for a thin screen with either the source or receiver located at a distance a few wavelengths from the edge of the screen such that the wedge index, υ=1/2 and the length ratio remains finite. In such a case, the integral given in (9.18c) can be simplified and expressed in terms of Fresnel integrals. Furthermore, the four diffraction terms can be grouped leading to the compact formula,

283 Predicting the performace of outdoor noise barriers 269 (9.19a) where X + =X(θ 0 +θ r ), X =X(θ θ 0 ), (9.19b) (9.19c) (9.19d) A D (X)=sgn(X)[f( X ) ig( X )]. (9.19e) The function A D (X) is known as the diffraction integral that can expressed in terms of the auxiliary Fresnel integrals f(x) and g(x) given by and (9.20a) (9.20b) The Fresnel integrals C(X) and S(X) are defined in (9.11 a) and (9.11b) respectively. Note that the first and fourth terms (with i=1 and 4) in (9.16) are combined to give the term A D (X ) in (9.19a). On the other hand, the second and the third terms in (9.16) are grouped to yield the term A D (X + ). Figure 9.5 shows a comparison of the two exact solutions with experimental data. The insertion loss (IL) of the screen (sometimes known as the attenuation, Att), defined by (9.21)

284 Predicting outdoor sound 270 where p w and p w/o are the total sound fields with or without the barrier, is used often to assess the acoustic performance. Figure 9.5 Comparison of predictions of the MacDonald solution and Hadden and Pierce formulation with laboratory data. The insertion loss of the thin half plane is plotted against the Fresnel number, N Approximate analytical formulation Many approximate analytical solutions for diffraction by a half plane include the physical interpretation of diffraction suggested by Young and Fresnel. The most common one, the Fresnel-Kirchhoff approximation, is a mathematical representation of the Huygens- Fresnel principle and is widely used in optics. The wavelengths of optical light are always smaller than obstacles, such as screens. However, the Fresnel-Kirchhoff approximation represents a high-frequency approximation in acoustics. It results from solving the Helmholtz equation with the aid of Green s theorem. The sound field behind a screen is expressed as the surface integral over the open aperture above the screen. Alternatively, by applying Babinet s principle, the same result can be obtained if the surface integral is performed on the barrier surface. The details of Young s and Fresnel s studies and the Fresnel-Kirchhoff approximation can be found in various textbooks by Hecht [14] and Born and Wolf [2]. Skudrzyk [15] transformed Kirchhoff s solution to a new diffraction formula, which is now known as Rubinowics-Young formula. He intended to investigate waves diffracted

285 Predicting the performace of outdoor noise barriers 271 by a screen containing an aperture. The diffracted field was expressed as a line integral along the aperture edge instead of a surface integral as stated in the Kirchhoff s solution. In addition, Rubinowics was able to show that the Kirchhoff solution for plane or spherical incident waves can be decomposed into two portions: the unobstructed portion of incident wave and the edge-scattered wave. Embleton [16] derived a formula for sound diffracted by a two-dimensional barrier based on the Rubinowics-Young formula. In his studies, Embleton assumed that the line integral was along the edge of the barrier and a semi-circular arc at infinity that connects the two ends of the barrier edge. Embleton s formula is convenient for numerical implementation because the integration variable is reduced to one dimension only. In this section, we present the Fresnel-Kirchoff approximation scheme only. Consider the configuration shown in Figure 9.6. The dashed line represents an infinite plane surface between the source and receiver. Again, the time-dependent factor e iωt is understood and suppressed throughout. According to the Fresnel-Kirchoff diffraction formula [2], the sound pressure at the receiver can be written as (9.22) where the integral is to be evaluated over a surface Γ of infinite extent. The variables d 0 and d r are the distances from a point on the surface Γ to the source and to the receiver respectively. The angles and are defined in Figure 9.6. Of course, the integral of (9.22) can be evaluated exactly to yield the free field sound pressure as Suppose that a semi-infinite thin screen is interposed between the source and receiver as shown in Figure 9.7. The screen is represented by the surface Γ 2 and the aperture is represented by the surface Γ 1. (9.23)

286 Predicting outdoor sound 272 Figure 9.6 Geometry and notation used for the Fresnel-Kirchhoff approximation for the sound field with a transparent screen in front of a source. Figure 9.7 Geometry and notation used for the Fresnel-Kirchhoff approximation for the sound field with an aperture above a rigid screen.

287 Predicting the performace of outdoor noise barriers 273 The advantage of adopting (9.22) to calculate the sound field is that the effect of a screen can be modelled by assuming that there is no contribution to the pressure at the receiver point from areas of the surface corresponding to the screen. Thus for the situation in Figure 9.7, the pressure at the receiver can be calculated approximately from (9.24) Invoking the Fresnel-Kirchoff diffraction integral for an approximate solution, we implicitly assume that the thin screen transmits and reflects no sound. The expression (9.24) can be evaluated readily by standard numerical quadrature techniques that provide a far field approximation of the total field. If the opening is small compared to the distances of from the surface Γ to the source and to the receiver, the factor does not change appreciably over the surface and can be moved out of the integral. Under this condition (9.24) can be simplified and expressed in terms of the Fresnel integrals as, (9.25) where the distances a, b and h are defined in Figure 9.7. When h in (9.25) is positive, the receiver is located in the illuminated zone. An interesting case is when h, that is the sound field in the absence of the screen. Then, according to the Fresnel-Kirchhoff model [cf. (9.25)], the total (free) sound pressure is ie ik(a+b) /4π(a+b), which is different from the exact solution by a phase factor of π/2. Despite the fact that the condition for a small aperture in the study of Fresnel diffraction is violated in the problem of the half plane, the approximation given in (9.25) appears to give reasonable predictions of barrier attenuation. However there are many reported discrepancies of the predictions due to the Fresnel-Kirchhoff model with other more accurate theoretical models and experimental data [17]. Using (9.23) and (9.25), the insertion loss of the screen, defined by (9.21) can be calculated from (9.26) To give a better perspective of different approximate solutions, the comparison of the Fresnel-Kirchhoff formula with the experimental data will be deferred until the next section because it is more desirable to discuss a well-known empirical scheme first.

288 Predicting outdoor sound Empirical formulations for studying the shielding effect of barriers The most direct way to investigate the acoustical performance of noise barrier is to conduct full-scale field measurements. Alternatively, it can be determined through indoor scale model experiments. The first known graph for sound attenuation in the shadow zone of a rigid barrier due to a point source was developed by Redfearn in the 1940s [18]. The noise attenuation was given as a function of two parameters in his graphs, namely, (i) the effective height of barriers normalized by the wavelength and (ii) the angle of diffraction. Nearly 30 years later, Maekawa [19] measured the attenuation of a thin rigid barrier, with different source and receiver locations. In his experimental measurements, a pulsed tone of short duration was used as the noise source. He presented his experimental data in a chart that the attenuation was plotted against a single parameter known as the Fresnel number. The Fresnel number is the numerical ratio of path difference (the distance difference between the diffracted path and direct path of sound) to the half of a sound wavelength. In the same period, Rathe presented an empirical table for the sound attenuation by a thin rigid barrier due to a point source [20]. His empirical table was based on a large number of experimental data. The attenuation was given in octave frequency bands normalized by the reference frequency with the Fresnel number of 0.5. Kurze and Anderson [21, 22] reviewed diffraction theory from Keller [23], and used Maekawa s and Redfearn s experimental data to derive empirical formulas for the barrier attenuation. The attenuation is expressed as the function of relative locations of source and receiver, including the diffracted angles at the source and receiver side. There are some common features of the experimental investigations of these studies. First, a point source was used to generate incident waves in these early experimental studies. Second, the Fresnel number is an elementary parameter for determining the barrier attenuation. As a result of their comparative simplicity, Kurze-Anderson s formulas and Maekawa s chart are used extensively for engineering calculations. Although there are several other empirical models and charts to predict the acoustic performance of a thin barrier, we shall only discuss the Maekawa chart and other related developments in the following sections. Maekawa described the attenuation of a screen using an empirical approach based on the important parameters affecting the screening. An important parameter which appears in the treatments described earlier is the difference in path lengths from the source to the receiver via the top of the barrier (r 0 +r r in Figure 9.7) and the direct path length from source to receiver, R 1 =a+b. This is a measure of the depth of shadow produced by the screen for the given source and receiver positions. The path difference, δ 1, is given by δ 1 =(r 0 +r r ) R 1. (9.27) The other important parameter is the wavelength of the sound, λ. The longer the wavelength, the greater is the diffracted wave amplitude.

289 Predicting the performace of outdoor noise barriers 275 These parameters may be combined into the Fresnel number N 1 associated with the source, that is (9.28) In the 1960s, Maekawa obtained a comprehensive set of data for the insertion loss, which corresponds to the attenuation in the absence of ground. The measurements were made in a test room using a spherically spreading pulsed tone of short duration. The solid line in Figure 9.8 shows predictions according to Maekawa s chart for predicting the attenuation of sound by a semi-infinite plane screen as a function of positive N 1 compared with data from measurements made in an anechoic chamber. Various geometrical configurations were used and the frequency range varied from 500 to 10 khz. Despite the wide range of Fresnel numbers associated with these experimental data, they are consistent with the measurements shown in Maekawa s chart [19] and hence with the design curve. Also shown are predictions of the MacDonald solution (9.9) and the Fresnel-Kirchoff approximation (9.24). Maekawa obtained experimental data not only in the shadow zone but also in the illuminated zone where the source can see the receiver. In his notation, a negative value of N 1 signifies that the receiver is located in the illuminated zone although the same definition (9.28) is used. For N 1 >1, the abscissa is logarithmic. Below this value it has been adjusted to make the design curve linear. For N 1 =0 the attenuation is 5 db. At this point the source is just visible over the top of the barrier. For N 1 <0, the receiver is in the illuminated zone and the attenuation quickly drops to zero. In Figure 9.8, we only show N 1 ranging from 0.01 to 9. A function which fits this curve quite well is Figure 9.8 Comparison between laboratory data for the sound attenuation by a semi-infinite hard screen in free space and predictions of the Maekawa Chart (thick solid

290 Predicting outdoor sound 276 Att=10 log 10 (3+20N) db. line), the Fresnel-Kirchoff approximation (broken line) and the MacDonald solution (thin solid line). (9.29) This formula was originally defined only for N>0 but is often used for N 1 > 0.05 [24]. More accurate formulas have been proposed by Yamamoto and Takagi [25]. Based on Maekawa s original chart, they developed four different approximations. (9.30a) (9.30b) (9.30c) (9.30d) where the function G(N 1 ) in (9.30a) is given by (9.30e) Any one of these expressions gives good agreement with the data obtained from Maekawa s chart. The maximum difference is less than 0.5 db in (i) and is no more than 0.3 db in cases (ii), (iii) and (iv). Although formula (i) leads to a marginally greater error, it has the advantage that only one formula is required to describe the whole chart. Using the data shown in Figure 9.8 again, we display their comparison with the Yamamoto and Takagi formulae and predictions using the MacDonald solution in Figure 9.9. Kurze and Anderson [21] have derived another simple formula that has been used widely. Maekawa s curve can be represented mathematically by,

291 Predicting the performace of outdoor noise barriers 277 (9.31) The empirical formulae associated with Maekawa s chart only predict the amplitude of the attenuation and no wave interference effects are predicted [19]. In the next section, we shall discuss how they can be extended to model the interference effects of contributions from different diffracted wave paths. Recently, Menounou [26] has modified Maekawa s chart from a single curve with one parameter to a family of curve with two Fresnel numbers. The first Fresnel number is the conventional Fresnel number and is associated with the relative position of the source to the barrier and the receiver. The second Fresnel number is defined similarly to the first Fresnel number. It represents the relative position of the image source and the receiver. Menounou also modified the Kurze-Anderson empirical formula and the Kirchhoff Figure 9.9 Comparison between laboratory data and predictions of the different empirical formulae of Yamamoto and Takagi (Y&T) (formulas (i) (iv) correspond to (9.30a) (9.30d)) and of the MacDonald solution (9.30). The insertion loss of the thin screen is plotted against the Fresnel number, N 1.

292 Predicting outdoor sound 278 solution. Unlike earlier studies, Menounou considered the plane, cylindrical and spherical incident waves in her analyses. Her study combines simplicity of use with the accuracy of sophisticated diffraction theories. Without providing the details of the derivation, we quote an improved Kurze-Anderson formula that allows a better estimation of the barrier attenuation by including the effect of image source on the total field. The improved Kurze-Anderson formula is given by Att=Att s +Att b +Att sb +Att sp (9.32a) where (9.32b) (9.32c) (9.32d) (9.32e)

293 Predicting the performace of outdoor noise barriers 279 Figure 9.10 Comparison between laboratory data and predictions of the Kurze and Anderson (K&A) empirical formula. Predictions of Menounou s modification on the Kurze and Anderson formula is also shown. The insertion loss of the thin half plane is plotted against the Fresnel number, N 1. The term Att s is a function of N 1 which is a measure of the relative position of the receiver from the source. The second term depends on the ratio of N 2 /N 1. It is a measure of the proximity of either the source or the receiver from the half plane. The third term is only significant when N 1 is small. It is a measure of the proximity of the receiver to the shadow boundary. The last term is a function of the ratio R /R 1 which is used to account for the diffraction effect due to spherical incident waves. Figure 9.10 shows the predicted attenuation according to the Kurze and Anderson formula and the Menounou modification. Again, both formulae appear to give predictions in agreement with the data for a thin screen. 9.4 The sound attenuation by a thin barrier on an impedance ground The diffraction of sound by a semi-infinite rigid plane is a classical problem in wave theory and dates back to the nineteenth century. There was renewed interest in the problem between the 1970s and early 1980s in connection with the acoustic effectiveness

294 Predicting outdoor sound 280 of outdoor noise barriers. Since outdoor noise barriers are always on the ground, the attenuation is different from that discussed in sections 9.2 and 9.3 for a semi-infinite plane screen. Extensive literature reviews on the pertinent theory and details on precise experimental studies have been carried out by Embleton and his co-workers [27, 28]. Figure 9.11 shows the situation of interest. A source S g is located at the left side of the barrier, a receiver R g at the right side of the barrier and O is the diffraction point on the barrier edge. The sound reflected from the ground surface can be described by the source s mirror image, S i. On the receiver side, sound waves will also be reflected from the ground and this effect can be considered in terms of the image of the receiver, R i. The pressure at the receiver is the sum of four terms which correspond to the sound paths S g OR g, S i OR g, S g OR i and S i OR i. If the surface is a perfectly reflecting ground, the total sound field is the sum of the diffracted fields of these four paths, P T =P 1 +P 2 +P 3 +P 4 (9.33) where P 1 =P(S g, R g, O), P 2 =P(S i, R g, O), P 3 =P(S g, R i, O), P 4 =P(S g, R i, O), (9.34a) (9.34b) (9.34c) (9.34d) and P(S, R, O) is the diffracted sound field due to a thin barrier for given positions of source S, receiver R and the point of diffraction at the barrier edge O.

295 Predicting the performace of outdoor noise barriers 281 Figure 9.11 Ray paths for a thin barrier placed on the ground between the source and receiver. If the ground has a finite impedance (such as grass or a porous road surface) then the pressure corresponding to rays reflected from these surfaces will need to be multiplied by the appropriate spherical wave reflection coefficient(s) to allow for the change in phase and amplitude of the wave on reflection as follows: P T =P 1 +Q s P 2 +Q R P 3 +Q s Q R P 4, (9.35) where Q s and Q R are the spherical wave reflection coefficients at the source and receiver side, respectively. The spherical wave reflection coefficients can be calculated according to (2.40c) for different types of ground surfaces and source/receiver geometries. For a given source and receiver position, the acoustic performance of the barrier on the ground is normally assessed by use of either the excess attenuation (EA) or the insertion loss (IL). They are defined as follows: EA=SPL b SPL f IL=SPL b SPL g, (9.36a) (9.36b) where SPL f is the free field noise level, SPL g is the noise level with the ground present and SPL b is the noise level with the barrier and ground present. Note that in the absence of a reflecting ground, the numerical value of EA (see also Att defined by (9.21) in section 9.2.3) is the same as IL. If the calculation is carried out in terms of amplitude only, as described in [19], then the attenuation Att n for each ray path can be directly determined from the appropriate Fresnel number F n for that path. The excess attenuation of the barrier on a rigid ground is then given by

296 Predicting outdoor sound 282 (9.37) The attenuation for each path can either be calculated by the empirical or analytical formulas listed in sections 9.2 and 9.3 depending on the complexity of the model and the required accuracy. If the calculation demands an accurate estimation on both the amplitude and phase of the diffracted waves, then the MacDonald solution or the Hadden and Pierce solution should be used although the latter solution will give a more accurate solution for both source and receiver locating very close to the barrier [28]. Nevertheless, the former solution is equally accurate for most practical applications where both the source and receiver are far from the barrier edge. Figure 9.12(a) displays predicted insertion loss spectra for propagation from a point source for the source-receiver-barrier geometry indicated. The sound pressure was calculated using the four sound paths described above and the MacDonald expression (9.9). Curves plotted with a thin line is for a rigid ground surface and a dashed line is for an absorbing surface such as grassland. For the grassland impedance, the Delany and Bazley one-parameter model with an effective flow resistivity of 300 kpa s m 2 has been used. The oscillations are due to interference between the sound waves along the four ray paths considered. The curve plotted as a thick line is the result using Maekawa s method to determine the attenuation of each ray path. The total insertion loss is obtained by summing all four paths incoherently by assuming that the point source is located above a rigid ground. The same method is used to predict the insertion loss above the absorbing surface. The predicted results (dotted line) are presented in Figure 9.12(a). Neither of the incoherent predictions oscillates since the phases of the waves are not considered. Nevertheless, these lines give a reasonable approximation to the results of more complex calculations [29]. Figure 9.12(b) shows results for a similar geometry except that in this case a line of incoherent point sources, separated by approximately 1.5 m along the nominal road line, has been assumed and the calculation has been carried out for each source. The averaging over all the predictions for the different sources has smoothed the curves. Note that the IL for the grassland case becomes negative around 500 Hz indicating that the sound absorption due to the grass is greater than the screening effect of the barrier for these conditions. In both Figure 9.12(a) and (b), the height of the barrier is taken as 3 m. The source is assumed to be 5 m from the barrier and 0.3 m above the ground. The receiver is assumed to be 30 m from the barrier and 1.2 m above the ground. Lam and Roberts [30] have suggested a relatively simple approach for modelling full wave effects. In their model the amplitude of the diffracted wave, Att1, may be calculated using a method such as that described in section 9.3. However, the phase of the wave at the receiver is calculated from the path length via the top of the screen assuming a phase change in the diffracted wave of π/4. It is assumed that the phase change is constant for all source, barrier and receiver heights and locations. For example, the diffracted wave for the path S g OR g would be given by (9.38)

297 Predicting the performace of outdoor noise barriers 283 It is straightforward to compute the contributions for other diffracted paths and hence the total sound field can be calculated through the use of (9.35) for impedance ground and (9.33) for a hard ground. This approach provides a reasonable approximation for many conditions normally encountered in practical barrier situations where source and receiver are many wavelengths from the barrier and the receiver is in the shadow zone. Figure 9.12 Comparison of predictions resulting from summing the contributions of different ray paths incoherently and coherently for the sound field (a) insertion loss due to a point source and (b) excess attenuation from a coherent line source behind an outdoor noise barrier placed on a ground surface.

298 Predicting outdoor sound 284 The scheme works well for a barrier located on a hard ground but less so if the ground has finite impedance. To illustrate this point, Figure 9.13 compares data obtained in an anechoic chamber near a thin barrier placed on an absorbing surface of thickness m with predictions of the Hadden and Pierce Figure 9.13 Comparison of laboratory data (points) with predictions obtained from the Hadden and Pierce solution (solid line) and the Lam and Roberts approximate scheme (broken line) for a barrier place on an impedance ground. The source is located at m from the barrier and at m above the ground. The receiver is placed at m from the barrier and m above the ground. The ground is modelled as a Delany and Bazley hard backed layer (3.1, 3.2) with effective flow resistivity 9000 Pa s m 2 and (known) layer thickness of m. formula and Lam and Roberts approximate scheme. It is obvious that predictions obtained by using the Hadden and Pierce formula agree well with the experimental observations. On the other hand, while the Lam and Roberts model predicts the general trend of the insertion loss spectrum, it does not predict its magnitude as well, particularly in frequency intervals where the ground effects are strong.

299 Predicting the performace of outdoor noise barriers Noise reduction by a finite length barrier All outdoor barriers have finite length and, under certain conditions, sound diffracting around the vertical ends of the barrier may be significant. The MacDonald solution (see section 9.2.2) has been adapted to include the sound fields due to rays diffracted around the two vertical edges of a finite length barrier [31]. There are other diffraction theories for the vertical edge effects which have met with varying degrees of success when compared with data [32, 33]. In this section, Figure 9.14 Eight ray paths associated with sound diffraction by a finite length barrier. a more practical method is described [30]. The accuracy of this approach has been verified [34]. Figure 9.14 shows eight diffracted ray paths contributing to the total field behind a finite length barrier. In addition to the four normal ray paths associated with diffraction at the top edge of the barrier (see Figure 9.7), four more diffracted ray paths from the vertical edges, including two ray paths each from either side, have been identified. In principle, there are reflected-diffracted-reflected rays also which reflect from the ground twice but these are ignored. This is a reasonable assumption if the ground is acoustically soft. The two rays at either side are, respectively, the direct diffracted ray and the diffracted-reflected ray from the image source. The reflection angles of the two diffracted-reflected rays are independent of the barrier position. These rays will either reflect at the source side or on the receiver side of the barrier depending on the relative positions of the source, receiver and barrier. We illustrate both possibilities in Figure The total field is given by P T =P 1 +Q s P 2 +Q R P 3 +Q s Q R P 4 +P 5 +Q R P 6 +P 7 +Q R P 8, (9.39)

300 Predicting outdoor sound 286 where P 1 P 4 are those given in (9.34 a d), for diffraction at the top edge of the barrier. The contributions from the two vertical edges E 1 and E 2 are P 5 P(S g, R g, E 1 ), P 6 P(S i, R g, V 1 ), P 7 P(S g, R g, V 2 ) and P 8 P(S i, R g, E 2 ) respectively. The spherical wave reflection coefficient Q R is used for P 6 and P 8 as the reflection is assumed to take place at the receiver side but Q s should be used instead if the reflection happens at the source side. Although we can use any of the more accurate diffraction formulas described in section 9.2 to compute P 1 P 8, a simpler approach, following Lam and Roberts [30], is to assume that each diffracted ray has a constant phase shift of π/4 regardless of the position of source, receiver and diffraction point. Thus the empirical formulations described in section 9.3 can also be used to compute the amplitudes of the diffracted rays. Indeed, Figure 9.15 Comparison between laboratory data for the insertion loss of a barrier (0.3 m high and 1.22 m long) on hard ground and predictions using the Lam and Roberts approximate scheme [30]. The source was m high and m from the barrier. The receiver was on the ground and m from the barrier. Muradali and Fyfe [34] compared the use of the Maekawa chart, the Kurze-Anderson empirical equation and the Hadden-Pierce analytical formulation with a wave-based Boundary Element Method (BEM). Predictions of the ray-based approaches show excellent agreement with those of the relatively computationally intensive BEM formulation.

301 Predicting the performace of outdoor noise barriers 287 In Figure 9.15, data obtained by Lam and Roberts (figure 3 in [30]) are compared with predictions using their proposed approximation scheme. In their studies, the measurements were conducted over a hard ground with the barrier height of 0.3 m and length of 1.22 m. The source was located at m from the barrier at the source side and m above the ground. The receiver was situated at m from the barrier at the receiver side with the microphone placed on the ground. Although the data shown in Figure 9.15 are the same as used by Lam and Roberts, the predicted spectrum of the insertion loss is calculated by our own computer program. Predictions based on other numerical schemes detailed in sections 9.2 and 9.3 are not shown because they tend to give similar results to the Lam and Roberts predictions. 9.6 Adverse effect of gaps in barriers When the transmission of sound through the barrier is negligible, the acoustic field in the shadow region is mainly dominated by the sound diffracted around the barrier. However, in some cases, leakage will occur due to shrinkage, splitting and warping of the panels, and weathering of the acoustic seals. The problems of shrinkage and splitting are particularly acute for noise barriers made of timber. Sometimes gaps are unavoidable since spaces are required, for example, for the installation of lamp posts in urban districts. Watts [35] has investigated the resulting sound degradation of screening performance due to such leakage. He used a 2-D numerical model based on the BEM to predict the sound fields behind barriers of various heights with different gap widths and distributions. The A-weighted sound leakage increases as the gap size increases. Also, he reported that (a) sound leakage is more significant in region closed to barrier and (b) there are no significant leakage effects of noise from heavy vehicles. It should be noted that only horizontal gaps can be included in his two-dimensional boundary element model. Here we outline a recent theory developed by Wong and Li [36] and show that their predictions of the effects of barrier leakages agree reasonably well with data from laboratory and outdoor experiments. Consider the barrier diffraction problem shown in Figure Based on the Helmholtz integral formulation, Thomasson [37] derived an approximate scheme for the prediction of the sound field behind an infinitely long barrier. The solution for a thin rigid screen is given by (9.40)

302 Predicting outdoor sound 288 Figure 9.16 The coordinate system used in the Helmholtz integral formulation for a sound diffracted by a rigid infinitely long barrier of height H. where ds is the differential element of the barrier surface, r S, r R and r 0 are, respectively, the position of source, receiver and barrier surface. The symbol signifies the sound field at receiver points (which can either be or ) due to the source located at (which can either be or ). The computation of is straightforward if the Weyl-Van der Pol formula is used (see (2.40a)). The sound field from source to receiver in the absence of a barrier can be determined by noting that the sound field vanishes when the rigid screen, Γ say, occupies the infinite plane of y=0. Then (9.41a) or (9.41b) The sound field behind a rigid screen can be computed by substituting (9.41a) into (9.40) to yield

303 Predicting the performace of outdoor noise barriers 289 (9.42) Although the Thomasson approach (9.42) provides an accurate method for predicting the sound field behind a rigid screen, the use of (9.40) in conjunction with the Weyl-Van der Pol formula is preferable because the area required for integration is generally smaller than that required in (9.42). Hence, less computation is required for the evaluation of the sound fields behind a thin barrier. Since (9.40) is the most general formula for the computation of a sound field behind a thin barrier, it can be generalized to allow any arbitrary gaps in the barrier. Specifically, if there are gaps in the barrier, then their areas should not be included in the computation of the integral. For instance, suppose there is a vertical gap of width 2d in an otherwise infinitely long barrier of height H. Without loss of generality, we assume that the gap extends from y= d to y=d. The sound field behind such a barrier can be computed by (9.43) Note that the term can be regarded as the contribution due to horizontal dipoles on the surface of the barrier. The asymptotic solution for a horizontal dipole above an impedance plane is detailed in Chapter 5. In many practical situations, the thin barrier may be assumed to be infinitely long and to have a constant cross-section along the y-direction. In view of (9.40), the sound field can therefore be expressed as (9.44) where l B is the constant barrier height. It follows immediately that the y-integral of the second term can be evaluated asymptotically to yield a closed-form analytical expression. The evaluation of the integral is fairly straightforward but involves tedious algebraic manipulations. The sound field can be simplified to (9.45)

304 Predicting outdoor sound 290 where d 1, d 2, d 3 and d 4 are defined in Figure 9.17 and X R is the shortest distance measured from the receiver to the barrier plane. The spherical wave reflection coefficients Q s and Q R are calculated on the basis of ground impedance on the source side and receiver side respectively. In our case, the ground impedance is the same for both sides but, as demonstrated by Rasmussen [38], they can be different. The integral limits in (9.45) specify the height of the thin barrier. If there are horizontal gaps in the barrier, the height of the barrier can be adjusted accordingly. Consider the thin barrier of height H with a horizontal gap of size (h 2 h 1 ) shown in Figure The integral term in (9.45) is broken down into two parts where the integral limits for l B range from z=0 to z=h 1 and from z=h 2 to z=h as follows: (9.46) It is straightforward to generalize (9.46) to allow for multiple horizontal gaps in the barrier by splitting the integral into appropriate smaller intervals representing the barrier surface. Figure 9.17 Definition of path lengths and coordinates for evaluating (9.45).

305 Predicting the performace of outdoor noise barriers 291 Figure 9.18 An infinitely long thin barrier with a horizontal gap. It is important to note that the sound field behind a barrier with horizontal gaps can also be calculated by means of a simple ray method, thus eliminating the need for the timeconsuming numerical integration procedure. Based on either the analytical or the empirical formulation of the diffraction theory outlined earlier, we can use either one of the expressions to evaluate the required sound field in the presence of horizontal gaps. Recognizing that P T (H) may be regarded as the asymptotic solution for the right side of (9.45) and combining the two terms in the equation, we can state the integral as (9.47) In the case of a barrier of height H with a horizontal gap starting from z=h 1 extending to z=h 2 (see Figure 9.18), the sound field can be calculated by using (9.46) to yield

306 Predicting outdoor sound 292 (9.48) With the use the above equation, we can compute the sound field behind a barrier that has a horizontal gap as follows: (9.49) The first term of (9.49) corresponds to the sound field diffracted by a thin rigid barrier without any gaps. Grouping the second and third terms, we can interpret (9.49) as the sound field due to the leakage through the barrier gap. That is to say, the leakage of sound through a single gap can be represented by the difference of two barriers with different heights, h 1 and h 2 (9.50) The total sound field behind a barrier with n horizontal gaps can be generalized to give (9.51) where is the sound field behind the thin barrier (with no gaps) and is the leakage of sound through gaps. The leakage of sound at each gap can be calculated by using (9.51) with the appropriate barrier heights.

307 Predicting the performace of outdoor noise barriers 293 Figure 9.19 Comparison of measured (broken line) and predicted transmission loss for a m high barrier with a wide gap at ground level (thick solid line). Also shown is the predicted transmission loss for a barrier of the same dimensions but with no gap (thin solid line). The receiver is located at horizontal distances of (a) 0.5 m, (b) 0.6 m, (c) 0.7 m and (d) 0.8 from the barrier. Figure 9.19 shows data from a laboratory experiment in which the barrier top edge was m from the ground with a horizontal gap of m starting from the ground level [36]. The total area of the gap was approximately 6% of the total area of the barrier surface. The source height was located at 0.07 m above the ground and at 0.49 m from the barrier. The receiver was located at 0.11 m above the ground, and at horizontal distances of (a) 0.5 m, (b) 0.6 m, (c) 0.7 m and (d) 0.8 m from the barrier. Measurements were conducted on a hard ground. Experimental data denoted as Transmission Loss (TL) in Figure 9.19 are presented as the relative sound pressure levels with respect to the reference sound pressure measured at 1 m in free field. Numerical predictions are in general agreement with these experimental results. Hence, it is possible to use the proposed model to assess the degradation of acoustic performance of the barrier due to presence of a horizontal gap. 9.7 The acoustic performance of an absorptive screen The application of sound absorbent materials on barrier surfaces for increasing the insertion loss of a barrier has been a subject of interest in the past few decades. Although

308 Predicting outdoor sound 294 the theory of the sound diffracted by a rigid half plane has been studied with considerable success, there are still uncertainties about the usefulness of covering barriers with sound absorption materials. An early theoretical study [39] has suggested that an absorbent strip of one wavelength width should be adequate to provide a similar insertion loss as the corresponding barrier totally covered by the same sound absorption material. However, this study does not provide the required information about the effect of placing this absorbent strip on either the source side or the receiver side of the barrier. In addition, the experimental and theoretical studies of Isei [40] indicated that the absorptive properties of the barrier surface did not significantly increase the acoustic performance of barriers. However, his conclusion is in contrast with previous experimental and theoretical studies by Fujiwara [41]. In this section, we outline an analytical solution which is based on a heuristic extension of MacDonald s solution. We also give a heuristic solution based on Hadden and Pierce [13], but a corresponding heuristic extension of the line integral approach of Embleton [16] is not pursued here. Fujiwara [41] suggested the use of a complex reflection coefficient, R p into the second term of the MacDonald solution (see (9.9)) to obtain an approximate solution for the sound field behind an absorbent thin screen. Clearly, the diffracted field could be modified to allow for reflection of spherical wave fronts. Hence (9.52) where Q b is the spherical wave reflection coefficient at the barrier façade facing the source. This model takes no account of the impedance on the side of the barrier facing the receiver. It is remarkable that neither of the diffraction models developed by Isei [40] and Fujiwara [41] satisfy the reciprocity condition, that is the predicted sound fields are different if one exchanges the position of source and receiver behind an absorbent barrier with different impedance on the source and receiver sides. To satisfy the reciprocity condition, L Espérance et al. [42] extended the Hadden and Pierce diffraction model to include consideration of the impedance on both sides of the barrier in the model. In the presence of a rigid wedge, the diffracted field is given by (9.16), re-stated here as p d =V 1 +V 2 +V 3 +V 4, (9.53) where V i V(ς i ) for i=1, 2, 3 and 4. The values of V 1, V 2, V 3 and V 4 can be obtained by evaluating the integrals of (9.18c) numerically using the standard Laguerre technique. These four terms can be interpreted physically as the contributions from the diffracted rays travelling from the source to the edge to the receiver, from the image source to the edge to the receiver, from the source to the edge to the image receiver and from the image source to the edge to the image receiver respectively. As a result, the effect of the impedance boundary conditions on both sides of the wedge can be incorporated into the Hadden and Pierce formulation as p d =V 1 +Q b V 2 +Q f V 3 +Q b Q f V 4 (9.54)

309 Predicting the performace of outdoor noise barriers 295 where Q b Q(r 0 +r r, β b, π/2 θ 0 ) and Q f =Q(r 0 +r r, β f, π/2 θ r ) are, respectively, the spherical wave reflection coefficients of the barrier façades facing the source and receiver. The corresponding parameters β b and β f are the effective admittances of the source and receiver sides of the wedge. Using the formula specified in section 9.4, we can determine the insertion of loss of an absorbent barrier resting on an impedance ground. The implementation is straightforward and the details will not be given here. Figure 9.20 shows the comparison between laboratory data obtained by L Espérance et al. [scaled from figure 4(b) of ref. 43] with predictions of the heuristic modification of the Hadden and Pierce model. In the experiments, the height of the barrier was m, the source was placed at 1.2 m from the barrier and at m above the ground. The receiver was located at 0.6 m from the barrier and m above the ground. A 0.04 m thick fibreglass layer was used to cover either the source side or both sides of the barrier surface. In Figure 9.20, the data corresponding to these two cases are shown as the open circles and open triangles respectively. Also shown are the measured results for a rigid Figure 9.20 Comparison of laboratory data [43] with the predictions of a heuristic modification of the Hadden-Pierce model [13] for an absorptive screen. The solid circles and solid line represent theoretical and experimental results, respectively, for the hard barrier. The open circles and short-dashed line represent theoretical and experimental results, respectively, when the surface of the barrier facing the source is covered with fiberglass. The open triangles and long-dashed line represent theoretical and experimental results, respectively, when both surfaces of the barrier are covered with fiberglass.

310 Predicting outdoor sound 296 barrier (small solid circles). The impedance of the fibreglass layer was predicted by using the Delaney and Bazley one-parameter model (3.2) and assuming an effective flow resistivity of 40 kpa s m 2. Good agreement between the predicted and measured insertion losses is evident. In this way, L Espérance et al. have shown that the application of sound absorption materials on the barrier façade can lead to a significant improvement in the barrier s performance. However, Hayek [44] has suggested that barrier absorption is less effective in reducing the diffracted noise in the shadow zone where ground absorption is the most dominant factor in determining the acoustic performance of the barrier. It remains important to investigate the effectiveness of using absorptive screens in protecting receivers from excess noise particularly for the case in urban environments where multiple reflections and scattering between surfaces are prevalent. 9.8 Other factors in barrier performance Although there are many other factors that may affect the barrier effectiveness in shielding receivers from excessive noise, in this section we shall address the two issues concerning meteorological effects and the influence of barrier shape that are of current research interest and give a brief review of the salient points Meteorological effects In the barrier diffraction models described earlier, atmospheric conditions have been assumed to be steady and uniform. However, meteorological effects, such as temperature and wind velocity gradients, play a very important role in determining the acoustic performance of barriers. As a result of atmospheric refraction, barrier effectiveness is expected to be reduced in the downwind direction but enhanced in the upwind direction. By means of full-scale measurements conducted in the 1970s, Scholes et al. [45] demonstrated such wind effects on the acoustic performance of a noise barrier. Salomons [46] was among the first to study the effect of an absorbing barrier in a refracting medium by using a Parabolic Equation (PE) formulation. He assumed an absorbing barrier in his PE model and set the pressure at the grid points representing the barrier to zero. Thus, the part of the sound that falls on the barrier is stopped. A minor conflict appears in his method. The enforced zero pressure on grid implies that the normal velocity on barrier is zero too. This is the boundary condition for rigid surface. On the other hand the PE, which assumes only forward propagation from source to receiver, does not allow any backscattering of waves by the barrier. Thus, the reflection from the barrier is ignored when computing the total diffracted field. Rasmussen [47] studied the diffraction of barrier over an absorbing ground and in an upwind condition both theoretically and experimentally. He used an approach rather similar to the fast field formulation to estimate effective distances from source to the barrier edge and from the barrier edge to receiver in an upwind condition. Then the uniform theory of diffraction [48] was used to compute the diffracted sound field in the shadow zone. It is questionable whether the barrier diffraction can be de-coupled from the wind gradient effects in this way. Nevertheless, Rasmussen s numerical results agreed reasonably well with his 1:25 scale

311 Predicting the performace of outdoor noise barriers 297 model experiments. Muradali and Fyfe [49] have proposed a heuristic method for modelling barrier performance in the presence of a linear sound speed profile. Their model combines the effects of the sound refracted over a barrier and the sound diffracted by a barrier. Geometrical ray theory is used to determine the ray paths and the diffracted wave effect is incorporated by using the modified Hadden-Pierce formula. Li and Wang [50] have proposed a novel method to simulate the effects of a noise barrier in a refracting atmosphere. They mapped the wave equation in a medium where the speed of sound varies exponentially with height above a flat boundary to that for a neutral fluid above a curved surface. The barrier problem in an upward-refracting medium is then solved by applying the standard Boundary Element Method over a curved surface in a homogeneous medium. Recently, Taherzadeh et al. [51] developed a hybrid scheme for predicting barrier in a downwind condition. Their numerical scheme is based on a combination of the boundary integral and FFP techniques in which the Green s function required in the boundary integral equations is evaluated by using the FFP formulation. Atmospheric turbulence is an important factor in barrier performance since it causes scattering of sound into the shadow zone. Although experimental studies of the effects of atmospheric turbulence on the acoustic performance of the barrier are rare, theoretical studies have been conducted by Daigle [52], Forssén and Ogren [53] and others quoted in these references. Some progress in modelling the acoustic performance of barriers in a turbulent atmosphere using a substitute sources method has been reported by Forssén [54, 55]. Hybrid models have been proposed, involving combinations of the BEM with the PE, that enable effects of barriers, range-dependent impedance and turbulence to be taken into account [56, 57]. Rasmussen and Arranz [58] have applied the Gilbert et al. [59] approach, assuming weak atmospheric turbulence with a simple Gaussian distribution in the fluctuating velocity field, in a PE formulation to predict the sound field behind a thin screen in a scale model experiment. The indoor experiments based upon a 1:25 scaling ratio were conducted in a wind tunnel in which the agreement of numerical results and experimental measurements was very good. They concluded that the flow pattern associated with a specific noise barrier could be an important design parameter for improving the acoustic performance of barriers. Improvements in computational speeds and resources have encouraged wider use of time domain models in acoustics. Recent work has been devoted to combining calculations employing time-dependent computational fluid dynamics with PE-based frequency domain calculations [60]. However, there are few studies to measure or predict the velocity profiles and their associated turbulence accurately in the region close to a barrier on the ground and it remains a challenge to combine fluid dynamics and diffraction models to establish an accurate scheme for predicting the acoustic effectiveness of screens under the influence of wind Effects of barrier shape Most of the theoretical and experimental studies found in literature are focused on a straight-edge barrier or a wedge. In fact, the shapes of barrier edges can affect their shielding efficiencies significantly. Variations in the barrier shapes may be used to improve the shielding performance without increasing the height of barriers.

312 Predicting outdoor sound (a) Ragged-edge barrier Wirts [61] introduced a type of barrier with non-straight top edge called Thnadners. The saw-tooth Thnadners barrier consists of a number of large triangular panels placed side by side. Flattop Thnadners are similar to sawtooth Thnadners but each of the sharp corners at the saw-tooth top is flattened. Wirts suggested using the simplified Fresnel theory to calculate the sound fields behind Thnadners barriers. Results of his scale model experiments agreed with the proposed theory. The most interesting conclusions of his study are additional improvements on the shielding performance if some parts of the barrier are removed. Improvements are obtained at some receiver locations even if over 50% of the barrier surface areas are removed. Ho et al. [62] conducted many scale model experiments and concluded that the acoustic performance of a barrier can be improved at high frequencies by introducing a random fluctuation to the height of the barrier, producing the so-called ragged-edge barrier. However, the poorer performance of the raggededge barrier at low frequencies was not explained. An empirical formula for the insertion loss of the ragged-edge barrier was obtained from the experimental results. The formula involves four parameters which are the frequency of the incident sound, the Fresnel number, the maximum height fluctuation from the average barrier height and the horizontal spacing between the height fluctuations. Shao et al. [63] have proposed a 2-D model for a random edge barrier. Their models were based on the Rubinowics-Young formula which is a line integral method. The integration is performed along the random edge. They also conducted scale model experiments to validate their theoretical models. They concluded that the improvement could be obtained by introducing a random edge profile, especially at high frequencies. The insertion loss is controlled by the parameter of average barrier height and the variations of the randomness. In some cases, an improvement in the insertion loss can be obtained if the average height of a random edge barrier is smaller than the original straight-edge barrier (b) Multiple-edge barriers After reviewing relevant previous experimental studies by May and Osman [64] and Hutchins et al. [65], Hothersall et al. [66] have used the BEM approach to study T-profile, Y-profile and arrow-profile barriers theoretically (see Figure 9.21). They concluded from their numerical simulation results and previous work that a multiple-edge profile barrier has better acoustical performance than a normal thin plane barrier of the same height. However, in most conditions, the Y-profile and arrow-profile barriers perform less efficiently than the T-profile barrier. The use of T-profile barrier can benefit

313 Predicting the performace of outdoor noise barriers 299 Figure 9.21 T-profile, Y-profile, arrow-profile and cranked (top-bended) barriers. regions close to the barrier and close to ground. On the other hand, the height of the barrier is a dominant factor for the receiver located far away from the barrier and the top profile plays a less important role. The application of absorbent materials on the cap of T- profile has also been studied. A T-profile barrier with an absorptive cap can increase the insertion loss as long as the thickness of absorbent materials is thin. An excessively thick layer of sound absorbing materials can lead to a less efficient shielding. At present, only numerical tools, such as BEM, are used for study of the acoustic performance of the T-profile and multiple-edge barriers. Consequently, most of the theoretical studies are restricted to two dimensions. Another type of multiple-edge barrier is the cranked or top-bended barrier (see Figure 9.21) which is used increasingly in high-rise cities. Jin et al. [67] have proposed an analytical method for the prediction of sound behind a top-bended barrier. They used a theoretical model based on the uniform geometrical theory of diffraction [48]. The total diffracted sound is calculated by summing multiple diffractions occurring at the top and corner points, at which convex and concave edges are formed. They concluded that the third and higher orders of multiple diffraction did not contribute significantly to the total diffracted sound and so these higher-order diffracted terms were ignored in their model. They compared the predicted insertion loss spectra between finite and infinite length barriers but some ripples caused by edge effects are evident in their predictions for the finite length barrier. Jin et al. used a commercially available BEM programme to compare with their analytical solutions for the 2-D case and obtained good agreement. Okubo and Fujiwara [68] have suggested another design in which an acoustically soft waterwheel-shaped cylinder is placed on the top of a barrier. They studied its effects by using a 2-D BEM approach and conducted 3-D scale model experiments. They concluded that the insertion loss of barrier is improved by the cylinder but the effect is strongly dependent on the source frequency. Since the depth and opening angle of each channel influence the centre frequency, a suitable variation of the channel depth can flatten the frequency dependency and improve the overall performance of the A-weighted noise levels behind the barrier (c) Other barrier types Wassilieff [69] has introduced the concept of picket barrier suggesting that traditional Fresnel-Kirchhoff diffraction is not applicable to a picket barrier at low-frequencies

314 Predicting outdoor sound 300 because the wavelengths of the low-frequency sound are comparatively larger than the widths of the vertical gaps in the picket barrier. He added terms for a mass-layer effect to the Fresnel-Kirchhoff diffraction theory to increase the accuracy of the model at low frequencies. Nevertheless, the original Fresnel-Kirchhoff diffraction theory gives better accuracy at the high-frequency region. He concluded that a careful adjustment of the picket gap can result in an improvement at certain low frequencies compared with a solid barrier. However, the overall acoustic performance of barriers is reduced at other frequencies. Lyons and Gibbs [70] have studied the acoustic performance of open screen barriers, mainly the performance of louvre-type barrier at low frequency. Their model barrier consisted of vertical pickets, having a sound absorbing surface on one side, which were in two offset rows. The cavity left between two offset pickets is designed for better ventilation. They concluded that the side with the open area has less influence on transmission loss at low-frequency sound. They also derived an empirical formula for the calculation of transmission loss of his louvre-type barrier with the dependent parameters being the overlapped area and the cavity volume between two offset rows of pickets. They found that the leakage loss through two offset rows of pickets is relatively small at low frequency. As a refinement of this study, Viverios et al. [71] have developed a numerical model that was based on the Fresnel-Kirchoff approximate scheme to estimate the transmission loss of a louvre barrier. After incorporating the amplitude and phase changes due to the absorbing materials within the louvre bladders, their model agreed reasonably well with measurements. Watts et al. [72] have used the BEM formulation to study the acoustic performance of louvred noise barriers. Nevertheless, it seems that more theoretical and experimental studies are required to develop a simple scheme for predicting the sound field behind open screen barriers. A comprehensive summary of the efficiency of noise barriers of different shapes can be found in the paper by Ishizuka and Fujiwara [73]. The authors used the boundary element method and standard traffic noise spectrum to determine the broadband efficiency of several popular noise barrier shapes with respect to that predicted for an equivalent 3 m high plain noise screen. Their work shows that a complex noise barrier edge, for example a Y-shape barrier or a barrier with a cylindrical edge, can yield a 4 5 db improvement in the insertion loss. This is important when the increase in the barrier height is not practical or impossible. Figure 9.22 reproduced from [73] illustrates the change in the mean insertion loss relative to a standard 3 m high, plane screen noise barrier. If a source of noise is close to the barrier, then the positive effect of the complex barrier shape can become small in comparison with the effect of absorbing barrier treatment. This phenomenon has been investigated, experimentally and numerically, by Hothersall et al. [74] who studied the performance of railway noise barriers. Reflecting

315 Predicting the performace of outdoor noise barriers 301 Figure 9.22 Predicted acoustic performance of traffic noise barriers of various shapes assuming a standard traffic noise spectrum. The figures indicate the relative change in the mean insertion loss relative to a 3 m plane screen [73]. Reprinted with permission from Elsevier. and absorbing railway noise barriers of different shapes were positioned as close as possible to the train structure, within the limitations of the structure gauge (see Figure 9.23). The barrier shapes included a plain screen (a), a cranked screen (b), a parabolic screen (c), a multiple-edge noise screen (d) and a corrugated barrier (e). It was found that the insertion loss values for a rigid screen were between 6 and 10 db lower than those for a similar screen with absorbing treatment.

316 Predicting outdoor sound 302 Figure 9.23 Configuration of the model and barrier cross-sections in a comparative study [74]. In a particular case, where a 1.5 m high plain screen was installed 1.0 m away from a passing railway carriage, the maximum predicted improvement in the average insertion loss due to the absorbing barrier treatment was 14 db. On the other hand, any of the

317 Predicting the performace of outdoor noise barriers 303 modifications in the barrier shape, resulted in a less than 4 db gain in the insertion loss compared with that of the basic plain screen of equivalent height [74]. 9.9 Predicted effects of spectral variations in train noise during pass-by Models for predicting the effects of noise control treatments rely heavily on the knowledge of the reference noise source spectrum. In road traffic noise studies, a standardized spectrum is used widely to account for the non-linearity in the emitted acoustic spectrum [75]. No similar standard exists at present to regulate the noise spectrum emitted by high-speed trains. Unlike noise from motorway traffic, which is a relatively stationary process, the noise spectrum of a train can vary significantly during its pass-by. The results of the work by Horoshenkov et al. [76] suggest that during the pass-by of a high-speed intercity train the variation of the sound pressure level (SPL), in the frequency range between 500 and 5000 Hz, can be as high as 20 db sec 1. It is common in many practical calculations to use the average spectrum for the whole passby. Indeed, several railway noise spectra have been compiled for such use [77, p. 175]. However, a likely drawback in adopting an average spectrum is the failure to model the distinctive time-dependent performance of noise control elements (e.g. noise barriers) during the pass-by of a train. In practice, the engineering measure of the in situ efficiency of a railway noise barrier, the broadband insertion loss, can vary considerably during the pass-by so that the perceived (subjective) acoustic efficiency of a noise control element is somewhat reduced. In this sense, one can argue that the noticeable fluctuations in the insertion loss during the train pass-by are subjectively more significant than the limited average barrier performance. This phenomenon may need to be given a careful consideration when the efficiency of expensive railway noise barrier schemes is assessed. Here we demonstrate that the variation in the reference noise source spectrum can result in a perceivable fluctuation of the noise efficiency of some noise barrier designs. Figures 9.24 and 9.25 present typical narrow-band spectrograms measured for the passage of the Intercity 125 (diesel) and Intercity 225 (electric) trains respectively. The distances on the vertical axes in these Figures correspond to the time-dependent separation between the fixed receiver position on the ground and the varying position of the front locomotive in the train. We have used the measured narrow-band spectrograms to determine the 1/3-octave band reference, time-dependent noise spectra which are required to model numerically the time-dependent insertion loss of three different barrier shapes: a plane screen, a screen with a quadrant of cavities at its upper edge ( spiky top ) and a T-shape noise barrier. The spiky top barrier has been designed to achieve its maximum efficiency

318 Predicting outdoor sound 304 Figure 9.24 The spectrogram of noise emitted by a passing Intercity 125 (diesel) travelling at 184 km h 1 Figure 9.25 The spectrogram of noise emitted by a passing Intercity 225 (electric) travelling at 176 km h 1. around Hz, that is in the part of the spectrum where the maximum noise emission occurs (see Figures 9.24 and 9.25). The supposed noise control configurations are shown in Figure 9.26(a), (b) and (c). Two sources of noise are elevated 0.3 m above

319 Predicting the performace of outdoor noise barriers 305 the ballast and their positions coincided with the areas of rail-to-wheel interactions (see Figure 9.26). Figure 9.26 Schematic representation of the terrain, train body and the noise barrier profiles assumed for calculations with the BEM model: (a) plane screen, (b) spiky top and (c) T- shape. Predictions have been made using the incoherent line source boundary element model proposed by Duhamel [78]. The model requires the discretization of the surfaces in two dimensions only since it assumes that the barrier and the source are infinitely long. The pseudo-3-d pressure field is calculated from (9.55) where is the attenuation and p 2D (x, y, k) is the 2-D, frequencydependent acoustic field predicted, for example, using the boundary integral equation

320 Predicting outdoor sound 306 method detailed in Chapter 8 (8.2.4). In ideal circumstances, a full 3-D model would be required to predict the time-dependent insertion loss of a finite noise barrier length for a moving train of finite dimensions. A practical realization of such a model is beyond the scope of many numerical methods and, certainly, beyond the power of modern PCs. Using the measured time-dependent spectral data and predicted values of the insertion loss, the broadband time-dependent noise barrier insertion loss is calculated using (9.56) where L 0 (f n, t) is experimentally determined time-dependent, 1/3-octave railway noise spectra, L B (f n, t)=l 0 (f n, t) IL P (f n ) and IL P (f n ) is the predicted 1/3-octave insertion loss. The values of f n are the standard 1/3-octave band frequencies defined in the range between 63 and 3150 Hz. The resulting predictions of the time-dependent insertion loss are shown in Figures 9.27(a) and (b) for the diesel and electric trains respectively. The noise barrier surface and the body of the train have been assumed to be rigid so that multiple reflections are included. The acoustic impedances of the porous ballast Figure 9.27 Predicted time-dependent, broad band insertion losses for three barrier designs: (a) Intercity 125 (diesel) and (b) Intercity 225 (electric).

321 Predicting the performace of outdoor noise barriers 307 and the porous grassland have been modelled using the four-parameter model and the parameters listed in Table 5.1. The results for the Intercity 125 train demonstrate that the performance of the plane screen barrier is stable throughout the entire passage of the train. The average performance of the T-shape barrier is superior to that of the plane screen; however its dependence upon the instantaneous spectrum of the emitted noise is more pronounced. Although the average performance of the spiky top barrier is similar or lower to that of the plane screen during the pass-by, the variation in the performance of this barrier during the pass-by is considerable. With the Intercity 225 train pass-by spectral variation, the insertion loss for the plane screen is inferior, but remarkably stable in comparison with the more complex barrier shapes. The fluctuation in the insertion loss of the spiky top barrier is between 5 and 7 db with a maximum fluctuation around the time of the train arrival and departure. This level of fluctuations might be perceived to be more significant than the predicted gain of between 1 and 2 db in the average barrier performance. These effects may need to be given careful consideration when the efficiency of expensive railway noise barriers is assessed. References 1 A.Sommerfeld, Mathematische Theorie der Diffraktion, Math. Ann., 47: (1896). 2 M.Born and E.Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light, Pergamon Press, Cambridge, UK (1975). 3 H.S.Carslaw, Some multiform solutions of the partial differential equations of physical mathematics and their applications, P. Lond. Math. Soc., 30: (1899). 4 H.S.Carslaw, Diffraction of waves by a wedge of any angle, P. Lond. Math. Soc., 18: (1920). 5 H.M.MacDonald, A class of diffraction problems, P. Lond. Math. Soc., 14: (1915). 6 E.T.Copson, On an integral equation arising in the theory of diffraction, Q.J. Math., 17:19 34 (1946). 7 H.Levine and J.Schwinger, (a) On the theory of diffraction by an aperture in an infinite plane screen. J, Phys. Rev., 74: (1948) (b) On the theory of diffraction by an aperture in an infinite plane screen. II, Phys. Rev., 75: (1949). 8 D.G.Crighton, A.P.Dowling, J.E.Ffowcs Williams, M.Heckl and F.G.Leppington, Modern Methods in Analytical Acoustics Lecture Notes, Springer-Verlag, London, Ch. 5 (1996). 9 M.C. M.Wright (ed.), Lecture Notes on the Mathematics of Acoustics, Part II, Ch. 5, Imperial College Press, London (2005). 10. I.Tolstoy, Diffraction by a hard truncated wedge and a strip, IEEE J. Ocean Eng., 14:4 16(1989). 11 I.Tolstoy, Exact, explicit solutions for diffraction by hard sound barriers and seamounts, J. Acoust. Soc. Am., 85: (1989). 12 J.J.Bowman and T.B.A.Senior, Electromagnetic and Acoustic Scattering by Simple Shapes, edited by J.J.Bowman, T.B.A.Senior and P.L.E.Uslenghi, North-Holland, Amsterdam (1969) (revised edition, Hemisphere Publishing Corp., New York, 1987). 13 W.J.Hadden and A.D.Pierce, Sound diffraction around screens and wedges for arbitrary pointsource locations, J. Acoust. Soc. Am., 69(5): (1981). See also the Erratum, J. Acoust. Soc. Am., 71:1290 (1982). 14 E.Hecht, Optics, Addison-Wesley Publishing Company, Wokingham, fifth edition (1998). 15 E.J.Skudrzyk, The Foundations of Acoustics, Springer-Verlag, New York (1971).

322 Predicting outdoor sound T.F.W.Embleton, Line integral theory of barrier attenuation in the presence of the ground, J. Acoust. Soc. Am., 67:42 45 (1980). 17 G.M.Jebsen and H.Medwin, On the failure of the Kirchhoff assumption in backscatter, J. Acoust. Soc. Am., 72: (1982). 18 S.W.Redfearn, Some acoustical source-observer problems, Philos. Mag., 30: (1940). 19 Z.Maekawa, Noise reduction by screens, Appl. Acoust., 1: (1968). 20 E.J.Rathe, Note on two common problems of sound propagation, J. Sound Vib., 10: (1969). 21 U.J.Kurze and G.S.Anderson, Sound attenuation by barriers, Appl. Acoust., 4:35 53 (1971). 22 U.J.Kurze, Noise reduction by barriers, J. Acoust. Soc. Am., 55: (1974). 23 J.B.Keller, The geometrical theory of diffraction, J. Opt. Soc. Am., 52: (1962). 24 R.B.Tatge, Barrier-wall attenuation with a finite sized source, J. Acoust. Soc. Am., 53: (1973). 25 K.Yamamoto and K.Takagi, Expressions of Maekawa s chart for computation, Appl. Acoust., 37:75 82 (1992). 26 P.Menounou, A correction to Maekawa s curve for the insertion loss behind barriers, J. Acoust. Soc. Am., 110: (2001). 27 T.Isei, T.F.W.Embleton and J.E.Piercy, Noise reduction by barriers on finite impedance ground, J. Acoust. Soc. Am., 67:46 58 (1980). 28 J.Nicolas, T.F.W.Embleton and J.E.Piercy, Precise model measurements versus theoretical prediction of barrier insertion loss in presence of the ground, J. Acoust. Soc. Am., 73:44 54 (1983). 29 E.M.Salomons, Reduction of the performance of a noise screen due to screen-induced windspeed gradients. Numerical computations and wind tunnel experiments, J. Acoust. Soc. Am., 105: (1999). 30 Y.W.Lam and S.C.Roberts, A simple method for accurate prediction of finite barrier insertion loss, J. Acoust. Soc. Am., 93: (1993). 31 A.L Espérance, The insertion loss of finite length barriers on the ground, J. Acoust. Soc. Am., 86: (1989). 32 R.Pirinchieva, Model study of the sound propagation behind barriers of finite length, J. Acoust. Soc. Am., 87: (1990). 33 H.Medwin, Shadowing by finite noise barriers, J. Acoust. Soc. Am., 69: (1981). 34 A.Muradali and K.R.Fyfe, A study of 2D and 3D barrier insertion loss using improved diffraction-based methods, Appl. Acoust., 53:49 75 (1998). 35 G.R.Watts, Effects of sound leakage through noise barriers on screening performance, 6th International Congress on Sound and Vibration, Copenhagen, Denmark, (1999). 36 H.Y.Wong and K.M.Li, Prediction models for sound leakage through noise barriers, J. Acoust. Soc. Am., 109: (2001). 37 S.I.Thomasson, Diffraction by a screen above an impedance boundary, J. Acoust. Soc. Am., 63: (1978). 38 K.B.Rasmussen, A note on the calculation of sound propagation over impedance jumps and screens, J. Sound Vib., 84(4): (1982). 39 A.D.Rawlins, Diffraction of sound by a rigid screen with an absorbent edge, J. Sound Vib., 47: (1976). 40 T.Isei, Absorptive noise barrier on finite impedance ground, J. Acoust. Soc. Jpn. (E), 1:3 9 (1980). 41 K.Fujiwara, Noise control by barriers, Appl Acoust., 10(2): (1977). See also part II of this paper K.Fujiwara, Y.Ando and Z.Maekawa, Noise control by barriers part 2: noise reduction by an absorptive barrier, Appl. Acoust., 10(3): (1977). 42 A.L Espérance, J.Nicolas and G.A.Daigle, Insertion loss of absorbent barriers on ground, J. Acoust. Soc. Am., 86: (1989).

323 Predicting the performace of outdoor noise barriers V.E.Ostashev and G.H.Goedecke, Interference of direct and ground-reflected waves in a turbulent atmosphere, Proceedings of the eigth LRSPS, Penn State, (1998). 44 S.I.Hayek, Mathematical modeling of absorbent highway noise barriers, Appl. Acoust., 31: (1990). 45 W.E.Scholes, A.C.Salvidge and J.W.Sargent, Field performance of a noise barrier, J. Sound Vib., 16: (1971). 46 E.M.Salomons, Diffraction by a screen in downwind sound propagation: a parabolicequation approach, J. Acoust. Soc. Am., 95: (1994). 47 K.B.Rasmussen, Sound propagation over screened ground under upwind conditions, J. Acoust. Soc. Am., 100: (1996). 48 R.G.Kouyoumjian and P.H.Pathak, A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface, Proc. IEEE (Communications and radar), J. Sound Vib., 62: (1974). 49 A.Muradali and K.R.Fyfe, Accurate barrier modeling in the presence of atmosphere effects, Appl Acoust., 56: (1999). 50 K.M.Li and Q.Wang, A BEM approach to assess the acoustic performance of noise barriers in the refracting atmosphere, J. Sound Vib., 211: (1998). 51 S.Taherzadeh, K.M.Li and K.Attenborough, A hybrid BIE/FFP scheme for predicting barrier efficiency outdoors, J. Acoust. Soc. Am., 110: (2001). 52 G.A.Daigle, Diffraction of sound by a noise barrier in the presence of atmospheric turbulence, J. Acoust. Soc. Am., 71: (1982). 53 J.Forssen and M.Ogren, Thick barrier noise-reduction in the presence of atmospheric turbulence: measurements and numerical modeling, Appl. Acoust., 63: (2002). 54 J.Forssén, Calculation of noise barrier performance in a turbulent atmosphere by using substitute-sources above the barrier, Acust. Acta Acust., 86: (2000). 55 J.Forssén, Calculation of noise barrier performance in a three-dimensional turbulent atmosphere using the substitute-sources method, Acust. Acta Acust., 88: (2002). 56 E.Premat and Y.Gabillet, A new boundary-element method for predicting outdoor sound propagation and application to the case of a sound barrier in the presence of downward refraction, J. Acoust. Soc. Am., 108(6): (2000). 57 Y.W.Lam, A boundary element method for the calculation of noise barrier insertion loss in the presence of atmospheric turbulence, Appl Acoust. 65(6): (2004). 58 K.E.Gilbert, R.Raspet and X.Di, Calculation of turbulence effects in an upward refracting atmosphere, J. Acoust. Soc. Am., 87: (1990). 59 K.B.Rasmussen and M.G.Arranz, The insertion loss of screens under the influence of wind, J. Acoust. Soc. Am., 104(5): (1998). 60 T.Van Renterghem, E.M.Salomons and D.Botteldooren, Efficient FDTD-PE model for sound propagation in situations with complex obstacles and wind profiles, Acust. Acta Acust. (in press). 61 L.S.Wirt, The control of diffracted sound by means of thnadners (shaped noise barriers), Acustica, 42:73 88 (1979). 62 S.S.T.Ho, I.J.Busch-Vishniac and D.T.Blackstock, Noise reduction by a barrier having a random edge profile, J. Acoust. Soc. Am., 101: (1997). 63 W.Shao, H.P.Lee and S.P.Lim, Performance of noise barriers with random edge profiles, Appl. Acoust., 62: (2001). 64 D.N.May and M.M.Osman, (a) The performance of sound absorptive, reflective, and T-profile noise barriers in Toronto, J. Sound Vib., 71:65 71 (1980). (b) Highway noise barriers new shapes, J. Sound Vib., 71: (1980). 65 D.A.Hutchins, H.W.Jones and L.T.Russell, (a) Model studies of barrier performance in the presence of ground surfaces. Part I thin, perfectly reflecting barriers, J. Acoust. Soc. Am., 75: (1984). (b) Model studies of barrier performance in the presence of ground surfaces. Part II different shapes, J. Acoust. Soc. Am., 75: (1984).

324 Predicting outdoor sound D.C.Hothersall, D.H.Crombie and S.N.Chandler-Wilde, The performance of T-profile and associated noise barriers, Appl. Acoust., 32: (1991). 67 B.J.Jin, H.S.Kim, H.J.Kang and J.S.Kim, Sound diffraction by a partially inclined noise barrier, Appl. Acoust., 62: (2001). 68 T.Okubo and K.Fujiwara, Efficiency of a noise barrier on the ground with an acoustically soft cylindrical edge, J. Sound Vib., 216: (1998). See also T.Okubo and K.Fujiwara, Efficiency of a noise barrier with an acoustically soft cylindrical edge for practical use, J. Acoust. Soc. Am., 105: (1999). 69 C.Wassilieff, Improving the noise reduction of picket barriers, J. Acoust. Soc. Am., 84: (1988). 70 R.Lyons and B.M.Gibbs, Investigation of open screen acoustic performance, Appl. Acoust., 49(3): (1996). 71 E.B.Viveiros, B.M.Gibbs and S.N.Y.Gerges, Measurement of sound insulation of acoustic louvers by an impulse method, Appl. Acoust., 63: (2002). 72 G.R.Watts, D.C.Hothersall and K.V.Horoshenkov, Measured and predicted acoustic performance of vertically louvred noise barriers, Appl. Acoust., 632: (2001). 73 T.Ishizuka and K.Fujiwara, Performance of noise barriers with variable edge shape and acoustical conditions, Appl. Acoust., 65: (2004). Reprinted with permission from Elsevier. 74 P.A.Morgan, D.C.Hothersall and S.N.Chandler-Wilde, Influence of shape and absorbing surface a numerical study of railway noise barriers, J. Sound Vib., 217: (1998). 75 British Standard BS EN , Road traffic noise reducing devices test methods for determining the acoustic performance. Part 3 normalized traffic noise spectrum (1998). 76 K.V.Horoshenkov, S.Rehman and S.J.Martin, The influence of the noise spectra on the predicted performance of railway noise barriers, CD-ROM Proceedings of the 9th International Congress on Sound and Vibration, Orlando, USA, July (2002). 77 P.A.Morgan, Boundary element modelling and full scale measurement of the acoustic performance of outdoor noise barriers, PhD Thesis, University of Bradford, November (1999). 78 D.Duhamel, Efficient calculation of the three-dimensional sound pressure field around a noise barrier, J. Sound Vib., 197(5): (1996).

325 Chapter 10 Predicting effects of vegetation, trees and turbulence 10.1 Effects of vegetation and crops on excess attenuation spectra The only published information on the effect of crops on sound propagation is that collected by Aylor [1] over maize and hemlock. In discussing his data, Aylor concentrated on the extra attenuation of sound at higher frequencies associated with the presence of crops, rather than on their influence on ground effect. In section 10.2 we discuss these data for higher frequency (>1 khz) attenuation of sound through crops. In this section we demonstrate, on the basis of measurements at fairly short ranges, that the presence of plants appears to disturb the ground effect (<1 khz). First we consider the observed effects on short-range measurements of excess attenuation using a source-receiver geometry that might be used for ground impedance deduction (see Chapter 4). Compare the excess attenuation spectra measured with source and receiver at 0.1 m height and separated by 1 m over grassland and a cultivated surface shown in Figures 10.1 and The grass site was part of a well-kept lawn-like plot. The cultivated area was a seedbed (evesham series clay) with a fairly rough surface containing sparse emerging bean plant seedlings. The spectra in Figure 10.1(b) appear to be more ragged than those in Figure 10.1(a). In this case the surface roughness might be contributing to this raggedness as well as the plants. Figure 10.3 shows short-range measurements using the same geometry in a wheat field. The wheat plants were 0.55 m high so that source and receiver were below the plant tops. The presence of crops appears to influence the coherence of the ground-reflected sound so that the relatively clean destructive interference that is evident, for example in the relative SPL spectrum above grass (Figure 10.1(a)) is disrupted as well as being altered in magnitude and frequency. Next we consider such effects at a little longer range. Figure 10.3 shows relative sound level spectra deduced from measurements at 1, 10 and 20 m range over medium clay soil containing oil seed rape plants which were approximately 1 m high and had recently flowered. The centre of the Electro-Voice loudspeaker source was at 1.65 m height. At 10 and 20 m range, the microphones were placed at 1.2 m height. The excess attenuation at

326 Predicting outdoor sound 312 Figure 10.1 Excess attenuation spectra obtained with (point) source and receiver at 0.1 m height and separated by 1 m (a) over established grass and (b) over a seed bed containing bean plants. Each graph shows the results of two consecutive measurements. Figure 10.2 Excess attenuation spectra obtained from two consecutive measurements with (point) source and receiver at 0.1 m height and separated by 1 m in 1 m high wheat. 10 and 20 m has been deduced from the level measured at 1 m range (source and receiver height=1.65 m) after allowing for spherical spreading only. Using a hand-held anemometer, it was observed that the wind speed was <2 m/s during the measurements. This implies that the effects of wind gradients and of any change in wind gradients between measurements can be ignored at the relatively small ranges of interest.

327 Predicting effects of vegetation, trees and turbulence 313 The variation in the data below 200 Hz is spurious and probably caused by high background noise. The measured main ground effect dip (excess attenuation maximum) occurs between 200 and 600 Hz for both ranges, being slightly greater at 10 m. The Figure 10.3 Measured sound levels relative to that at 1 m obtained at 10 m (solid line) and 20 m (broken line) from a loudspeaker source over 1 m high oil-seed rape plants at the Silsoe Research Institute, Beds. Figure 10.4 Predicted excess attenuation spectra at 10 m (solid line) and 20 m (broken line) over a surface with impedance given by the values shown in Figure 10.5.

328 Predicting outdoor sound 314 measured excess attenuation at frequencies above 1500 Hz at 20 m tends to be greater than that at 10 m. Figure 10.4 shows predictions of the excess attenuation (excluding turbulence and atmospheric absorption) at 10 and 20 m over a surface with the average impedance shown in Figure 10.5 obtained from data taken at the edge of the field where there were no plants. Referring to Figure 10.4, first it should be noted that below 200 Hz, the 20 m data is unreliable because of poor signal-to-noise ratio. However between 200 and 600 Hz, Figure 10.5 Averaged relative impedance data deduced from complex excess attenuation measurements made at edge of oil seed rape field on 11 th July The point source and receiver were separated by a horizontal distance of 1 m. Source and receiver heights were respectively 0.2 m and 0.1 m. There were no oil seed rape plants in the area of the measurements. the measured data show that an increase in range from 10 to 20 m results in a slight reduction of the amplitude of the main ground effect dip (excess attenuation maximum) whereas the predictions obtained with the measured (bare soil) impedance show a distinctly contrary effect. Since turbulence has relatively little effect below 500 Hz, at these short ranges, it is likely that scattering by vegetation is the cause of the measured decrease in the main dip. The measured data at 20 m departs from the predicted behaviour also above 2000 Hz. At these frequencies turbulence and air absorption may be factors. However, another possibility is the attenuation caused by the plant foliage. This will be discussed in the next section.

329 Predicting effects of vegetation, trees and turbulence Propagation through trees and tall vegetation A typical statement in many noise control texts is that Trees and hedges are not effective noise barriers [2]. There is increasing evidence that this is not always true. For example, measurements [3] made with broadband source 2 m high and receiver height 1.5 m through 500 m of coniferous woodland have shown significant extra attenuation compared with CONCAWE predictions (see Chapter 12) particularly in 63 Hz (3.2 db), 125 Hz (9.7 db), 2 khz (21.7 db) and 4 khz (24.7 db) octave bands. Indeed the United States Department of Agriculture National Agroforestry Center [4] has suggested guidelines for the planting of trees and bushes for noise control based on extensive data collected in the 1970s [5]. A mature forest may have three types of influence on penetrating sound [6 8]. First is ground effect. This is particularly significant if there is a thick litter layer of partially decomposing vegetation on the forest floor. This forms a thick highly porous layer with rather low flow resistivity lying, usually, on higher flow resistivity soil, and gives rise to a primary excess attenuation maximum at lower frequencies than observed over typical grassland. This is consistent with the data collected over ranges of 500 m [3] and is similar to the effect, mentioned earlier, over snow. In respect of traffic noise, an unfortunate consequence of the lower-frequency ground effect is that many existing tree belts alongside roads do not offer much additional attenuation compared with the same distances over open grassland. A Danish study, which has been the basis for the foliage correction in ISO-9613 part 2, found relative attenuation of only 3 db in the A-weighted L eq due to traffic noise for tree belts between 15 and 41 m wide [9]. On the other hand, through 100 m of red pine forest, Heisler et al. [10] have found 8 db reduction in the A-weighted L eq due to road traffic compared with open grassland. The edge of the forest was 10 m from the edge of the highway and the trees occupied a gradual downward slope from the roadway extending about 325 m in each direction along the highway from the measurement site. In the United Kingdom, an extra reduction of 6 db in the A-weighted L 10 level due to traffic noise through 30 m of dense spruce has been found compared with the same depth of grassland [11]. This study found also that the effectiveness of the vegetation was greatest closest to the road. A relative reduction of 5 db in the A-weighted L 10 level was found after 10 m of vegetation. If (a) there is sufficient coherency between ground-reflected and direct sound through the belt, and (b) there is no leaf litter and the belt is not particularly wide, the lowfrequency destructive interference resulting from the relatively soft ground between the trees is likely to be similar to that over grassland and associated with a constructive interference maximum at important frequencies (near 1 khz) for traffic noise. On the other hand, for a narrow tree belt to give extra attenuation of traffic noise compared with the same width of grassland, it is important that the extra attenuation through scattering is significant. Physical conditions that will achieve these requirements are discussed later. Price et al. [6] have measured spectra of the difference in high-frequency levels between microphones at 2 m and various distances through three different woodlands (mixed deciduous, spruce monoculture and mixed coniferous) using a two-way loudspeaker source of broadband random noise. They corrected their data for wavefront spreading and air absorption. The mixed deciduous woodland contained alternating bands of mature Norway spruce (Picus abies) and oak. The oak had dense undergrowth consisting of hawthorn (Crataegus monogyna), rose and honey suckle. The spruce monoculture

330 Predicting outdoor sound 316 contained trees uniformly between 11 and 13 m height, with dead branches only below 4 m. The third woodland consisted of a young mixed coniferous stand containing alternate rows of red cedar, Norway spruce and Corsican pine (Pinus nigra). There was foliage along the whole length of the pine trunks, that is, down to ground level. The wind speed and direction and temperature at two heights were recorded for each measurement. Measurements were carried out only on days with little or no measurable wind (<1.5 m/s) or temperature gradients. Measurements were taken at 1.2 m high receivers positioned 12, 24, 48 and 72 m from the loudspeaker source of broadband sound, with the centre of its low-frequency cone at 1.3 m height, in the mixed deciduous wood; 12, 24, 48 and 96 m range in the spruce monoculture and 12, 24, 26 and 40 m in the mixed conifers. Data were averaged over three receiver positions at a given range (laterally displaced by ±0.5 m from a central location) and each measurement represented a time average over approximately 90 s. Huisman et al. [7, 8] have measured the excess attenuation through woodland consisting of a 29-years-old monoculture of Austrian pines (Pinus nigra subsp. nigra) located in an area (polder) reclaimed from the sea in the Netherlands. A loudspeaker, with its centre at a height of 1 m, was used as the source of swept sine waves, and simultaneous measurements were made of wind and temperature gradients. Receivers were 100 m from the source and at three different heights (1 m, 2.5 m and 4.5 m). Figure 10.6 shows the spectra of measured level difference and excess attenuation respectively (a) for two woodlands with receiver separations of 48 and 96 m and (b) for the pine forest at receivers 1, 2.5 and 4.5 m high placed at a horizontal distance of 100 m. Also shown are corresponding predictions of ground effect. Ground effect alone predicts the frequency and receiver-height dependence of the data up to approximately 1 khz. At higher frequencies the data have significantly different magnitudes and frequency dependence to those predicted by ground effect alone. This can be attributed (a) to scattering of the sound out of the path between source and Figure 10.6 (a) Level difference spectral data [6] obtained in mixed conifers with receiver separation of 48 m (boxes) and a spruce monoculture at a separation of 96 m (circles) (b) excess attenuation data [7] obtained at three different receiver heights (boxes=4.5 m,

331 Predicting effects of vegetation, trees and turbulence 317 circles=2.5 m, diamonds=1 m), 100 m from a loudspeaker source in a pine forest. Corresponding ground effect predictions, shown as continuous and broken lines, have made use of a 2-parameter variable porosity model (see Chapter 3) with parameters of 15 kpa s m 2, 30 m 1 and 7.5 kpa s m 1, 50 m 1 respectively. receiver by trunks and branches and (b) to attenuation of the sound by viscous friction in the foliage. Much of the data shows the foliage attenuation to be dominant. Price et al. [6b] give their results in db 25 m 1. However, to enable comparison with empirical prediction schemes, the results have been amended to db m 1 in Figure 10.7(a). These data suggest approximately linear relationships between attenuation and either distance or log(frequency). The attenuation rates measured through the coniferous sites are rather greater than through the mixed deciduous wood even in summer. Above 1500 Hz, all of the measured attenuation rates are significantly greater than predicted for foliage according to ISO (see Chapter 12, section 2), shown as the broken line in Figure 10.7(a). Figure 10.7(a) also shows empirical fits (for frequencies >1 khz) given by Attenuation in db m 1 through mixed conifers=0.7(log f) 2.03 (10.1) Attenuation in db m 1 through mixed oak and spruce (summer) =0.4(log f) 1.2 (10.2) Attenuation in db m 1 through spruce monoculture=0.26(log f) (10.3) Figure 10.7 (a) Measured variation of corrected level difference with frequency above 1 khz through three woodlands: mature

332 Predicting outdoor sound 318 mixed deciduous forest in summer (circles), Norway spruce monoculture (diamonds) and a mixed conifers (boxes) [6]. Also shown are empirical best-fit equations (solid and broken lines) and the predicted foliage attenuation according to ISO (broken line), (b) Measured excess attenuation through 100 m of pine forest for three different receiver heights; 4.5 m (boxes), 2.5 m (circles), 1 m (diamonds) and mean values (crosses). The continuous line represents a straight line fit to a restricted portion (2 khz<f<6 khz) of these data. The attenuation rate measured through the mixed deciduous site is comparable with that obtained over long range (300 m) through dense evergreen woods by Wiener and Keast [12]. Figure 10.7(b) shows results obtained by Huisman [8] through a pine forest. These do not give a linear relationship between attenuation in db m 1 and log(frequency) except in a limited frequency range (2 khz<f<6 khz). Within this range the empirical fit is given by Attenuation in db m 1 through pine forest=0.39(log f) (10.4) There are few data on propagation through tall crops. Aylor [1] has made a classical series of measurements that demonstrate significant attenuation of sound above 1 khz through tall vegetation including corn, hemlock, brush and pine. Figure 10.8 shows data obtained by Aylor through a dense planting of field corn (Zea mays) and reeds (Phragmites). Good fits to these data are provided by Attenuation in db m 1 through corn=0.95(log f) 2.3 (f>500 Hz) (10.5) Attenuation in db m 1 through reeds=2.3(log f) 12.2 (f>2 khz). (10.6) Note that the attenuation measured at 250 Hz through the corn is likely to be influenced by ground effect, whereas that measured at 4000 Hz was influenced by background noise. Again these attenuation data imply higher rates of attenuation than predicted by the foliage correction in ISO (see Chapter 11), shown as joined diamonds in Figure Analytical modelling of propagation through trees and tall vegetation has been of interest to several authors. Bullen and Fricke [13] developed a diffusion model for sound intensity which involves adjustable parameters for scattering and absorption cross-

333 Predicting effects of vegetation, trees and turbulence 319 sections. Embleton [14] was first to suggest the use of Twersky s multiple-scattering theory [15] for propagation through an idealized array of identical parallel impedancecovered cylinders to explain attenuation through forests. The propagation constant (k b ) through the array of cylinders may be deduced from (10.7) where N is the average number of cylinders per unit area normal to their axes, k=2πf/c 0, g and g are forward and backward scattering amplitudes given, respectively, by A n are scattering coefficients given by Figure 10.8 Measured variation of attenuation with frequency through corn (circles) and reeds (boxes) [1] and linear fits (solid and broken lines respectively). Also shown (joined diamonds) is the attenuation due to foliage predicted by ISO A n =[ij n (ka)+zj (ka)]/[ih n (ka)+zh (ka)] (10.8)

334 Predicting outdoor sound 320 where Z is the surface normal impedance of the cylinder of radius a and J n and H n are Bessel and Hankel functions of the first kind and order n: primes indicate derivatives. Price et al. [6] modelled each of the three woodlands that they investigated as an array of large cylinders (trunks) with finite impedance plus an array of small rigid cylinders (branches and foliage) over an impedance ground. As a first approximation Price et al. simply added the attenuations calculated for the large cylinders, the small cylinders and the ground. Their calculations are in qualitative agreement with their measurements but they found it necessary to adjust several parameters to obtain quantitative agreement. The theory due to Twersky, used by Embleton et al. describes the energy of the squared mean of the transmitted sound field rather than the quadratic mean of the transmitted energy The latter is the value of interest since it characterizes the global attenuation [16]. Twersky s approach neglects the incoherent scattering between cylinders and thus introduces an error which is of importance at high frequency. Laboratory experiments with wooden dowel rods as scatterers [6b] have indicated that the multiple-scattering results (equations (10.15) and (10.16)) tend to over-predict the attenuation and that an empirical reduction in the known density by 60% is required to obtain agreement with data. It has been shown that, after the empirical correction and parameter adjustment, scattering theory (e.g. equation (10.15)) for the sum of two independent cylinder arrays predicts attenuation rates comparable to the measured values. However, these predictions give rise to a non-linear behaviour with log(frequency), similar to that shown in Figure 10.7(b), whereas the data such as those in Figure 10.7(a) are fitted fairly well by linear relationships. Such linear relationships with log(frequency) and with concentration are predicted and measured for sound attenuation through aerosols containing wide particle size distributions [17]. Huisman et al. [7, 8] used a stochastic particle-bounce approach to model attenuation and reverberation through an array of parallel cylinders. In conjunction with a ground effect model and an assumption about the dependence of the proportion of incoherent scattering on distance and frequency, good agreement was obtained with data from a pine forest. Another important result of this work was that even at 100 m through trees, the sound field is mainly formed by the direct and ground-reflected fields and the contribution of the reverberant field is minimal. This argues against modelling sound propagation through trees as a diffusion phenomenon. Past work considering the scattering mechanism (a) has been restricted to infinitely long cylindrical scatterers with best-fit dimensions; (b) has represented surface absorption as an effective impedance, which is a relatively crude way of describing scattering losses; (c) has assumed that scattering and ground effects are simply additive and (d) has largely ignored meteorological effects. Attenuation rates through trees, tall crops and other vegetation remain somewhat uncertain both as a result of lack of data and deficiencies in propagation modelling. Improvement in multiple-scattering models of attenuation through leaves or needles might result from more sophisticated treatment of the boundary conditions or by allowing for scattering by shapes other than infinite cylinders. Attenborough and Walker [18] have formulated a model for viscous and thermal scattering in an array of parallel semi-infinite cylinders with radii much smaller than a wavelength. In the case of corn and wheat plants, there are a large number of long ribbon-like leaves that scatter sound. A ribbon may be modelled as an elliptic cylinder with a minor axis of length approaching zero.

335 Predicting effects of vegetation, trees and turbulence 321 Another possibility is to treat leaves as oblate spheroids with a minor axis approaching zero. A further possibility is to treat the foliage as a suspension in the visco-inertial regime for scattering by sound with wavelengths significantly larger than the maximum dimensions of the particles and amenable to a coupled-phase approach [19]. Several researchers have considered the possibility that resonance induced in the leaves and branches may affect sound propagation [20 22]. However, there are few data to suggest that the resonance mechanism is as important as the others discussed previously Meteorological effects on sound transmission through trees Compared with open grassland, Huisman [8] has predicted an extra attenuation of 10 db(a) for road traffic noise through 100 m of pine forest. He has remarked also that whereas downward-refracting conditions lead to higher sound levels over grassland, the levels in woodland are comparatively unaffected. This suggests that extra attenuation obtained through use of trees should be relatively robust to changing meteorology. Defrance et al. [23] have presented results from both numerical calculations and outdoor measurements and have compared them for different meteorological situations. Their numerical method is based on a Green s Fast Parabolic Equation (GFPE) method. A 2-D GFPE code has been developed [24] and adapted to road traffic noise situations [25] where road line sources are modelled as series of equivalent point sources of height 0.5 m. To allow for the non-uniform wind conditions due to the forest, GFPE calculations were made for each equivalent point source by projecting the wind profile on the sourcereceiver direction (see Figure 10.9). For each equivalent source, the acoustic field was initialized by an analytical expression [26] on a vertical axis containing the source point, and then propagated step by step to the receiver. The experimental values were obtained through a pine forest with an average tree height of m, an average circumference of 53.1 cm and a density of trunks m 2. For different range-dependent sound speed profiles (downwardrefracting, homogeneous or upward-refracting situations), the acoustic attenuation between highway source and receivers at 150 m distance, that is on the other side of the 100 m wide forest, was compared with that obtained in the absence of the trees. During the measurement period, the mean flow was of 2200 vehicles per day, for each of the two lanes, with 5% of heavy trucks. The mean speed of light vehicles was 85 km h 1. The data showed a reduction in A-weighted L eq due to the trees of 3 db during downwardrefracting conditions, 2 db during homogeneous conditions and 1 db during upwardrefracting conditions.

336 Predicting outdoor sound 322 Figure 10.9 Range-dependent sound speed profiles used in simulations by Defrance et al. [23]. The rectangle shows the region occupied by trees. Reprinted with permission from ICSV 9 Procs, Orlando, Figure Predicted acoustical efficiency of a 100 m wide and 10.5 m high pine forest relative to the situation without forest, in db(a), in the case of: (a) typical day