In-situ measurements of the Complex Acoustic

Transcription

1 MASTER S THESIS 2006 In-situ measurements of the Complex Acoustic Impedance of materials for automobile interiors Jorge D. Alvarez B. Ørsted DTU Acoustic Technology TECHNICAL UNIVERSITY OF DENMARK Ørsteds Plads, bldg. 352 DK-2800 Kgs. Lyngby Denmark

2 In-situ measurements of the Complex Acoustic Impedance of materials for automobile interiors c Jorge D. Alvarez B., 2006 Master s Thesis 2006 Ørsted DTU Acoustic Technology Technical University of Denmark Ørsteds Plads, bldg. 352 DK-2800 Kgs. Lyngby Denmark Tel: (+45) info@oersted.dtu.dk Printed at the Department of Acoustic Technology DK-2800 Kgs. Lyngby, Denmark 2006

3 Abstract This project is developed in collaboration with the automotive company Volkswagen AG in Wolfsburg, Germany. The acoustical representation of materials used in automobile interiors in terms of their complex acoustic impedance is of interest. On the other hand, the evaluation of measurement techniques and the viability of implementing them for in-situ evaluation of materials constitutes the main goal in the project. Over the years, several methods of measuring the complex acoustic impedance of materials have been presented, but none of them is well-established for practical in-situ measurements. In this context, two main methods were analyzed throughout this document. On one hand, the transfer function method, considered as the more stable and confident method to obtain such data (ISO 10534), employs ideal sound field conditions such as the propagation of plane waves inside of a tube. On the other hand, the free field method is applied on a more complex spherical sound field. A portable impedance tube was constructed and analyzed on in-situ measurements. Leakage in the interface tube-material constitutes the breaking point of this application. By pressing the tube into the materials, their properties might be changed; however, this may not be the case for rigid stiff materials. Furthermore, no reference is available when measuring the acoustic impedance of stiff materials; mounting based errors become critical on the transfer function method on a typical impedance tube. The small size of the microflown intensity probe makes the measurement of the sound pressure and normal component of the particle velocity possible at very small distances from a surface. However, according to simulations, the ratio of them does not characterize properly the material in terms of its acoustic impedance. The application of models to approximate the actual acoustic impedance is needed. Three different models are analyzed for different location of source and receiver. Furthermore, the intrinsic uncertainty on the probe due to calibration has been included artificially in simulations and it has been demonstrated that it still constitutes an important issue. Finally, in-situ measurements were carried out in an ordinary room. The results have shown that the modes of the room unexpectedly influence the measurement of the acoustic impedance. The analysis of the methods has been carried out with samples of typical porous materials. The outcome of them has been applied on the Volkswagen AG material measurements and the results are presented in appendices. i

4 Acknowledgements The following work is submitted as the culmination of my two years of study in the M.Sc. Engineering Acoustics program at the Technical University of Denmark (DTU). I would like to specifically recognize the following individuals and institutions: Finn Jacobsen, Ørsted DTU Acoustic Technology, for his guidance throughout all my career as acoustician in Denmark, in the areas of linear acoustics and other matters Leonardo Miranda, Volkswagen AG Dipl.Ing. Acoustics - NVH Simulation - Group Research/Vehicle Concepts - CAE Methods, for his input during the development of project goals and his influential support Jørgen Rasmussen, Ørsted DTU Acoustic Technology, for his assistance whenever needed in all the setups for the conduction of measurements Aage Sonesson, Ørsted DTU Acoustic Technology, for his valuable help on the construction of mechanic structures Albert Allen, Kostantinos Angelakis, Yi Shen and other colleagues, Ørsted DTU Acoustic Technology, for their general support and source of ideas My parents, Teresa Balderrama and Nataniel Alvarez, for their general support throughout my professional career My wife, Tania Castedo, for her endless patience and comprehension Volkswagen AG for providing financial support during this project ii

5 Table of Contents Abstract Acknowledgements i ii 1 Introduction Scope of the project Structure of the thesis Acoustic impedance Acoustical representation of porous materials Properties and categories of porous materials Reflection of sound waves Constructive and destructive interference Plane-wave reflections Spherical-wave reflections Laplace transform formulation by Lindell and Alanen Grazing incidence approximation by Chien and Soroka Brief simulation analysis Means to measure the acoustic impedance Intensity Probe as a tool to measure Z of porous materials Microflown and its application to acoustic impedance measurements Transfer Function Method Introduction Description of the method Experimental investigation Portable impedance tube analysis iii

6 TABLE OF CONTENTS iv 6 Free field method Introduction Sound Field approximations Model 1: Plane-wave approximation Model 2: Image source model with plane-wave reflection coefficient Model 3: Image source model with spherical reflection coefficient Simulated Investigation Simulation 1: On the acoustical representation of porous materials through a measurement right above the surface Simulation 2: On the acoustical representation of porous materials through approximation models Simulation 3: The effect of intrinsic uncertainty on the microflown probe Experimental investigation Summary and conclusions Future work A From simulation 1 - Stiff material 94 B From simulation 2 Receiver at 4.5 cm above the surface 97 C From simulation 3 Receiver at 4.5 cm above the surface 102 D Appendix 4 Vokswagen materials 106 D.1 Floor foam D.2 Ceiling D.3 Floor carpet D.4 Front panel

7 List of Figures 2.1 Normalized acoustic impedance for a spherical sound field. The dashed line represents the real part of the normalized acoustic impedance, the dash-dot line shows the imaginary part, and the continuous line gives the phase D representation of the normalized acoustic impedance of a spherical sound field as a function of frequency and distance Normalized Acoustic impedance of the porous material - Delany and Bazley model with σ = 50000Kgm 3 s Effect of the surface in the sound pressure. A totally reflecting surface is taking into account in order to show only constructive and destructive interference Plane-wave sound field hitting a perfectly flat surface at oblique incidence angle Point source and receiver above an infinite plane surface Reflection Coefficients. The blue color represents the plane-wave reflection coefficient, the green color shows Q with the grazing incidence model, whereas the red color shows Q calculated using the La place transform model; h s = 1.5m, h r = 1.5m, and horizontal distance d = 100m. The dotted lines are, for all three colors, for a flow resistivity of Kgm 3 s 1. The dashed lines are for a flow resistivity of Kgm 3 s 1. Finally the dash-dot lines are for a flow resistivity of Kgm 3 s Effect of the surface on the sound pressure level at the receiver position. The blue color represents the use of the plane-wave reflection coefficient, the green color the use of Q calculated with the grazing incidence model, whereas the red color the use of Q calculated using the La place transform model; h s = 1.5m, h r = 1.5m, and horizontal distance d = 100m. The dotted lines are, for all three colors, for a flow resistivity of Kgm 3 s 1. The dashed lines are for a flow resistivity of Kgm 3 s 1. Finally the dash-dot lines are for a flow resistivity of Kgm 3 s Reflection Coefficients. The blue line represents the plane-wave reflection coefficient, the green line shows Q with the grazing incidence model, whereas the red line shows Q calculated using the La place transform model; h s = 0.6m, h r = 0.01m, and horizontal distance d = 0m Effect of the surface on the sound pressure level at the receiver position. The blue color represents the use of the plane-wave reflection coefficient, the green color the use of Q calculated with the grazing incidence model, whereas the red color the use of Q calculated using the La place transform model; h s = 0.6m, h r = 0.01m, and horizontal distance d = 0m v

8 LIST OF FIGURES vi 4.1 Bruel&Kjær sound intensity probe kit Type PU regular intensity probe - microflown Calibration factor for the microflown probe. The amplitude and phase are presented for two different sound field analysis. The continuous line was obtained in a standing tube, whereas the dashed line was obtained in free field conditions with a monopole and distance source-receiver of 60 cm Calibration factors for the microflown probe. A monopole is located in partial free field conditions, at 1 m above a rigid surface. Several receiver positions were used to measure the response of the microflown probe and obtain calibration factors for this particular sound field. As reference, the correction factor obtained in complete free field conditions is included Calibration factor for the microflown probe. This curve was obtained on the combination of the standing wave tube calibration factor at low frequencies (below 400 Hz) and the free field calibration factor at high frequencies (above 400 Hz) Calibration factors for the microflown probe. The calibration factor obtained in this procedure is compared with a reference, where a monopole on sphere was used to calculate the correction factor Impedance tube Absorption and Reflection properties. Comparison PP method and PU method for a 50mm Rockwool A-Batt Normalized Acoustic Impedance. Comparison PP method and PU method for a 50mm Rockwool A-Batt Normalized acoustic impedance variation when errors in the distance sample-probe are committed while doing the calibration of the probe. Variance of ± 10mm are considered Normalized acoustic impedance variation when errors in the distance sample-probe are committed while doing the actual measurement. Variance of ± 10mm are considered (a) Portable impedance tube on a handy tripod; (b) Rubber ring on the termination of the portable impedance tube Reflection and Absorption properties - Open tube case. The blue curve Levine and Schwinger model, whereas the red dotted curve new portable tube Normalized Acoustic Impedance - Open tube case. The blue curve Levine and Schwinger model, whereas the red dotted curve new portable tube Reflection and Absorption properties - Rigid surface case. The blue curve Reflection coefficient for a rigid surface in an isothermal process, whereas the red dotted curve new portable tube Normalized Acoustic Impedance - Rigid surface case. The blue curve normalized acoustic impedance for a rigid surface in an isothermal process, whereas the red dotted curve new portable tube Reflection and Absorption properties - porous material placed in the tubes (50 mm Rockwool A-Batt). The blue curve reference tube, whereas the red dotted curve new portable tube Normalized Acoustic Impedance - porous material placed in the tubes (50 mm Rockwool A-Batt). The blue curve reference tube, whereas the red dotted curve new portable tube Reflection and Absorption properties - stiff material placed in the tubes. The blue curve reference tube, whereas the red dotted curve portable tube

9 LIST OF FIGURES vii 5.14 Normalized Acoustic Impedance - stiff material placed in the tubes. The blue curve reference tube, whereas the red dotted curve new portable tube Reflection and Absorption properties - Schematic analysis of portable tube response for in-situ evaluations. The continuous line represents the reference tube measurement, the dashed line shows the results when the tube is located at 1 cm above the material, the dotted line is the result when the tube is on the surface, not pressing the material, and finally the dash-dot line shows the result when the tube on the surface, pressing the material. The porous material is a large sample of 40 mm Rockfon Samson Normalized acoustic impedance - Schematic analysis of portable tube response for in-situ evaluations. The continuous line represents the reference tube measurement, the dashed line shows the results when the tube is located at 1cm. above the material, the dotted line is the result when the tube is on the surface, not pressing the material, and finally the dash-dot line shows the result when the tube on the surface, pressing the material. The porous material is a large sample of 40 mm Rockfon Samson Reflection and Absorption properties - Big and small sample comparison. The dashed line shows the big sample results, whereas the dash-dot curve exhibits the results with the small sample Normalized acoustic impedance - Simulation on the effect of a small leakage on the interface tube-material when measuring the acoustic impedance of materials. Holes of 0, 0.2, 0.4, 0.6, 0.8 and 1 mm of radius are considered Reflection and absorption properties - Simulation on the effect of a small leakage on the interface tube-material when measuring the reflection and absorption coefficient. Holes of 0, 0.2, 0.4, 0.6, 0.8 and 1 mm of radius are considered Reflection and Absorption properties - stiff piece of wood. The blue continuous line corresponds to the measurement with the reference tube - the sample has been cut an introduced in the tube. The red-dashed line corresponds to the measurement in-situ with the portable tube, which is been only pushed into the material as an attempt to avoid leakage Normalized acoustic impedance - stiff piece of wood. The blue continuous line corresponds to the measurement with the reference tube - the sample has been cut an introduced in the tube. The red-dashed line corresponds to the measurement in-situ with the portable tube, which is been only pushed into the material as an attempt to avoid leakage Source above an infinite plane surface Monopole above an infinite plane surface Comparison of the surface acoustic impedance (Delany and Bazley model) with the ratio of pressure and particle velocity measured right above the material at 5, 10, 15 and 20 millimeters. The material is exposed to plane-wave sound field and the plane-wave reflection coefficient is used for the calculation of the acoustic impedance at the receiver position Effect of the surface in the sound pressure and particle velocity component. A totally reflecting surface is taking into account in order to show only constructive and destructive interference and its corresponding shifting on frequency when the receiver position is changed

10 LIST OF FIGURES viii 6.5 Comparison of the surface acoustic impedance (Delany and Bazley model) with the ratio of pressure and particle velocity measured right above the material at 5, 10, 15 and 20 millimeters. The material is exposed to spherical sound field and the plane-wave reflection coefficient is used for the calculation of the acoustic impedance at the receiver position. Source located at 0.5 meters above the surface Comparison of the surface acoustic impedance (Delany and Bazley model) with the ratio of pressure and particle velocity measured right above the material at 5, 10, 15 and 20 millimeters. The material is exposed to spherical sound field and the spherical reflection coefficient is used for the calculation of the acoustic impedance at the receiver position. Source located at 0.5 meters above the surface Spherical reflection coefficient and its variation with respect to the frequency and receiver position Spherical reflection coefficient and its variation with respect to the receiver position for specific frequencies Comparison of reflection coefficients. The sound field has been defined at 5 mm above the surface, by the calculation of the spherical reflection coefficient Q (continuous line - Sound Field given). On the calculation of Q, an acoustic impedance given by Delany and Bazley model with a flow resistivity σ = 50000Kgm 3 s 1 was introduced. This specific position becomes afterwards a measurement receiver point, where models 1 ( ) and 2 ( ) have been applied to approximate the reflection coefficient Normalized surface impedance comparison. The actual surface impedance given by Delany and Bazley model is compared with approximations to it according to model 1 ( ) and 2 ( ). This material can be categorized as limp. Receiver position on the measurement: 5 mm above the surface Normalized surface impedance comparison. The actual surface impedance given by Delany and Bazley model is compared with the approximations to the surface impedance by model 2 and 3. In model 3, the calculation of Q is been done by introducing the surface impedance approximated by model 2. This material is categorized as limp. Receiver position on the measurement: 5 mm above the surface Differences on the sound field at the receiver position. These differences are produced by the calculation of Q with the surface impedance given by model 2 instead of the actual surface impedance. The continuous line must be assumed as the representation of an actual measurement, whereas the dashed line represents the prediction at the receiver position, with differences due to the error on the Q calculation. Receiver position on the measurement: 5 mm above the surface Comparison of reflection coefficients. The sound field has been defined at 5 mm above the surface, by the calculation of the spherical reflection coefficient Q (continuous line - Sound Field given). On the calculation of Q, an acoustic impedance given by Delany and Bazley model with a flow resistivity σ = Kgm 3 s 1 was introduced. This specific position becomes afterwards a measurement receiver point, where models 1 ( ) and 2 ( ) have been applied to approximate the reflection coefficient Normalized surface impedance comparison. The actual surface impedance given by Delany and Bazley model is compared with approximations to it according to model 1 ( ) and 2 ( ). This material can be categorized as stiff. Receiver position on the measurement: 5 mm above the surface

11 LIST OF FIGURES ix 6.15 Normalized surface impedance comparison. The actual surface impedance given by Delany and Bazley model is compared with the approximations to the surface impedance by model 2 and 3. In model 3, the calculation of Q is been done by introducing the surface impedance approximated by model 2. This material is categorized as stiff. Receiver position on the measurement: 5 mm above the surface Differences on the sound field at the receiver position. These differences are produced by the calculation of Q with the surface impedance given by model 2 instead of the actual surface impedance. The continuous line must be assumed as the representation of an actual measurement, whereas the dashed line represents the prediction at the receiver position, with differences due to the error on the Q calculation. Receiver position on the measurement: 5 mm above the surface The effect of amplitude mismatch on the approximation to the normalized acoustic impedance on the surface by using model 2. The material is considered limp with a flow resistivity σ = 50000Kgm 3 s The effect of phase mismatch on the approximation to the normalized acoustic impedance on the surface by using model 2. The material is considered limp with a flow resistivity σ = 50000Kgm 3 s The effect of amplitude and phase mismatch on the approximation to the normalized acoustic impedance on the surface by using model The effect of amplitude mismatch on the approximation to the normalized acoustic impedance on the surface by using model 2. The material is considered stiff with a flow resistivity σ = Kgm 3 s The effect of phase mismatch on the approximation to the normalized acoustic impedance on the surface by using model 2. The material is considered stiff with a flow resistivity σ = Kgm 3 s The effect of amplitude and phase mismatch on the approximation to the normalized acoustic impedance on the surface by using model 2. The material is considered stiff with a flow resistivity σ = Kgm 3 s Normalized surface impedance comparison - Rockwool 50mm A-batt. Results are shown for measurements in the kundt s tube and in free-field conditions, using Bruel&Kjær and microflown intensity probes. Models 1 and 2 have been applied to approximate the surface impedance out of a measurement at 4.5 cm above the surface Normalized surface impedance comparison - Rockwool 50mm A-batt. Results are shown for measurements in the kundt s tube and in free-field conditions, using the microflown intensity probe. Models 1 and 2 have been applied to approximate the surface impedance out of a measurement at 5 mm above the surface. Model 3 is used by calculating Q with the approximation to the surface impedance provided by model Effect of the surface (Rockwool 50mm A-batt) in the sound pressure and particle velocity component at the receiver position. A deep interference minima in the particle velocity component is observed around 100 hz Comparison of reflection coefficients. The plane-wave reflection coefficient is calculated with the transfer function method on tube measurements, as well as through the free field method, using models 1 and 2. Furthermore, the calculation of the spherical reflection coefficient Q is done by introducing in the calculation the surface impedance approximated by model

12 LIST OF FIGURES x 6.27 Normalized surface impedance comparison - Rockwool 50mm A-batt. Results are shown for measurements in the kundt s tube (continuous line), in the large anechoic chamber (dashed line) and in the small anechoic chamber (dash-dot line), using the microflown intensity probe. Models 2 has been applied to approximate the surface impedance out of a measurement at 5 mm above the surface Normalized surface impedance comparison - Rockwool 50mm A-batt. Results are shown for measurements in the kundt s tube and in-situ, using the microflown intensity probe. Model 2 has been applied to approximate the surface impedance out of a measurement at 5 mm above the surface Effect of the room in the sound pressure, particle velocity and the ration of them (acoustic impedance) at the receiver position. The data is not treated and the measurement is conducted 5 mm above a large sample of Rockwool 50mm A-batt Coherence between sound pressure and particle velocity component channels on in-situ measurement of a large sample of Rockwool 50mm A-batt Normalized surface impedance comparison - 40 mm Rockfon Samson. Results are shown for measurements in the kundt s tube and in-situ, using the microflown intensity probe. Model 2 has been applied to approximate the surface impedance out of a measurement at 2.5 cm above the surface A.1 Comparison of the surface acoustic impedance (Delany and Bazley model) with the ratio of pressure and particle velocity measured right above the material at 5, 10, 15 and 20 millimeters. The material is exposed to plane-wave sound field and the plane-wave reflection coefficient is used for the calculation of the acoustic impedance at the receiver position A.2 Comparison of the surface acoustic impedance (Delany and Bazley model) with the ratio of pressure and particle velocity measured right above the material at 5, 10, 15 and 20 millimeters. The material is exposed to spherical sound field and the plane-wave reflection coefficient is used for the calculation of the acoustic impedance at the receiver position. Source located at 0.5 meters above the surface A.3 Comparison of the surface acoustic impedance (Delany and Bazley model) with the ratio of pressure and particle velocity measured right above the material at 5, 10, 15 and 20 millimeters. The material is exposed to spherical sound field and the spherical reflection coefficient is used for the calculation of the acoustic impedance at the receiver position. Source located at 0.5 meters above the surface A.4 Spherical reflection coefficient and its variation with respect to the frequency and receiver position. 96 A.5 Spherical reflection coefficient and its variation with respect to the receiver position for specific frequencies B.1 Comparison of reflection coefficients. The sound field has been defined at 4.5 cm above the surface, by the calculation of the spherical reflection coefficient Q (continuous line - Sound Field given). On the calculation of Q, an acoustic impedance given by Delany and Bazley model with a flow resistivity σ = 50000Kgm 3 s 1 was introduced. This specific position becomes afterwards a measurement receiver point, where models 1 ( ) and 2 ( ) have been applied to approximate the reflection coefficient

13 LIST OF FIGURES xi B.2 Normalized surface impedance comparison. The actual surface impedance given by Delany and Bazley model is compared with approximations to it according to model 1 ( ) and 2 ( ). This material can be categorized as limp. Receiver position on the measurement: 4.5 cm above the surface B.3 Normalized surface impedance comparison. The actual surface impedance given by Delany and Bazley model is compared with the approximations to the surface impedance by model 2 and 3. In model 3, the calculation of Q is been done by introducing the surface impedance approximated by model 2. This material is categorized as limp. Receiver position on the measurement: 4.5 cm above the surface B.4 Differences on the sound field at the receiver position. These differences are produced by the calculation of Q with the surface impedance given by model 2 instead of the actual surface impedance. The continuous line must be assumed as the representation of an actual measurement, whereas the dashed line represents the prediction at the receiver position, with differences due to the error on the Q calculation. Receiver position on the measurement: 4.5 cm above the surface. 99 B.5 Comparison of reflection coefficients. The sound field has been defined at 4.5 cm above the surface, by the calculation of the spherical reflection coefficient Q (continuous line - Sound Field given). On the calculation of Q, an acoustic impedance given by Delany and Bazley model with a flow resistivity σ = Kgm 3 s 1 was introduced. This specific position becomes afterwards a measurement receiver point, where models 1 ( ) and 2 ( ) have been applied to approximate the reflection coefficient B.6 Normalized surface impedance comparison. The actual surface impedance given by Delany and Bazley model is compared with approximations to it according to model 1 ( ) and 2 ( ). This material can be categorized as stiff. Receiver position on the measurement: 4.5 cm above the surface B.7 Normalized surface impedance comparison. The actual surface impedance given by Delany and Bazley model is compared with the approximations to the surface impedance by model 2 and 3. In model 3, the calculation of Q is been done by introducing the surface impedance approximated by model 2. This material is categorized as stiff. Receiver position on the measurement: 4.5 cm above the surface B.8 Differences on the sound field at the receiver position. These differences are produced by the calculation of Q with the surface impedance given by model 2 instead of the actual surface impedance. The continuous line must be assumed as the representation of an actual measurement, whereas the dashed line represents the prediction at the receiver position, with differences due to the error on the Q calculation. Receiver position on the measurement: 4.5 cm above the surface. 101 C.1 The effect of amplitude mismatch on the approximation to the normalized acoustic impedance on the surface by using model 2. Receiver at 4.5cm above the material C.2 The effect of phase mismatch on the approximation to the normalized acoustic impedance on the surface by using model 2. Receiver at 4.5cm above the material C.3 The effect of amplitude and phase mismatch on the approximation to the normalized acoustic impedance on the surface by using model 2. Receiver at 4.5cm above the material C.4 The effect of amplitude mismatch on the approximation to the normalized acoustic impedance on the surface by using model 2. Receiver at 4.5cm above the material

14 LIST OF FIGURES xii C.5 The effect of phase mismatch on the approximation to the normalized acoustic impedance on the surface by using model 2. Receiver at 4.5cm above the material C.6 The effect of amplitude and phase mismatch on the approximation to the normalized acoustic impedance on the surface by using model 2. Receiver at 4.5cm above the material D.1 Reflection coefficient - Floor foam Volkswagen Golf. The foam does not have any irregularity on either side. Three measurements of the reflection coefficient in plane-wave sound field are shown. 107 D.2 Normalized acoustic impedance - Floor foam Volkswagen Golf. The foam does not have any irregularity on either side. Two measurements of the acoustic impedance in plane-wave sound field are shown D.3 Reflection coefficient - Floor foam Volkswagen Golf. Effect of irregularities on the bottom side of the material D.4 Normalized acoustic impedance - Floor foam Volkswagen Golf. Effect of irregularities on the bottom side of the material D.5 Reflection coefficient - Ceiling Volkswagen Golf. The material is analyzed with a rigid backed surface D.6 Normalized acoustic impedance - Ceiling Volkswagen Golf. The material is analyzed with a rigid backed surface D.7 Reflection coefficient - Ceiling Volkswagen Golf. The effect of a small cavity behind the ceiling. 110 D.8 Normalized acoustic impedance - Ceiling Volkswagen Golf. The effect of a small cavity behind the ceiling D.9 Reflection coefficient - Floor carpet Volkswagen Golf. The carpet does not have any irregularity on either side D.10 Normalized acoustic impedance - Floor carpet Volkswagen Golf. The carpet does not have any irregularity on either side D.11 Reflection coefficient - carpet above the floor foam. Differences of having the carpet with a rigid backed surface or in the top of the floor foam D.12 Normalized acoustic impedance - carpet above the floor foam. Differences of having the carpet with a rigid backed surface or in the top of the floor foam D.13 Reflection coefficient - Front panel Volkswagen Golf. The material is been measured in several positions and the averaged response is included D.14 Normalized acoustic impedance - Front panel Volkswagen Golf. The material is been measured in several positions and the averaged response is included

15 Chapter 1 Introduction During the last two decades, several computational methods have been developed to simulate the sound field inside of cavities. Among the input data, the acoustical representation of surface materials plays a critical role for the accurate estimation of the sound field behavior. In most of the available prediction methods, the surfaces are characterized by the absorption coefficient, which provides amplitude information disregarding phase content of the response of the material. In contrast, the acoustical response of materials should be characterized by its acoustic impedance, which not only includes information about the absorbing acoustical energy, but also information regarding non-dissipative processes that occurs when the sound field strikes the surface. There is actually much data available for absorption coefficients of different materials, but hardly good information concerning real and imaginary parts of the acoustic impedance. This may be related with the difficulty involved in measuring the acoustic impedance. Regarding experimental methods to measure the acoustic impedance of materials, the more stable and confident method requires ideal sound field conditions, such as the propagation of plane waves inside of a tube (ISO-10534). Furthermore, the application of this method requires the mounting of a small sample of material inside of the standing wave tube; this is not always possible to accomplish for instance with complicated structures such as a car seat. Similar methods on more complicated sound fields, such as the one produced by a point source in free field con- 1

16 1.1. Scope of the project 2 ditions, have been studied by several authors. Severe complications have been found specially at low frequencies. Moreover, none of them have been well-established for practical in-situ measurements of acoustic impedance. 1.1 Scope of the project This project is developed in collaboration with the automotive company Volkswagen AG in Wolfsburg, Germany. The company is concerned with the accurate characterization of interior surfaces of VW vehicles in order to produce cavity models that may help to predict the behavior of the sound field before the actual construction. Moreover, the models may improve the quality of the products in terms of human comfort. On the other hand, both the author and the academic supervisor are concerned mainly with the measurement techniques themselves and the development of techniques for in-situ measurements of the complex acoustic impedance. Therefore, the main objectives of the present project are listed below: The characterization of interior surfaces of VW vehicles in terms of the complex acoustic impedance by means of experimental evaluations. The evaluation of different methods to obtain the reflection and absorption properties, and thus the complex acoustic impedance of different materials. The development of techniques for the evaluation of materials in-situ in terms of the complex acoustic impedance with reasonable accuracy. In this context, two main methods to measure the acoustic impedance have been analyzed: the transfer function method in a standing wave and the free field method. In both cases, the viability of in-situ measurements has been analyzed in detail.

17 1.2. Structure of the thesis Structure of the thesis This thesis is organized as follows: Chapter 2 provides briefly the relevant theory concerning the concept of acoustic impedance and the acoustical representation of porous materials. Chapter 3 presents the theory regarding reflection of both plane and spherical waves, as well as a comparison of two typical models to represent the reflection of spherical waves. Chapter 4 provides information about the transducers used throughout the investigation and explanation about calibration processes. Chapter 5 describes in short term the transfer function method, presents an error analysis on the sensitivity of the p-u intensity probe results to location and calibration errors, and includes the schematic analysis conducted in order to analyze the response of the portable impedance tube. In chapter 6, the free field method is introduced and the response of the method is analyzed through simulation as well as experimental investigation. Moreover, it includes the analysis of in-situ measurements using this method. Some of the results of this chapter are presented in appendices 1-3. Conclusions are drawn and presented in chapter 7. Finally, and particularly of interest to Volkswagen AG, appendix 4 provides information on the complex acoustic impedance of VW vehicle materials.

18 Chapter 2 Acoustic impedance Specific acoustic impedance is a mathematically complex quantity defined as the ratio of the complex values of the sound pressure and particle velocity of a single frequency component of a sound field at a point in space: Z s = P/U, where P and U are, respectively, the complex values of sound pressure and a vector component of particle velocity. This quantity is commonly measured in Rayls which in MKS units corresponds to [ Pa s ] [Rayl] = = m [ N s ] [ Kg ] m 3 = m 2 s (2.1) Z s is clearly a vector-like quantity since its value depends upon the direction of the associated particle velocity component. For a plane wave sound field formed, for instance, inside of a long uniform tube with rigid walls, where the sound waves are constrained by the presence of the walls to take a particularly simple form at frequencies for which the acoustic wavelength exceeds about half the peripheral length of the tube cross-section, the sound pressure and particle velocity component in the x-direction are given by P(x, t) = A e j(ωt±kx) (2.2) U(x, t) = A ρc e j(ωt±kx) (2.3) 4

19 5 where A is the amplitude. Thus, the specific acoustic impedance at a point in space is given by Z s = P U = ±ρc (2.4) where c is the speed of sound and ρ is the mass density of air. Hence, the specific acoustic impedance in a plane wave sound field is purely real. The quantity in equation (2.4) is known as the characteristic impedance of air Z air and is equal to 410 Pa s/m at a temperature of 25 C. The normalized acoustic impedance Z is defined as the specific acoustic impedance with respect to Z air : Z = Z s Z air = Z s ρc = P ρcu (2.5) The dimensionless feature of this complex quantity is definitely an advantage with regard to the graphic representation of material properties. On the other hand, for a spherical sound field formed, for example, by a point source in free field conditions, where waves propagate away from the source uniformly in all directions, the sound pressure is given by P(r, t) = A r e j(ωt kr) (2.6) The particle velocity component in the radial direction can be calculated as U r (r, t) = 1 ρc [ ] [ ] A 1 r jkr ± 1 e j(ωt kr) P(r, t) 1 = ρc jkr ± 1 (2.7) Therefore, the ratio between the acoustic pressure and the particle velocity gives Z s = P U r = ρc jkr 1 ± jkr (2.8) If the distance r from the origin is very large, the quantity jkr will be sufficiently large compared to 1. In this case, the characteristic impedance becomes ±ρc. The sign is dependent on whether one is looking at inward-going or outward-going waves. Therefore, at a large enough distance

20 6 from the origin of a spherical wave, the curvature of any part of the wave finally becomes negligible and the characteristic impedance becomes just as in plane waves. On the contrary, when kr << 1, the particle velocity component is almost in quadrature with the sound pressure and near field conditions are given. In order to analyze the sound field in much more detail, equation (2.8) is written as Z s = ρc(kr)2 1 + (kr) 2 + jω ρc 1 + (kr) 2 (2.9) The real part of the acoustic impedance is known as the acoustic resistance, whereas the imaginary part is called the acoustic reactance. Figure 2.1 shows the real and imaginary parts of the normalized acoustic impedance as well as the phase as a function of the factor kr. Furthermore, figure 2.2 shows the real and imaginary part of the normalized acoustic impedance of the sound field as a function of the distance and frequency. Figure 2.1: Normalized acoustic impedance for a spherical sound field. The dashed line represents the real part of the normalized acoustic impedance, the dash-dot line shows the imaginary part, and the continuous line gives the phase. When the distance from the source is only a small fraction of a wavelength (kr << 1), the phase difference between the complex pressure and particle velocity component is large,

21 7 Figure 2.2: 3-D representation of the normalized acoustic impedance of a spherical sound field as a function of frequency and distance. whereas the acoustic resistance and reactance tend to small values. When Kr = 1, both the acoustic resistance and reactance are equal to ρc/2, the phase is 45, and the acoustic reactance has its maximum value. When the distance from the source corresponds to a considerable number of wavelengths, the sound pressure and particle velocity are very nearly in phase and the spherical wave then assumes the characteristics of plane waves. Analogous to electrical quantities, where the real part of the impedance is associated with a force that tends to resist the flow of electrical current, converting electrical energy into heat, the acoustic resistance is associated with dissipative losses that occur when there is a viscous flow of air through, for instance, a fine mesh screen, and acoustical energy is converted into heat. On the other hand, the imaginary part of the electric impedance is related with the presence of capacitors and inductors, which alternately absorb energy from the circuit and return energy to the circuit. That is, unlike a resistance, a reactance does not dissipate power. The acoustic reactance is formed by acoustic compliance and acoustic mass. The former is associated with any volume of air that is compressed by an applied force without an acceleration of its center of gravity, whereas the latter is associated with any volume of air that is accelerated without being

22 2.1. Acoustical representation of porous materials 8 compressed. For materials, the acoustic impedance can be related with the opposition to the flow of sound through the material, which is formed by the resistance, representing dissipative losses, and the reactance, which is an effect of standing waves within the material. 2.1 Acoustical representation of porous materials The sound pressure and particle velocity component depend of course of the sound field. In other words, if the amplitude of the sound field is doubled, for instance, the amplitudes of the sound pressure, as well as the particle velocity component are also doubled. However, the ratio of the sound pressure to the particle velocity component is independent of the amplitude of the sound field, and it is therefore a convenient quantity for the acoustic characterization of porous materials [3]. Let us consider a harmonic sound field which strikes a surface, where part of the wave is reflected and part is transmitted into the material. The specific acoustic impedance on the surface is defined as the ratio Z s = P/U c,n, where P is the complex pressure amplitude and U c,n is the normal component of the complex particle velocity (normal component in the direction of the wave). In other words, the acoustic impedance on a given surface is defined as the ratio of sound pressure to the mean molecular velocity passing through it. This ratio is continuous at the surface, i.e. it has equal values just above and just below the surface; the latter is related to the characteristic impedance of the material Z m. The acoustical properties that are inherent to a porous material are its characteristic impedance Z m and wave number k 2. The characteristic impedance Z m of a limp or stiff porous medium is the ratio of pressure P to particle velocity U c in the direction of wave propagation in a plane wave propagating in an infinitely extended piece of material. The wave number k 2 of a plane wave propagating freely within a porous material is complex and is a measure of both the speed of wave propagation (and hence the wavelength within the material) and the rate at which a

23 2.1. Acoustical representation of porous materials 9 propagating wave attenuates. Porous materials, such as glass fibers or foams, are characterized as being either locally reacting or non-locally reacting. Physically, a porous layer can be assumed to be locally reacting when the speed of wave propagation in the porous layer is much less than the speed of sound in the ambient medium, with the result that the transmitted waves propagate normal to the material surface regardless of incident angle. Since the propagation direction of the transmitted wave is normal to the surface, U c = U c,n and, therefore, the specific impedance on the surface is equal to the characteristic impedance of the material (Z s = Z m ). The surface impedance of a locally reacting porous material is independent of the incidence angle, with the important practical consequence that the reflection coefficient of a locally reacting porous material may be obtained using a simple one-dimensional analysis. Thus, it is very convenient from a computational point of view if a porous material may be considered as locally reacting. Note that non-locally reacting materials may be made locally reacting by segmenting the material into small elements by using rigid partitions. In this case the partitions prevent sound propagation parallel to the surface making the material effectively locally reacting. On the other hand, when a material is non-locally reacting (i.e. when the sound speed in the material is not small compared to that of air), its surface impedance varies with angle of incidence. This means that the incident wave causes a reaction at a point on the material that depends on the pressure distribution in a region around this point. Obviously, from a theoretical point of view, it would be convenient if all porous materials could be treated as surfaces of local reaction; however, in most situations the latter assumption is not strictly valid. As mentioned previously, the characteristic impedance of a porous material Z m is, generally, a complex function of frequency as the pressure and particle velocity component are not in phase with each other. Several authors have worked on the modeling of porous materials, such as glass fibers, mineral wools, etc. An empirical impedance model for fibrous porous materials was developed by Delany and Bazley in the 70 s[3]. The fact that the model is only dependent on the flow resistivity σ of the material, has constituted an advantage over all the other models

24 2.1. Acoustical representation of porous materials 10 presented along the years. According to Delany and Bazley, the acoustic impedance Z m and wave number k 2 can be approximated as ( ) 0.75 ( ) 0.73 σ σ Z m = ρc j f f (2.10) k 2 = ω c ( σ f ) 0.7 j ( ) 0.59 σ f (2.11) These empirical relations are based on experimental data for values of f /σ ranging from 0.01 to 1.0 [m 3 Kg 1 ]. However, it is been demonstrated that the relations also give good agreement with experimental data for grassland, with the flow resistivity σ varying between 100 and 300 [Kgm 3 s 1 ]. Since the main concern of this project is the obtainment of the acoustic impedance through practical measurements, more computational models will not be described in this document. Figure 2.3: Normalized Acoustic impedance of the porous material - Delany and Bazley model with σ = 50000Kgm 3 s 1. The response of the model is shown in figure 2.3 for a flow resistivity of σ = 50000Kgm 3 s 1. According to Delany and Bazley, both the real and imaginary part of the acoustic impedance of porous materials are inversely proportional to the frequency. Concerning the real part, the

25 2.2. Properties and categories of porous materials 11 acoustic impedance decreases as the frequency increases, having a minimum value ρc. The imaginary part, due to the negative sign of j, is a negative function that increases as the frequency increases, approaching to zero at high frequencies. Concerning the impedance of acoustically rigid surfaces, in theory, the impedance goes to infinity since normal movements of air are not allowed. However, there will always be measured a finite value that is the product of thermal losses on the surface. This heat transfer is due to temperature differences between the material and the gas where the waves propagate. Moreover, accurate measurements with repeatability are much more difficult to conduct. All the results for the acoustic impedance of materials throughout this document will be presented with respect to the characteristic impedance of air, and therefore on a dimensionless scale. 2.2 Properties and categories of porous materials Most porous materials are anisotropic, i.e. their physical properties vary with direction. This is particularly true for many fibrous materials that have a preferred fiber direction resulting from the process by which they are manufactured. It is also common experience that most porous materials are spatially inhomogeneous, i.e. their macroscopic properties vary randomly throughout the material. As this effect is acknowledged, the properties of porous materials are usually estimated be averaging the results for a number of individual samples. Nevertheless, it is most often assumed that materials are macroscopically homogeneous and isotropic. Nonlinearities can also be observed in porous materials, resulting from both nonlinear inertial effects related to pore fluid flow and nonlinear frame elasticity[11]. However, these phenomena usually appear at high excitation levels only. The most important macroscopic physical properties of a porous material are: flow resistivity, connected porosity, pore tortuosity (these first three together constituting the material s fluid-

26 2.2. Properties and categories of porous materials 12 acoustical properties), bulk density, in vacuo bulk modulus, shear modulus and loss factor (the latter four properties comprising the elastic properties of a porous material). The acoustical performance of the various types of porous media can be predicted with a knowledge of all or a sub-set of these properties. However, procedures to obtain these properties were not studied in this project, since one of the main concerns is related with the analysis of methodologies to measure the acoustic impedance in different sound fields. It should be pointed out that there is a direct link between the microscopic structure of a porous media (i.e. fiber radius, fiber shape, fiber orientation, fiber material density, number of fibers per unit material volume, etc.) and its macroscopic properties. Porous materials are categorized as being rigid, limp or elastic. Rigid and limp porous materials differ from elastic porous materials in the number of wave types that can propagate within them. Rigid or limp material can only support a single longitudinal wave type, whereas elastic materials can support three wave types simultaneously: airborne, frame and shear waves. Examples of elastic porous materials are polyurethane and polyimide foams.

27 Chapter 3 Reflection of sound waves 3.1 Constructive and destructive interference The reflection of sound waves in perfectly flat surfaces follows the law that states angle of incidence equals angle of reflection. The same behavior is observed with light and other waves, and by the bounce of a billiard ball off the bank of a table. The reflected waves interfere with the incident waves producing patterns of constructive and destructive interference. This leads to several acoustic phenomena, such as standing waves in rooms. The phase of the reflected sound waves from hard surfaces determines whether the interference of the reflected and incident waves will be constructive or destructive at a given point away from the surface. A wall or surface is described as having a higher specific acoustic impedance than air, and when a wave encounters a medium of higher acoustic impedance, there is no phase change upon reflection. That is, when the positive pressure part of a sound wave hits the wall, it will be reflected as a positive pressure. On the other hand, if a sound wave in a solid strikes an air boundary, the pressure wave which is reflected back into the solid will experience a phase reversal. That is, reflections off a lower impedance medium will be reversed in phase. For a totally reflecting surface, the sound pressure level is enhanced by 6 db when constructive interference occur. On the contrary, destructive interference will produce very deep interference 13

28 3.1. Constructive and destructive interference 14 Figure 3.1: Effect of the surface in the sound pressure. A totally reflecting surface is taking into account in order to show only constructive and destructive interference. minima whenever there occurs a difference in path length, meaning the difference in distance between the direct and reflected path that the sound travels from the source is an odd-numbered multiple of half a wavelength. In order to describe in more detail these phenomena, Figure 3.1 shows the effect of the surface on the sound pressure as a function of frequency for a specific position of source and receiver. As mentioned before, the position of the source or receiver do not affect the destructive interference location in frequency; what affects it is the difference between the direct and reflected path length. If the surface is no longer totally reflecting and some of the acoustical energy is lost when the sound waves strike the surface, a factor which informs about the amount of lost energy has to be included in the formulation. Such a factor is known as the reflection coefficient. As the material becomes softer, the interference minima will be shifted in frequency and its effect will not be so pronounced.

29 3.2. Plane-wave reflections Plane-wave reflections The plane-wave reflection coefficient is defined as the ratio of the sound pressure of the reflected wave to the sound pressure of the incident wave evaluated at the surface: R = P r P i (3.1) This coefficient is mainly dependent on the resistance that the surface offers to normal movement of air; however, it also contains information regarding standing waves on the material surface, and therefore it is usually a complex number. Figure 3.2: Plane-wave sound field hitting a perfectly flat surface at oblique incidence angle. When a harmonic plane-wave sound field strikes an infinite, plane, locally reacting surface at an arbitrary angle of incidence as in Figure 3.2, the sound pressure and the normal component of the particle velocity can be written as P(x, y, t) = P o (e j(ωt kx sin θ+ky cos θ) + R e j(ωt kx sin θ ky cos θ)) (3.2)

30 3.3. Spherical-wave reflections 16 U y = P o ρc ( cos θ e j(ωt kx sin θ+ky cos θ) + R cos θ e j(ωt kx sin θ+ky cos θ)) (3.3) As the surface is locally reacting, the characteristic impedance of the material is equal to the ratio of sound pressure to the normal component of the particle velocity evaluated at the surface (y = 0), and therefore P(x, 0) U y (x, 0) = ρc ( e jkx sin θ + R e jkx sin θ) cos θ (e jkx sin θ R e jkx sin θ) = ρc cos θ 1 + R 1 R = Z s = Z m (3.4) which implies that R = Z s cos θ ρc Z s cos θ + ρc (3.5) Thus, the plane-wave reflection coefficient R varies with the angle of incidence. R approaches 1 if θ tends to π/2 (grazing incidence). For a rigid surface, Z s is a big number and R becomes Spherical-wave reflections When sound waves from a point source strike a plane wall, they produce reflected circular wavefronts as if there were an image of the sound source at the same distance on the other side of the wall. If something obstructs the direct sound from the source from reaching your ear, then it may sound as if the entire sound is coming from the position of the image behind the wall. This kind of sound imaging follows the same law of reflection as one s image in a plane mirror. The most important and best reference work on reflection of spherical waves is still, without a doubt, the 72 pages paper by Sommerfeld [2]. Although he was interested in electromagnetic wave propagation, the solution to this problem has also applicability in propagation of sound waves. Several authors have worked in this subject in order to provide solution to the so-called Sommerfeld problem, which involves an arguably difficult integral. Various assumptions and

31 3.3. Spherical-wave reflections 17 approximations have been adopted by different authors [28-31], who where mainly searching for practical applications with regard to outdoor sound propagation. Chien and Soroka [26] presented an approximate solution for grazing incidence which is frequently used and even implemented in several engineering sound propagation prediction methods. In contrast, an exact method based on an image source distribution was originally proposed by Lindell and Alanen for vertical electric and magnetic dipoles in radio-wave propagation and it was applied in sound propagation by Di and Gilbert [34]. By representing the plane-wave reflection coefficient as the Laplace transform of an image source distribution, a well-behaved image integral, instead of the usual Sommerfeld integral, is obtained. This approach is valid for both locally and extended reacting surfaces. The exact method by Lindell and Alanen, as well as the approximation by Chien and Soroka are presented in this document and compared for different setups. However, the reader is referred to [26-27] and [34] for the derivation of the solutions. Note that the equations are based on the e jwt sign convention and thus it provides the complex conjugated of the used convention in this document. Figure 3.3: Point source and receiver above an infinite plane surface.

32 3.3. Spherical-wave reflections Laplace transform formulation by Lindell and Alanen For a locally reacting surface, the complex sound pressure at the receiver position when a point source is located above a perfectly flat surface is given by (see Figure 3.3) p 1 = e jkr 1 R 1 + Q e jkr2 R 2 (3.6) where Q = 1 2 k Z R 2 exp( jkr 2 ) 0 exp ( qk ) exp( jk x 2 + y 2 + (z + z s + jq) 2 ) dq (3.7) Z x 2 + y 2 + (z + z s + jq) 2 The first and second terms in equation (3.6) represent the direct sound field and the reflected sound field, respectively. The spherical reflection coefficient expresses the strength of the image source. As shown in equation (3.7), the spherical reflection coefficient Q is basically dependent on the geometry of the case, meaning the position of source and receiver and the normalized acoustic impedance on the surface (Z). For numerical calculations, the upper limit in the integral can be set to q max = λ Re(Z)2 + Im(Z) 2 Re(Z) (3.8) where λ is the wavelength and Re(Z) and Im(Z) are the real and imaginary parts of Z respectively Grazing incidence approximation by Chien and Soroka In outdoor sound propagation applications, the source and receiver are frequently separated by large horizontal distances, and approximations, such as R 2 >> (z + z s ) or θ π/2, can be applied. Particularly for these cases, the spherical reflection coefficient Q is given by Q = R(ω) + (1 R(ω)) F(d) (3.9)

33 3.3. Spherical-wave reflections 19 where F(d) = 1 + jd π exp( d 2 ) er f c( jd) (3.10) is referred to as the boundary loss factor. The factor d is known as the numerical distance and is given by d = jkr2 2 ( ) 1 Z + cos(θ) (3.11) and the factor er f c is called the complementary error function, which can be computed as er f c(x) = 2 π x exp( t 2 ) dt (3.12) Hence, the grazing incidence approximation is again dependent on the acoustic impedance on the surface, as shown in equation (3.11) Brief simulation analysis Simulations have been carried out in order to analyze the behavior and differences of the methods presented previously to calculate the spherical reflection coefficient Q. In addition, the plane wave reflection coefficient is included in all the results to compare the reflections of plane and spherical waves. The Delany and Bazley model has been used to calculate the surface acoustic impedance of the material for different values of flow resistivity. First, a typical outdoor sound propagation setup is analyzed, with 100 m as the horizontal distance between the source and receiver, and with the source and receiver at 1.5 m above the material. The spherical reflection coefficient for both models are shown in Figure 3.4 in comparison with the plane wave reflection coefficient for three different values of flow resistivity: 50000, and Kgm 3 s 1. Big differences between plane-wave and spherical-wave reflection coefficients are noticed at low frequencies, being broader the frequency range as the material is stiffer. Even for the stiffest

34 3.3. Spherical-wave reflections 20 Figure 3.4: Reflection Coefficients. The blue color represents the plane-wave reflection coefficient, the green color shows Q with the grazing incidence model, whereas the red color shows Q calculated using the La place transform model; h s = 1.5m, h r = 1.5m, and horizontal distance d = 100m. The dotted lines are, for all three colors, for a flow resistivity of Kgm 3 s 1. The dashed lines are for a flow resistivity of Kgm 3 s 1. Finally the dash-dot lines are for a flow resistivity of Kgm 3 s 1. Figure 3.5: Effect of the surface on the sound pressure level at the receiver position. The blue color represents the use of the plane-wave reflection coefficient, the green color the use of Q calculated with the grazing incidence model, whereas the red color the use of Q calculated using the La place transform model; h s = 1.5m, h r = 1.5m, and horizontal distance d = 100m. The dotted lines are, for all three colors, for a flow resistivity of Kgm 3 s 1. The dashed lines are for a flow resistivity of Kgm 3 s 1. Finally the dash-dot lines are for a flow resistivity of Kgm 3 s 1.

35 3.3. Spherical-wave reflections 21 material with a flow resistivity σ = Kgm 3 s 1, the plane-wave reflection coefficient becomes similar to the spherical reflection coefficient above, say, 2 khz. Both models for the calculation of the spherical reflection coefficient predict the same spherical reflection coefficient as expected, since a grazing incidence setup is under analysis (the curves are superposed). No matter the kind of material under analysis, the curves are always superposed for this particular grazing incidence configuration. On the other hand, the effect of the surface on the sound pressure level at the receiver position is shown in Figure 3.5. The stiffer the material, the more important is the effect of the surface in both constructive and destructive interference. The softer the material, the larger the differences in the effect of the surface between the calculation with the plane-wave reflection coefficient or the spherical reflection coefficient at low frequencies. This is clearly due to larger differences in the reflection coefficients as the material is softer. Figure 3.6: Reflection Coefficients. The blue line represents the plane-wave reflection coefficient, the green line shows Q with the grazing incidence model, whereas the red line shows Q calculated using the La place transform model; h s = 0.6m, h r = 0.01m, and horizontal distance d = 0m.

36 3.3. Spherical-wave reflections 22 The next step would be to analyze an extreme case, where the grazing incidence approximation for the calculation of Q should present problems. A normal incidence setup with height of the source h s =60 cm and height of receiver h r = 1 cm has been chosen for this analysis. This setup is only analyzed for a specific kind of surface with a flow resistivity σ = 50000Kgm 3 s 1. The resulting reflection coefficients are shown in Figure 3.6. Concerning the plane-wave reflection coefficient, it seems that larger discrepancies are found with the spherical reflection coefficient when source and receiver are separated by long distances. For this particular setup, the plane-wave reflection coefficient presents differences, but not as much as in Figure 3.5. In comparing the two models for the calculation of the spherical reflection coefficient Q, there is a very small difference throughout the whole frequency range. Nevertheless, these kind of differences produce important effects for some applications. More deviation was expected since one of the models is an approximation for grazing incidence, however, the Chien and Soroka model seems to present still good approximations for normal incidence. This was reported by authors such as Salomons [3]. Figure 3.7: Effect of the surface on the sound pressure level at the receiver position. The blue color represents the use of the plane-wave reflection coefficient, the green color the use of Q calculated with the grazing incidence model, whereas the red color the use of Q calculated using the La place transform model; h s = 0.6m, h r = 0.01m, and horizontal distance d = 0m.

37 3.3. Spherical-wave reflections 23 Finally, the effect of the surface in the sound pressure level at the receiver position is shown in Figure 3.7. At such a small distance from the source to the receiver, the surface tend to produce constructive interference in most of the frequency range. Several different setups could have been analyzed. However, it is thought that the presented analysis is enough to cover what is required in this project.

38 Chapter 4 Means to measure the acoustic impedance As discussed in Chapter 2, given a locally reacting surface, the measurement of the complex sound pressure and the normal component of the particle velocity on the surface would represent the acoustic impedance of materials. However, for non-locally reacting materials, the acoustic impedance varies with the angle of incidence of the sound field and, therefore, it might be necessary to average over some measurements in order to more accurately represent the acoustic characteristics of the material. In this context, the behavior of pressure microphones has been studied for several years and they seem to offer accurate and stable measurements in a broad dynamic range. On the other hand, the measurement of particle velocity components has been a subject for several years and still the available methods offer the measurement of this physical quantity with certain restrictions. The most successful method of measuring particle velocity in air is the two microphone method (or p-p method ), which makes use of two closely spaced pressure microphones and relies on a finite difference approximation to the sound pressure gradient[5]. 24

39 4.1. Intensity Probe as a tool to measure Z of porous materials Intensity Probe as a tool to measure Z of porous materials Figure 4.1: Bruel&Kjær sound intensity probe kit Type Since the specific acoustic impedance is defined as the ratio of the complex amplitudes of sound pressure and particle velocity of a single frequency component at a point in space (Z = P/U), and the conventional two microphone intensity probe measures/calculates these components, this equipment can be used to obtain the acoustic impedance anywhere in a sound field (see Figure 4.1). The following approximations are used to calculate the complex pressure and particle velocity when using the intensity probe P = P 1 + P 2 2 U = P 1 P 2 jwρd (4.1) (4.2)

40 4.2. Microflown and its application to acoustic impedance measurements 26 here, P 1 and P 2 are the outputs of microphone 1 and 2, and d is the separation distance between microphones. The transfer function H pu can be calculated as H pu = U P = P U P 2 (4.3) where P is the complex conjugated of the sound pressure. By introducing equation (4.1) and (4.2) into equation (4.3) H pu = S 11 S 22 2 jim(s 12 ) S 11 + S Re(S 12 ) (4.4) where S 11 and S 22 are the auto-spectra of microphone 1 and 2 respectively and S 12 is the crossspectrum. Therefore, H pu can be also obtained in a relatively simple manner with the intensity probe by using a FFT analyzer. 4.2 Microflown and its application to acoustic impedance measurements Figure 4.2: PU regular intensity probe - microflown A new particle velocity transducer has recently become available (see Figure 4.2). This equipment is denominated as Microflown and consists of two very closely spaced heat wires. The

41 4.2. Microflown and its application to acoustic impedance measurements 27 measurement principle is based on the detection of the temperature difference between these two resistive sensors. A traveling acoustic wave causes a time-varying heat transfer from one sensor to the other. The subsequent temperature difference results in a time-dependent difference between both electrical resistance values, which quantifies the particle velocity in a linear matter[13]. The fact that the sensitivity of the microflown decreases with the frequency due to the effects of diffusion and heat capacity makes important the calibration of the transducer. This can be performed by exposing the transducer to a sound field with a known relationship between the sound pressure and the particle velocity. The measured response of the device is compared with the theoretical expression, and a correction factor can be obtained as CF = H pu theoretical H pu measured (4.5) This correction factor should be multiplied with the response of the consecutive measurements. In this context, measurements have been carried out in three different well-known sound fields in order to obtain the correction factor CF. PULSE system Labshop was used in all the measurements, and the analyzer settings were assigned to include a frequency range of 0 to 3.2 khz with a frequency resolution of 1 Hz. Furthermore, Hann windowing was used during measurements due to its high spectral accuracy with stationary signals. The theoretical sound field expressions at specific positions are described as follow: Inside of a standing wave tube with a rigid surface on the termination, the ratio of the particle velocity to the sound pressure at a distance l from a rigid termination is given by H pu tube (l) = U(l) P(l) = j tan (kl) ρ c (4.6) Monopole in free field conditions, where the complex relation between the particle veloc-

42 4.2. Microflown and its application to acoustic impedance measurements 28 ity and the sound pressure at any point in the space is given by H pu f f = 1 ( ρ c ) jkr (4.7) and where r is the distance between source and receiver. Monopole on partial free field condition, where the presence of a rigid surface can be represented as an imaginary source that contributes to the particle velocity and sound pressure at the receiver position. The relation between the particle velocity and the sound pressure in this case is thus given by H pu p f f = 1 ( ) ρc jkr 1 1 ( 1+ jkr 2 1+ jkr 1 ) ( R1 R 2 ) 2 e jk(r 1 R 2 ) 1 + ( R 1 R 2 ) e jk(r 1 R 2 ) (4.8) where R 1 and R 2 are the direct and reflected paths, respectively. The response of a monopole is generated by a 60 mm diameter two-way coincident-source loudspeaker unit produced by KEF, mounted in a rigid plastic sphere with a diameter of 270mm. An analysis of this source has shown that it presents good agreement with the sound field produced by a monopole; much better than a common two-ways loudspeaker, where the different location of diaphragms produces interference [18]. Figure 4.3 illustrates the free field calibration factor in comparison with the one obtained in the standing wave tube. It is clear the agreement between both curves and the fact that the free field calibration factor is not affected by the tangent function in the theoretical expression for the standing wave tube. It is important to mention that the measurement on the standing tube was conducted at approximately 13 cm from the rigid surface. In the free field measurements, the microflown probe was located at 60 cm from the source.

43 4.2. Microflown and its application to acoustic impedance measurements 29 Figure 4.3: Calibration factor for the microflown probe. The amplitude and phase are presented for two different sound field analysis. The continuous line was obtained in a standing tube, whereas the dashed line was obtained in free field conditions with a monopole and distance source-receiver of 60 cm. On the contrary, Figure 4.4 shows how difficulties may arise in the calibration of the PU probe while using partial free field conditions with a rigid surface. The source in this case was located at 1 m above the rigid surface and the receiver was located at different heights, as shown in the legend of Figure 4.4. The result is compared with the correction factor obtained in the free field conditions. This was attempted in order to determine the applicability of in-situ calibrations. Really close to the rigid surface, the amplitude of the normal particle velocity is minimized; this is also possible to observe in the figure. The best result could be the one obtained at 11 cm from the surface, which gets close to the free field calibration. Since the surface is finite, reflections on the boundaries are critical factors on the validation of the image source model.

44 4.2. Microflown and its application to acoustic impedance measurements 30 Figure 4.4: Calibration factors for the microflown probe. A monopole is located in partial free field conditions, at 1 m above a rigid surface. Several receiver positions were used to measure the response of the microflown probe and obtain calibration factors for this particular sound field. As reference, the correction factor obtained in complete free field conditions is included. Figure 4.5: Calibration factor for the microflown probe. This curve was obtained on the combination of the standing wave tube calibration factor at low frequencies (below 400 Hz) and the free field calibration factor at high frequencies (above 400 Hz).

45 4.2. Microflown and its application to acoustic impedance measurements 31 A combined calibration factor has been obtained by using the data of the standing wave tube at low frequencies (below 400 Hz) and the data from the free field conditions at high frequencies ( Hz). The result of this combination is given in Figure 4.5, and constitutes the used calibration factor in the consecutive measurements. Furthermore, a comparison with the calibration factor presented in reference [] is shown in Figure 4.6. Good agreement is found in the whole frequency range.. Figure 4.6: Calibration factors for the microflown probe. The calibration factor obtained in this procedure is compared with a reference, where a monopole on sphere was used to calculate the correction factor. It is important to mention that the reference calibration obtained with the monopole on a sphere is given in 1/12 octave bands, whereas the experimental calibration factor obtained in this procedure is presented with a resolution of 1 hz.

46 Chapter 5 Transfer Function Method 5.1 Introduction The transfer function method was developed by Chung and Blaser in the 80 s [20]. In modern times, this procedure to measure the acoustic impedance of materials has already been established as an International Standard [4]. The method is well-known as the Transfer Function Method, since it is based on the measurement of the complex acoustic transfer function between two pressure microphones located along a standing wave tube (See Figure 5.1). Figure 5.1: Impedance tube. 32

47 5.1. Introduction 33 Here, a test sample is mounted at one end of a straight, rigid, smooth, and airtight impedance tube. Plane waves are generated in the tube by a sound source in the other extreme and normal incidence properties of materials are obtained out of this method. In order to correct the measured transfer function data for channel mismatch, the method requires repeated measurements with interchanged channels. There have been a number of studies dealing with the accuracy of the transfer function method, concerning mainly with the effect of the mounting conditions on the acoustic impedance measurement. An interesting result was reported by Cummings, who studied the effects of air-gaps around the sample on the measurement of the acoustic impedance [21]. He reported that the effects of air-gaps increase as frequency decreases and media of high flow resistivity tend to be more susceptible to measurement errors incurred by air-gaps than those with low flow resistivity. Moreover, Bodén and Åbom came up with some suggestions in the 90 s, which can help to ensure accurate results when using the transfer function method [22]. They conclude that sources of error are first uncertainty in the input data, and second the sensitivity of the calculation formulas to errors in the input data. In order to minimize errors in the transfer function, they suggest that the overall length of the tube should be kept small, in practice L 5 10 duct diameters, the source end of the duct should be as non reflective as possible, and the microphone near to the surface should be placed as close as possible to the sample; however, care must be taken since the microphone might be located within the nearfield. The minimum sensitivity to errors in the input data occurs when the distance between microphones is equal to a quarter of a wavelength, where the maximum difference between the sound pressure at each microphone is reached. Bodén and Åbom suggest that, in order to avoid high sensitivity to errors in the input data, the method should only be used when 0.1π < ks < 0.8π. Following this conclusion, the reference tube in this project should provide results with poor sensitivity to errors in a range of frequency 430Hz < f < 3440Hz. On the other hand, a Round-Robin test has been carried out by several authors inside Europe meaning the transfer function, the distance between microphones and the distance between the sample under test to the further pressure microphone

48 5.2. Description of the method 34 [23]. The same samples of material were analyzed in different laboratories using the transfer function method. Considerable variations in the measured spectra for the acoustic absorption coefficient have been observed both in results between individual samples and individual laboratories. The variations are attributed to: (1) inhomogeneity of the provided material sample; (2) methods of sample preparation; (3) mounting and structural conditions during test; (4) diameter of the standing wave tube; (5) signal processing method. This shows that even for this method, which provides the most stable results, the variance may be considerable. 5.2 Description of the method The main equations to calculate the surface acoustic impedance using the transfer function method are presented in this section. The reader is referred to [4] for a complete description of the method. The sound field inside of the tube consists of one incident wave and one reflected wave and the sound pressure at any position along the tube is given by the addition of two waves traveling in opposite directions. According to the following sketch: The sound pressure in microphones 1 and 2 are given by P(x 1,2 ) = P i (e jkx 1,2 + R e jkx 1,2 ) (5.1) where R represents the plane-wave reflection coefficient. The transfer function from the microphone at position one to the microphone at position two is defined by the complex ratio

49 5.2. Description of the method 35 P(x 1 )/P(x 2 ) and is denoted as H 12. This can also be calculated as H 12 = P(x 1) P(x 2 ) = S 12 S 11 = S 22 S 21 (5.2) where S 12 or S 21 are the cross-spectra between channels, whereas S 11 and S 22 are auto-spectra of channels 1 and 2 respectively. If the transfer function is acquired, the plane-wave reflection coefficient as a function of H 12 can be calculated as R = H 12 e jks e jks H 12 e 2 jkx 2 (5.3) where s is the distance between microphones and x 2 is the distance from the further microphone to the sample under analysis. Once R is calculated, the specific acoustic impedance of the material is given by Z s = ρc 1 + R 1 R (5.4) and the absorption coefficient is calculated as α = 1 R 2 (5.5) Concerning the working frequency range, two clear things are defined. First, plane-wave propagation has to be ensured, and therefore the dimensions of the tube defines the upper working frequency f u ; the shorter considered wavelength should be approximately two time the larger cross-section dimension of the tube. Second, the method will break down whenever the microphone separation is equal to multiples of half a wavelength, since the microphones acquire the same information. Regarding the lower working frequency, this is dependent on the quality of microphones; for a pair of well-matched transducers, the differences on amplitude, as well as in phase should be possible to notice even for very low frequencies. Furthermore, no swapping technique would be needed. In spite of that, the standard intends to avoid any kind of problem suggesting a series of guidelines in order to set the working frequency range, and ensure accu-

50 5.3. Experimental investigation 36 rate results. Variations of this method, by using the microflown intensity probe, which makes the measurement of sound pressure and particle velocity possible at the same point, have been attempted (referred to as PU method). The method in this case would be even simpler; the sound pressure is equal to equation (5.1), whereas the particle velocity component is given as U x (x) = P i ρc (e jkx R e jkx ) (5.6) It follows that the reflection factor out of a measurement at the position x 1 is given by R = 1 ρch pu 1 + ρch pu e 2 jkx 1 (5.7) where H pu is the transfer function from the sound pressure to the particle velocity (U/P) and x 1 is the distance from the sample to the PU probe. 5.3 Experimental investigation For experimental determination of acoustic properties, the transfer function method using the two-microphone technique is currently the most stable and confident method available and thus there is no clear need to achieve several measurements of different materials. In contrast, the performance of the microflown probe is of interest and it has been analyzed on a tube with rectangular cross-section, which ensures plane-wave propagation up to around 2.5 khz (see Figure 5.1). A pair of microphones Bruel&Kjær - type 4192 and a half inch regular PU microflown probe have been used. The separation between microphones is 4 cm and the further microphone is located at approximately 8.5 cm from a sample of 50mm Rockwool A-Batt. Once the measurement with the microphones was accomplished, the microflown probe was located at the same position as the further microphone. The hole where the other microphone was placed was

51 5.3. Experimental investigation 37 then closed with a metal plug. PULSE system Labshop was used in all the measurements, and the analyzer settings were assigned to include a frequency range of 0 to 3.2 khz with a frequency resolution of 1 Hz. Furthermore, Hann windowing was used during measurements due to its high spectral accuracy with stationary signals. Figure 5.2: Absorption and Reflection properties. Comparison PP method and PU method for a 50mm Rockwool A-Batt. Calibration on a standing wave tube with rigid surface has been used for the microflown as in equation (4.6). The distance between the PU probe and the rigid termination is approximately 13.5 cm. The calibration of the PU probe produces irregularity around 630 Hz where x 1 is equal to a quarter of a wavelength. This is an effect of the tangent function on the calibration, which goes to infinity whenever x 1 is equal to an odd multiple of a quarter of a wavelength. This phenomenon is shown on Figure 5.2, where the absorption and reflection properties of this particular material are exhibited for both PP and PU method. The results show that the PU method can be as accurate as the PP method. However, the calibration of the PU probe seems

52 5.3. Experimental investigation 38 Figure 5.3: Normalized Acoustic Impedance. Comparison PP method and PU method for a 50mm Rockwool A-Batt. still an important issue, since no swapping technique is applicable. The normalized acoustic impedance for the same material is presented in Figure 5.3 and the same effect is certainly exposed. Figure 5.4: Normalized acoustic impedance variation when errors in the distance sample-probe are committed while doing the calibration of the probe. Variance of ± 10mm are considered. In addition, an error analysis has been carried out in order to find the sensitivity of the PU probe results when distance errors are committed. In this sense, acoustic impedance measure-

53 5.3. Experimental investigation 39 Figure 5.5: Normalized acoustic impedance variation when errors in the distance sample-probe are committed while doing the actual measurement. Variance of ± 10mm are considered. ments are presented in Figure 5.4 with an introduced variance of ±10mm in the surface-to-probe distance, while doing the calibration. Figure 5.5 presents the same variance in the same distance, but while doing the actual material measurement instead. The error in the calibration seems to have an effect mainly at low frequencies. This error is introduced in the tangent function of the correction factor. The correction factor has been analyzed separately and negligible changes in amplitude are found when varying the distance to the rigid surface. However, it seems that these differences in amplitude cause severe differences in the real part of the acoustic impedance. The error in the actual measurement puts the estimation of the surface impedance at a different position along the x-axis, and thus a larger or lower particle velocity component would be acquired with regard to the estimation at the correct position. Therefore, the acoustic impedance curve should be shifted in amplitude. This shifting is observed in Figure 5.5; however, it seems to effect the real more than the imaginary part.

54 5.4. Portable impedance tube analysis Portable impedance tube analysis (a) (b) Figure 5.6: (a) Portable impedance tube on a handy tripod; (b) Rubber ring on the termination of the portable impedance tube. The unique performance of the transfer function method leads to the implementation of a portable impedance tube, which, under certain circumstances, may make possible the measurement of acoustic impedance in-situ. Few papers have been found regarding portable standing wave tubes for in-situ applications. In the early 60 s, Berendt and Schmidt presented a portable impedance tube which had a narrow operational frequency range of 400 to 900 Hz.[24]. The open end of the tube was terminated with an annular gasket of soft neoprene rubber to provide an airtight seal against and prevent marring of the acoustical material under test. The measurements of the maximum-minimum sound pressure ratio for a given sample were repeatable within ±0.1dB. On the other hand, Allan J. Zuckerwar presented in the early 80 s an acoustic ground impedance meter which was able to measure specific ground impedance at normal incidence up to 50 times the specific acoustic impedance of the air [25]. The method was based on direct pressure and volume velocity measurement by means of a microphone and an optical monitor

55 5.4. Portable impedance tube analysis 41 of the piston displacement which permits measurement of the phase angle between the volume velocity and the sound pressure. From this, the real and imaginary parts of the impedance can be evaluated. In this context, a portable impedance tube has been built and installed in a tripod. This configuration allows one to place the termination of the tube in any direction (see Figure 5.6(a)). The response of the new tube is analyzed in this section and comparisons with theoretical expressions in specific cases, such as open tube and rigid termination, are presented. Subsequently, the results for specific materials are compared with the results given by the reference tube (see Figure 5.1). The new portable tube has a circular cross-section with 4 cm diameter and the separation between microphones is 3 cm. Hence, the upper working frequency is approximately 4.2 khz and the first frequency where the transfer function methods breaks down occurs at around 5.5 khz. Figure 5.7: Reflection and Absorption properties - Open tube case. The blue curve Levine and Schwinger model, whereas the red dotted curve new portable tube. Initially, the performance of the portable tube is analyzed in two extreme cases: open tube,

56 5.4. Portable impedance tube analysis 42 where no resistance to movements of air exists in the termination of the tube, and rigid surface, where the particle velocity at the termination should be close to zero. Figure 5.7 shows the reflection and absorption properties of the open portable tube in comparison with Levine and Schwinger model [8]. As it can be seen, the portable tube presents good agreement in most of the frequency range with the approximation to the reflection coefficient of an unflanged circular pipe, presented by Levine and Schwinger. This approximation is valid as long as ka < 1, where a is the radius of the tube. At ka = 1 the absolute value of the reflection coefficient is in excess of the correct value by less than 3%. Particularly for this tube, it occurs around 2.7 khz. The real part of the reflection coefficient is 1 as it should be, since a release is found at the termination. Concerning the absorption coefficient, thermal losses explain the increase of absorption at high frequencies. The same just presented comparison, but instead for the normalized acoustic impedance is shown in Figure 5.8. The normalized acoustic impedance on the termination of the tube should be minimal since the air does not represent any resistance. Good agreement is found between the portable tube results and the Levine and Schwinger model below 1 khz. Figure 5.8: Normalized Acoustic Impedance - Open tube case. The blue curve Levine and Schwinger model, whereas the red dotted curve new portable tube. On the other hand, Figure 5.9 shows the reflection and absorption properties of a rigid sur-

57 5.4. Portable impedance tube analysis 43 face analyzed with the portable tube in comparison with the reflection coefficient when assuming an isothermal process [6]. The real part of the reflection coefficient is quit close to 1 as expected, while the imaginary part keeps consistently close to 0 in most of the frequency range. The absorption offered by the surface is minimal since most of the sound field is reflected back into the tube. Figure 5.9: Reflection and Absorption properties - Rigid surface case. The blue curve Reflection coefficient for a rigid surface in an isothermal process, whereas the red dotted curve new portable tube. In reality, unsteady heat transfer at the rigid surface will act as a sink of sound reducing slightly the reflection coefficient, and thus some absorption will be always measured. This heat transfer is the result of a difference between the surface temperature, which remains typically constant, and the bulk temperature of the gas, which varies with the adiabatic pressure fluctuations. The experimental acoustic impedance obtained with the portable tube for a rigid surface is compared with the normalized acoustic impedance of a rigid surface under isothermal con-

58 5.4. Portable impedance tube analysis 44 ditions. This comparison is shown in Figure Concluding, the portable tube results are in reasonable agreement with the references in both extreme analyzed cases. Figure 5.10: Normalized Acoustic Impedance - Rigid surface case. The blue curve normalized acoustic impedance for a rigid surface in an isothermal process, whereas the red dotted curve new portable tube. Figure 5.11: Reflection and Absorption properties - porous material placed in the tubes (50 mm Rockwool A- Batt). The blue curve reference tube, whereas the red dotted curve new portable tube.

59 5.4. Portable impedance tube analysis 45 Following with the analysis, samples of porous materials (50 mm Rockwool A-Batt) have been cut and introduced in the portable tube, as well as in the standard reference tube. The results will be shown in a frequency range between 20 Hz and 2 khz, since the standard reference tube presents high-frequency modes above 2.5 khz. Figure 5.11 shows the reflection and absorption properties of this particular material. The portable tube results seem to be in quite good agreement with the reference tube. However, for the normalized acoustic impedance, the real part presents serious discrepancies in comparison with the reference tube (shown in Figure 5.12). This might be caused by air leakage at low frequencies. Figure 5.12: Normalized Acoustic Impedance - porous material placed in the tubes (50 mm Rockwool A-Batt). The blue curve reference tube, whereas the red dotted curve new portable tube. A rather stiff material is now introduced in the tubes and the same comparison is accomplished (see Figures 5.13 and 5.14). As the material becomes stiffer, the measurement becomes much more uncertain. Slight errors on the mounting procedure will affect more the accuracy on the acoustic impedance measurement. This will appear in any procedure which involves the mounting of samples inside structures. Air-gaps around the sample are just an example of what could occur. Furthermore, errors in geometric measurements bring much more consequences on the accurate measurement when analyzing stiff surfaces. The comparison between both tubes

60 5.4. Portable impedance tube analysis 46 present serious discrepancies and none of them could be used as a reference. Nevertheless, the orders of magnitude in the reflection coefficient, for instance, are similar. Figure 5.13: Reflection and Absorption properties - stiff material placed in the tubes. The blue curve reference tube, whereas the red dotted curve portable tube. Figure 5.14: Normalized Acoustic Impedance - stiff material placed in the tubes. The blue curve reference tube, whereas the red dotted curve new portable tube.

61 5.4. Portable impedance tube analysis 47 An analysis has been conducted in order to analyze the behavior of the new portable tube for in-situ measurements. The samples have not been mounted inside the portable tube from now on in all the results; instead, the samples are in-situ. For the interface, a rubber ring has been introduced at the termination, as an attempt to provide an airtight seal. This is shown in Figure 5.6(b). A large sample of porous material (40 mm Rockfon Samson) is analyzed with the portable tube and comparisons with the reference tube results for a small cut sample are presented. Figure 5.15 shows the absorption and reflection properties for measurements in both the reference and portable tube. Figure 5.15: Reflection and Absorption properties - Schematic analysis of portable tube response for in-situ evaluations. The continuous line represents the reference tube measurement, the dashed line shows the results when the tube is located at 1 cm above the material, the dotted line is the result when the tube is on the surface, not pressing the material, and finally the dash-dot line shows the result when the tube on the surface, pressing the material. The porous material is a large sample of 40 mm Rockfon Samson. If the measurement is performed while keeping the tube 1 cm above the surface, the behavior is similar to an open tube (dashed line). Special attention should be taken in the real part of

62 5.4. Portable impedance tube analysis 48 the reflection coefficient, which is close to minus one in most of the frequency range. Regarding the absorption, the curve drops down as expected, since only air is found at the termination. The results concerning the normalized acoustic impedance are shown in Figure 5.16 in comparison with the reference tube result. As it can be seen, for a measurement at 1 cm above the material, the measured acoustic impedance is similar to the result in the open tube, i.e. close to zero in most of the frequency range. Figure 5.16: Normalized acoustic impedance - Schematic analysis of portable tube response for in-situ evaluations. The continuous line represents the reference tube measurement, the dashed line shows the results when the tube is located at 1cm. above the material, the dotted line is the result when the tube is on the surface, not pressing the material, and finally the dash-dot line shows the result when the tube on the surface, pressing the material. The porous material is a large sample of 40 mm Rockfon Samson. If the measurement is performed with the tube on the surface while not pressing at all the material (dotted line in Figures 5.15 and 5.16), the real part of the reflection factor increases but still presents a negative value, which indicates undesired holes or openings, and thus the results approach those given with an open tube. It is clear that at the termination the sound waves are not obliged to reflect back into the tube and, instead, they are released less than in the open tube. The equation to calculate the absorption coefficient is only dependent on the absolute value of the reflection coefficient and therefore, the results on the bottom of Figure 5.15 are completely erroneous. About the measurement of the normalized acoustic impedance, the result seems to

63 5.4. Portable impedance tube analysis 49 improve and approaches the reference tube curve for high frequencies. However, for low frequencies the curve is still around zero. Now, if the measurement is performed with the tube on the surface while pressing the material (dash-dot line in figures 5.15 and 5.16), the real part of the reflection factor becomes positive but still quite different in comparison with the measurement given by the reference tube. This indicates the presence of leakage. Regarding the normalized acoustic impedance, the real part becomes similar in magnitude with that from the reference tube results above, say, 200Hz. However, the imaginary part is minimal and keeps around zero for the whole frequency range. Figure 5.17: Reflection and Absorption properties - Big and small sample comparison. The dashed line shows the big sample results, whereas the dash-dot curve exhibits the results with the small sample. On the other hand, propagation of waves in the material could constitute an important factor. In order to see whether the release is due to waves propagating in the medium or mainly due to leakage, the result given with a small sample is compared with the result given with a big sample

64 5.4. Portable impedance tube analysis 50 of the same material. The same height and pressure over the material is attempted. Although the measurement is completely erroneous, the comparison for reflection and absorption properties is shown in Figure 5.17 in order to exhibit this phenomenon. No considerable differences are observed. In both measurements, the same release is present at the termination and only slight differences are found. Therefore, propagation of waves in the medium does not seem to be the main reason why the results are given a release, and on the contrary the air leakage seems a more critical factor. The four-pole method has been used in order to study the effect of small undesired holes or openings on the interface between the termination of the tube and the sample under analysis [5]. The same dimensions of the portable tube have been considered for the simulations. The sound field inside of the tube with a sample of porous material at the termination has been simulated. The material has a flow resistivity σ = Kgm 3 s 1, and the acoustic impedance has been calculated as in the Delany and Bazley model. A small hole at the interface is represented by the radiation impedance of a circular piston of radius a and cross-section S in an infinite baffle. An approximation valid for low frequencies or for a small radius a is been used to calculate the radiation impedance. The expression is given by Z a ρc S ( (ka) j 8ka ) 3π (5.8) This approximation is valid as long as ka < 0.5. The biggest considered hole has a radius of 1 mm and thus, the approximation is valid in the whole frequency range. The effect of a small hole on the measurement of the acoustic impedance of the material is shown in Figure When the hole is closed the simulation gives perfect agreement with the Delany and Bazley model and the curves are over-written. When a small hole is found on the interface, the measurement of the acoustic impedance is seriously compromised in both real and imaginary components. Concerning the real part, the larger the hole, the broader the frequency range where the acoustic impedance drops down to minimal values at low frequencies. The smaller

65 5.4. Portable impedance tube analysis 51 Figure 5.18: Normalized acoustic impedance - Simulation on the effect of a small leakage on the interface tubematerial when measuring the acoustic impedance of materials. Holes of 0, 0.2, 0.4, 0.6, 0.8 and 1 mm of radius are considered. the hole, the larger the frequency range at high frequencies where good approximation to the acoustic impedance is obtained. Regarding the imaginary part, even the smallest hole produces already oscillations below, say, 1 khz. The oscillations are always around the zero value. The smaller the hole, the larger the frequency range where the hole does not affect the measurement, although the oscillation at low frequencies is larger. The larger the hole, the more the curve is shifted upward at high frequencies. This is basically what we have been able to see in Figure 5.16: the real part of the acoustic impedance drops down due to leakage at low frequencies and, the more effort is applied to reduce the leakage, the better approximation one obtains at high frequencies. In the imaginary part, all the measurements have given values around zero, and the more effort is done to close the leakage, the more the curve is shifted upward at high frequencies. Figure 5.19 exhibits the effect of leakage on the reflection and absorption properties. The larger the hole, the broader the low frequency range affected in the real part of the reflection factor, the broader the low frequency range where the imaginary part of the reflection coefficient is positive, and thus the broader the low frequency range where low absorption coefficient is found. Com-

66 5.4. Portable impedance tube analysis 52 paring with the measurements (given in Figure 5.15), the imaginary part in all three positions of the portable tube has shown positive values. The real part presents negative values in the first two steps and in the third it becomes positive but very small. More pressure had to be applied in order to measure the correct result; however, properties of the material may be changed. Figure 5.19: Reflection and absorption properties - Simulation on the effect of a small leakage on the interface tube-material when measuring the reflection and absorption coefficient. Holes of 0, 0.2, 0.4, 0.6, 0.8 and 1 mm of radius are considered. Hence, it can be concluded that air leakage constitutes a serious problem in the measurement of acoustic impedance of porous materials using the portable tube. The measurement is first influenced at low frequencies and the affected frequency range becomes broader as the leakage is larger. By pressing the tube on the material to avoid leakage, one can improve the results; however, the properties may be changed.

67 5.4. Portable impedance tube analysis 53 Figure 5.20: Reflection and Absorption properties - stiff piece of wood. The blue continuous line corresponds to the measurement with the reference tube - the sample has been cut an introduced in the tube. The red-dashed line corresponds to the measurement in-situ with the portable tube, which is been only pushed into the material as an attempt to avoid leakage. If the material is not affected by the added forcing of the tube as an attempt to avoid leakage, one would expect good results. The measurement with the rigid surface presented before demonstrates that. The sealing method constitutes the breaking point of the portable tube. Thus, one could argue that the portable tube could be used for stiff materials, rather than for limp materials. In this context, the properties of a stiff piece of wood have been measured in the reference tube as well as the in-situ. The comparison is shown for reflection and absorption properties in figure (5.20)), whereas for normalized acoustic impedance is shown in Figure Concerning the reflection factor, both measurements provide results close to 1 and thus very poor absorption is offered. A closer examination reveals important differences. Regarding the normalized acoustic impedance, these differences are quite important in the whole frequency range; however, the reference tube provides a reference only in the case of perfect mounting of the material into the tube, which is hardly possible to achieve. Since a rather stiff material is under analysis

68 5.4. Portable impedance tube analysis 54 and the uncertainty due to mounting based errors is quite high, the reference tube result does not provide any confidence and it is no longer a reference. On the other hand, the results with the portable tube seem reasonable, there is no important physical contradiction to point out and the measurement for hard surfaces will be, as in any method, difficult to make. Statistical analysis is recommended instead of direct measurements, in order to analyze the dispersion of measurements and find the uncertainty. Figure 5.21: Normalized acoustic impedance - stiff piece of wood. The blue continuous line corresponds to the measurement with the reference tube - the sample has been cut an introduced in the tube. The red-dashed line corresponds to the measurement in-situ with the portable tube, which is been only pushed into the material as an attempt to avoid leakage.

69 Chapter 6 Free field method 6.1 Introduction Free field measurements have been used by several authors to evaluate the surface acoustic impedance of materials for which Kundt s tube measurements are not possible or reliable. Numerous methods have been presented, all of them related to different approximations of the interference sound field, which is formed when a point source is located close to a surface. The transfer function method discussed in the previous chapter could be applied in free field conditions; however, the sound field is much more complicated to describe. Evaluating impedance from pressure and velocity measurements was first suggested by Kurze in 1968[6]. However, it was Allard in the 80 s who started to discuss this method [37], probably related with the fact that intensity meters became available and, therefore, the measurement of particle velocity was a lot easier. Preliminary results for the acoustic impedance on the surface were obtained out of the measurement of pressure and particle velocity very close to the material, say 2 centimeters above, while assuming plane wave propagation. Therefore, the point source was usually located at large distances from the sample. Differences could be expected at low frequencies since a spherical sound field was emitted by the source. Allard reported, in 1984, a comparison of this method 55

70 6.1. Introduction 56 with Kundt s tube measurements for a sample of glasswool. The result showed similar results for frequencies higher than 500 Hz [38,39]. He concluded that the method cannot be used at low-frequencies, arguing that the finite dimension of the material sample strongly modifies the weak flux of energy from the source toward the material. Locating the source closer would mean that the condition of a plane wave sound field would no longer be valid. Later on, improvements to the method were discussed by Allard and Champoux [40], which consisted of using pure tone signals instead of white noise, increasing microphone spacing, and improving the calibration precision. These improvements in the technique meant the extension of accurate measurement to a frequency as low as 250 Hz. Bias at low frequencies due to the assumption of a plane wave sound field was also demonstrated at this stage. After that, measurements have been attempted for normal and oblique incidence, as well as in-situ [41,43]. Summarizing the available information, if the distance from the source to the material is several meters, the acoustic field is approximately a plane wave near the microphones. At low frequencies, the measurements are less accurate due to finite dimensions of the sample and the reflections from the walls and the ceiling if the measurement is performed in-situ. Moreover, the output power of the sound source decreases at low frequencies, resulting in a decreasing signal to noise ratio. These effects are reduced when the source is set close to the sample. In this case, the plane-wave approximation becomes inaccurate and a more elaborate modeling of the sound field is needed. For this particular project, where methods to measure inside vehicles are analyzed, it is not practical to think about large distances source-to-surface. Thus, more elaborate approximations are implemented for a better description of the sound field. Furthermore, it does not make sense to complicate the measurement by using oblique incidence measurements. Hence, only normal incidence is taken into account. The performance of microflown for this application is of interest. The small size of this transducer makes it possible to set the transducer quite close to the surface and measure both the sound pressure and particle velocity at the same point. Thus, some improvement may be achieved in the acoustic impedance measurement with regard to the conventional Bruel&Kjær

71 6.2. Sound Field approximations 57 intensity meter, where the shape of the intensity probe does not allow one to set the microphones close to the surface; the closest possible distance is around 5 centimeters. On the other hand, the fact that the normal component of the particle velocity becomes smaller at the surface as the materials is stiffer could produce problems when measuring stiff materials. 6.2 Sound Field approximations Model 1: Plane-wave approximation Already mentioned before, the simplest approximation assumes a plane wave sound field and the calculation of the acoustic impedance on the surface is performed with ease as with the standing wave tube. Figure 6.1: Source above an infinite plane surface. The sound pressure ˆP and the normal component of the particle velocity U n at any point above the surface can be written as ˆP = P i (e jkz + R e jkz ) e jωt (6.1) U n = P i ρc (e jkz R e jkz ) e jωt (6.2)

72 6.2. Sound Field approximations 58 Thus, the dimensionless normalized acoustic impedance at any position would be given by Z = e jkz + R e jkz e jkz R e jkz (6.3) At the receiver position, the plane-wave reflection coefficient R can be calculated as R = Z 1 Z + 1 e2 jkd (6.4) where d is the distance from the surface to the receiver. The exponential in equation (6.4) represents the fact that the measurement is done at a certain distance from the surface. Therefore, from an above the surface impedance measurement, the plane-wave reflection coefficient R can be calculated using equation (6.4). Then, the approximation to the normalized acoustic impedance at the surface (z = 0) can be achieved by equation (6.3) Model 2: Image source model with plane-wave reflection coefficient A slightly more complicated model makes use of the image source model. The presence of a monopole above an infinite plane surface gives rise to an image source: Figure 6.2: Monopole above an infinite plane surface.

73 6.2. Sound Field approximations 59 The sound pressure at the receiver point can be written as ˆP = jωρg ( ) 1 4π e j(ωt kr1) 1 + R e j(ωt kr 2) R 1 R 2 (6.5) Here, the first term represents the direct sound field, and the second represents the reflected sound field, (from the image source). G is the volume velocity of the monopole, R is the planewave reflection coefficient, and R 1 and R 2 are the direct path and reflected path respectively. The normal component of the particle velocity at the same position is given by U n = jkg 4π [ 1 R 1 ( ) e j(ωt kr1) R(ω) jkr 1 1 R 2 ( ) ] e j(ωt kr 2) jkr 2 (6.6) Therefore, the dimensionless normalized acoustic impedance at the receiver position in this case would be given by Z = ( 1 1 R 1 1 R 1 e j(ωt kr1) + R(ω) 1 R 2 e j(ωt kr 2) jkr ) ( e j(ωt kr1) 1 R(ω) 1 R 2 jkr ) e j(ωt kr 2) (6.7) and the plane-wave reflection coefficient can be written as Z ( 1 jkr R(ω) = ) 1 Z ( 1 jkr ) + 1 ( R2 R 1 ) e jk(r 2 R 1 ) (6.8) Again as in the plane wave approximation, by measuring the ratio of the sound pressure and the normal component of the particle velocity somewhere above the material, the plane-wave reflection coefficient is obtained. As noticed, this model considers a spherical sound field, but assumes that the waves are reflected in the same way as plane waves. Therefore, the approximation to the acoustic impedance on the surface can be achieved by letting R 1 = R 2 = h s in equation (6.7). This results in Z = 1 + R(ω) 1 R(ω) 1 ( ) (6.9) jkh s where h s is the height of the source.

74 6.2. Sound Field approximations Model 3: Image source model with spherical reflection coefficient A third approach, which is far more complicated, includes the calculation of the spherical reflection coefficient Q. There have been numerous studies into the calculation of the spherical reflection coefficient. This problem has been solved and it has been presented previously in section (3.2). For plane waves, the calculation of the reflection coefficient R can be achieved by measurements at any distance from the surface as shown previously. However, the measurement above the surface does not provide enough accurate information for the calculation of Q since it depends on the position of source and receiver, and on the surface impedance. Hence, in order to calculate the spherical reflection factor, the acoustic impedance on the surface, impossible to obtain as a direct measurement, is needed. For measurement procedures, where no information about the material is known, the assumption of introduce approximations to the acoustic impedance for the calculation of Q has to be taken. Afterwards, measurements and predictions of the sound field can be compared in order to analyze the differences. Since only normal incidence is taken into account in this analysis, the calculation of equation (3.7) for the spherical reflection coefficient is reduced to Q = 1 2 k Z R 2 exp( jkr 2 ) 0 exp ( q k ) exp( jk (R 2 + iq)) dq (6.10) Z R 2 + iq Similar to equation (6.5), the sound pressure at the receiver point can be in this case written as ˆP = jωρg ( 1 4π e j(ωt kr1) + Q(ω) 1 ) e j(ωt kr 2) R 1 R 2 (6.11) In order to obtain the normal component of the particle velocity at the receiver position, the differential of the sound pressure with respect to the height has to be found. Since the spherical reflection coefficient varies with respect to the geometry of the problem, the normal component

75 6.3. Simulated Investigation 61 of the particle velocity is given by the following expression U n = jkg 4π [ 1 R 1 ( ) e j(ωt kr1) Q jkr 1 1 R 2 ( ) e j(ωt kr2) + Q jkr 2 z e j(ωt kr2) ] jkr 2 (6.12) Therefore, the dimensionless normalized acoustic impedance in this model would be given by Z = ( 1 1 R 1 1 R 1 e j(ωt kr1) 1 + Q R 2 e j(ωt kr 2) jkr ) e j(ωt kr 1) Q 1 R 2 ( 1 jkr ) e j(ωt kr 2) + Q z e j(ωt kr 2 ) jkr 2 (6.13) and the acoustic impedance on the surface can be estimated as Z = 1 + Q ( ) ( ) jkh s Q jkh s + Q z 1 jk (6.14) 6.3 Simulated Investigation Simulation 1: On the acoustical representation of porous materials through a measurement right above the surface As mentioned previously, the small size of the microflown makes the measurement of the sound pressure and the normal component of the particle velocity possible at small distances above a surface. When laying the probe down on the surface, the proper transducer-to-surface distance is not clear, and it is not indicated in the specifications of the probe. However, one estimates that both pressure and particle velocity are measured at approximately 5-6 mm. At such small distance from the surface, the ratio of sound pressure to the normal component of particle velocity could correctly express the impedance that the surface offers to normal movements of air particles. In other words, the measurement of the acoustic impedance at 6 mm above the surface could be enough to characterize the surface with its acoustic impedance. In this context, simu-

76 6.3. Simulated Investigation 62 Figure 6.3: Comparison of the surface acoustic impedance (Delany and Bazley model) with the ratio of pressure and particle velocity measured right above the material at 5, 10, 15 and 20 millimeters. The material is exposed to plane-wave sound field and the plane-wave reflection coefficient is used for the calculation of the acoustic impedance at the receiver position. lations have been carried out in order to discern the differences of a given acoustic impedance to the measured acoustic impedance right above the surface. Let us assume that the acoustic impedance of a specific material is given, according to the Delany and Bazley model, for a flow resistivity σ = 50000Kgm 3 s 1. The material is exposed to a plane-wave sound field and the ratio of sound pressure to the normal component of the particle velocity is predicted at 5, 10, 15 and 20 millimeters above the surface. These predictions constitute, from now on, measurements at the receiver position and they are compared with the actual acoustic impedance of the material as given by the Delany and Bazley model in figure 6.3. Even for the closest position, the measurement at the receiver position presents important differences. The real part of the acoustic impedance seems to be more affected than the imaginary part. This is related with the fact that the Delany and Bazley model is inversely proportional to the frequency and, particularly, that the imaginary part contains a negative sign. Such a function, which is included in the predictions through the reflection coefficient, varies less than the real component. Furthermore, the real part includes a minimum value ρc, as can be seen in equation (2.10). Therefore, the differences are inherent on the Delany and Bazley model. this will be

77 6.3. Simulated Investigation 63 Figure 6.4: Effect of the surface in the sound pressure and particle velocity component. A totally reflecting surface is taking into account in order to show only constructive and destructive interference and its corresponding shifting on frequency when the receiver position is changed. discussed in more detail later. The further away from the surface the receiver is located, the larger the normal component of the particle velocity is and therefore, the curves are shifted down. At high frequencies, some fluctuations are observed; that correspond to the first destructive interference of the particle velocity. In order to describe in more detail these phenomena, figure 6.4 is included, showing the effect of the surface on the sound pressure and particle velocity as a function of the frequency; a totally reflecting surface is used. The particle velocity s normal component is greatly effected in amplitude at low frequencies when the surface is strongly reflecting. This is the case for most of the materials, even for porous materials where the wavelength is large compared to the complexities found on the porous surface. Therefore, the stiffer the material, the more difficult it is to measure the particle velocity component on the surface, as it becomes small in magnitude. On the other hand, as the prediction is obtained further away from the surface, the normal component of the particle velocity is larger and it can be acquired easier. However, care must be taken since destructive interference occurs at lower frequencies. The position of the source does not

78 6.3. Simulated Investigation 64 Figure 6.5: Comparison of the surface acoustic impedance (Delany and Bazley model) with the ratio of pressure and particle velocity measured right above the material at 5, 10, 15 and 20 millimeters. The material is exposed to spherical sound field and the plane-wave reflection coefficient is used for the calculation of the acoustic impedance at the receiver position. Source located at 0.5 meters above the surface. affect the destructive interference location in frequency; what affects it is the difference between the direct and reflected path length. As the material becomes limper, the normal component of the particle velocity is larger and the measurement on the surface is less problematic and the destructive interference becomes negligible. Now, if the material is exposed to a more complicated sound field, such as a spherical sound field, two approximations can be used to predict the pressure and particle velocity at the receiver position: using either the plane-wave or spherical reflection coefficient. If one assumes that spherical waves are reflected in the same manner as the plane-waves, the measurements on the receiver position present enormous differences at low frequencies in comparison with the actual surface acoustic impedance (as shown in Figure 6.5). At high frequencies, however, the same phenomenon as in the plane-wave sound field is noticed (Figure 6.3): shifting curves down due to the larger normal component of particle velocity as the measurement is performed farther away from the surface. If the spherical reflection coefficient Q is included instead, as the correct representation of the reflection of spherical waves, the differences between the surface acoustic impedance and the

79 6.3. Simulated Investigation 65 Figure 6.6: Comparison of the surface acoustic impedance (Delany and Bazley model) with the ratio of pressure and particle velocity measured right above the material at 5, 10, 15 and 20 millimeters. The material is exposed to spherical sound field and the spherical reflection coefficient is used for the calculation of the acoustic impedance at the receiver position. Source located at 0.5 meters above the surface. measurements at receiver positions above the material are definitely important (shown in Figure 6.6). The calculation of the normal component of the particle velocity is far more time consuming since Q has to be calculated for several positions of receiver in order to obtain the differential with respect to the height. The variance of Q with respect to the receiver position and frequency is shown in Figure 6.7. Moreover, Figure 6.8 shows Q for specific frequencies. Both real and imaginary part shows small variations when the calculation of Q is made, for instance, with the receiver position at 10 cm and 40 cm above the surface. The variation of the real part seems to decrease as the frequency increases. On the contrary, the variation of the imaginary part seems to not be influenced by the frequency. The simulation has shown that the measurement of the ratio of the sound pressure to the normal component of the particle velocity right above the material is not reliable enough to represent acoustically porous materials in terms of their acoustic impedances. This assumption is not even valid for simple sound fields such as the plane-wave sound field. In spherical sound fields, the assumption would create more discrepancies on the characterization of materials.

80 6.3. Simulated Investigation 66 Figure 6.7: Spherical reflection coefficient and its variation with respect to the frequency and receiver position. Figure 6.8: Spherical reflection coefficient and its variation with respect to the receiver position for specific frequencies.

81 6.3. Simulated Investigation 67 This procedure has been repeated for a stiffer material (flow resistivity σ = Kgm 3 s 1 ) and the figures are shown in Appendix A. As the material becomes stiffer, the differences between the surface impedance and the directly measured impedance right above the material becomes larger as can be seen in Figures A.1 and A.3. This is basically due to the smaller normal component of the particle velocity. The spherical reflection coefficient seems to become almost independent of the receiver position as the material becomes stiffer (to be examined at Figures A.4 and A.5). Therefore, even for a limp material with a flow resistivity σ = 50000Kgm 3 s 1, the measured acoustic impedance right above the surface does not characterize the material properly. It is important to mention that all the figures are normalized with respect to Z air and, therefore, small variances in the figures are significant on the specific acoustic impedance of materials Simulation 2: On the acoustical representation of porous materials through approximation models Simulations have been carried out in order to analyze the response of models in the acoustical representation of porous materials in terms of their acoustic impedances. The models (described in section 6.2), approximate the surface acoustic impedance by measuring the ratio of sound pressure and particle velocity somewhere above the surface. Therefore, the simulation includes in its first step the definition of the sound field at a specific position, which will afterwards be taken as a receiver position, where the models are applied to estimate the surface acoustic impedance. Let us assume that the acoustic impedance of a specific material is given according to the Delany and Bazley model for a flow resistivity σ = 50000Kgm 3 s 1. A large sample of this material is exposed to a spherical sound field, which is produced by a monopole located 50 cm above the surface. The sound field can be defined at a specific position by the calculation of the spherical reflection coefficient Q. Let us take the position 5 mm above the surface. Once the sound field is

82 6.3. Simulated Investigation 68 Figure 6.9: Comparison of reflection coefficients. The sound field has been defined at 5 mm above the surface, by the calculation of the spherical reflection coefficient Q (continuous line - Sound Field given). On the calculation of Q, an acoustic impedance given by Delany and Bazley model with a flow resistivity σ = 50000Kgm 3 s 1 was introduced. This specific position becomes afterwards a measurement receiver point, where models 1 ( ) and 2 ( ) have been applied to approximate the reflection coefficient. defined at this specific position, it can be taken as the simulation of a measurement at the receiver position, where the models can be applied to approximate the surface acoustic impedance given by Delany and Bazley. Figure 6.10: Normalized surface impedance comparison. The actual surface impedance given by Delany and Bazley model is compared with approximations to it according to model 1 ( ) and 2 ( ). This material can be categorized as limp. Receiver position on the measurement: 5 mm above the surface.

83 6.3. Simulated Investigation 69 Models 1 and 2 have been used to approximate the reflection coefficient as well as the surface acoustic impedance and the results are shown in Figures 6.9 and 6.10, respectively. Regarding the reflection coefficient approximations, it is clear that model 1 presents important discrepancies as expected, since the monopole is located nearby. Model 2 approximates better, but with considerable discrepancies at low frequencies. Concerning the approximation of the acoustic impedance on the surface (Figure 6.10), the model 2 provides a good estimation, which is almost on top of the Delany and Bazley model. Close examination reveals small differences at low frequencies, more noticeable in the imaginary part. Moreover, above 500 Hz the plane-wave approximation produces an accurate approximation for this specific source and receiver position. Figure 6.11: Normalized surface impedance comparison. The actual surface impedance given by Delany and Bazley model is compared with the approximations to the surface impedance by model 2 and 3. In model 3, the calculation of Q is been done by introducing the surface impedance approximated by model 2. This material is categorized as limp. Receiver position on the measurement: 5 mm above the surface. Previously, discussions concerning the calculation of the spherical reflection coefficient Q were presented. As mentioned, the accurate calculation of Q can be only achieved by measuring the acoustic impedance on the surface, which is impossible. Therefore, the surface acoustic impedance approximated with either model 1 or 2 can be assumed for the calculation of Q. Undoubtedly, model 2 provides a better approximation to the surface impedance and thus it is

84 6.3. Simulated Investigation 70 used. Once Q is calculated, the surface acoustic impedance can be approximated by model 3 as in equation (6.14). It is important to mention that the calculation of Q is needed for more than one receiver position since the differential with respect to the height has to be found. In this way, the approximation to the surface impedance by model 3 is compared with the approximation by model 2 and the actual surface impedance given by the Delany and Bazley model as exhibited in Figure The differences are only noticeable at low frequencies. In fact, model 3 has approximated the actual surface impedance with more discrepancies than model 2. Hence, model 3 does not present a considerable advantages over model 2 for this specific position of source and receiver. Figure 6.12: Differences on the sound field at the receiver position. These differences are produced by the calculation of Q with the surface impedance given by model 2 instead of the actual surface impedance. The continuous line must be assumed as the representation of an actual measurement, whereas the dashed line represents the prediction at the receiver position, with differences due to the error on the Q calculation. Receiver position on the measurement: 5 mm above the surface. Clarifying, at the beginning of this simulation the sound field was defined at a specific position by introducing the acoustic impedance given by the Delany and Bazley model while calculating the spherical reflection coefficient Q. In the just mentioned step, a slightly different sound field should be obtained since the calculation of Q was achieved by an approximation to the acoustic impedance on the surface. A clear evidence is the sound field produced at the receiver

85 6.3. Simulated Investigation 71 position (shown in Figure 6.12), which shows a small variation only appearing at rapid changes of slope. However, in the approximation to the surface acoustic impedance, this variation is enough to produce large discrepancies at low frequencies. The modification of the acoustic impedance on the surface (dashed line on Figure 6.11) to obtain the minimum difference between the sound fields on the receiver point was proposed in reference [15]. Hence, the acoustic impedance on the surface would be obtained when the minimum difference is reached. However, the procedure to obtain the minimum difference is not explained in detail. If the same source and receiver position is used to analyze a stiffer material such as the one given by the Delany and Bazley model with a flow resistivity σ = Kgm 3 s 1, the following comments must be pointed out. For stiffer materials, the plane-wave reflection coefficient calculated as in model 2 seems to be closer to the actual spherical reflection coefficient, that is, the one that was used to define the sound field. On the contrary, model 1 presents more discrepancies (see Figure 6.13). Figure 6.13: Comparison of reflection coefficients. The sound field has been defined at 5 mm above the surface, by the calculation of the spherical reflection coefficient Q (continuous line - Sound Field given). On the calculation of Q, an acoustic impedance given by Delany and Bazley model with a flow resistivity σ = Kgm 3 s 1 was introduced. This specific position becomes afterwards a measurement receiver point, where models 1 ( ) and 2 ( ) have been applied to approximate the reflection coefficient.

86 6.3. Simulated Investigation 72 On the approximation of the surface impedance, model 2 seems to provide a good approximation to the actual acoustic impedance as given by the Delany and Bazley model. However, model 1 presents discrepancies in a broader frequency range; it could only approximate the surface impedance above, say, 1kHz (see Figure 6.14 and compare with Figure 6.10). Figure 6.14: Normalized surface impedance comparison. The actual surface impedance given by Delany and Bazley model is compared with approximations to it according to model 1 ( ) and 2 ( ). This material can be categorized as stiff. Receiver position on the measurement: 5 mm above the surface. Concerning the approximation to the surface impedance by using model 3, the behavior seems to neither improve nor become worse when the approximation is performed with a stiffer material. Model 2 seems to be a better approximation for this specific source and receiver position(see Figure 6.15). Regarding the differences between the measured and predicted sound fields, they are only present at rapid changes of slope (see Figure 6.16). The performance of models 2 and 3 seems to not be greatly affected by the kind of material under analysis. However, model 1 presents more serious discrepancies as the material becomes stiffer.

87 6.3. Simulated Investigation 73 Figure 6.15: Normalized surface impedance comparison. The actual surface impedance given by Delany and Bazley model is compared with the approximations to the surface impedance by model 2 and 3. In model 3, the calculation of Q is been done by introducing the surface impedance approximated by model 2. This material is categorized as stiff. Receiver position on the measurement: 5 mm above the surface. Figure 6.16: Differences on the sound field at the receiver position. These differences are produced by the calculation of Q with the surface impedance given by model 2 instead of the actual surface impedance. The continuous line must be assumed as the representation of an actual measurement, whereas the dashed line represents the prediction at the receiver position, with differences due to the error on the Q calculation. Receiver position on the measurement: 5 mm above the surface.

88 6.3. Simulated Investigation 74 The use of this method on stiff materials is of interest. As discussed previously, in this case the normal component of the particle velocity on the surface is strictly zero and the acquisition of data becomes much more difficult. Therefore, for stiff materials, the measurement should be done at a certain distance from the surface in order to ensure a considerable magnitude of the particle velocity component. On the other hand, destructive interference is more important and care should be taken. As evidenced in Figure 6.6 for instance, the direct ratio of pressure and particle velocity at, say, 5 mm above the material presents serious discrepancies with respect to the surface impedance and therefore the models have to be applied. The simulation procedure has been repeated with the receiver position at 4.5 cm above the surface. The sound source was kept at 50 cm above the material and the sound field at the receiver position was defined by using the Delany and Bazley model with a flow resistivity σ = 50000Kgm 3 s 1, and calculating the spherical reflection coefficient. Now, thinking the other way around, the receiver position is the measurement point and models are applied in order to estimate the surface acoustic impedance. The Figures are shown in appendix B and the following conclusions are obtained: Considerable differences in the performance of model 2 when estimating the reflection coefficient due to changes of the receiver position have not been found. On the contrary, the performance of model 1 is worse when the measurement is made further away from the material. (compare figure 6.9 with figure B.1). As the measurement is made further away from the surface, the representation of spherical wave reflections becomes more important and model 2 presents problems at low frequencies. Therefore, the further away from the surface the measurement is conducted, the more necessary is the calculation of a spherical reflection coefficient (see figure (B.2)). On the other hand, model 1 provides good approximation to the surface impedance at high frequencies in a narrower frequency band, as the measurement is conducted further away from the surface.

89 6.3. Simulated Investigation 75 The performance of model 3, by the introduction of the surface impedance approximated by model 2 on the calculation of Q, seems to be again worse than model 2 itself. It seems that the differences are amplified through the Q calculation procedure. Thus, a better procedure to implement the spherical reflection coefficient has to be applied (see figure B.3). Furthermore, the closer to the surface the measurement is conducted, the narrower the frequency band affected with discrepancies at low frequencies when approximating the surface impedance. Minimum differences are again found on the sound field at the receiver position (see figure B.4), again only at rapid changes of slope. The figure provides also a clear representation of the destructive interference on the particle velocity component. Now, the same location of receiver and source is used to repeat the procedure for the stiffer material (σ = Kgm 3 s 1 ). The results are shown in the second part of appendix B, and the conclusions follow as: The better performance of model 2 on the approximation to the reflection coefficient when the material is stiff seems to be independent of whether the measurement is conducted at 4.5 cm above the surface or at 5 mm above the surface. On the contrary, model 1 performs worse as the material becomes stiffer and as the measurement is conducted further away from the surface. (compare figures B.5, B.1 and 6.13). The performance of model 2 on the approximation to the surface impedance seems to have discrepancies of the same order at low frequencies and only in the imaginary part. However, since the normalized acoustic impedance is shown, the discrepancies are larger in amplitude (see figure B.7) as the material is stiffer. The response of model 3 has not been changed due to the analysis of a stiffer material.

90 6.3. Simulated Investigation Simulation 3: The effect of intrinsic uncertainty on the microflown probe Accurate calibration methods for the microflown probe have been reported in reference [18]. The required accuracy of the calibration depends on the application and, particularly for measurements of absorption coefficient, reference [19] has concluded that reliable results with such a device call for calibration errors within 0.5 db and 2. Furthermore, phase mismatch was reported as more serious than amplitude mismatch. Both errors are much more critical as the material under test becomes stiffer, and thus more strongly reflecting. Figure 6.17: The effect of amplitude mismatch on the approximation to the normalized acoustic impedance on the surface by using model 2. The material is considered limp with a flow resistivity σ = 50000Kgm 3 s 1. In this context, the effect of the intrinsic uncertainty in the microflown probe on the approximation to the surface impedance by the models described in section 6.2 is studied in this section. Thus, 0.5 db amplitude mismatch has been introduced in the simulated measurement, more precisely on the normal component of the particle velocity, and the resultant approximation to the normalized acoustic impedance on the surface is calculated with model 2. As a reminder, model 2 approximates the surface impedance by using a measurement above the material and assuming that the waves are reflected in the same manner as plane waves. The simulated measurement has been acquired similar to the previous simulations by using the

91 6.3. Simulated Investigation 77 Delany and Bazley model with a flow resistivity σ = 50000Kgm 3 s 1 and the calculation of the spherical reflection coefficient for a specific source and receiver position. The source is located at 50 cm above the surface whereas the receiver lies down on the surface at 5 mm above. Therefore, the intrinsic uncertainty of the transducer can be evaluated and compared with an ideal measurement. This comparison is shown in Figure 6.17 and it exhibits that amplitude mismatch produces slight frequency dependant variations in the real and imaginary part of the approximated acoustic impedance on the surface. Close examination reveals deviations in the whole frequency range, that become smaller as the frequency increases. In the same manner, 2 phase mismatch has been introduced on the normal component of the particle velocity and the approximation to the acoustic impedance on the surface is calculated as in model 2. Similar comparison is presented in Figure The effect of phase mismatch on the approximation to the surface impedance is, for this particular limp surface, less important than the amplitude mismatch, which is opposite to what was reported in [19] for the calculation of absorption coefficients. Main discrepancies show up at very low frequencies and mainly in the imaginary part of the acoustic impedance. Figure 6.18: The effect of phase mismatch on the approximation to the normalized acoustic impedance on the surface by using model 2. The material is considered limp with a flow resistivity σ = 50000Kgm 3 s 1.

92 6.3. Simulated Investigation 78 Figure 6.19: The effect of amplitude and phase mismatch on the approximation to the normalized acoustic impedance on the surface by using model 2. If both kinds of uncertainty are presented and the same procedure is followed in order to approximate the acoustic impedance on the surface, the errors seem to compensate each other at low frequencies to reduce differences in the real part; however, the errors in the imaginary part are added (shown in figure 6.19). Figure 6.20: The effect of amplitude mismatch on the approximation to the normalized acoustic impedance on the surface by using model 2. The material is considered stiff with a flow resistivity σ = Kgm 3 s 1. If the effect of the intrinsic uncertainty in the microflown probe is now analyzed for a stiffer

93 6.3. Simulated Investigation 79 Figure 6.21: The effect of phase mismatch on the approximation to the normalized acoustic impedance on the surface by using model 2. The material is considered stiff with a flow resistivity σ = Kgm 3 s 1. material with an acoustic impedance given by the Delany and Bazley model with a flow resistivity σ = Kgm 3 s 1, the errors seem to be much more important. Figure 6.20 presents the effect of a 0.5 db amplitude mismatch on the approximation to the acoustic impedance on the surface. Much more important discrepancies on the approximation to the surface impedance are noticed at low frequencies, when comparing with figure The effect of a 2 phase mismatch is presented in Figure The discrepancies are larger in comparison with Figure However, they still seem to be less important than the amplitude mismatch error. The effect of both kinds of error on the approximation to the surface impedance is shown for this stiff surface in Figure The errors in the calibration of the probe definitely are more important as the material becomes stiffer or more strongly reflecting. The amplitude mismatch still seems to be more critical than the phase mismatch. Finally, when both kinds of error are included, a compensation phenomenon seems to occur in the real part of the approximation to the surface impedance, which reduces the discrepancies; however the errors in the imaginary part are added (compare for instance figures 6.20, 6.21, and 6.22). The same procedure has been conducted, but, instead, the receiver is located at 4.5 cm above

94 6.3. Simulated Investigation 80 Figure 6.22: The effect of amplitude and phase mismatch on the approximation to the normalized acoustic impedance on the surface by using model 2. The material is considered stiff with a flow resistivity σ = Kgm 3 s 1. the surface and two kinds of surfaces, one limp and one stiff, are analyzed. The results are shown in Appendix C. It is not possible to draw direct conclusions concerning the changes in the errors when the measurement is conducted further away from the surface since the approximation to the surface impedance by model 2, which is used for comparison, includes already some discrepancies with the actual surface impedance at low frequencies. However, the errors are definitely more important as can be seen comparing for instance figures 6.17 with C.1.

95 6.4. Experimental investigation Experimental investigation Experimental measurements of the acoustic impedance on the surface using the models described in section (6.2) were carried out in both an anechoic room that provides a good approximation to free-field conditions down to 150 Hz and in an ordinary room of about 48 m 3 with a reverberation time of about 2 s. Two kinds of devices were used to obtain the sound pressure and particle velocity above the surface: a conventional face-to-face Bruel&Kjær intensity meter with 12 mm microphone separation distance and a half inch microflown intensity probe. The source was a 60 mm diameter two-way coincident-source loudspeaker unit produced by KEF, mounted in a rigid plastic sphere with a diameter of 270 mm. Moreover, PULSE system Labshop was used in all the measurements and the analyzer settings were assigned to include a frequency range of 0 to 3.2 khz with a frequency resolution of 1 Hz. Hann windowing was used during measurements due to its high spectral accuracy with stationary signals. A large sample of Rockwool 50mm A-batt is analyzed first in the anechoic chamber. The source is located at 50 cm above the material and the first measurements are carried out with both the Bruel&Kjær and microflown probes located at the same position, thus approximately 4.5 cm above the surface. For reference, the same material has been cut and analyzed in the impedance tube. The ratio of pressure and particle velocity using the Bruel&Kjær intensity meter is calculated as shown in section 4.1. For the microflown device, the ratio of pressure and particle velocity is obtained directly in the field and only corrections are applied according to what was mentioned in section 4.2. Models 1 and 2 have been applied in order to approximate the surface acoustic impedance. The results with both devices are compared with the surface acoustic impedance given by the transfer function method on the tube, which shows reliable results up to approximately 2.2 khz (see Figure 6.23). The Bruel&Kjær device is undoubtedly equipped with two precisely matched microphones able to notice differences in sound pressure even at very low frequencies. However, the procedure to obtain the particle velocity component involves an integral, which produces noise in the channel,

96 6.4. Experimental investigation 82 Figure 6.23: Normalized surface impedance comparison - Rockwool 50mm A-batt. Results are shown for measurements in the kundt s tube and in free-field conditions, using Bruel&Kjær and microflown intensity probes. Models 1 and 2 have been applied to approximate the surface impedance out of a measurement at 4.5 cm above the surface. and that seems to make the difference when surface impedance is approximated. Figure 6.23 shows the characteristic erroneous decrease in the real part of the impedance toward zero at low frequencies caused by noise in the velocity channel. This was also reported by F.J. Fahy in reference [1]. The direct measurement of the particle velocity component on the microflown probe is clearly an advantage on this particular application, and thus the following measurements are achieved only with this device. Concerning the results with microflown probe, the models show discrepancies at low frequencies, which were also observed in the simulations for this particular receiver and source position (see figure B.2). The peak between 2 and 3 khz corresponds to destructive interference on the particle velocity for the results on the free field method, whereas in the result with the transfer function method could correspond to the first mode on the cavity. Measurements are now conducted with the microflown probe on the surface, say, at 5 mm above the material. The application of the models to approximate the acoustic impedance shows better agreement with the result given in the tube, as can be seen in Figure 6.24; however, this is expected to happen only for limp materials, such as the one under analysis. In this figure, the result

97 6.4. Experimental investigation 83 Figure 6.24: Normalized surface impedance comparison - Rockwool 50mm A-batt. Results are shown for measurements in the kundt s tube and in free-field conditions, using the microflown intensity probe. Models 1 and 2 have been applied to approximate the surface impedance out of a measurement at 5 mm above the surface. Model 3 is used by calculating Q with the approximation to the surface impedance provided by model 2. with model 3, using the approximation to the surface impedance by model 2 on the calculation of Q, has been also included. It does not represent an advantage over model 2. Figure 6.25: Effect of the surface (Rockwool 50mm A-batt) in the sound pressure and particle velocity component at the receiver position. A deep interference minima in the particle velocity component is observed around 100 hz. At low frequencies, the normal component of the particle velocity is seriously affected by

98 6.4. Experimental investigation 84 Figure 6.26: Comparison of reflection coefficients. The plane-wave reflection coefficient is calculated with the transfer function method on tube measurements, as well as through the free field method, using models 1 and 2. Furthermore, the calculation of the spherical reflection coefficient Q is done by introducing in the calculation the surface impedance approximated by model 2. the surface and thus the acoustic impedance is over-estimated; this phenomenon is shown in Figure 6.25, which provides information about the effect of the surface in the amplitude of both sound pressure and particle velocity. Furthermore, the reflection coefficients are compared in figure 6.26, where is also included as reference the plane-wave reflection coefficient measured in the tube. The closest reflection coefficient to the reference is the one given by the model 1 as it should. However, the features of model 2, which includes spherical sound field but plane wave reflection coefficient, produce differences in its plane-wave reflection coefficient estimation. A similar setup was simulated in section 3.3, more precisely in Figure 3.6, and clear differences are shown between the spherical and plane wave reflection coefficients. However, since Q is approximated with a slightly different surface impedance, important discrepancies with respect to model 2 are not found. In addition, the fact that the anechoic room provides good approximation to free field conditions down to 150 Hz might affect the results at low frequencies. Therefore, the sample has been moved to a larger anechoic chamber, where free field conditions are ensured down to 50 Hz. Similar setup has been used and the approximation to the surface impedance is accom-

99 6.4. Experimental investigation 85 Figure 6.27: Normalized surface impedance comparison - Rockwool 50mm A-batt. Results are shown for measurements in the kundt s tube (continuous line), in the large anechoic chamber (dashed line) and in the small anechoic chamber (dash-dot line), using the microflown intensity probe. Models 2 has been applied to approximate the surface impedance out of a measurement at 5 mm above the surface. plished only using model 2. Figure 6.27 shows a comparison only for low frequencies between the large and small anechoic chamber results; the tube data is also included. Some improvement due to the free field condition down to 50 Hz is exhibited only in the real part of the acoustic impedance, extending the valid frequency range, say, down to 120 Hz. However, no advantage is noticeable in the imaginary part. Concluding, the measurement on the surface with the microflown probe and the application of model 2 in order to estimate the acoustic impedance of the material seems to provide quite reasonable results between 200 and 1500 Hz approximately. The accurate measurement has to be restricted to a frequency range where the destructive interference does not affect the particle velocity component. Moreover, not only the poor amplitude of the particle velocity component, but also the non-existence of free field conditions will produce important uncertainties at low frequencies. Now the same sample of material is moved to the ordinary room and measurements are con-

100 6.4. Experimental investigation 86 Figure 6.28: Normalized surface impedance comparison - Rockwool 50mm A-batt. Results are shown for measurements in the kundt s tube and in-situ, using the microflown intensity probe. Model 2 has been applied to approximate the surface impedance out of a measurement at 5 mm above the surface. ducted on the surface with the microflown probe. Following the conclusions from the simulated investigation, model 2 should provide a good estimation to the surface acoustic impedance. Therefore, only model 2 is been applied for the measurements in-situ of this particular material. This approximation is compared with the surface impedance provided by the transfer function method in Figure The in-situ data shows oscillations around the tube data. A closer examination has revealed that they correspond to the effect of the room. The modes of the room should have an effect on the measurement of sound pressure, as well as the normal component of the particle velocity. However, for locally reacting materials, the modes should not affect the measurement and the properties of the surface should be acquired. Figure 6.29 shows some of the modes of the room in the pressure, as well as in the particle velocity component; some of these modes are also possible to recognize in the ratio of them. At low frequencies, the amplitude of signals is really small and the coherence between the pressure and velocity channels becomes lower than 1 (as shown in Figure 6.30). However, that is not the case above 80 Hz, and modes are possible to recognize. This explains the oscillations in Figure Indirectly, this provides a method to analyze the locally reacting of materials.

101 6.4. Experimental investigation 87 Figure 6.29: Effect of the room in the sound pressure, particle velocity and the ration of them (acoustic impedance) at the receiver position. The data is not treated and the measurement is conducted 5 mm above a large sample of Rockwool 50mm A-batt. Figure 6.30: Coherence between sound pressure and particle velocity component channels on in-situ measurement of a large sample of Rockwool 50mm A-batt.

102 6.4. Experimental investigation 88 The in-situ measurements were carried out in purpose at an ordinary room with quite a high reverberation time. This has shown that the room influences the approximation to the acoustic impedance of the material, which was not clearly expected. Time selectivity might be applied to avoid the reverberant field. Figure 6.31: Normalized surface impedance comparison - 40 mm Rockfon Samson. Results are shown for measurements in the kundt s tube and in-situ, using the microflown intensity probe. Model 2 has been applied to approximate the surface impedance out of a measurement at 2.5 cm above the surface. Continuing with the analysis, a smaller sample of porous material (40 mm Rockfon Samson) has been analyzed in the small anechoic chamber. The sample is half as large as the first analyzed material and it can be considered stiffer. The source and receiver have been located at 50 and 2.5 cm above the material respectively. The further away from the source the receiver is located, the more important the calculation of the spherical reflection coefficient is as concluded

103 6.4. Experimental investigation 89 in the simulation study. However, since it is not possible to calculate Q without knowing the actual surface impedance, only model 2 has been applied to approximate the surface impedance. Figure B.2, which shows the approximation to the surface impedance using the model 2 with a similar setup and material, demonstrates good accuracy in the estimation when using model 2 above 200 Hz. Therefore, below 200 Hz, two extra sources of uncertainty are expected in the approximation to the surface impedance: the one intrinsic in the model 2, and the one produced by more possible reflections from the edges of the sample. Figure 6.31 shows the approximation to the surface impedance in comparison with the result obtained in the standing wave tube. The intrinsic error in this model would appear as a shifting in the results; on the contrary, mainly large oscillations are found below 200 Hz. Therefore, the problems below 200 Hz are attributed to reflections from the edges of the sample rather than problems with the approximation model. Moreover, the differences are found more in phase rather than in amplitude. The application of the method has been attempted in order to analyze stiffer surfaces for several positions of source and receiver, and enormous differences have been found in comparison with the results given by the transfer function method on the standing wave tube. For such materials, the sources of error in the tube are much more, and therefore it does not constitute anymore a valid reference. Moreover, according to the simulations on porous materials, the application of the models to approximate the acoustic impedance in stiffer materials is viable. However, the material must be porous-like, such as glass fibers or foams. Therefore, the method should be restricted to porous materials, no matter how stiff or limp they are.

104 Chapter 7 Summary and conclusions Two methods to conduct measurements of the acoustic impedance were analyzed throughout this document and their viability for in-situ evaluation of materials has been analyzed in detail. The simulated investigation has provided important information on the application of such methods, as well as the error analyses that were carried out mainly for the use of the microflown intensity probe for accurate acoustical representation of materials. Therefore, the following conclusions are drawn: Concerning the models to calculate the spherical reflection coefficient, the grazing incidence model by Chien and Soroka provides slightly different results in comparison with the Laplace transform formulation by Lindell and Alanen, even for normal incidence. On the idea of produce an equipment to measure the acoustic impedance of materials in-situ, where a small trans-to-surface distance is convenient, either model to calculate the spherical reflection coefficient would be applicable. However, since the surface impedance is needed to calculate Q, an iterative procedure to find the minimum differences between measured and predicted sound fields may be applied in order to approximate the surface impedance as described in reference []. Regarding the transfer function method, it undoubtedly constitutes the most accurate 90

105 91 method to measure acoustic impedance. The swapping technique makes the accurate measurement possible even when the tube is equipped with poorly matched microphones. However, mounting based errors are definitely important and become critical when stiff materials are analyzed. The response of a portable impedance tube has been analyzed in detail. Air leakage at the termination constitutes a critical problem in the measurement of acoustic impedance. The measurement is first influenced at low frequencies and the affected frequency range becomes broader as the leakage is larger. Applicability of the portable tube is found only for materials where properties are not changed due to the applied forcing of the tube against the material. In regard to the free field method and the characterization of materials through a measurement of sound pressure and normal component of particle velocity at, say, 5 or 6 mm above the surface with the microflown probe, the simulations have shown important discrepancies, and therefore the measurement does not represent properly the impedance of the material. This assumption is not even valid for simple sound fields such as the plane-wave sound field. In spherical sound fields, the assumption would create more discrepancies in the characterization of materials. Furthermore, as the material is stiffer, the differences between the actual acoustic impedance and the directly measured impedance near the surface become larger. Typical values of flow resistivity for foams of porous materials lie in a range between [Kgm 3 s 1 ]. Since one of the simulations was carried out for a material with flow resistivity σ = 50000Kgm 3 s 1, the assumption does not provide accurate measurements even for typical porous materials. With regard to the effect of the intrinsic uncertainty in the microflown probe on the ap-

106 92 proximation to the surface impedance, the effect of amplitude mismatch seems to be more critical than phase mismatch and both become more important as the material becomes stiffer. If the approximation of the surface impedance is made out of a measurement farther away from the surface, errors are definitely more critical. Concerning the Bruel&Kjær intensity meter, the intrinsic noise in the particle velocity channel has important consequences at low frequencies. The direct measurement of the particle velocity with the microflown probe is clearly an advantage. On the acoustical representation of porous materials through the approximation models, the simulations have shown that when measuring near the surface, model 2 provides an accurate estimation of the actual acoustic impedance no matter the kind of material under analysis. However, the models do not consider the fact that the experimental acquisition of the normal component of the particle velocity might be difficult as the surface becomes stiffer. Therefore, if the sample under analysis presents limp characteristics, the measurement and application of model 2 does provide an accurate measurement. In contrast, if the material presents stiff characteristics, the measurement has to be performed farther away from the surface in order to let the transducer acquire the normal component of the particle velocity. As the measurement is performed farther away, the calculation of the spherical reflection coefficient becomes more important and the application of model 3 should provide better approximation to the acoustic impedance. However, problems in the approximation by model 2 occur only at low frequencies, i.e. the farther away the measurement is conducted, the broader the low frequency range effected in the application of model 2 to the actual surface impedance. Particularly for the measurement conducted at 4.5 cm above the surface, model 2 correctly approximates the actual surface impedance above 200 Hz.

107 7.1. Future work 93 The experimental investigation has exhibited how difficult the acquisition of good results is below 200 Hz. Accurate measurement of the acoustic impedance has to be restricted to a frequency range where the destructive interference does not affect the particle velocity component. Moreover, not only the poor signal amplitude of the particle velocity component, but also the non-existence of free field conditions and the finite size of the sample will produce important uncertainties at low frequencies. On the other hand, in-situ evaluations of materials have shown that the room influences the approximation of the actual surface impedance. The modes of the room should have an effect on the measurement of sound pressure, as well as the particle velocity component. However, this should not, in principle, occur when measuring the ratio of them. 7.1 Future work In-situ measurements require a handy equipment that should not be large, neither uncomfortable. Therefore, the spherical reflection coefficient has to be calculated, and the iterative procedure mentioned previously must be improved. However, this does not mean that the problem will be solved at low frequencies; it basically will cancel one of the error sources. Furthermore, time selectivity might be a good starting point to eliminate reflections on in-situ evaluations; however, problems due to windowing are expected. If the sound field is constrained by the presence of the walls of a portable tube, plane-wave sound field is formed in an important frequency range and the case is much simpler. However, leakage on the interface tube-material is a critical point. As an attempt to avoid it by pressing the tube into the material, the properties might be changed. Therefore, it would be interesting to discover the relation leakage vs. force applied, using the Brinell Hardness relation. The Brinell scale characterizes the indentation hardness of materials through the scale of penetration of an indenter, loaded on a material test-piece. It is one of several definitions of hardness in materials science.

108 Appendix A From simulation 1 - Stiff material Figure A.1: Comparison of the surface acoustic impedance (Delany and Bazley model) with the ratio of pressure and particle velocity measured right above the material at 5, 10, 15 and 20 millimeters. The material is exposed to plane-wave sound field and the plane-wave reflection coefficient is used for the calculation of the acoustic impedance at the receiver position. 94

109 95 Figure A.2: Comparison of the surface acoustic impedance (Delany and Bazley model) with the ratio of pressure and particle velocity measured right above the material at 5, 10, 15 and 20 millimeters. The material is exposed to spherical sound field and the plane-wave reflection coefficient is used for the calculation of the acoustic impedance at the receiver position. Source located at 0.5 meters above the surface. Figure A.3: Comparison of the surface acoustic impedance (Delany and Bazley model) with the ratio of pressure and particle velocity measured right above the material at 5, 10, 15 and 20 millimeters. The material is exposed to spherical sound field and the spherical reflection coefficient is used for the calculation of the acoustic impedance at the receiver position. Source located at 0.5 meters above the surface.

110 96 Figure A.4: Spherical reflection coefficient and its variation with respect to the frequency and receiver position. Figure A.5: Spherical reflection coefficient and its variation with respect to the receiver position for specific frequencies.