A Nondimensional Parameterization for Sound Propagation 

in the Atmosphere 

by Michael Mungiole and D. Keith Wilson 

ARL-TR-2950 March 2003 

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Army Research Laboratory 

Adelphi, MD 20783-1197 

ARL-TR-2950 March 2003 

A Nondimensional Parameterization for Sound Propagation 

in the Atmosphere 

Michael Mungiole and D. Keith Wilson 

Computational and Information Sciences Directorate, ARL 

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A Nondimensional Parameterization for Sound Propagation in the Atmosphere 

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Michael Mungiole and D. Keith Wilson 

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14. ABSTRACT 

Parabolic equation (PE) techniques have been successfully used to obtain numerical solutions of sound pressure attenuation in 

which sound propagation is affected by turbulence and vertical gradients in wind and temperature. The PE models generally 

produce accurate attenuation values, but the execution time is excessive for applications when near real-time results are required. 

To obtain sound level attenuation predictions at selected locations more quickly, we are developing an artificial neural network. 

As a first step in this effort, the PE and boundary conditions were modified to obtain a nondimensional version, written in the

MATLAB code. This nondimensional version was developed to be used to train the artificial neural network because a fewer 

number of parameters (seven) would be required to be specified, resulting in a reduced number of model runs to develop the 

training algorithm. This report documents the derivation of the appropriate equations that are used in the modified 

(nondimensional) version of the acoustic propagation model. In addition, graphical data are provided that identify the sensitivity 

of sound pressure attenuation to each of the seven nondimensional parameters. 

15. SUBJECT TERMS

Sound propagation, parabolic equation techniques 

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Michael Mungiole 

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v

Contents 

List of Figures vi

List of Tables vi

1. Introduction 1

2. Derivation of the Nondimensional Equations 1

2.1 Parabolic Equation ..........................................................................................................2 

2.2 Ground Boundary Condition...........................................................................................3 

2.3 Upper Absorbing Boundary Condition ...........................................................................5 

2.4 Initial Condition ..............................................................................................................5 

2.5 Atmospheric Profiles.......................................................................................................6 

3. Results 8

4. Summary 11

5. References 13
vi

List of Figures 

Figure 1. Influence of normalized source height, receiver height, and horizontal separation 

between them on the normalized sound pressure attenuation..................................................10 

Figure 2. Influence of the angle between propagation and wind directions (β), and 

normalized friction velocity, scale for turbulent sound speed fluctuations, and static flow 

resistivity on the normalized sound pressure attenuation ........................................................10 

List of Tables 

Table 1. Range and baseline values for the seven normalized parameters (Actual 

[dimensional] values and their units are indicated in parentheses.).........................................11
1

1. Introduction

Parabolic equation (PE) methods have become very popular for the calculation of sound 

transmission along ground-to-ground paths in the atmosphere [1]. These methods can account for 

refraction of the propagating waves by atmospheric wind and sound speed gradients, diffraction 

into shadow regions, and forward scattering of sound by atmospheric turbulence. With the 

introduction of the Green’s function PE by Gilbert and Di [2], it became possible to perform 

sound field calculations with parabolic equations very rapidly. Nevertheless, even the Green’s 

function PE is too slow in many Army applications, where calculations are required at many 

frequencies in fractions of a second. Among these applications are sensor platforms with 

algorithmic adaption to the propagation environment, tactical decision aids, and combat 

simulations. 

To address the Army’s need for faster sound transmission calculations, we are developing an 

artificial neural network model to predict the transmission loss (diminishment in sound energy 

resulting from propagation in the atmosphere). If successful, the neural net will provide an 

extremely fast but still accurate determination of transmission loss. Although neural nets have 

been used previously in acoustics [3], [4], [5], the main application has been automatic target 

recognition (ATR). Here, we are attempting something completely different, namely, the 

prediction of propagation effects. In principle, the propagation neural net could be used to 

compensate signatures for the propagation effects, thereby improving robustness of ATR 

algorithms, whether based on neural nets or other methodologies. 

In the development of any neural net, an appropriate set of input (training) and output variables 

must be determined. The output variable for the propagation neural net is simply the sound 

pressure at the receiver position. The input variables and parameters would be the propagation 

geometry (target and sensor heights, horizontal separation, sound frequency), atmospheric 

conditions (wind velocity and sound speed profiles), and acoustical ground properties. 

Successful development of the neural net requires the output variable to have a fairly smooth 

dependence on the input variables. It is also desirable to keep the input variables as few in 

number as possible. For this purpose, we develop in this report a nondimensional version of the 

acoustic propagation model that describes the propagation with a minimal number of parameters. 

The propagation model includes the wave propagation equation (in this case, the PE) as well as 

equations for the boundary conditions, initial condition, and atmospheric fields. 

2. Derivation of the Nondimensional Equations 

Among the features of this model are input parameters that specify the acoustic properties of 

porous media using relaxation theory [6], the ability to handle both narrow and wide angle cases, 

attenuation of the reflection at the upper boundary, and a second order accurate ground boundary 
2

condition. The model was written in MATLAB1 [7] and follows the procedures that are outlined 

in West et al. [1], which use a finite difference numerical technique to solve the parabolic partial 

differential equation. 

2.1 Parabolic Equation 

We take as our starting point the following narrow angle parabolic equation for sound 

propagation in a moving, inhomogeneous atmosphere: 

P / x i /( k )( / y / z k k )P, 2

0

2

eff

2 2 2 2 ∂ ∂ = 2 0 ∂ ∂ + ∂ ∂ + − (1) 

in which p = P exp (ik0x)x

1/2 is the sound pressure, x is the horizontal coordinate in the 

propagation direction, y is the horizontal coordinate perpendicular to the propagation, z is the 

height, keff is the effective wave number, and k0 is the wave number at ambient conditions.2

Furthermore, keff = 2π f/ceff, in which f is the frequency and ceff = c + ux is the effective sound 

speed, c being the actual sound speed and ux the component of the wind velocity in the direction 

of propagation. This equation includes waves traveling outward from the source in a roughly 30-

degree beam. It was derived by Nghiem-Phu and Tappert [8] and is equivalent to Equation (2.88) 

in Ostashev [9] if the components of the wind perpendicular to the propagation are neglected. 

The PE code we use in this project provides a “pseudo three-dimensional” solution to Equation 

(1), as described by West et al. [1]. In actuality, the field is determined only in the vertical plane 

of the x- and z- coordinates. This is accomplished by neglecting P/x in comparison to ∂P/∂x (a 

far-field approximation) and dropping the differentiation in y, with the result 

P / x i /( k )( / z k k )P. 2

0

2

eff

2 2 ∂ ∂ = 2 0 ∂ ∂ + − (2) 

This is equivalent to Equation (15) in West et al. [1] (although with keff replacing k) and 

represents the standard narrow angle parabolic equation. Note that P in this equation does not 

have units of pressure but rather, units of pressure divided by the square root of length. We will 

refer to P as the surrogate sound pressure. In practice, we may set keff = 2π f/ceff + iα in the 

preceding equation, in which α is an attenuation coefficient that represents the conversion of 

sound energy to heat in the propagating waves. 

Equation (2) contains three fundamental physical dimensions: time, length, and mass. To 

develop a nondimensional version of this equation, we must normalize by three quantities that 

involve these dimensions. We choose here the ambient sound speed (c0), ambient air density 

(ρ0), and frequency (f). Indicating normalized variables with overbars, we have the following 

nondimensional quantities: 

( ) ( ) 2

0 0

1 2

0 P = P c / f / ρ c (3) 

0 x = xf / c (4) 

0 z = zf / c (5) 

 1

 MATLAB is a registered trademark of the MathWorks. 

2

Ambient in this report refers to an arbitrary reference value that is characteristic of the propagating environment. Typically, 

the value at the source height is used. 
3

k0 = koc0 / f = 2π (6) 

and 

k k c / f eff = eff 0 (7) 

Substituting these quantities into Equation (2) results in the following nondimensional version of 

the parabolic equation: 

P / x i /( )( / z k )P. 2 2

eff

2 2 ∂ ∂ = 4π ∂ ∂ + − 4π (8) 

This equation is solved in a manner similar to the procedure used for the dimensional version in 

West et al. [1], and the detailed steps can be obtained from this reference. First, the terms are 

rearranged so that the equation is written as 

∂P / ∂x = a∂ P / ∂z + bm P 2 2 (9) 

and the coefficients a and bm, after it is transformed into a finite difference equation, are found to 

be 

( ) 2

a = i / 4π∆z (10) 

m , , ,M

k b i m

m 1 for 1 2 K 2

2

= 















− 













 = π

π (11) 

in which M is the total number of vertical mesh points and ∆z is the nondimensional difference 

between two vertical mesh points 

m m ∆z = z − z +1 (12) 

The complex wave number, mk , is defined by 

m m k / c i m = 2π eff + α (13) 

In this equation, m ceff is the effective sound speed in layer m normalized by c0 and α m is the 

attenuation normalized by wavelength (c0/f). 

Once P has been determined, the actual nondimensional pressure follows from the equation 

p Pe / x . i2πx = (14) 

2.2 Ground Boundary Condition 

As discussed by West et al. [1], the ground impedance boundary condition for the parabolic 

equation is 

P / x if P / Z , = −2π ρ0 c ∂ ∂ (15) 
4

in which Zc is the ground impedance. Many models are available for the impedance of porous 

ground surfaces such as are encountered outdoors. A particularly simple one that nevertheless 

gives accurate results was developed by Wilson [10]. The impedance equation is 

( ) ( )

0 5

0 5 0 5

0 0 1

1 1

1

1 1

.

.

v

.

e

c

i i

q

c

Z −

































− − 















−

− +

Ω= ωτ ωτ

γ

ρ

 (16) 

in which 

( ) ν ν τ ρ σ τ τpr

2 2

0 = q / 2 s pΩ and e = N (17) 

In these equations, the τν is a relaxation time associated with viscous diffusion, τe is a relaxation 

time associated with thermal diffusion, γ is the ratio of specific heats, Npr is the Prandtl Number, 

q is the tortuosity, Ω is the porosity, sp is the pore shape factor, and σ is the static flow resistivity. 

We would like to nondimensionalize the impedance equation in a form using fewer parameters. 

To do so, we first note that 

= ( Ω) 2 2

0 2 2 p iωτ i ωρ q f / σs ν (18) 

and define a nondimensional flow resistivity, σ , which depends on parameters previously 

defined. 

= ( Ω) 0

2 σ σs p / fρ (19) 

Combining Equations (18) and (19) gives 

( ) ( ) 2 2

pr

2 2 iωτ ν = iπq / σΩ andiωτ e = iπN q / σΩ (20) 

Substituting them into Equation (16) gives 

( ) ( ) ,

i N q / i q /

Z q

.

. .

pr

c

0 5

0 5 2 0 5 2 1

1 1

1

1 1

−





































−

− 



















−

− = +

π σ π σ

γ (21) 

in which Zc = Z/(ρ0c0) and q = q/Ω. Considering that γ and Npr are determined once the 

temperature is specified, the only parameters that are required to calculate Zc are σ and q . 

Thus, the number of ground parameters has been reduced from four (σ, q, Ω, and sp) to two (σ

and q ). 

In nondimensional form, the boundary condition, Equation (15), is 

P / z iP / Z . = −2π c ∂ ∂ (22) 

Regarding the numerical implementation of ∂P / ∂z in spatially discretized form, the first order 

accurate case can easily be obtained (West et al. [1], Section 5). For the nondimensional case, the 

first order accuracy boundary condition is found to be 
5

( ) 1

0 1 1 2 −

= − ∆ Zc P P πi z / (23) 

and the respective nondimensional boundary condition for second order accuracy is 

( )( ) 1

0 4 1 2 3 4 −

= − − ∆ Zc P P P πi z / (24) 

2.3 Upper Absorbing Boundary Condition 

In order to avoid spurious numerical reflections of the sound wave at the upper boundary of the 

computational domain, an absorbing layer at the top of the domain is often added. Salomons [11] 

suggests the following quadratic form for the attenuation coefficient in this layer: 

abs

2

top abs

abs

z , z zz

z z

> 















−

− α = μ (25) 

in which μ is an adjustable parameter (dimensions of inverse length), zabs is the height where the 

absorbing layer begins, and ztop is the top of the computational domain. The absorbing layer is 

typically 30 to 100 wavelengths deep. Thus, we might rewrite Equation (25) in nondimensional 

form as 

abs

2

abs , z z

N

z z

> 













 − =

λ

α μ (26) 

in which Nλ is the depth of the layer in wavelengths, α = αc0/f, and μ = μc0/f. Salomons [11] 

suggests values for the parameter μ that, while frequency dependent, are only roughly 

proportional to f. Here, we propose setting μ = 1. This leads to values of μ that are fairly close 

to the ones in Salomons’ Table I, particularly near 100 Hz. 

2.4 Initial Condition 

The “initial condition” here refers to the sound pressure at x = 0, from where the solution is 

marched forward in distance, as opposed to the solution at some initial start time. (Keep in mind 

that the source is assumed to be harmonic, so that there is no start time.) Following Salomons 

[11], we use the following Gaussian starter function for the initial condition: 

( ) ( ) ( ) z z , k

z z R k P ,z S k s s





















 



  + − +



 



 = − − 2

2

2 0

2

0

0 2 exp 2 0 (27) exp

in which zs is the source height, S is the source strength (in units of pressure) and 

( )

( ) 1

1

0 0

0 0

+

− = Z / c

Z / c R

c

c

ρ

ρ (28) 
6

is the surface reflection coefficient. In nondimensional form, these equations become 

P( ) ,z S { [ (z z ) ] R [ (z z ) ]}, s s

2 2 2 2 0 = 2π exp − 2π − + exp − 2π + (29) 

in which 

2

0 0 c

S S

ρ = (30) 

and 

. Z

Z

R

c

c

1

1

+

− = (31) 

2.5 Atmospheric Profiles 

By the refraction of sound energy, the wind and temperature profiles near the ground can have a 

very significant impact on the pressure field. Therefore, we also need a nondimensional model 

for these profiles. Let us begin by writing the effective sound speed as 

c c ucosβ , eff = + (32) 

in which c is the actual sound speed, u is the horizontal wind, and β is the angle between the 

propagation direction and the wind. Neglecting the effect of humidity (which is usually small), 

the sound speed in air is given by 

, T

T

c c 













 ′ = +

0

0 2

1 (33) 

in which T = T +T′ 0 is the temperature and T′ a small perturbation about the ambient value T0. 

The following equations, based on Monin-Obukhov similarity [12, 13], are known to describe 

the near-ground temperature and wind profiles well over reasonably uniform terrain: 

() ( ) ( ) z z , L

z

L

z

z

z

k

P T

T z T z d r

o

r h

o

h

r

t r − Γ − 































+ Ψ 













 − = − Ψ ∗ ln

ν

 (34) 

and 

( ) , L

z

L

z

z

z

k

u

u z

o

m

o

m

































+ Ψ 













 = − Ψ ∗ 0

0

ln

ν

 (35) 

in which kν = 0.40 is von Kármán’s constant, Pt = 0.95 is the turbulent Prandtl number, Γd = g/Cp

= 0.0098 K/m is the dry adiabatic lapse rate (accounting for the decrease of temperature with 

height because of compression in the air column), g is gravitational acceleration, z0 is the surface 

roughness length, ∗ u is the friction velocity, ∗ > ∗ T = − < w′T′ s / u is a temperature scale, and 
7

w T >s < ′ ′ is the covariance of vertical velocity and temperature at the surface (kinematic heat 

flux). The variable zr is a reference height at which the temperature is measured; if we explictly 

set T0 = T (zr), then T′ in Equation (33) equals T(z) – T(zr). The Ψ’s are universal functions of 

the dimensionless ratio ζ = z/Lo, in which o g w T s L = u∗T0 / kν < ′ ′ > 3 is called the Obukhov length. 

The following equations for Ψh and Ψm are recommended based on Högström [14] and Wilson 

[15]: 

( ) ,

b ,

a / ,

b,m

h,m

h,m











− ≥

< 



 



 



 



 + + Ψ =

0

2 ln 1 1 2 0 2 3

ζ ζ

ζ ζ ζ (36) 

in which the a’s and b’s are constants with the values ah = 7.9, am = 3.6, bh = 8.4, and bm = 5.3. 

The case ζ < 0 corresponds to buoyantly unstable conditions, which typically occur when the sun 

heats the ground. Buoyantly stable conditions (ζ > 0) typically occur when the ground cools at 

night. In the limit ζ → 0, the Ψ-functions equal 0 and the profiles assume their familiar 

logarithmic forms for neutral conditions. 

Now, for the effective sound speed, we have 

( ) ( )

. L

z

L

z

z

z

k c

u

L

z

L

z

z

z

k T

PT

T

z z

c

c z

o

m

o

m

o

r

h

o

h

r

d r t

































+ Ψ 















+ − Ψ

































+ Ψ 















+ − Ψ

Γ − = −

∗

∗

0

0 0

0 0 0

eff

ln cos

ln

2 2

1

ν

ν

β

 (37) 

Using the thermodynamic equations Cp = Cν + R and γ = Cp/Cν, along with the ideal gas law 

0

2

c0 = γRT , we find 2

0 0 gT ( 1)/c Γd = γ − . This relationship allows the second term on the right of 

the previous equation to be rewritten as 2 2 0 g( 1)(z z )/ c − − r γ . Following Wilson [16], it is also 

convenient to define 0 2 0 c∗ = c T∗/ T . We then have 

( ) ( )( )

. u c

k gc z

u c

k gc z

z

z

k c

u

u c

k gc z

u c

k gc z

z

z

k c

P c

c

g z z

c

c z

m m

r

h h

r

t

r

































+ Ψ 















+ − Ψ

































+ Ψ 















+ − Ψ

− − = −

∗

∗

∗

∗ ∗

∗

∗

∗

∗ ∗

0

2

0

0

2

0 0

0

2

0

2

0

2

0 0

eff

2 2 ln cos

2 2 ln

2

1 1

ν ν

ν

ν ν

ν

β

γ

 (38) 

Let us next define the following nondimensional variables: 0 c c /c eff = eff , 0 c c /c ∗ = ∗ , 0 u u /c ∗ = ∗ , 

( ) 0 g = g / c f , 0 z = zf/c , 2

0 z z g/c r = r , and 20 0 0 z = z g/c : 
8

( ) ( )( )

. u

K c z

u

k gc z

z

gz

k

u

u

K c z

u

k gc z

z

gz

k

P c

gz z

c z

v

m m

v r

h h

t

r

































+ Ψ 















+ − Ψ

































+ Ψ 















+ − Ψ

− − = −

∗

∗

∗

∗ ∗

∗

∗

∗

∗ ∗

2

0

2

0

2 2

0

eff

2 2 ln cos

2 2 ln

2

1 1

ν

ν

ν

ν

β

γ

 (39) 

The profile c ( z ) eff therefore depends on the nondimensional parameters r z , 0z , g , ∗ c , u ∗ , cos β. 

It may initially seem peculiar to normalize zr and z0 with c / g

2

0 instead of c0/f, as was used for 

other lengths. The advantage of the c / g

2

0 normalization is that zr and z0 are fixed values in many 

applications; therefore, r z and 0z are constants and need not be used as parameters in the neural 

net. 

3. Results 

The development of these nondimensional equations results in 13 nondimensional parameters 

that need to be specified. These are 

1. s z , the source height 

2. c z , the receiver height 

3. s x , the horizontal separation between the source and receiver 

4. σ , the normalized static flow resistivity for the ground 

5. q , the tortuosity to porosity ratio for the ground 

6. Nλ, the number of wavelengths in the upper absorbing layer 

7. μ , the attenuation parameter for the upper absorbing layer 

8. r z , the reference height at which the temperature is measured 

9. 0 z , the roughness length 

10. g , the normalized gravitational acceleration 

11. ∗ c , the scale for turbulent sound speed fluctuations 

12. ∗ u , the scale for turbulent wind speed fluctuations (the friction velocity) 

13. cosβ, the cosine of the angle between the propagation and wind directions 
9

Many of these parameters would not be needed for the training of a neural net. If the calculations 

are done properly, the parameters for the upper absorbing layer, Nλ and μ , should not affect the 

solution. The reference height r z is typically a constant value. For a given ground surface, 0 z , 

q , and the ratio σ/g are constants. σ/g is a constant because the source frequency is eliminated 

when this ratio is taken, resulting in the necessity of only one of these parameters being 

specified. Thus, as few as seven parameters could be used to train the neural net. These include 

the three parameters that specify the source and receiver locations ( s z , c z , and s x ), the three 

parameters needed to calculate the normalized effective sound speed ( ∗ c , ∗ u , and cosβ), and the 

nondimensional flow resistivity (σ ), which is used to calculate the nondimensional ground 

impedance. 

In order to get an idea of the relative importance of each of these seven parameters in generating 

a training algorithm for an artificial neural network, it would be necessary to determine their 

sensitivity on the sound pressure attenuation at a specific location. This was performed for the 

seven parameters, with the results indicated in Figures 1 and 2. The parameter representing each 

respective curve in these figures was varied over a typical physical range, and the resulting sound 

pressure attenuation was determined from the nondimensional model. The smallest and largest 

values for each parameter represent 0% and 100%, respectively, in the figures. These values and 

the baseline values for the model runs in which the respective parameter was not varied are given 

in Table 1. For simulation runs in which the receiver height and horizontal separation were not 

the parameters being varied, the sound pressure attenuation was obtained at a location for which 

the value of these parameters were 1 m and 500 m, respectively. 

The effects of varying the geometric parameters are indicated in Figure 1 while the influence of 

the parameters used to calculate effective sound speed and ground impedance are shown in 

Figure 2. Although the range of the sound pressure attenuation is different in these two figures, 

their scales are equivalent. This allows a direct comparison of the sensitivity of the sound 

pressure attenuation to each of the seven manipulated parameters. As indicated in these figures, 

the attenuation is relatively insensitive to the source and receiver heights and friction velocity. 

For the source and receiver heights, this is attributable to the small range over which these 

parameters were manipulated. In general, varying static flow resistivity had little effect on sound 

attenuation. For small values of this parameter, however, there was a more substantial change in 

the sound pressure attenuation. This could be partly attributable to the large range over which 

this parameter was varied, which was almost two orders of magnitude. As the wind direction was 

varied from 0 to 180 degrees (i.e., from the same to the opposite direction of sound propagation), 

the sound attenuation varied by approximately 10 dB. The ranges over which the normalized 

horizontal separation and sound speed fluctuation scale were varied had the largest effects on 

attenuation, with both of them producing more than a 17-dB reduction in the sound pressure. The 

effect of the sound speed fluctuation scale on attenuation was unique in that it was the only 

parameter that did not exhibit a monotonically increasing or decreasing relationship. 

Because the sound pressure attenuation values are normalized to 2

0 0 ρ c , it is probably necessary 

to put the results in proper perspective. Dividing the sound pressure by the product of 2

0 0 ρ c

results in a decrement of approximately 103 dB; thus, one would need to add 103 dB to the 

values indicated in Figures 1 and 2 to obtain the actual sound pressure level attenuations in the 

figures. 
10

Figure 1. Influence of normalized source height, receiver 

 height, and horizontal separation between them 

 on the normalized sound pressure attenuation. 

Figure 2. Influence of the angle between propagation 

 and wind directions (β), and normalized friction 

 velocity, scale for turbulent sound speed fluctuations, 

 and static flow resistivity on the normalized sound 

 pressure attenuation. 
11

Table 1. Range and baseline values for the seven normalized parameters (Actual [dimensional] 

 values and their units are indicated in parentheses.) 

Parameter smallest value largest value baseline value 

source height, s z 0.0 0.74 0.148 

(m) (0.0) (5.0) (1.0) 

receiver height, c z 0.0 0.74 0.148 

(m) (0.0) (5.0) (1.0) 

horizontal separation, s x 14.8 133.4 74.1 

(m) (100) (900) (500) 

angle between propagation 

and wind direction, β 0 π 0 

friction velocity, u∗ 4 1 48 10− . ∗ 3 1 48 10− . ∗ 4 8 32 10− . ∗

(m/s) (0.05) (0.5) (0.28) 

turbulent sound-speed 

scale, ∗ c 3 3 5 10− . ∗ 3 0 296 10 − . ∗ – 3 0 593 10− . ∗

(m/s) (–1.18) (0.1) (–0.2) 

static flow resistivity, σ 200 16000 1605 

(mks rayls/m) (25000) (2000000) (200000 

4. Summary 

A system of nondimensional equations for sound propagation in the near-ground atmosphere was 

developed. They are based on the narrow angle parabolic equation for sound propagation in a 

moving medium, a relaxation model for the sound-absorbing lower boundary condition, and 

Monin-Obukhov similarity for the atmospheric profiles. 
12

INTENTIONALLY LEFT BLANK. 
13

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