Weak Scattering of Sound Waves 

in Random Media That Have 

Arbitrary Power-Law Spectra 

D. Keith Wilson 

ARL-TR-i 866 

- ....l.l_ I- --..--- ~ 

April 1999 

Approved for public release; distribution unlimited. 
The findings in this report are not to be construed as an 

official Department of the Army position unless so 

designated by other authorized documents. 

Citation of manufacturer’s or trade names does not 

constitute an official endorsement or approval of the use 

thereof. 

Destroy this report when it is no longer needed. Do not 

return it to the originator. 
Army Research Laboratory 

Adelphi, MD 20783-l 197 

ARL-TR-1866 March 1999 

Weak Scattering of Sound 

Waves in Random Media That 

Have Arbitrary Power-Law 

Spectra 

D. Keith Wilson 

Information Science and Technology Directorate 

Approved for public release; distribution unlimited. 
Abstract 

Log-amplitude and phase variances of a weakly scattered acoustic signal 

are calculated for line-of-sight propagation through a random medium. 

The spectrum of the index-of-refraction fluctuations in the random 

medium is assumed to scale in proportion to the wavenumber raised to an 

arbitrary power in the limit of large wavenumbers (small spatial scales). 

Both scalar and vector contributions to the index of refraction are 

considered. Most of the calculated results reduce to those given by 

Tatarskii (1971) and Ostashev (1994) when the power law exponent is. 

-5/3, which is the value characteristic of turbulence. However, the results 

do not exactly reduce to an equation given by Flatte et al (1979) for the 

log-amplitude variance in terms of strength and diffraction parameters. 

The equation from Flatte et al is shown to be an approximation, valid only 

when the spectral energy in the random medium is concentrated at a 

well-defined outer scale. 

ii 
Contents 

1 Introduction 1 

2 Review of Previous Formulations for Weak Scattering 

3 Turbulence Modeling-Assumptions and Parameters 

4 Derivation of the Log-Amplitude and Phase Variances 

4.1 Scalars, Plane Waves . . . . . . . . . . . . . . . . . . . . . . . 

4.2 Vectors, Plane Waves . . . . . . . . . . . . . . . . . . . . . . . 

4.3 Scalars and Vectors, Spherical Waves . . . . . . . . . . . . . . 

5 Summary 

3 

6 

8 

8 

11 

12 

14 

References 16 

Distribution 

Report Documentation Page 

17 

19 

. . . 

111 
Tables 

1 Log-amplitude variances for weak scattering and weak diffraction written in terms of structure-function parameter C$, 

acoustic wavenumber ko, and propagation distance X . . . . 15 

2 Log-amplitude variances for weak scattering and weak diffraction written in terms of diffraction parameter A, strength parameter a, and ratio of correlation length to integral length 

scale, L/L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 

iv 
1. Introduction 

Sound waves that have propagated through the atmosphere exhibit 

random behavior. The cause for this behavior in most cases is atmospheric 

turbulence, although other random atmospheric phenomena such as 

internal waves can also play a role. The random behavior of the sound 

wave can significantly impact acoustical systems used for target detection, 

identification, and location. Hence a capability to estimate the statistics of 

sound waves is needed to quantify and predict the performance of Army 

acoustical systems operating in atmospheric environments. 

This report concerns the calculation of the variances in the phase and the 

logarithm of the amplitude (log amplitude) of a received signal. These 

variances have been the subject of much research during the past several 

decades. Two contrasting formulations of the variance statistics can be 

found in the literature: one originating with the pioneering work on 

random scattering by Tatarskii (1971), and the other with the elegant 

treatment of propagation through the random ocean by Flatte et al (1979). 

Tatarslcii’s results are specifically for the scattering of waves by turbuIent 

phenomena; Flatte et al adopt a more general approach, although its 

application to turbulence is not obvious from their book. More recently, 

Ostashev (1994) pointed out that the results from researchers such as 

Tatarslcii and Flatte et al apply strictly to scattering driven by fluctuations 

in an isotropic scalar field, as opposed to fluctuations in an isotropic 

vector field. In the atmosphere, for example, this would mean that the 

previous results apply only to scattering by random temperature 

fluctuations, and not to scattering by wind velocity &.rctuafions. This is a 

significant concern, since sound scattering by atmospheric turbulence 

comes primarily from the velocity fluctuations. Fortunately, Ostashev 

(1994) was able to extend Tatarskii’s results to isotropic vector fields. 

This report bridges the approaches of Tatarskii and Flatte et al, and 

extends their results to velocity fluctuations. To this end, I devise a very 

general formulation of wave scattering by a random medium that has an 

arbitary power-law spectrum. The formulation is compared to the results 

of Tatarskii and FIatte et al. Specificially, I consider the scattering of plane 

waves by scalar and vector fields, and the scattering of spherical waves by 

scalar and vector fields. 

The report presumes that the reader has some familiarity with the 

literature on propagation through random media, such as the books by 

1 
Flat&? et al and Tatarskii, or Rytov et al (1989). The reader may also find 

my earlier work (Wilson, 1998b) a useful reference on modeling of 

turbulence spectra and correiations. 

2 
2. Review of Previous Formulations for Weak Scattering 

Flatte et al (1979) analyzed the scattering of waves by random media using 

two parameters: a strength parameter @ and a diffraction parameter A. 

These parameters are given by the equations 

Q2 = 2a2LCIciX, and (1) 

X A=-. 

6L2ko (2) 

In the preceding equations, a2 is the variance of the (normalized) 

sound-speed fluctuations,* .C is the integral length scale, L is the 

correlation length scale of the sound-speed fluctuations, ko is the acoustic 

wavenumber (= 2~f /Q, where f is the frequency and ~0 the mean sound 

speed), and X is the propagation distance. This formulation involves two 

distinct length scales: the integral and correlation length scales. The 

integral length scale is calculated by integrating the correlation function 

with respect to separation, and then dividing by the variance. As a result, 

L is representative of the size of the largest, most energetic motions in the 

flow. On the other hand, the correlation length scale is defined by an 

expansion of the correlation function for small arguments and is 

representative of the outer scale, where the power-law scaling becomes 

invalid. The correlation length scale will be defined more precisely in the 

next section. 

Those situations where the strength parameter is much less than 1 are 

termed weak scattering. A large strength parameter indicates strong 

scattering. Similarly, weak and strung dij%rcfion correspond to small and 

large values of the diffraction parameter, respectively. Only weak 

scattering is considered in this report, since the statistical properties of 

strong scattering are already well established (Flatte et al, 1979). (In strong 

scattering, the complex acoustic signal has independent real and 

imaginary parts possessing zero-mean Gaussian statistics.) Weak 

*For an acoustic wave propagating through the atmosphere, the normalized soundspeed fluctuation is 

IV+g+$ 

0 co 

in which T’ is the fluctuation in temperature about the mean temperature TO, and v’ is the 

fluctuation in the wind speed (along the direction of propagation) about the mean. 

3 
scattering is in many ways the more difficult problem. More importantly, 

it is also pertinent to most Army scenarios, where the frequencies are low 

and the propagation distances are moderate. 

For weak scattering and strong diffraction (@ < 1, A > l), Flatte et al gave 

the results 

(p) = $ (x2) = ;, 

where ($2) and (x2) are the phase and log-amplitude variances, 

respectively. For weak scattering and weak diffraction (a < 1, A < l), the 

phase variance is 

The calculation of the log-amplitude fluctuations for weak scattering and 

weak diffraction-the geometrical acoustics regime-is the most challenging 

problem. Flatte et al gave but did not present a derivation for the equation 

(their equation (8.3.2)) 

Following this equation, they state: 

The log-amplitude fluctuations in the geometrical-optics region are 

thus of second order, and a carg%l treatment yields C&3.2), where the 

exact cot$cient of Aa depends on the details of the 

ocean-jluctuation spectrum . . . but is of order unity. 

A primary purpose of this report is to explore the general validity of 

equation (5). 

One issue concerning the general validity of equation (5) is evident in 

Flatte et al’s remark that the coefficient (l/4 in the equation) depends on 

the “details of the ocean-fluctuation spectrum.” And so, we ask the 

following: How much does the coefficient actually vary? On what 

properties of the propagation medium does it depend? Can the coefficient 

be defined more precisely? 

A broader issue concerns whether the general relationship that (x2) is 

proportional to AQ2 holds for propagation through random media besides 

the ocean. The main interest of Flatte et al was propagation through ocean 

internal (buoyancy) waves. Does their result apply to propagation through 

4 
turbulence? An important clue to answering this question is in the 

authors’ discussion in section 6.2 of their book regarding the transition 

from the geometric acoustics to the partially saturated scattering regime. 

There they state that the boundary between these two regimes is given by 

f&2@’ = 1, where p is the exponent in the power-law spectrum for the 

random fluctuations (F (Ic) 0: Ic-P). Based on well-known arguments first 

made by Kolmogorov, p = 5/3 for turbulence. However, for the internal 

waves considered by Flatte et al, p = 2. Therefore, we might anticipate that 

( > x2 cx APi21D2. (6) 

As discussed earlier, equations for the log-amplitude variance have been 

derived for propagation through turbulence by Tatarskii (1971). For plane 

waves, 

( > x2 = 0.308C$k,7’6X11’6, 

and for spherical waves, 

(>, x2 2 716 = 0.124CNk, X 

in which C$ is called the strucfure-function par1 

l/f3 

, 

ameter for the 

index-of-refraction fluctuations. The results for plane and spherical waves 

differ only with regard to their numerical coefficients. If one substitutes 

the values for @ and A in equations (1) and (2) into equation (6), the 

dependence on frequency and propagation distance agrees with 

Tatarskii’s results. Nonetheless, there is an indication that something is 

amiss: equation (6) involves two parameters associated with the 

large-scale (energy-subrange) structure of the turbulence, a2 and ,C. In 

equation (7), Cg is strictly a property of the small-scale (inertial-subrange) 

structure of the turbulence. Unless the product a2L in equation (1) has a 

universal relationship to C,, 2 there is a fundamental disagreement 

between the results. The solution to this quandry, as well as to the other 

questions posed earlier, will become apparent in the following sections. 

5 
3. Turbulence Modeling-Assumptions and Parameters 

In this section, I discuss the statistical assumptions regarding the 

fluctuations in the propagation medium that will be used to derive the 

propagation statistics. Only two basic assumptions are made; additional 

assumptions are avoided so as not to restrict applicability of the results. 

The first assumption is simply that the medium has a finite variance and 

integral length scale. This leads to the requirement for the 

one-dimensional (1D) spectrum F (k) of the fluctuations in the 

propagation medium that (Wilson, 1998b) 

F(k=O)=$ 

The second assumption is that for large wavenumber k >> L, the spectrum 

obeys a power-law scaling: 

F (k) = yk-P, (10) 

where y is some constant (which may depend on the properties of the 

flow), and p is the power-law exponent. Equivalently, we can set 

F(k) = eS (kL) , 

where S (k,C) is a function equal to 1 for k = 0, and proportional to 

fl (kC)-* at large wavenumber, where 

An alternative formulation to equation (10) involves the structure 

function. The structure function for separations parallel to the direction of 

propagation is by definition 

q (r) = ([N(?-) - N (0)]2) = c;?J+ 

(When the preceding equation is applied to the velocity field, the 

separation T between the measurement points is implicitly in the same 

(13) 
direction as the velocity component being measured.) A relationship 

between y and Cg can be derived using the relationship 

1 - cos Icr) F (Ic) dk. 

The function (1 - cos kr) filters out the dependence of F (k) on small 

wavenumbers. Hence we can substitute equation (10) into (14). Integration 

then leads to 

c2 2 = -4lY (1 - p) sin ‘5. 

7 

A second alternative formulation to equation (10) was used by Flat@ et al. 

For the correlation function at small separations, they wrote 

where the subscripted I means that the separation between the 

measurement points is perpendicular to the direction of propagation. (In 

an isotropic scalar field, it does not matter in which direction the 

displacement is taken. For an isotropic vector field, however, the 

displacement direction is relevant.) For scalars, it follows directly from the 

definition of the structure function that 

C&=$-g. 

For a divergenceless vector field (such as velocity fluctuations in 

turbulence), 

Using this relationship, we derive for vector fields 

2 

c; = (p +“;; LP-1. 

(17) 

09) 
4. Derivation of the Log-Amplitude and Phase Variances 

Under the conditions of weak scattering, Rytov’s method can show that 

the variance of log amplitude and phase for a propagating plane wave are 

(Rytov et al, 1989) 

(x2,$2) =27r2t$jX Om 

11 

1F &sin?] Q(O,kJLldkl, (20) 

where the minus sign applies to the log-amplitude variance, and the plus 

sign to the phase variance. The 3D spectrum of the turbulence is indicated 

by 0 (0, JCL), where the first argument is the wavenumber component 

parallel to the direction of propagation, and the second argument is the 

wavenumber component perpendicular to it. For velocity fluctuations, the 

pertinent velocity component is the one parallel to the direction of 

propagation. 

4.1 Scalars, Plane Waves 

To determine the log-amplitude and phase variances, we need to 

determine the 3D spectrum 0 (0, Ic) from the 1D spectrum F (Ic). A helpful 

intermediate step is to calculate the energy spectrum E (Ic), which 

specifies the energy in the random fluctuations per unit wavenumber, 

from F (Ic). For scalar fluctuations, the needed relationship is 

E(k) = 4%. 

Hence we have 

(21) 

For the 3D spectrum, we multiply by 2 to go from energy to variance, and 

then divide by the area of a spherical shell in the wavenumber space, 

4rk2. Hence 

0 (0, Icl) = -&-$. 

I (23) 

8 
Therefore, for large wavenurnbers, 

(24) 

It follows from equations (20) and (23) that 

By changing the integration variable to the normalized wavenumber 

x = klL, one can rewrite this equation as 

(26) 

in which 

D = X/koC2. (27) 

By comparing the definition of D to Flat@ et al’s diffraction parameter in 

equation (2), we see that the two are proportional: D = 6 (L/Q2 A. For 

strong diffraction (D, A >> l), contributions from the second term in the 

square brackets in equation (26) become negligible, and we can verify 

equation (3). For weak diffraction (D, A < l), the second term is 

significant. In fact, assuming that most of the spectral energy is 

concentrated at low wavenumber (which is the case for most media of 

interest), the second term is approximately 1, and for the phase 

fluctuations we arrive at equation (4). The only nontrival case is the 

log-amplitude variance for weak diffraction. Here the two terms in the 

square brackets nearly cancel out at low wavenumber, so that the 

significant contribution to the integral occurs at high wavenumber. Hence 

it is reasonable to use the high-wavenumber limit of the spectrum, 

equation (10). With this substitution we find 

( > x2 _ Pm2 O” 1_ 

-- 

2 J [ 1 

- sin Ox2 %-‘-l dx. 

0 DE2 1 

Changing the integration variable to u = Dz2, we find 

(28) 

(29) 

9 
I could not find the integral in any standard table. It can be solved, 

however, by first replacing the difference in square brackets with an 

equivalent integral representation: 

1 

sin u 

-- = l-j&) 

U 

- cos (21 cos 6)] sin 8 d0 

(30) 

= lsin2 (F) sint?cM 

= 2~T’2sin2 (9) sinf3dO. 

In the first line, je is the spherical Bessel function. The second line follows 

from Poisson’s integral, equation (10.1.13) in Abramowitz and Stegun 

(1965). The third line follows from the trigonometric identity 

2 sin2 A = 1 - cos 2A. The fourth line is a consequence of the symmetry 

properties of the integrand. Substituting this result into equation (29), and 

reversing the order of integration between x and 19, we have 

( x2 ) = $Dp/202jf’2 [Jdmsin? (y) qJ,-P~2-1du] s~(jde. (31) 

The integral within the square brackets is given by Gradshteyn and 

Ryzhik’s (1994) equation (3.823), with the restriction 0 < p < 4. We now 

have 

COSP/~ 8 sin e de, (32) 

in which r (x) is the gamma function. Finally, the remaining integral is 

known from Gradshteyn and Ryzhik’s (1994) equation (3.621.5), leading to 

the result 

( > X2 = L-r (1 -p/2) cos $Dp’2@2. 

First we wilI verify that equation (33) reduces to Tatarskii’s result, 

equation (7). We start by replacing a, p, and D with their definitions 

(equations (1), (12), and (27), respectively), and this leads to 

( > X 

2 _ ;ypr (1 -p/2) cos ~~~-p~2x’+pi2- w 

10 
Next, using equation (15) to eliminate y in favor of C;, and performing 

some manipulations on the gamma functions, we have 

( x2 > = hs (P> ‘%k, 2 2-P/2xl+p/2 7 

where 

2p-2fir p -I- 1 camp/2 

h(P)=- 2+p ( 2 > sin rp/4 * 

(35) 

(36) 

This very general equation holds for any power-law exponent 0 < p < 4. 

For turbulence (p = 5/3), it does reduce exactly to equation (7). 

Similar manipulations on equation (33) produce 

= h, (p) #-‘i2 $ Ap/2@2. 

0 

This equation does have the basic form of equation (6). However, the 

factor (L/L) is not strictly a constant. The correlation length L is a 

property of the inertial subrange, whereas the integral length scale L 

depends on the large-scale turbulence structure. In fact, it can be shown 

for the atmosphere that the ratio depends strongly on the relative 

importance of shear and buoyancy forcings of the turbulence; for 

turbulence generated almost entirely by wind shear, the ratio is close to 1, 

whereas for turbulence generated by buoyancy, the ratio may be l/10 or 

less (Wilson, 1998a). If one sets p = 2, and assumes that C M 0.2L, as was 

done by Flat@ et al, one finds 

( > X2 M 12AQ2. (38) 

This result has the same form as equation (5), although the constant 12 

appears in place of l/4. The cause of this discrepency is unclear, since 

Flatte et al do not provide a derivation of their equation. The discrepency 

could be due, in part, to confusion in the definition of the diffraction 

parameter; in much of the literature it is defined with a factor of 4 in the 

numerator, rather than 6 in the denominator, as in equation (2). 

4.2 Vectors, Plane Waves 

The spectrum of fluctuations of an isotropic vector field differs somewhat 

from that of an isotropic scalar field. The energy spectrum follows from 

the equation (Batchelor, 1953) 

11 
Therefore, at large wavenumber, we have 

E W = P (P + 2) q (kL)-p. 

The spectral density of the ith velocity component can be determined 

using (Batchelor, 1953) 

where k/l is the wavenumber in the ith direction. Hence, in place of 

equation (24), we have 

0 (0, kl) = 

P (P + 2) Po2L 

4n2k2 (kIL)-p. 

I w 

This is the same as equation (24), except that it incorporates an extra factor 

(p + 2) /2. Hence the scalar results apply to vectors after multiplication by 

this extra factor. Equivalently one can replace h, (p) in equations (35) and 

(37) by 

h, (p) = -2P-3J;;r ‘+ F; ;;;. 

( > 

4.3 Scalars and Vectors, Spherical Waves 

For weakly scattered spherical waves, we start with (Rytov et al 1989) 

(43) 

(W 

instead of equation (20). In the preceding, the integration over t represents 

averaging over the propagation path. From this equation it can be shown 

that equations (3) and (4) are still valid for spherical waves. As with plane 

waves, the difficulty is in the log-amplitude variance for conditions of 

weak scattering and weak diffraction. Substituting equation (24) into (44), 

making the substituting u = Dk2L2, and using the trigonometric identity 

2 sin2 A = 1 - cos 2A, we find that 

( x2 > PP 

= -FDP 12~2 JJ [ 1 O3 sin2 ut (l - t> U-P/2-ldudt. 

2 1 (45) 0 0 

12 
Performing the integrations as with the plane waves results in 

(X2) = $1 -p/2)B(l t-p/2,1 +P/2)cos~DP’2@2, 

2 (46) 

where B (2, y) is the beta function. Comparing this equation with equation 

(33), one finds that 

(x2> sphere 

(x2> 

= (1 + P/2) I? (1+ P/&l + P/Q * 

plane 

This result is valid for vectors as well as scalars. 

(47) 

13 
5. Summary 

The log-amplitude variances for weak scattering and weak diffraction (the 

geometric acoustics regime) are summarized in tables 1 and 2. The 

equations in the two tables are equivalent, but the first table specifies the 

variances in terms of structure-function parameters, whereas the second 

specifies them in terms of the strength and diffraction parameters. These 

tables tie together and generalize the previous results of Tatarskii (1971), 

Flat@ et al (1979), and Ostashev (1994). 

An interesting conclusion of this study is that one cannot generally 

calculate the log-amplitude variance from the strength and diffraction 

parameters alone, as suggested by equation (5). Knowledge of the 

length-scale ratio L/L is also required. Alternatively, one could define a 

modified strength parameter based on L (i.e., Qt = 2a2Lk$X) and use Qi 

to calculate the log-amplitude variance in the geometric acoustics regime. 

The distinction between the length scales would have little practical 

importance if the scales were nearly equal. However, in many propagation 

media, including the atmosphere, it appears that L can be an order of 

magnitude or more longer than L (Wilson, 1998a). 

14 
. 

Table 1. Log-amplitude variances for weak scattering and weak diffraction (the geometric acoustics regime) written in 

terms of structure-function parameter C$, acoustic wavenumber Ica, and propagation distance X. The function h, (p> 

is defined in equation (36). 

Wave type Pluctuations General (0 < p < 4) Tiarbulence (p = 5/3) Internal waves (p = 2) 

Plane scalar h, (p) C$k,2-d2X’+P/2 0.307C$k;‘6X”/6 o.393c$kox2 

Plane Vector (1 + 5) h, (p) C; k;-p’2X1+P/2 0.563C; k;‘6X1”6 0.785c;kox~ 

Spherical Scalar (1-t 5) B (1 + 5 , 1 + g) h, (p) C$ k,2-p’2X’+p’2 2 7/6 O.l24C, k, X 11/6 0.131cj$kox2 

Spherical Vector (I+ $)” B (I+ f, 1 + 5) h, (p) C;k,2-p’2X1+p’2 0.228C$k,7’6X11’6 0.262C;koX’ 

Table 2. Log-amplitude variances for weak scattering and weak diffraction (the geometric acoustics regime) written in 

terms of diffraction parameter A, strength parameter @, and ratio of correlation length to integral length scale, L/L. 

The function hs (p) is defined in equation (36). 

Wave type Pluctuations General (0 < p < 4) Turbulence (p = 5/3) Internal waves (p = 2) 

Plane h, (p) $ (6A)p’2 a2 1.37$A5/6@2 2.36+’ 

Plane Vector (1 + g) h, (p) $ (6~9~‘~ ~3’ 2.51+~I~“%~ 4.71$A@’ 

Spherical Scalar (1 + $) B (1-t f, 1+ $) h, (p) $ (6A)p’2 Qi2 0.553+A5’692 0.785+A@‘. 

Spherical Vector (1s g)” B (1 + f, 1+ $) h, (p) + (6A)p’2 (P2 1.01+A5’6@2 1.57+Aa2 

15 
References 

[l] Abramowitz, M., and I. A. Stegun (1965): Handbook ofMathematical 

Functions. Dover, San Francisco, 675-679. 

[2] Batchelor, G. K. (1953): The Theo y of Homogeneous Turbulence. 

Cambridge University Press, Cambridge, UK. 

[3] Flatte, S. M., R. Dashen, W. H. Murk, K. M. Watson, and F. 

Zacharasen (1979): Sound Transmission Through a Fhxfuafing Ocean. 

Cambridge University Press, Cambridge, UK. 

[4] Gradshteyn, I. S., and I. M. Ryzhik (1994): Table oflnfegrals, Series, and 

Products. Academic Press, San Diego, CA. 

[5] Cktashev, V E. (1994): Sound propagation and scattering in media 

with random inhomogeneities of sound speed, density and medium 

velocity. Waves Random Media, 4,403-428. 

[6] Rytov, S. M., Y. A. Kravtzov, and V. I. Tatarskii (1989): Principles of 

Statistical Radio Physics, Parf 4: Wave Propagation Through Random 

Media. Springer, Berlin. 

[7j Tatarskii, V I. (1971): The Effects of the Turbulent Atmosphere on Wave 

Propagation. Keter, Jerusalem. 

[S] Wilson, D. K. (1998a): On the application of turbulence 

spectral/correlation models to sound propagation in the atmosphere. 

Eighth International Symposium on Long Range Sound Propagafion, 

Pennsylvania State University, State College, pp 296-312. 

[9] Wilson, D. K. (1998b): Turbulence ModeZs and fhe Synthesis of Random 

Fields for Wave Propagation and Scattering Calculafions. U.S. &my 

Research Laboratory, ARL-TR-1677. 

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REPORT DOCUMENTATION PAGE Form Approved 

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March 1999 Final, 1 O/1/98-1 l/l S/98 

A-ANDSIJBTIM Weak Scattering of Sound Waves in Random Media 5. FUNDING NUMBERS 

That Have Arbitrary Power-Law Spectra 

6. AUTHOR(S) D. Keith Wilson 

DA PR: B53A 

PE: 61102A 

7. PERFORMING ORGANlZATlON NAME(S) AND ADDRESS 

U.S. Army Research Laboratory 

Attn: AMSRL-lS-EP email: dkwilson@arl.mil 

2800 Powder Mill Road 

Adelphi, MD 20783-l 197 

6. PERFORMING DRGANlZAllON 

REPORT NUMBER 

ARL-TR-1866 

9. SPONSORINGMONlTORING AGENCY NAME(S) AND ADDRESS 

U.S. Army Research Laboratory 

2800 Powder Mill Road 

Adelphi, MD 20783-l 197 

10. SPONSORlNG!MONtTORlNG 

AGENCY REPORT NUMBER 

11. SUPPLEMENTARY NOTES 

ARL PR: 9FEJ61 

AMS code: 611102.53A 

12a. DlSlRlEVrtOWAVAllABlLITY STATEMENT Approved for public release; distribution 12b. DlSTRlBUllON CODE 

unlimited. 

13. ABSTf?ACT (Maximum 200 words) 

Log-amplitude and phase variances of a weakly scattered acoustic signal are calculated for line-of-sight 

propagation through a random medium. The spectrum of the index-of-refraction fluctuations in the 

random medium is assumed to scale in proportion to the wavenumber raised to an arbitrary power in 

the limit of large wavenumbers (small spatial scales). Both scalar and vector contributions to the index 

of’refraction are considered. Most of the calculated results reduce to those given by Tatarskii (1971) 

and Ostashev (1994) when the power law exponent is -5/3, which is the value characteristic of 

turbulence. However, the results do not exactly reduce to an equation given by Flatte et al (1994) for 

the log-amplitude variance in terms of strength and diffraction parameters. The equation from Flatte 

et al is shown to be an approximation, strictly valid only when the spectral energy in the random 

medium is concentrated at a well-defined outer scale. 

14. SUBJECTTERMS 

Acoustics, scattering, turbulence, internal waves, spectral functions 

15. NUMBER OF PAGES 

25 

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