Abstract Simulations of sound pressure levels are often used as a decision basis when planning location for e.g wind turbines. In previous work, an e

Transcription

1

2 Abstract Simulations of sound pressure levels are often used as a decision basis when planning location for e.g wind turbines. In previous work, an e cient model for computing sound propagation over irregular terrain has been developed. The model simulates the propagation using the time dependent wave equation. The purpose of this study was to improve this solver by finding a way to represent atmospheric attenuation in the time dependent wave equation by adding a damping term to the equation. This was done by numerically simulating wave propagation and comparing to measured data on the atmospheric attenuation. The study did not result in finding a satisfying way to model the attenuation. The results of the simulations are analyzed and suggestions for further research are given.

3 Contents 1 Introduction Introduction Atmospheric Attenuation Project definition Analysis The wave equation The SBP-SAT method/the semi discrete model Results Simulations using damping term u t Simulations using damping term u t and r 2 u t Simulations using damping term u t r 2 u t Conclusions and discussion 11

4 1 Introduction 1.1 Introduction Calculations of sound propagation are often used as decision basis when planning the construction of for example wind turbines, airports, tra c junctions and other places where extreme sound pressure levels occur and cause problems. This makes accurate modeling of sound propagation an important subject. To predict sound pressure levels there are several factors that should be included in the model. For example, the sound pressure levels depends strongly on meteorological data such as sound speed variations and atmospheric attenuation as well as the topography of the area over which the sound propagates [7]. There are several models that can be used to simulate sound propagation, varying in computational cost and accuracy. A common model used in Sweden today is NORD2000, based on ray tracing theory. Ray tracing theory models do not take complicated topography or meteorology into account [5]. More accurate models such as the finite di erence time domain method (FDTD) are often too time consuming to be used in three dimensional calculations. A good model for sound propagation is the time-dependent acoustic wave equation: u tt = c 2 u (1) Solving the wave equation can result in high computational cost if not using an e cient numerical method. The computation time can be reduced by using the Helmholtz equation instead. Helmholtz equation is a time independent, frequency domain representation of the wave equation. Transforming the wave equation to frequency domain results in less computational cost, but modeling in frequency domain demands sources consisting of a single frequency. Wind turbines are broadband sources and require a time-dependent model for e cient simulation of sound pressure levels. In previous work, an e cient numerical solver for solving wave propagation problems on large domains has been implemented [3]. This solver uses fourth order accurate finite di erence methods and runs in parallel in MATLAB. The solver can be used to model propagation over complicated terrains, but does not take into account meteorological data such as atmospheric attenuation. 1

5 1.2 Atmospheric Attenuation Atmospheric attenuation is a damping of the sound waves caused by transport processes and molecular relaxtion losses in the atmosphere. As sound waves propagate through the atmosphere the amplitude decreases due to absorption as p t = p i e s, (2) where p t is the sound pressure amplitude at a distance s from the initial pressure p i and is the attenuation coe cient dependent on frequency, background pressure, humidity and temperaure. More detailed information about atmospheric attenuation and measured values for for varying atmospherical data is given in [2]. Figure 1: Example of table with values on the attenuation constant for varying atmospherical data as given in [2]. 1.3 Project definition In this project we have studied the possibility to use a damping term to model the atmospheric attenuation in the time dependent wave equation. This would be a way to improve the model developed in [3]. The idea was to add a physical damping term to the right hand side of (1) of the form r ru t,wherethecoe cient is a function of the atmosphere, i.e. dependent on temperature, humidity and pressure, u tt = c 2 u + r ru t. (3) 2

6 The damping term should depend on frequency correspondingly to the frequency dependence in the atmospheric attenuation. Ideally, this would lead to a model that could accurately simulate the atmospheric attenuation of broad band sources, which as discussed above is desired. The main object of the project was, thus, to study the wave equation with the added damping term with the aim to find an explicit form of the damping term. This is done by simulating the wave propagation and testing it against -values from [2] inserted in the analytic solution given by (2). To begin with, a fixed atmosphere (i.e fixed values on humidity, temperature and background pressure) was examined. The aim was to later extend the modeling to more general atmospheres. In the following sections we first describe the numerical model and how the sound propagation was simulated. We then present the results of our simulations, followed by a discussion about the results and recommendations for further research. 2 Analysis 2.1 The wave equation In order to investigate the behavior of the damping term we perform simulations of damped waves and compare them to analytical data. Sound from a point source propagates with spherical symmetry in free space. Using spherical coordinates in one dimension will therefore lead to a simplified model that will still simulate e ects relevant to propagation in three dimensions. Adding the damping term as described above to the one-dimensional wave equation in spherical coordinates and setting in (3) to be independent of r yields r 2 u tt = c r r By numerically modeling this wave and comparing it to the analytical attenuated wave equation the damping term can be adjusted to fit the analytical damping. The analytical form of the damped wave is given by applying the damping described in (2) to the analytical solution of the one dimensional spherical wave equation u d (r, t) = a r cos(!t! c r) e r (5) 2.2 The SBP-SAT method/the semi discrete model The SBP-SAT method is a finite di erence method that combines Summation By Part operators (SBP) with the simultaneous approximation terms 3

7 method (SAT) to weakly impose boundary conditions. Proving stability for SBP-SAT representations is done by analyzing the energy using the semi discrete energy method as demonstrated later in this section. In order to present the SBP-SAT method we first introduce some notation and definitions (following [4]). Discretization The domain r 0 apple r apple r N is discretized using M points given by r i = r 0 + ih, h = r N r 0, i =1, 2,...,M. M 1 The numeric solution at r i is denoted by v i and the corresponding analytic solution is denoted by u(r i ). These vectors are used when introducing the simultaneous approximation terms ê 1 =[1, 0,..., 0] T, ê N =[0,..., 0, 0] T. Definition 1 A di erence operator D 1 = H 1 Q is a first derivative SBP operator if H = H T > 0, and Q + QT = B = diag( 1, 0..., 0, 1). Definition 2 The operator D (b) 2 = H 1 ( M (b) + BS) where b(r) > 0, H is diagonal and positive definite, M (b) is symmetric and positive semidefinite, S approximates the first derivative operator at the boundaries and B = diag( b 0, 0,...,0,b N ),. Definition 3 An inner product for discrete real-valued vector functions v, w 2 R M is given by (v, w) H = v T Hw where H = H T > 0, with corresponding norm v 2 H = vt Hv. The continuous term u r, for example, would have the semi-discrete approximation D 1 v using the first derivative SBP operator introduced above. Below we apply the SBP-SAT method to the wave equation given in spherical coordinates. A nonreflecting boundary conditions for spherical coordinates as described by Grote in [1] is applied on the right boundary in order to prevent non-physical reflections. The left (inflow) boundary is implemented by strongly enforcing the sound pressure levels, sometimes referred to as injection: 4

8 8 r 2 u tt = c r r 0 apple r apple r N, t > u r + 1 c u t + u r =0, r = r N >: u(r, t) = a r cos(!t! c r), r = r 0 (6) (7) (8) 1 Analytic injection Non-reflective boundary Amplitude 0-1 Computational domain 0 r 0 r N >r N Distance Figure 2: The simulation set up. The actual computational domain is between r 0 and r N. The semi-discrete approximation of (6) using the second order SBP operator is given by Bv tt = c 2 D (b) 2 v (9) where B = diag(b 0,b 2 1,...,b2 N ) and b(r) =r2. The nonreflecting boundary condition is imposed weakly by adding a simultaneous approximation term to the right hand side of the equation Bv tt = c 2 D (b) 2 v H 1 ê N {(Sv + v t c + v r ) N} (10) The SAT method can be seen as a force pulling the solution on the boundary towards the boundary condition. To prove stability we use the semi discrete energy method, given by multiplying the equation by v T t H and adding the transpose. This yields d dt ( v t 2 HB + c 2 v T M (b) v)=c 2 v T t BSv + c 2 (v T t BSv) T 2 (v T t ) N {(Sv + v t c + v r ) N} (11) 5

9 Where we have used that v T t HBv + v T HBv t = d dt vt HBv = d dt v HB. The semi discrete energy estimate for the wave equation is given by E = v t 2 HB +c2 v T M (b) v, which gives that the left hand side in (11) corresponds to de/dt. d dt E = c2 v T t BSv + c 2 (v T t BSv) T 2 (v T t ) N {(Sv + v t c + v r ) N} (12) The right hand side is further simplified using the definition of B and S d dt E = 2c2 b 0 (vt T ) N (Sv) N + 2(c 2 b N )(vt T ) N (Sv) N 2 2 (vt T ) N v N (13) c r N To prove stability we must have de/dt apple 0. In order to cancel out the term 2c 2 b N (vt T ) N (Sv) N that could lead to non-physical growth in energy we must set the SAT coe cient to = b N c 2 = r 2 Nc 2. (14) Adding the damping term u t to the equation introduces a second SAT term to the semi discrete representation. By the same method as above the penalty term is set to 2 = rn 2, and the semi discrete approximation is given by 3 Results Bv tt = c 2 D (b) 2 v + D(b) 2 v t r 2 Nc 2 H 1 ê N {(Sv + v t c + v r ) N} r 2 NH 1 ê N {(Sv t + v tt c + v t r ) N} (15) In this section we present results of simulations for a fixed atmosphere, 50% relative humidity and 15 C (see Table 1 in [2] for exact values of ). All the testing is done against -values for this specific atmosphere. The initial intention was to further extend the study to more general atmospheric data, but this was omitted given the nature of the results obtained. The simulations began with trying to adapt the damping term u t to the atmospheric attenuation data for the given atmosphere. It showed that it was possible to mimic the attenuation for a specific frequency by choosing a suitable value on, but that the damping term and the attenuation did not follow the same frequency dependence. We then moved on to testing other damping operators and combinations of operators. Below the results from these simulations are presented. 6

10 3.1 Simulations using damping term u t The simulations were started by trying to find the correct damping for a fixed frequency. This was possible, which is shown in Figure 3 where the maximum values of the amplitude for both the analytical and the numerical solution are shown for a damped wave with frequency 1000 Hz. The numerical and analytical solution follows the same decay Amplitude for frequency 1000 Hz over distance Analytic Numeric: β(1000 Hz) 0.7 Amplitude Distance (m) Figure 3: The amplitude of a signal with a frequency of 1000 Hz with optimized for 1000 Hz coincide with the measured values of the attenuation. However, it was found that signals with di erent frequencies demand different values of. The damping operator u t a ects signals with higher frequencies much stronger than the atmospheric attenuation does, which results in that a lower value on is needed for higher frequencies in order to fit the damping. The optimal -values for frequencies ranging between 50 and 1000 Hz are shown in Figure 4. Since di erent frequencies demand different values, a signal containing several frequencies can not be modelled using just one damping term as it would only give the correct damping for one frequency. This is illustrated in Figure 5 and Figure 6. An incorrect results in an incorrect damping and an incorrect modeling of the sound. 7

11 β Frequency (Hz) Figure 4: The value of as a function of frequency. Since di erent frequencies demand di erent values on, this damping term can not be used to model signals containing multiple frequencies Amplitude for frequency 1000 Hz over distance Analytic Numeric: β(1000 Hz) Numeric: β(50 Hz) Amplitude Distance (m) Figure 5: The amplitude of a signal with a frequency of 1000 Hz with optimized for 50 Hz compared to the analytical solution. 8

12 Amplitude ratio β(50 Hz) β(250 Hz) β(1000 Hz) Distance (m) Figure 6: Amplitude ratio between the numerical and the analytical solution. A signal with frequency 250 Hz is modelled using three values of. An incorrect value of leads to a increasing error as the sound wave propagates. 3.2 Simulations using damping term u t and r 2 u t Proceeding our project, we tried a damping term of a di erent form, namely u t, resulting in the equation u tt = c 2 u u t. (16) As in the previous subsection, we started out by trying to find the correct damping for a signal of fixed frequency. This did not succeed using the damping term u t. The numerical solution did not follow the same exponential decay as the analytical solution. In Figure 7, the amplitude ratio between the numerical and the analytical solution is shown. What is desired is a straight line, meaning that the numerical solution follows the same damping relation as the analytical. In the case with the damping term u t this was not observed. We found that we needed to add a factor of r 2 to the damping term in order to obtain the correct damping, u tt = c 2 u r 2 u t (17) As seen in Figure 7, the r 2 u t -term gives a decay coinciding with the analytical solution. However, this damping did not give better results than the 9

13 u t -term when simulating varying frequencies. Here too it was found that di erent frequencies demand di erent values of (see Figure 7 and Figure 8) γ Undamped γ r 2 Amplitude ratio Distance (m) Figure 7: The amplitude ratio between the numerical and analytical solution for a 250 Hz signal. Using a damping term u t, the numerical solution did not follow the same damping as the analytical solution. Using r 2 u t it was possible to obtain a correct damping. The undamped solution is included for comparison γ Frequency (Hz) Figure 8: The value of as a function of frequency. As in the previous simulations it was found that di erent frequencies demanded di erent values on. 10

14 3.3 Simulations using damping term u t r 2 u t As the damping was stronger for high frequencies and the r 2 damping was stronger for low frequencies a combination of the two was tested, namely u tt = c 2 u + u t r 2 u t. (18) The idea was that a combination of the two damping terms could result in a fair approximation of the damping ranging over various frequencies. The two damping terms were adjusted to optimize the damping for two di erent frequencies, 50 Hz and 1000 Hz, i.e the values of and were chosen to give correct damping for signals of 50 Hz and 1000 Hz. When this setup was tested against frequencies lying between 50 Hz and 1000 Hz, it did not result in a correct damping. This is shown in Figure 9, for the amplitude ratios for frequencies 50, 250 and 1000 Hz Hz 250 Hz 1000 Hz 250 Hz Undamped Amplitude ratio Distance (m) Figure 9: The amplitude ratio for the frequencies 50, 250 and 1000 Hz using the modeling explained above. Satisfactory damping was obtained for signals near 50 Hz and 1000 Hz, but not for signals in between. The amplitude ratio for an undamped signal is shown as a reference. 4 Conclusions and discussion It was proved hard to find a damping term able to model the atmospheric attenuation by using the damping terms investigated above. The aim was to find a way to model signals containing multiple frequencies, but no satisfactory way to do this was found. The reason for this was that the atmospheric attenuation followed a more complicated frequency dependence than expected. A closer look at the physics describing the attenuation reveals some 11

15 useful information about this. The ISO-paper [2] describes the attenuation as a sum of di erent types of absorption derived to several physical mechanisms having distinct frequency dependencies: Classical ( cl ) and rotational absorption ( rot ) caused by transport processes described by classical mechanics and molecular rotational relaxation. The classical absorption is caused by compression and expansion of the air leading to friction between particles [6]. These types of absorptions follow a quadratic frequency dependence given by cr = cl + rot = (T/T 0 ) 1/2 f 2 p a /p r, where T 0 and p r represent reference temperature and pressure. Vibrational relaxation damping caused by vibrational relaxation of oxygen( vib,o ) and nitrogen( vib,n ).The attenuation caused by these factors follow a more complicated frequency dependence given by vib,o = 2[( ) max] cf ro f 2 [1 + (f/f ro ) 2 ] 1 vib,n = 2[( ) max] cf rn f 2 [1 + (f/f rn ) 2 ] where the terms f r represent relaxation frequencies. The relaxation frequencies are properties of the nitrogen and oxygen, dependent on temperature and humidity of the air. Based on these factors the frequency dependence in the vibrational relaxation damping can change significantly (see figure 10 and figure 11). The total attenuation is thus given by = cr + vib,o + vib,n.sincethese di erent types of attenuation are derived from di erent physical mechanisms and follow varying frequency dependence, it seems logical to separate them into di erent damping terms in the time dependent wave equation as well. As can be seen in Figure 12, the frequency dependence in varies as the properties of the air changes. As the damping terms we investigated all have a fixed frequency dependence it is easy to see that they will not be suitable to model the attenuation as it will not be able to represent these air property changes. 1 12

16 Vibational relaxational damping Humidity: 10% Humidity: 50% Humidity: 90% Frequency (Hz) Figure 10: Vibrational relaxation damping vib,o caused by oxygen for three di erent atmospheres. The frequency dependence varies significantly. Vibrational relaxation damping 8 x Humidity: 10% Humidity: 50% Humidity: 90% Frequency (Hz) Figure 11: Vibrational relaxation damping vib,n caused by nitrogen for three di erent atmospheres. The frequency dependence varies significantly. Total damping Humidity: 10% Humidity: 50% Humidity: 90% Frequency (Hz) Figure 12: Total damping for three di erent atmospheres. Since the damping is caused by several factors it is di cult to model it using a single damping term. 13

17 Further research should include trying to find damping terms corresponding to the di erent types of attenuation described above. By finding representation for all three terms in the time dependent wave equation it would be easier to make the model suitable for varying atmospheric data. 14

18 References [1] M. J. Grote, Nonreflection Boundary Conditions for Time Dependent Wave Propagation, Research Report , Seminar für Angewandte Mathematik, Zürich, Switzerland, April [2] ISO b, Acoustics - Attenuation of sound during propagation outdoors. [3] Martin Almquist, Numerical wave propagation in large-scale 3-d environments, [4] Ken Mattsson, Frank Ham, Gianluca Iaccarino, Stable and accurate wave-propagation in discontinuous media, Journal of Computional Physics, 277: , [5] DELTA (Danish Electronics, Lights & Acoustics), Nord2000. Comprehensive Outdoor Sound Propagation Model. Part 2: Propagation in an Atmosphere with Refraction, Journal no. AV 1851/00, Project no. A550054, 31 December 2001, Revised 31 March [6] Gustav Myhrman, The impact on tra c voice by roundabouts, ISSN , December [7] Conny Larsson, Olof Öhlund, Variations of sound from wind turbines during di erent weather conditions. 15