Low-frequency Sound Propagation 

Modeling Over a Locally-reacting Boundary 

Using the Parabolic Approximation 

by 

J.S. Robertson, W.L. Siegmann 

and M.J. Jacobson 

/v&-/- 9A4 

Department of Mathematical Sciences 

Rensselaer Polytechnic Institute 

Troy, New York 12180-3590 

RPI Math. Fkp. No. 175 

August 31, 1989 

This work is sponsored by 

National Aeronautics and Space Administration 

Langley Research Center 

Research Grant No. NAG-1-929 

This document has been approved for public release and sale; its distribution is unlimited. 
ABSTRACT 

There is substantial interest in the analytical and numerical modeling of low-frequency, 

long-range atmospheric acoustic propagation. Ray-based models, because of frequency limitations, do not always give an adequate prediction of quantities such as sound pressure 

or intensity levels. However, the parabolic approximation method, widely used in ocean 

acoustics, and often more accurate than ray models for lower frequencies of interest, can 

be applied to acoustic propagation in the atmosphere. Modifications of an existing implicit 

finite-difference implementation for computing solutions to the parabolic approximation are 

discussed. A locally-reacting boundary is used together with a one-parameter impedance 

model. Intensity calculations are performed for a number of flow resistivity values in both 

quiescent and windy atmospheres. Variations in the value of this parameter are shown to 

have substantial effects on the spatial variation of the acoustic signal. 
INTRODUCTION 

The propagation of low-frequency sound through the earth's atmosphere over long distances near the ground surface is a problem with numerous applications. For example, the 

propagation through winds of a low-frequency noise field generated by large wind turbines has 

been extensively studied both theoretically and experimentally.' In addition, low-frequency 

sound can be used to detect, locate and track aircraft or vehicles with passive acoustic 

sensors. 2r3 

In many instances, acoustic propagation occurs in environments which may be characterized by winds, atmospheric turbulence, extremes of weather, and other natural and manmade atmospheric variations, as well as irregular topography and terrain structure. These 

environmental variations are typically range- as well as height-dependent, and can profoundly 

affect the behavior of sound waves. Geometrid acoustics, or classical ray theory, is one ap 

proach that has been applied widely to the study of atmospheric acoustics. Unfortunately, 

the approximations under which the ray equations hold are valid only for sufficiently high 

source frequencies. At lower frequencies where diffraction effects are especially important, 

the use of other mathematical models can provide more accurate and useful results. Models, 

such as parabolic approximations4 and the fast-field program' are more appropriate for use at 

frequencies where dif€ractive effects are non-negligible. Of these two approaches, the method 

of parabolic approximation is well-suited for handling range-dependent environments and 

has been applied to examine atmospheric propagation in a number of studies.6-10 Originally 

formulated to examine tropospheric radio wave propagation, and successfully applied to a 

broad variety of problems in ocean acoustics, parabolic approximations exploit characteristic 

features of the propagation medium associated with the formation of a waveguide. Atmospheric acoustic waveguides can be created by certain meteorological conditions either with 

or without boundary interaction. Within such a waveguide, sound waves may propagate to 

relatively large distances with significant amplitudes. 

While many studies have focussed attention on calculating the sound field propagation 

1 
over an impedance boundary, (e.g. see Refs. 11 and 12 and the references contained therein), 

relatively few have been concerned with low-frequency (10 to 100 Hz), long distance (5 or 

more km) propagation along the ground surfa~e.'~-'~ Reference 13 examined the sensitivity 

of attenuation variations to changes in both the atmospheric boundary layer and ground 

surface composition while phase effects were considered in Ref.14. An exact analytical 

model using normal modes was obtained in Ref. 15. For two examples of the extensive 

literature treating long-range propagation of infrasound away from the ground, see Refs. 16 

and 17. References 18 and 19 examined high-frequency sound fields through a wind over 

an impedance plane, while Ref. 20 formulated an analytical solution to long-distance, lowfrequency propagation, but over a perfectly reflecting ground. In Ref. 7, an asymptotic 

solution to the parabolic approximation is developed and analyzed along an impedance 

boundary, but no numerical implementation was discussed. On the other hand, Ref. 9 included refraction from both temperature and wind effects, and developed a finite element 

PE implementation, but the emphasis in that study was on relatively high frequencies and 

moderate ranges. 

In this paper, we are concerned principally with developing a numerical model which can 

be used to conduct parametric studies of low-frequency cw propagation over an impedance 

boundary in a windy atmosphere over substantially long ranges. As noted above, relatively 

little computational work has addressed this problem. In Sec. I we briefly review the development of the parabolic approximation in a moving atmosphere, and then we discuss 

implementation details of the boundary conditions and other technical issues. Numerical 

simulations for propagation in quiescent and windy atmospheres are discussed in Sec. 11, 

and the results of the paper are summarized in the last section. 

I. MODEL FORMULATION 

Let p(r, z) be the acoustic pressure caused by the presence of a point source in a stratified, 

moving atmosphere, where P and z denote the range and height in cylindrical coordinates. 

We will confine our attention to a vertical plane containing the source, and parallel to the 

2 
wind motion. In addition, we assume that the sound speed is independent of azimuth, so 

that we deal only with two-dimensional sound propagation. The time-independent wave 

field, denoted as A(r, z), is obtained by assuming that the source is harmonic with frequency 

f, so that p = Aexp(2xift), where t is time. In the source receiver plane, it can be shown21 

that A satisfies the reduced wave equation 

V IC0 dv 

Q Q dz 

V2A + kin2A + 2ikin2-4 - 2i--Az, = 0, 

where Q is a reference sound speed, ko = 2xf/~ is a reference wave number, c(r,z) is 

the sound speed, n(r, z) = Q/C(P, z) is the index of refraction, and v(z) is the wind speed. 

Furthermore, it can be shown that away from the source, the quantity A takes on the 

asymptotic form 

Equation (2) is an essential feature of the parabolic approximation, where the quantity $ 

is related to the slow-scale (Le. many-wavelength) variation in the acoustic pressure. In 

addition, through careful scaling and asymptotic arguments, it can also be shown that II, 

satisfies one of a family of parabolic equations (PES). Details of the derivation of this family 

of PES in an inhomogeneous moving medium can be found in Refs. 21 and 22. For the 

numerical examples considered in the next section, the appropriate member of this family is 

given by 

(3) 2 -2- 2iko$r + $22 + ko(n I)$ = 0, 

where 

fi = Q/E, 

with 

E=c+v. 

(4) 

(5) 

The quantity E is called the eflective sound speed profile (ESSP).21 

We assume that the ground surface is a locally reacting boundary. Although not strictly 

true in many cases, nevertheless this assumption has been shown repeatedly to be reasonably 

3 
a~curate.'~ Such a boundary, when horizontal, is modeled by the following equation: 

where @ is the normalized surface admittance (reciprocal of impedance). For the remainder 

of this paper, we will use the Delaney-Bazley-Chessell (DBC) impedance model:23 

p-' = 1 + 9.08(f/0)-~.~~ - ill.9(f/a)-0*n, 

where Q is the flow resistivity in cgs units (g cm-3 s-l). We emphasize that any alternative 

single or multi-parameter impedance model could be used as well.27 The DBC model has the 

advantage of being simple to implement and widely used. 

The usual PE starting field requires some modification. In contrast to the method used 

in Ref. 9, we insert an image source beneath the locally-reacting ground surface and require 

that the sum of the image field and source field satisfy the boundary condition given by 

Eq. (6). The point source and its image in the plane are modeled here by the function 

G( z, z,) where 

G(z, 2,) = $jexp(-[ko(z - 4/212), (8) 

where z, is the height above the bottom boundary. 

rG(z, -2,). Substituting this into Eq. (6) yields 

Let g(z) = $(O,z) = G(z,z,) + 

As @ + 0,7 -+ 1, which yields an image source with the expected amplitude over a perfectlyreflecting boundary. As @ + 00, 7 -+ -1, and this produces the correct behavior of the 

image source across a pressure-release boundary. [In fact, this special case of the boundary 

condition has been previously implemented for ocean applications in which the source was 

close to the sea ~urface.'~] 

II. SIMULATION RESULTS 

Our implementation of Eqs. (3) and (6) is based upon IFD, the implicit finite difference 

model described in Ref. 24. That model, together with the numerous modifications and 

4 
enhancements described by us here and el~ewhere,~'~~ shall be referred to as the NASA 

Implicit Finite Difference (NIFD) model. The Crank-Nicholson scheme used to march the 

solution forward in range is well-suited for many propagation situations, for example, those 

involving locally reacting surfaces or irregularly-shaped boundaries. From this algorithm, we 

determine $, then A from Eq. (2) is the complex-valued pressure field, and finally relative 

intensity I(r, z), defined as 

where pnf is the pressure at 1 m from the source. Figure 1 depicts an idealized atmospheric 

acoustic waveguide. We note here that this waveguide is similar to one used as a model in 

Ref. 1, a study of the downwind propagation of low frequency noise from a wind turbine 

located at a test site in Wyoming. A cw sound source is located hs = 40 m above a horizontal, 

locally-reacting ground surface. The receiver will be located on the ground surf- for all 

examples. The air is assumed to be isospeed with Q = 330 m s-l. The atmosphere will be 

taken in Subsection B to move within the indicated plane with a logarithmic velocity profile, 

a modeling assumption often used for the vertical structure of winds: 

where = 14 m s-l is a strong but reasonable value for the free-stream wind speed, and 

zo = 0.1 m is characteristic thickness of the boundary layer (a similar profile was used in 

Refs. 1 and 9). Once the free stream velocity is achieved, the wind speed is held fixed for 

further increases in height z. As shown, the channel is bounded above by a horizontal, 

artificial, pressure-release surface of height h, beneath which is an artificial absorbing layer 

of thickness 500 m. This absorbing layer is designed to eliminate reflections that would 

otherwise occur from the pressure-release surface at the top of the waveguide. This technique 

is used to simulate bottom boundary conditions in ocean acoustics,24 and, modified by us 

as described here, is a feature of NIFD, the numerical implementation which we use for our 

calculations. 

A. No Wind 
An exact solution to the sound field of a point source over an impedance plane in a 

quiescent, isothermal (Q = 330 m s”) atmosphere12 can be used to check the accuracy 

of our implementation. Figure 2 depicts the exact solution calculated with flow resistivity 

Q = 1000 (an acoustically hard surface) compared against our PE solution in two ways. The 

solid curve represents the solution when the artificial boundary is at a height of 1000 m 

above the ground, while the dashed curve is computed when it is 2000 m above the ground. 

The dotted curve is the exact solution. Note that near the ground, which is that part of 

the atmosphere with which we are concerned here, all three calculations are in very close 

agreement. For the solid curve, the absorbing layer begins at height 500 m, and the effect of 

this layer is seen at this height. The higher artificial surface, whose absorbing layer begins 

at height 1500 m, extends the accuracy even higher as seen in the dashed me. For very 

low values of flow resistivity, the PE is still accurate as seen in Fig. 3. In this example, 

u = 10 (an acoustically soft surface). For several hundred meters above the ground, the 

three solutions are in very close agreement, which is where we require the accuracy. Figures 

2 and 3 illustrate the accuracy of our locally-reacting boundary implementation as well as 

the effectiveness of the upper boundary condition. 

A detailed picture of the sound field from a source with frequency 10 Hz when no wind is 

present can be seen in Fig. 4. In this and all remaining examples, Q = 330 m s-’. Relative 

intensity as a function of range T is shown for four values of the flow resistivity Q. Values of 

Q shown have been measured by researchers in the field.28929 Roughly, u = 1000 corresponds 

to very hard-packed earth, while a flow resistivity of 250 might be encountered over a soil 

consisting of sandy loam. A value of u = 50 might be found on ground surfaces under 

cultivation, while a new-fallen snow cover could yield values of u near 10. Our choices of 

flow resistivity values attempt to capture a variety of ground surfaces. In the figure, note 

that intensity decreases as u decreases. Equivalently, a cw signal experiences additional 

attenuation as the surface impedance (admittance) decreases (increases). At longer ranges, 

even a 10 Hz signal can encounter substantial additional attenuation over very soft gound 

surfaces. As seen in Fig. 5, the effect of ground impedance obtained from the DBC model 

6 
is stronger for a signal one octave up, at f = 20 Hz. For a very hard surface, the change 

in intensity is small, but as D decreases, the intensity decrease for the 20 Hz signal ca,n be 

much stronger than for the 10 Hz one. For example, with u = 250, the intensity decreases 3 

dB with a doubling of frequency. When u = 10, the 20 Hz signal is nearly 10 dB lower. We 

note here that this general trend continues as frequency increases. 

B. Downwind 

For the remaining examples, the receiver is located downwind from the source using the 

wind profile in Eq. (11) with parameters discussed previously. With this geometry, the effective sound-speed increases with height, so that sound waves tend to be refracted towards 

the ground surface. As a consequence, a waveguide is formed at the ground surface, and 

intensity levels can be enhanced significantly. In Fig. 6, we see a 10 Hz signal propagating 

over four different impedance boundaries in the wind profile. For large values of flow resistivity, there is very little structure present in the intensity curve. However, as Q decreases, 

an interference pattern begins to emerge, and it most pronounced when u = 10. 

Impedance effects on the interference structure are even more pronounced in the next 

illustration in Fig. 7. For f = 20 Hz, we begin to see substantial multipath effects occur even 

over very hard ground surfaces as depicted in Fig. 7. As the flow resistivity decreases, there 

is not only a decrease in overall intensity level (as was seen in Fig. 5), but the interference 

structure shifts markedly. In fact, if the interference pattern is though of as beating modes, 

it is evident from Fig. 7 that the higher-order mode is being attenuated at a faster rate 

for smaller values of the flow resistivity. In fact for u = 10 multipath effects seem to be 

very weak. These kinds of shifts can have interesting practical consequences. For example, 

acoustic systems designed to detect and track low-flying aircraft could find their performance 

affected as the system is moved from place to place. Even systems deployed at a fixed 

installation could encounter a substantial degradation in effectiveness after a heavy snowfall. 

In applications for which source-receiver ranges are fixed, such as the design of noise control 

barriers, the acoustic propagation pattern could be “tuned” in some sense, by altering the 

7 
ground surface through cultivation or planting. In any case, it is apparent that winds 

(or other focusing phenomena) can influence sound interaction with locally-reacting ground 

surfaces in interesting ways. 

As the source frequency increases, more complex dependence of the received field on 

impedance emerges. As displayed in Fig. 8, when f = 40 Hz an additional propagation 

mode is clearly seen to emerge for u = 1000. As the flow resistivity decreases, the intensity 

curves not only experience a decrease in level, but a noticeable smoothing also occurs. This 

stripping of the highest-order mode(s) appears to be strongest within the transition from 

Q = 1000 to u = 250. Also, note that the intensity peaks and nulls appear to be less sharply 

pronounced at 40 Hz than at 20 Hz. 

At f = 80 Hx the tendency to strip modes seems to be lessened as flow resistivity decreases. In Fig. 9 there is a very strong interference pattern present over the acousticallyhardest ground surface. Peaks and nulls me very well-defined, and bear similarities more 

with Fig. 7 than with Fig. 8. As u decreases to 250 there is a drop in the intensity level but 

the interference structure seems to remarkably well-preserved At u = 50 there is a noticeable change in pattern, while for the softest ground surface, there appears to be a noticeable 

change in the interference pattern, and some obvious mode stripping has occurred. 

SUMMARY 

We have described an implementation of an implicit finite-difference method for solving 

the parabolic approximation in the atmosphere over a locally-reacting boundary. This model 

handles upper the radiation condition, and utilizes the Delaney-Bazley-Chessell impedance 

model. We present calculations in both quiescent and windy atmospheres. Solutions obtained 

from our implementation are shown to agree well with exact solutions for the sound field 

over an impedance plane in an isospeed, quiescent atmosphere. As the source frequency 

increases in octaves from 10 Hz to 80 Hz, variations in the flow resistivity can produce marked 

changes in the multipath interference structure of the downwind sound field. In general, lower 

resistivities (higher ground admittances) tend to strip off higher-order modes, which can 

8 
cause substantial changes in both the location and magnitude of intensity peaks and nulls. 

We emphasize that this numerical implementation has the potential to handle significantly 

more complicated environments, which might include irregular ground topography and range 

dependent sound speed conditions in the medium. 

9 
J.A. Hawkins, “Application of ray theory to propagation of low frequency noise from wind 

turbines,” NASA Langley Res. Center, Hmpton, VA (1987), CR-178367. 

F. Dommermuth and J. Schiller, “Estimating the trajectory of an accelerationless aircraft 

by means of a stationary acoustic sensor,” J. Acoust. SOC. Am. 76, 1114-1122 (1984). 

I?. Dommermuth, ”The estimation of target motion parameters from cpa time measurements in a field of acoustic sensors,” J. Acoust. SOC. Am. 83, 1476-1480 (1988). 

S.M. Candel, “Sound source radiation in two-dimensional shear flow,” AXAA J. 21,221- 

227 (1983). 

R. Raspet, S.W. Lee, E. Kuester, D.C. Chang, W.F. Richards, R. Gilbert, and N. Bong, 

“A fast-field program for sound propagation in a layered atmosphere above an impedance 

ground,” J. Acoust. SOC. Am. 77, 345-353 (1985). 

M.K. Myers and G.L. McAninch, “Parabolic approximation for sound propagation in the 

atmosphere,” AIAA J. 16, 836-842 (1978). 

G.L. McAninch and G.K. Myers, “Propagation of quasiplane waves along an impedance 

boundary,” in Proceedings, AIAA 26th Aerospace Sciences Meeting (AIAA, Washington, 

1988). 

J.S. Itobertson, M.J. Jacobson, and W.L. Siegmann, “Mathematical modeling of sound 

propagation in the atmosphere using the parabolic approximation,” Transactions of the 

Sixth Army Conference on Applied Mathematics and Computing (1989) 

K. Gilbert and M. White, “Application of the parabolic equation to sound propagation 

in a refracting atmosphere,” J. Acoust. SOC. Am. 85, 630-637 (1989). 

lo P. Malb6qui and F. Jouaillec, “Acoustic propagation over the ground using the parabolic 

approximation,” to appear in Proceedings, INTERNOISE 88. 

10 
l1 C.F. Chien and W.W. Soroka, “Sound propagation along an impedance plane,” J. Sound 

Vib. 43, 9-20 (1975). 

l2 M.A. Nobile and S.I. Hayek, “Acoustic propagation over an impedance plane,” J. Acoust. 

SOC. Am. 78, 1325-1336 (1985). 

l3 A.I. Otrezov and I.P. Chunchuzov, “The amplitude distribution of a low-frequency sound tfield near the surface of the earth,” Izv. Atmos. Ocean. Phys. 23, 76-78 (1987). 

l4 T.N. Nesterova, A.I. Otremv, A.S. Smirnov, and I.P. Chunchuzov, “Effect of temperature and wind stratification in the bottom layer of the atmosphere on the phase of 

low-frequency sound waves,” Izv. Atmos. Ocean. Phys. 23, 157-158 (1987). 

l5 W.E. Zorumski and W.L. Willshire, “The acoustic field of a point source in a uniform 

boundary layer over an impedance plane,” in Proceedings, AIAA 26th Aerospace Sciences 

Meeting (AIAA, Washington, 1988). 

l6 L. Liszka, “Long-distance propagation of infrasound from artificial sources,” J. Acoust. 

SOC. Am. 56, 1383-1388 (1974). 

l7 L. Liszka, “Temporal variations of long distance propagation of infrasound,” J. Low Freq. 

Noise Vib. 7, 1-6 (1988). 

l8 D.C. Pridmore-Brown, “Sound propagation in a temperature- and wind-stratified 

medium,” J. Acoust. SOC. Am. 34,438-443 (1962). 

l9 V. Ye. Ostashev, “High-frequency acoustic field of a point source lying above an impedance 

surface in a stratified moving medium,” Izv. Atmos. Ocean. Phys. 23, 370-377 (1987). 

2o I.P. Chunchuzov, “Field of a low-frequency point source of sound in an atmosphere with 

a nonuniform wind-height distribution,” Sov. Phys. Acoust. 30, 323-327 (1984). 

11 
21 J.S. Robertson, W.L. Siegmann, and M.J. Jacobson, “Current and current shear effects 

in the parabolic approximation for underwater sound channels,” J. Acoust. SOC. Am. 77, 

1768-1780 (1985). 

22 J.S. Robertson, W.L. Siegmann, and M.J. Jacobson, “Acoustical effects of ocean current 

shear structures in the parabolic approximation,” J. Acoust. SOC. Am. 82,559-573 (1987). 

23 C.I. Chessell, “Propagation of noise along a finite impedance boundary,” J. Acoust. SOC. 

Am. 62, 825-834 (1977). 

24 D. Lee and S.T. McDaniel, “Ocean acoustic propagation by finite difference methods,” 

Comp. & Maths. with Appls. 14, 305423 (1987). 

25 J.S. Robertson, D.C. Arney, W.L. Siegmann, and M.J. Jacobson, “An efficient enhance 

ment of finitedifference implementations for solving parabolic equations,” J. Acoust. SOC. 

Am. 86, 252-260 (1989). 

26 J.S. Robertson, D.C. Arney, W.L. Siegmann, and M.J. Jacobson, ”Adaptive numerical 

enhancements to finitedifference algorithms for parabolic approximations,” to appear in 

Proceedings, Second IMA CS Symposium on Computational Acoustics. 

27 K. Attenborough, “Acoustical impedance models for outdoor ground surfaces,” J. Sound 

Vib. 99, 521-544 (1985). 

28 R. &pet, H.E. Bass, and J. Ezell, “Effect of finite ground impedance on the propagation 

of acoustic pulses,” J. Acoust. SOC. Am. 74, 267-274 (1983). 

29 K. Attenborough, “Review of ground effects on outdoor sound propagation from continuous broadband sources,” Appl. Acoust. 24, 289-319 (1988). 

12 
FIGURE LEGENDS 

FIG. 1. Atmospheric sound channel. 

FIG. 2. Height z vs. relative intensity I for exact solution and two NIFD computations; no 

wind is present; f = 10 Hz, u = 1000. 

FIG. 3. Height z vs. relative intensity I; Same as Fig. 2 except u = 10. 

FIG. 4. Relative intensity I vs. range r for different values of flow resistivity a; no wind 

present; f = 10 Hz. 

FIG. 5. Relative intensity I vs. range r; same as Fig. 4 except f = 20 Hz. 

FIG. 6. Relative intensity I vs. range r; 14 m s-l wind present, receiver is downwind; 

otherwise same as in Fig. 4. 

FIG. 7. Relative intensity I vs. range r; same as in Fig. 6 except f = 20 Hz. 

FIG. 8. Relative intensity I vs. range r; same as in Fig. 6 except f = 40 Hz. 

FIG. 9. Relative intensity I vs. range r; same as in Fig. 6 except f = 80 Hz. 

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