Abstract

Simulations of sound pressure levels are often used as a decision ba-sis when planning location for e.g wind turbines. In previous work, ane�cient model for computing sound propagation over irregular terrainhas been developed. The model simulates the propagation using thetime dependent wave equation. The purpose of this study was to im-prove this solver by finding a way to represent atmospheric attenuationin the time dependent wave equation by adding a damping term to theequation. This was done by numerically simulating wave propagationand comparing to measured data on the atmospheric attenuation. Thestudy did not result in finding a satisfying way to model the attenua-tion. The results of the simulations are analyzed and suggestions forfurther research are given.

Contents

1 Introduction 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Atmospheric Attenuation . . . . . . . . . . . . . . . . . . . . 21.3 Project definition . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Analysis 3

2.1 The wave equation . . . . . . . . . . . . . . . . . . . . . . . . 32.2 The SBP-SAT method/The semi discrete model . . . . . . . . 3

3 Results 6

3.1 Simulations using damping term ��ut . . . . . . . . . . . . . 73.2 Simulations using damping term �ut and �r2ut . . . . . . . . 93.3 Simulations using damping term ��ut � �r2ut . . . . . . . . 11

4 Conclusions and discussion 11

1 Introduction

1.1 Introduction

Calculations of sound propagation are often used as decision basis whenplanning the construction of for example wind turbines, airports, tra�cjunctions and other places where extreme sound pressure levels occur andcause problems. This makes accurate modeling of sound propagation animportant subject. To predict sound pressure levels there are several factorsthat should be included in the model. For example, the sound pressure levelsdepends strongly on meteorological data such as sound speed variations andatmospheric attenuation as well as the topography of the area over whichthe 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 Swe-den today is NORD2000, based on ray tracing theory. Ray tracing theorymodels 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 cal-culations.

A good model for sound propagation is the time-dependent acoustic waveequation:

utt = c2�u (1)

Solving the wave equation can result in high computational cost if not us-ing an e�cient numerical method. The computation time can be reduced byusing the Helmholtz equation instead. Helmholtz equation is a time indepen-dent, frequency domain representation of the wave equation. Transformingthe wave equation to frequency domain results in less computational cost,but modeling in frequency domain demands sources consisting of a single fre-quency. Wind turbines are broadband sources and require a time-dependentmodel for e�cient simulation of sound pressure levels.

In previous work, an e�cient numerical solver for solving wave propagationproblems on large domains has been implemented [3]. This solver uses fourthorder accurate finite di↵erence methods and runs in parallel in MATLAB.The solver can be used to model propagation over complicated terrains, butdoes not take into account meteorological data such as atmospheric attenu-ation.

1

1.2 Atmospheric Attenuation

Atmospheric attenuation is a damping of the sound waves caused by trans-port processes and molecular relaxtion losses in the atmosphere. As soundwaves propagate through the atmosphere the amplitude decreases due toabsorption as

pt = pie�0.1151↵s, (2)

where pt is the sound pressure amplitude at a distance s from the initialpressure pi and ↵ is the attenuation coe�cient dependent on frequency,background pressure, humidity and temperaure. More detailed informationabout atmospheric attenuation and measured values for ↵ for varying atmo-spherical data is given in [2].

Figure 1: Example of table with values on the attenuation constant ↵ forvarying atmospherical data as given in [2].

1.3 Project definition

In this project we have studied the possibility to use a damping term tomodel the atmospheric attenuation in the time dependent wave equation.This would be a way to improve the model developed in [3]. The ideawas to add a physical damping term to the right hand side of (1) of theform r · �rut, where the coe�cient � is a function of the atmosphere, i.e.dependent on temperature, humidity and pressure,

utt = c2�u+r · �rut. (3)

2

The damping term should depend on frequency correspondingly to the fre-quency dependence in the atmospheric attenuation. Ideally, this would leadto a model that could accurately simulate the atmospheric attenuation ofbroad band sources, which as discussed above is desired.

The main object of the project was, thus, to study the wave equation withthe added damping term with the aim to find an explicit form of the damp-ing term. This is done by simulating the wave propagation and testing itagainst ↵-values from [2] inserted in the analytic solution given by (2). Tobegin with, a fixed atmosphere (i.e fixed values on humidity, temperatureand background pressure) was examined. The aim was to later extend themodeling to more general atmospheres.

In the following sections we first describe the numerical model and howthe sound propagation was simulated. We then present the results of oursimulations, followed by a discussion about the results and recommendationsfor further research.

2 Analysis

2.1 The wave equation

In order to investigate the behavior of the damping term we perform simula-tions of damped waves and compare them to analytical data. Sound from apoint source propagates with spherical symmetry in free space. Using spher-ical coordinates in one dimension will therefore lead to a simplified modelthat will still simulate e↵ects relevant to propagation in three dimensions.Adding the damping term as described above to the one-dimensional waveequation in spherical coordinates and setting � in (3) to be independent ofr yields

r2utt = c2@

@r

✓r2

@u

@r

◆+ �

@

@r

✓r2

@ut@r

◆(4)

By numerically modeling this wave and comparing it to the analytical atten-uated wave equation the damping term can be adjusted to fit the analyticaldamping. The analytical form of the damped wave is given by applying thedamping described in (2) to the analytical solution of the one dimensionalspherical wave equation

ud(r, t) =a

rcos(!t� !

cr) · e�0.1151↵r (5)

2.2 The SBP-SAT method/The semi discrete model

The SBP-SAT method is a finite di↵erence method that combines Summa-tion By Part operators (SBP) with the simultaneous approximation terms

3

method (SAT) to weakly impose boundary conditions. Proving stability forSBP-SAT representations is done by analyzing the energy using the semidiscrete energy method as demonstrated later in this section. In order topresent the SBP-SAT method we first introduce some notation and defini-tions (following [4]).

Discretization The domain r0 r rN is discretized using Mpoints given by

ri = r0 + ih, h =rN � r0M � 1

, i = 1, 2, ...,M.

The numeric solution at ri is denoted by vi and the correspondinganalytic solution is denoted by u(ri). These vectors are used whenintroducing the simultaneous approximation terms

e1 = [1, 0, ..., 0]T , eN = [0, ..., 0, 0]T .

Definition 1 A di↵erence operator D1 = H�1Q approximating@/@r is a first derivative SBP operator if H = HT > 0, andQ+QT = B = diag(�1, 0..., 0, 1).

Definition 2 The operator D(b)2 = H�1(�M (b) + BS) approximates

@/@r(b(r)@/@r), where b(r) > 0, H is diagonal and positive definite,M (b) is symmetric and positive semidefinite, S approximates the firstderivative operator at the boundaries and B = diag(�b0, 0, ..., 0, bN ),.

Definition 3 An inner product for discrete real-valued vectorfunctions v, w 2 RM is given by (v, w)H = vTHw where H = HT > 0,with corresponding norm ||v||2H = vTHv.

The continuous term ur, for example, would have the semi-discrete approxi-mation D1v using the first derivative SBP operator introduced above. Belowwe apply the SBP-SAT method to the wave equation given in spherical co-ordinates. A nonreflecting boundary conditions for spherical coordinatesas described by Grote in [1] is applied on the right boundary in order toprevent non-physical reflections. The left (inflow) boundary is implementedby strongly enforcing the sound pressure levels, sometimes referred to asinjection:

4

8>>>>>><

>>>>>>:

r2utt = c2@

@r

✓r2

@u

@r

◆, r0 r rN , t > 0

ur +1

cut +

u

r= 0, r = rN

u(r, t) =a

rcos(!t� !

cr), r = r0

(6)

(7)

(8)

0 r0 rN > rN

-1

0

1

Distance

Amplitude

Non-reflectiveboundaryAnalytic injection

Computational domain

Figure 2: The simulation set up. The actual computational domain is be-tween r0 and rN .

The semi-discrete approximation of (6) using the second order SBP operatoris given by

Bvtt = c2D(b)2 v (9)

where B = diag(b0, b21, ..., b2N ) and b(r) = r2. The nonreflecting boundary

condition is imposed weakly by adding a simultaneous approximation termto the right hand side of the equation

Bvtt = c2D(b)2 v �H�1eN⌧{(Sv + vt

c+

v

r)N} (10)

The SAT method can be seen as a force pulling the solution on the boundarytowards the boundary condition. To prove stability we use the semi discreteenergy method, given by multiplying the equation by vTt H and adding thetranspose. This yields

d

dt(||vt||2HB + c2vTM (b)v) = c2vTt BSv + c2(vTt BSv)T�

� 2⌧(vTt )N{(Sv + vtc+

v

r)N} (11)

5

Where we have used that vTt HBv + vTHBvt = ddtv

THBv = ddt ||v||HB.

The semi discrete energy estimate for the wave equation is given by E =||vt||2HB+c2vTM (b)v, which gives that the left hand side in (11) correspondsto dE/dt.

d

dtE = c2vTt BSv + c2(vTt BSv)T � 2⌧(vTt )N{(Sv + vt

c+

v

r)N} (12)

The right hand side is further simplified using the definition of B and S

d

dtE = �2c2b0(v

Tt )N (Sv)N + 2(c2bN � ⌧)(vTt )N (Sv)N

� 2⌧

c� 2⌧

rN(vTt )NvN (13)

To prove stability we must have dE/dt 0. In order to cancel out the term2c2bN (vTt )N (Sv)N that could lead to non-physical growth in energy we mustset the SAT coe�cient to

⌧ = bNc2 = r2Nc2. (14)

Adding the damping term ��ut to the equation introduces a secondSAT term to the semi discrete representation. By the same method as abovethe penalty term is set to ⌧2 = �r2N , and the semi discrete approximation isgiven by

Bvtt = c2D(b)2 v + �D

(b)2 vt�r2Nc2H�1eN{(Sv + vt

c+

v

r)N}

��r2NH�1eN{(Svt +vttc

+vtr)N} (15)

3 Results

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 ↵). Allthe testing is done against ↵-values for this specific atmosphere. The ini-tial intention was to further extend the study to more general atmosphericdata, but this was omitted given the nature of the results obtained. Thesimulations began with trying to adapt the damping term ��ut to the at-mospheric attenuation data for the given atmosphere. It showed that itwas possible to mimic the attenuation for a specific frequency by choosing asuitable value on �, but that the damping term and the attenuation did notfollow the same frequency dependence. We then moved on to testing otherdamping operators and combinations of operators. Below the results fromthese simulations are presented.

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3.1 Simulations using damping term ��ut

The simulations were started by trying to find the correct damping for afixed frequency. This was possible, which is shown in Figure 3 where themaximum values of the amplitude for both the analytical and the numericalsolution are shown for a damped wave with frequency 1000 Hz. The numer-ical and analytical solution follows the same decay.

10 15 20 25 30 35 40 45 50 55 60

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Distance (m)

Amplitude

Amplitude for frequency 1000 Hz over distance

AnalyticNumeric: !(1000 Hz)

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 dif-ferent values of �. The damping operator �ut a↵ects signals with higherfrequencies much stronger than the atmospheric attenuation does, which re-sults in that a lower value on � is needed for higher frequencies in order tofit the damping. The optimal �-values for frequencies ranging between 50and 1000 Hz are shown in Figure 4. Since di↵erent frequencies demand dif-ferent values �, a signal containing several frequencies can not be modelledusing just one damping term as it would only give the correct damping forone 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

0 200 400 600 800 10000

2

4

6

8

9

Frequency (Hz)

·10!3

!

Figure 4: The value of � as a function of frequency. Since di↵erent frequen-cies demand di↵erent values on �, this damping term can not be used tomodel signals containing multiple frequencies.

10 15 20 25 30 35 40 45 50 55 60

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Distance (m)

Amplitude

Amplitude for frequency 1000 Hz over distance

AnalyticNumeric: !(1000 Hz)Numeric: !(50 Hz)

Figure 5: The amplitude of a signal with a frequency of 1000 Hz with �optimized for 50 Hz compared to the analytical solution.

8

10 20 30 40 50 60 70 80 900.985

0.99

0.995

1

1.005

1.01

1.015

Distance (m)

Amplituderatio

!(50 Hz)!(250 Hz)!(1000 Hz)

Figure 6: Amplitude ratio between the numerical and the analytical solution.A signal with frequency 250 Hz is modelled using three values of �. Anincorrect value of � leads to a increasing error as the sound wave propagates.

3.2 Simulations using damping term �ut and �r2ut

Proceeding our project, we tried a damping term of a di↵erent form, namely�ut, resulting in the equation

utt = c2�u� �ut. (16)

As in the previous subsection, we started out by trying to find the cor-rect damping for a signal of fixed frequency. This did not succeed usingthe damping term �ut. The numerical solution did not follow the sameexponential decay as the analytical solution. In Figure 7, the amplituderatio between the numerical and the analytical solution is shown. What isdesired is a straight line, meaning that the numerical solution follows thesame damping relation as the analytical. In the case with the damping term�ut this was not observed.

We found that we needed to add a factor of r2 to the damping term inorder to obtain the correct damping,

utt = c2�u� �r2ut (17)

As seen in Figure 7, the �r2ut-term gives a decay coinciding with the ana-lytical solution. However, this damping did not give better results than the

9

��ut-term when simulating varying frequencies. Here too it was found thatdi↵erent frequencies demand di↵erent values of � (see Figure 7 and Figure8).

10 20 30 40 50 60 70 80 90 100 1100.9995

1.0005

1.0015

1.0025

1.0035

1.004

Distance (m)

Amplitu

dera

tio

!

Undamped

! r2

Figure 7: The amplitude ratio between the numerical and analytical solutionfor a 250 Hz signal. Using a damping term �ut, the numerical solution didnot follow the same damping as the analytical solution. Using �r2ut it waspossible to obtain a correct damping. The undamped solution is includedfor comparison.

0 200 400 600 800 10000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency (Hz)

!

Figure 8: The value of � as a function of frequency. As in the previoussimulations it was found that di↵erent frequencies demanded di↵erent valueson �.

10

3.3 Simulations using damping term ��ut � �r2ut

As the � damping was stronger for high frequencies and the �r2 dampingwas stronger for low frequencies a combination of the two was tested, namely

utt = c2�u+ ��ut � r2�ut. (18)

The idea was that a combination of the two damping terms could result ina fair approximation of the damping ranging over various frequencies. Thetwo damping terms were adjusted to optimize the damping for two di↵erentfrequencies, 50 Hz and 1000 Hz, i.e the values of � and � were chosen togive correct damping for signals of 50 Hz and 1000 Hz. When this setupwas tested against frequencies lying between 50 Hz and 1000 Hz, it did notresult in a correct damping. This is shown in Figure 9, for the amplituderatios for frequencies 50, 250 and 1000 Hz.

10 12 14 16 18 20 22 24 26 28 300.9995

1

1.0005

1.001

1.0015

1.002

1.0025

Distance (m)

Amplitu

dera

tio

50 Hz250 Hz1000 Hz250 Hz ! Undamped

Figure 9: The amplitude ratio for the frequencies 50, 250 and 1000 Hzusing the modeling explained above. Satisfactory damping was obtainedfor signals near 50 Hz and 1000 Hz, but not for signals in between. Theamplitude 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 atmosphericattenuation by using the damping terms investigated above. The aim was tofind a way to model signals containing multiple frequencies, but no satisfac-tory way to do this was found. The reason for this was that the atmosphericattenuation followed a more complicated frequency dependence than ex-pected. A closer look at the physics describing the attenuation reveals some

11

useful information about this. The ISO-paper [2] describes the attenuationas a sum of di↵erent types of absorption derived to several physical mecha-nisms having distinct frequency dependencies:

• Classical (↵cl) and rotational absorption (↵rot) caused by trans-port processes described by ”classical” mechanics and molecular rota-tional relaxation. The classical absorption is caused by compressionand expansion of the air leading to friction between particles [6]. Thesetypes of absorptions follow a quadratic frequency dependence given by

↵cr = ↵cl + ↵rot =1.6 · 10�10(T/T0)1/2f2

pa/pr,

where T0 and pr represent reference temperature and pressure.

• Vibrational relaxation damping caused by vibrational relaxationof oxygen(↵vib,O) and nitrogen(↵vib,N ).The attenuation caused by thesefactors follow a more complicated frequency dependence given by

↵vib,O =2[(↵�)max]

cfrO· f2[1 + (f/frO)

2]�1

↵vib,N =2[(↵�)max]

cfrNf2[1 + (f/frN )2]�1

where the terms fr represent relaxation frequencies. The relaxationfrequencies are properties of the nitrogen and oxygen, dependent ontemperature and humidity of the air. Based on these factors the fre-quency dependence in the vibrational relaxation damping can changesignificantly (see figure 10 and figure 11).

The total attenuation is thus given by ↵ = ↵cr+↵vib,O+↵vib,N . Since thesedi↵erent types of attenuation are derived from di↵erent physical mechanismsand follow varying frequency dependence, it seems logical to separate theminto 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 theproperties of the air changes. As the damping terms we investigated allhave a fixed frequency dependence it is easy to see that they will not besuitable to model the attenuation as it will not be able to represent theseair property changes.

12

0 100 200 300 400 500 600 700 800 900 10000

0.002

0.004

0.006

0.008

0.01

0.012

0.014

Frequency (Hz)

Vibationalre

laxationaldamping

Humidity: 10%Humidity: 50%Humidity: 90%

Figure 10: Vibrational relaxation damping ↵vib,O caused by oxygen for threedi↵erent atmospheres. The frequency dependence varies significantly.

0 100 200 300 400 500 600 700 800 900 10000

1

2

3

4

5

6

7

8x 10

!4

Frequency (Hz)

Vibra

tionalre

laxationdamping

Humidity: 10%Humidity: 50%Humidity: 90%

Figure 11: Vibrational relaxation damping ↵vib,N caused by nitrogen forthree di↵erent atmospheres. The frequency dependence varies significantly.

0 100 200 300 400 500 600 700 800 900 10000

0.002

0.004

0.006

0.008

0.01

0.012

0.014

Frequency (Hz)

Tota

ldamping↵

Humidity: 10%Humidity: 50%Humidity: 90%

Figure 12: Total damping ↵ for three di↵erent atmospheres. Since thedamping is caused by several factors it is di�cult to model it using a singledamping term.

13

Further research should include trying to find damping terms corre-sponding to the di↵erent types of attenuation described above. By findingrepresentation for all three terms in the time dependent wave equation itwould be easier to make the model suitable for varying atmospheric data.

14

References

[1] M. J. Grote, Nonreflection Boundary Conditions for Time DependentWave Propagation, Research Report 2000-04, Seminar fur AngewandteMathematik, Zurich, Switzerland, April 2000.

[2] ISO9613 1996b, Acoustics - Attenuation of sound during propagationoutdoors.

[3] Martin Almquist, Numerical wave propagation in large-scale 3-d envi-ronments, 2012.

[4] Ken Mattsson, Frank Ham, Gianluca Iaccarino, Stable and accu-rate wave-propagation in discontinuous media, Journal of ComputionalPhysics, 277:8753-8767, 2008.

[5] DELTA (Danish Electronics , Lights & Acoustics), Nord2000. Com-prehensive Outdoor Sound Propagation Model. Part 2: Propagation inan Atmosphere with Refraction, Journal no. AV 1851/00, Project no.A550054, 31 December 2001, Revised 31 March 2006.

[6] Gustav Myhrman, The impact on tra�c voice by roundabouts, ISSN1401-5765, December 2009.

[7] Conny Larsson, Olof Ohlund, Variations of sound from wind turbinesduring di↵erent weather conditions.

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