Shadow Zone Boundary

Limitation of the Effective

Acoustical Turbulence

Scattering Volume Using the

Turbule Ensemble Model

ARL-TR-2234 September 2000

Harry J. Auvermann and George H. Goedecke

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official Department of the Army position unless so

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thereof.

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

return it to the originator.
ARL-TR-2234 September 2000

Army Research Laboratory

Adelphi, MD 20783-1197

Shadow Zone Boundary

Limitation of the Effective

Acoustical Turbulence

Scattering Volume Using the

Turbule Ensemble Model

Harry J. Auvermann

Computational and Information Sciences Directorate, ARL

George H. Goedecke

Department of Physics, New Mexico State University

Approved for public release; distribution unlimited.
The Turbule Ensemble Model (TEM) was developed to handle

acoustical scattering from anisotropic inhomogeneous turbulence. A

turbule is a localized atmospheric inhomogeneity. TEM, then,

represents a turbulent region by a collection of turbules of different

sizes. Since acoustic sources and sensors are omnidirectional, the

scattering volume of the TEM region is ill defined. Scattering

properties of individual turbules in TEM show that the majority of

scattering originates from a constricted volume, called the effective

scattering volume. This is true for a region of homogeneous

turbulence and is anticipated to be true for a region of

inhomogeneous turbulence, although no calculations have been

made. Estimates have been given of the size and shape of the

effective scattering volume as a function of turbule size for an

experiment conducted in homogeneous turbulence and a uniform

atmosphere. In this report, homogeneous turbulence is retained, but

an upwardly refracting atmosphere is assumed, resulting in a shadow

zone. Inclusion of the shadow zone boundary further limits turbule

sizes and locations from which significant signals reach the detector.

The finding is that large turbules are less effective scatterers and

that effective scattering volumes are large enough for the number of

small turbules to be large. Implications of this finding are discussed.

Abstract

ii
iii

Contents

1. Introduction 1

2. Background 2

3. Turbule Spacing and Size Increment Parameters 5

4. Derivation of Shadow Zone Boundary Equation 7

5. Scattered Signal at Detector Equation 9

6. Imposition of Significance Criteria 10

7. Determination of ESV 11

8. Conclusions 13

References 14

Distribution 15

Report Documentation Page 19

Figures

1. Scattering volume shape with no shadow zone ........................... 3

2. Calculation geometry ............................................................ 3

3. Shadow zone boundary surfaces ............................................. 8

4. Effective scattering volume surface for size parameter of 4.0

superimposed on shadow zone boundaries .............................. 8

Tables

1. Selection of significance criteria ................................................ 11

2. Scattering volume results ...................................................... 11

3. Scattering volume constituents ................................................ 12
1

1. Introduction

Six papers were presented at the Battlefield Atmospherics Conferences and

other conferences that deal with acoustic scattering by atmospheric turbulence. The sixth paper (Auvermann and Goedecke, 1995a) contains a summary of the first five papers; this report is an extension of that paper and the

fifth paper in the series (Auvermann et al, 1994). Here we refer to the fifth

paper in that series (Auvermann et al, 1994) as paper A and the sixth paper

(Auvermann and Goedecke, 1995a) as paper B. Experimental evidence shows

that atmospheric turbulence near the ground is neither homogeneous nor

isotropic, two conditions required for the usual statistical model of turbulence to be valid. An alternate model, termed the Turbule Ensemble Model

(TEM), consists of a collection of turbules of different sizes. A turbule is an

isolated inhomogeneity of either temperature or velocity. In TEM, the scattering pattern of individual turbules is assumed known. The analysis proceeds by the assumption of a distribution function for the sizes, and then

location of the turbules of each size, randomly, within the atmospheric region of interest. The shadow zone signal is then the summation of the contributions from each turbule.

Acoustical signals of interest to the Army are, in general, low frequency.

The importance of this is that wavelengths are large compared to the

dimensions of either source or detector. Therefore, both source and detector

are nearly omnidirectional and thus cannot serve to define a scattering volume. Paper A (Auvermann et al, 1994) addressed the problem of determining the volume from which significant scattering can occur (called the effective scattering volume (ESV)) for an elementary spherical shell volume.

Paper B (Auvermann and Goedecke, 1995a) addresses the determination of

ESV for a homogeneous turbulence region isolated from the ground. In this

report, a shadow zone boundary (SZB) is introduced into the scenario, with

the remaining properties of the medium assumed to be homogeneous. The

full power of TEM for addressing scattering from anisotropic inhomogeneous turbulence has not been exploited as yet. The signal from anisotropic

inhomogeneous turbulence may be calculated by populating the ESV with

randomly oriented and located turbules and summing the scattered power.

Presently, the summation process is carried out with the use of integrals

over mathematically defined distributions. A distinction must be made between isotropic turbulence and isotropic scattering. Isotropic turbulence, in

the context of TEM, refers to an ensemble of turbules that has uniformly

distributed orientation vectors. Such an ensemble has anisotropic scattering properties that delimit ESV as a function of turbule size. Only velocity

turbulence will be considered in this report.
2

2. Background

In this section, the results obtained in papers A and B are recounted. Paper A

was the first step in the process of using the scattering properties of turbules

to define ESV. The calculation there was simply of the total scattering cross

section of the turbules in a spherical shell surrounding the detector. The

source was assumed to be a long distance away, so that the incident signal

was a plane wave. The turbulence had parameters appropriate for a 10-m

height above the ground. However, in the simplified calculation, the source

and detector were assumed to be far enough above ground as to be in a free

atmosphere. Sound speed gradients were assumed to be zero, so that no

shadow zones were present. The largest turbule considered had a 10-m

radius, which was appropriate for the 10-m height of the scenario. The

frequency and wavelength were 500 Hz and 0.688 m, respectively. The scattering pattern of the large turbules was found to greatly limit the region of

space from which the scattered signal could reach the detector. In spite of

this small region of space, the scattering properties of these large turbules

dominated the scattering cross section. This scenario (a plane wave from

the source in a medium of infinite size) cannot be extended to calculate the

signal from an infinite space, because the scattered signal at the detector

would be infinite. Although the signal from more distant spherical shells

falls off as 1/r2, the total volume of the shell increases by r2, leaving the

signal from each shell the same. When the total is computed for an increasing distance r, the total increases without bound.

In paper B, the distant source scenario was abandoned in favor of an experimental scenario, with the source and detector finitely separated. The

1/r2 loss from source to scatterer, coupled with the 1/r2 loss from scatterer

to detector, served to produce a finite signal at the detector. ESV was found

to vary from 4.4 m3 for the largest turbule to 2311 m3 for a turbule of 0.44 m

radius. Figure 1 (taken from paper B) shows the intersection locus of the

scattering volume limit with the vertical plane containing the source and

detector for three different turbule sizes.

Figure 2 depicts the scenario of this report. The coordinates are x, y, and z,

with the origin on the ground beneath the source. The source and detector

heights are hS and hD, respectively. The wind flows from the direction of the

detector toward the source. The simplest wind speed profile will be used to

produce a shadow zone, namely, a uniform gradient model. The plan—the

execution of which is discussed in section 5—is to introduce a wind-driven

shadow zone boundary into the scenario of figure 2 and then calculate the

ESV in a manner similar to the calculation in paper B. A reasonable wind

speed gradient will show an appreciably sized shadow zone over the experiment distance chosen (320 m). Two matters not addressed in paper B

must be dealt with first: (1) calculating parameters (turbule spacing and
3

10

8

6

4

2

0 0 10 20 30

Range, 10’s of m

χ = 4

Height, 10’s of m

4

3

2

1

0 0 10 20 30

Range, 10’s of m

χ = 16

Height, 10’s of m

5

4

3

2

1

0 0 10 20 30

Range, 10’s of m

χ = 92

Height, m

x

D

z

y

P

D = Detector

P = Scatterer point

S = Source

Wind direction

S

Figure 1. Scattering

volume shape with

no shadow zone.

Figure 2. Calculation

geometry.
4

size increment) for a representative turbulence distribution and (2) developing an equation for the SZB surface. These matters are discussed in the

next two sections.

Symbols for some of the variables and parameters used in this report are

summarized in the following list:

a = turbule characteristic size (m)

c = sound speed, 3.44 m⋅s–1

ED

v = total irradiance at detector

eD

v = relative irradiance at the detector

f = acoustic wave frequency = 500 Hz

θ = spherical coordinate system polar angle

φ = spherical coordinate system azimuthal angle

k = 2π/λ

l = ratio of turbule average spacing to turbule characteristic size a

μ = turbule size ratio increment parameter

λ = wavelength, m = 0.688 m

N1 = total number of turbules of the largest size

r = spherical coordinate system radial distance

RS = source irradiance reference distance = 1.0 m

RSD = source detector distance = 320 m

RSP = source (point P) distance

RPD = (point P) detector distance

σv(ψ, χ) = velocity turbule total scattering cross section

va0 = turbule flow velocity parameter

va(r, θ) = turbule velocity function

Vs = scattering volume

χ = size parameter = ka

Xu = upper limit of size parameter considered = 91.325367

Xm = maximum in the size parameter integration

Xn = minimum in the size parameter integration

ψ = scattering angle

Ω = magnitude of the angular velocity vector
5

3. Turbule Spacing and Size Increment Parameters

The turbule spacing and size increment parameters are the l and μ, respectively, from the list of parameters in section 2. The size increment parameter

will be determined by energy conservation considerations for a turbule

velocity distribution of (Goedecke, 1992)

v r = Ω z × r exp – r2/a2 ,

va r, θ = va0/a r sin θ exp – r2/a2 , (1)

where the rotation axis has been set along the z-axis. Because of rotational

symmetry, the latter expression shows the dependence on the spherical coordinates of r and θ. The total kinetic energy of the turbule T(a) will be the

integral of one-half the density ρ times the square of this velocity:

T(a) = ρ va0

2

2 a2 dr r4 exp –2 r2/a20

∞

dθ sin θ 3

0

π

dφ

0

2π

= π ρ Γ (5/2, 0, ∞)

3 × 21/2 va0

2 a3 , (2)

where Γ is the Generalized Incomplete Gamma Function (Wolfram, 1991).

The ratio of the energy of the largest turbule to that of the next smaller one

using the former velocity relation (Auvermann, 1994) is the following:

T a1

T a2

= v1

2 a13

v2

2 a23 = a1

a2

11/3

= exp μ 11/3 = 2; μ = (3/11) ln (2) = 0.1890 . (3)

The first expression in equation (3) results if the largest turbules break up

into two turbules of the next lower size and split energy equally. There are

49 size classes in this distribution, from the largest at 10 m (denoted by the

subscript 1) to the smallest at 1.146 mm (denoted by the subscript 49).

Next, recall the expression for the velocity structure constant (Goedecke et

al, 1998), which is

Cv

2 = 0.69

μ

N1 a1

3

Vs

v1

a1

1/3

2

J14/3

v (0, ∞) ,

J14/3

v (0, ∞) = dy y14/3 exp – y2

/2 = 211/6Γ (17/6, 0, ∞) = 6.1455

0

∞

.

(4)

For fractal scaling, each turbule size class has the same l (Goedecke et al,

1998). N1/Vs is the concentration of the largest turbules, so that the second

bracket is l

–3. The value of v1 that we have been using is 3.44 m⋅s–1. To obtain

a value for the structure parameter, we resort to the literature (Brown and

Clifford, 1976). The velocity structure constant expression is given as

Cv

2 = 0.04 + 0.33 z– 2/3 , (5)
6

where z is the height in meters. Substitution of 10 for z gives a value for the

velocity structure parameter of 0.1111 m4/3⋅s–2. Rewriting equation (4) gives

the value of l:

l = 0.69 v1

2 J

14/3

v (0, ∞)

Cv

2 μ a1

2/3

1/3

= (0.69) 3.44 2 (6.1455)

(0.1111) (0.1890) 10 2/3

1/3

= 8.0148 . (6)

If the centers of vortices are closer than 2.8 times the diameter of an individual vortex, the mutual distortion can be expected to cause the vortices to

disintegrate (Moore and Saffman, 1975, eq (13), p 468). This number of 2.8

would translate into a value of 5.6 if the spacing were compared to the vortex radius. Thus, the value of 8 from equation (6) is reasonable, although

the exact relation of the vortex diameter from Moore and Saffman’s work to

the turbule characteristic size (a) of the model is not known.
7

4. Derivation of Shadow Zone Boundary Equation

For a wind speed of 3.44 m⋅s–1 at a height of 10 m and zero at the ground,

the gradient g(0) is 0.344 s–1. A uniform gradient profile declining with increasing x produces a circular ray path (Pierce, 1989, p 385), whose radius is

the length for the sound speed to extrapolate to zero or 1000 m. Negligibly

small transverse curvature of ray paths is assumed, so that a ray launched

in a vertical plane making an angle β with the x-z plane will remain in that

plane. The sound-speed gradient is related to the component of the wind

velocity in the launch plane or g(β) = g(0) cos(β). The SZB is the limit ray

circle that contains the source point and is tangent to the ground. For β = 0,

the equation of this circle is given in the following:

xS – rc0

2

+ z – zc0

2 = rc0

2 ; rc0 = c/g(0) ; zc0 = 2 rc0 h S – h S

2 1/2 .

xS = rc0 – rc0

2 – z – zc0

2 1/2 . (7)

The last form in equation (7) is the useful one, since the height of the SZB at

a variable z will later determine a limit for an integral. For the circle in a

plane at an angle β with the x-z plane, in equation (7), replace g(0) with g(β)

and (z, zc0) by analogous distances, say, (ρβ, ρcβ). Then, replace cos(β) and

(ρβ, ρcβ) with the appropriate functions of the variable coordinates (y, z):

xS – rcβ

2

+ ρ β – ρ cβ

2 = rcβ

2 ; ρ β= y2 + z2 1/2

; cos β = z/ρ β .

rcβ= c/(g(0)cos(β)) ; ρ cβ= 2 r cβ h S – h S

2 1/2 .

xS = rcβ – rcβ

2 – ρβ – ρcβ

2 1/2 , z >0 .

= 0, z ≤ 0 .

(8)

The last expression for xS, where z is less than or equal to 0, recognizes that

propagation in the negative direction (down wind) will not experience a

shadow zone. This is unimportant, except for algorithm purposes. There is

also an SZB associated with the detector. Signal scattered from turbules below the detector SZB will not reach the detector. This boundary will be of

the same form, except that the z coordinate will turn to –zD, and then the zD

will be replaced by (RSD – z)

xD – rcα

2

+ ρ α – ρ cα

2 = rcα

2 ; ρ α = y2 + RSD – z

2 1/2

; cos α = RSD – z /ρ α .

rcα = c/(g(0)cos(α)) ; ρ cα = 2 r cα hD – hD

2 1/2 .

xD = rcα – rcα

2 – ρα – ρcα

2 1/2 , 0 < z <RSD

= 0, z ≥ RSD .

(9)

The integral limit mentioned above will be determined for a y-z plane point

(y, z) by choice of the larger of xS or xD. The larger of xS or xD is referred to as
8

SZB hereafter. Figure 3 is a three-dimensional plot of equations (8) and (9),

showing the SZB surfaces. Figure 4 shows the ESV surface from paper B for

a size parameter of 4.0 superimposed on these SZBs.

Figure 4. Effective

scattering volume

surface for size

parameter of 4.0

superimposed on

shadow zone

boundaries.

–200

–100

0

100

200

0

100

200

300

400

0

20

40

60

80

100

Source at

{z = 0, y = 0, x = 1.0}

Detector at

{z = 320, y = 0, x = 1.5}

z

y

x

0

100

200

0

100

200

300

400

0

20

40

60

80

100

–200

–100

Source at

{z = 0, y = 0, x = 1.0}

Detector at

{z = 320, y = 0, x = 1.5}

z

y

x

Figure 3. Shadow

zone boundary

surfaces.
9

5. Scattered Signal at Detector Equation

Since we make the assumption that the only influence of the wind gradient

and the presence of the ground is to produce the SZB of section 4, the equation for the scattered signal at the detector will be set up in terms of the

distances and angles relative to the source-detector line of figure 2. And

since we also assume homogeneous, isotropic turbulence, the scattering will

have rotational symmetry with respect to this line. Imposition of the SZB

will produce only a rotational angle in the φ integration. For the velocity

distribution of equation (22), the orientation averaged cross section is

(Goedecke, 1992), with c∞ as the asymptotic sound speed,

σv(ψ, χ) = π

3

Ω a χ4

4 k c∞

2

sin ψ cos ψ

2

exp – χ2 1 – cos ψ . (10)

The total signal will be the differential signal multiplied by a differential

volume with a space integration. Including integration over size parameter, the final expression is in equation (8) (Auvermann et al, 1994). The

differential volume is a partial ring around the source-detector line,

ED

v = N1 ES k a1

1/3 eμ

24 eμ – 1

π v1 a1 RS

c∞

2

dz

Zn

Zm

dy 0

Ym

2 rr φ cos γ RSP RPD

– 2

sin ψ cos ψ

2 d χ χ14/3

Xn

Xm

exp – χ2 1 – cos ψ , (11)

with the ring ending on the SZB on each side. By symmetry of the SZB with

respect to the x-z plane, the differential volume is 2 rr(y, z) φ(y, z) dz dy,

where rr is the radius of the ring and φ is the ring angle between the x-z

plane and the SZB intersection point. In equation (11), (rr, φ, RSP, RPD, ψ) are

all functions of (y, z) through the SZB height hZ. The cos(γ) accounts for the

inclination of the source-detector line [tan(γ) = (hD – hS)/RSD]. ES is the reference irradiance.
10

6. Imposition of Significance Criteria

Expressions for the relative signals of interest are

I Zn, Zm, Ym, Xn, Xm = d χ Xn

Xm

dz

Zn

Zm

dy 0

Ym

2 rr φ RSP RPD

–2

sin ψ cos ψ

2

χ14/3 exp – χ2 1 – cos ψ ,

(12a)

I – ∞, ∞, ∞, 0, Xu = dz ∞

∞

dy 0

∞

2 rr φ RSP RPD

–2

sin ψ cos ψ

2 dχ χ14/3

0

Xu

exp – χ2 1 – cos ψ ,

(12b)

eD

v = I Zn, Zm, Ym, Xn, Xm

I – ∞, ∞, ∞, 0, Xu

= Σ . (12c)

(Zn, Zm, Ym, Xn, Xm) are the integration limits for two conditions. The first

condition is that for all space for which these limits are infinity, zero, or Xu.

The objective of this report is to find limits subject to the condition that the

signal is a large fraction of the infinite limit signal, equation (12b). Thus the

expressions may dispense with the constant in front of the integrals in

equation (11). The relative signal eD

v

 is defined in terms of the integrals, and

the significance criterion Σ is set equal to it. The meaning is that the limits

are to be found that make eD

v

 at least as great as Σ, which is thought of as

being a number such as 0.99.
11

7. Determination of ESV

The procedure to be used is the following. The reference integral (eq 12(b))

will be calculated for the parameters specified in this report. Then, lower

and upper limits for the χ integration (Xn, Xm) will be determined such that

the integral will differ in each case by no more that 0.002 from the reference

integral, with all other limits the same. Then the y integral will be evaluated

as a function of (χ, z) to determine a table of Ym(χ, z) values such that this

integral differs no more than 0.002 from its value when the upper limit is

infinity. A function that approximates this table will be determined. Finally,

the integral with these limits will be calculated to show that the relative

signal is greater than the criterion Σ. These criteria are summarized in table

1. Table 2 summarizes the calculated results. Table 3 has been constructed

for five of the sizes of equation (3). From table 2, it is seen that the consequence of this rather simple criteria scheme was that Ym approximated a

cylinder, and the total Σ was too large. This implies that a more sophisticated scheme in which the χ dependence of the z limits is required to improve the accuracy of the results. The SZB is likely to be more like that of

figure 4 than a cylinder. However, important conclusions can be drawn

using the results of table 3. The SZB length lS has been estimated from other

data, and the concentration Cχ has been calculated for an l (eq (6)) of 8.

Associated with the table are some numbers calculated for the second entry. This entry is the maximum contributor to the total signal. Dividing lS by

Nχ for this size gives an average spacing, as shown. Dividing the average

I (–∞, ∞, ∞, 0, Xu) = 0.456394

I (Zn, Zm, Ym, Xn, Xm) = 0.452400

Σ = 0.9912

Xn = 1.3249

Xm = 52.4223

Zn = 44.5736 m

Zm = 272.8049 m

Ym = 816.4478 (χ)–1.31

Table 1. Selection of

significance criteria.

Goal: Σ = 0.99

Σ (Xn) = 0.998

Σ (Xm) = 0.998

Σ (Zn) = 0.998

Σ (Zm) = 0.998

Σ (Ym) = 0.998

Table 2. Scattering

volume results.
12

spacing by the wind speed gives an estimate of the period that might be

expected as turbules of this size are tracked through the ESV. This period of

1.4 s is close to the period from experimental results.* This fact is important

because it suggests that a more accurate calculation is warranted. However,

a more accurate calculation will need a realistic estimate of the size parameter division point between that part of the signal that can be obtained by

integration (a continuum calculation) and those parts that must be addressed

as discrete.

χ aχ rs ls Vs Cχ Nχ

51.80 5.67 4.63 40.00 1.350 × 103 1.072 × 10–5 1.440 × 10–2

16.67 1.83 20.47 120.00 7.898 × 104 3.187 × 10–4 2.517 × 101

7.83 0.86 55.12 182.78 8.723 × 105 3.071 × 10–3 2.678 × 103

2.52 0.28 243.51 220.49 2.054 × 107 8.897 × 10–2 1.827 × 106

1.43 0.16 511.84 228.23 9.392 × 107 4.768 × 10–1 4.478 × 107

χ = size parameter Dominant scatterer has

aχ = turbule radius, m size parameter = 16.67

rs = scattering volume radius, m Spacing in ESV = 4.767 m

ls = scattering volume length, m Wind speed = 3.44 m⋅s–1

Vs = scattering volume, m3 Period = 1.4 s

Cχ = turbule concentration, m–3 Experimental period = 4 s

Nχ = number of turbules

Table 3. Scattering

volume constituents.

*Shadow zone data reported by Havelock, Stenson, and Daigle (1992) were examined

privately by the authors to arrive at this estimate.
13

8. Conclusions

The ultimate utility of the work on ESV may be that it will afford a physical

explanation of the shadow zone scattered signal fluctuations. The data from

table 3 of section 7 lead to the belief that we can conclude that this variability is caused by the motion of a limited number of moderately sized turbules

through the ESV. The next step in the investigation will require a more sophisticated method for determining the (z, y) limits of the scattering volume. Of equal, or even greater, importance will be the determination of the

minimum population of the scattering volume as a function of size parameter that will result in insignificant fluctuations. This minimum population

will determine the turbule size, which divides the sizes that must be summed

over from the sizes that can be integrated.
14

References

Auvermann, H. J., G. H. Goedecke, and M. D. DeAntonio (1994), “The influence

of scattering volume on acoustic scattering by atmospheric turbulence,” Proc.

1994 Battlefield Atmospherics Conf., 29 November–1 December 1994, White

Sands Missile Range, NM.

Auvermann, H. J., and G. H. Goedecke (1995a), “Calculation of scattering volume limits for acoustical scattering from homogeneous turbulence using

the turbule ensemble model,” Proc. Sixth Annual Ground Target Modeling and

Validation Conf., 22–24 August 1995, Houghton, MI.

Auvermann, H. J., and G. H. Goedecke (1995b), “Shadow zone boundary limitation of the effective acoustical turbulence scattering volume using the turbule

ensemble model,” presented at 1995 Battlefield Atmospheric Conference,

5–7 December 1995, White Sands Missile Range, NM.

Brown, E. H., and S. F. Clifford (1976), “On the attenuation of sound by turbulence,” J. Acoust. Soc. Am. 60 (4), pp 788–794, October 1976.

Goedecke, G. H. (1992), Scattering of Acoustical Waves by a Spinning Atmospheric

Turbule, U.S. Army Atmospheric Sciences Laboratory, CR-92-0001-2, White

Sands Missile Range, NM.

Goedecke, G. H., M. DeAntonio, and H. J. Auvermann (1998), Structure Function

Spectra and Acoustic Scattering Due to Homogeneous Isotropic Atmospheric

Turbule Ensembles, U.S. Army Research Laboratory, ARL-TR-1518, August

1998.

Havelock, D. I., M. R. Stenson, and G. A. Daigle (1992), “Phase and amplitude

fluctuations in a refractive shadow,” J. Acoust. Soc. Am. 92 (4), part 2, p 2405,

October 1992.

Moore, D. W., and P. G. Saffman (1975), “The density of organized vortices in a

turbulent mixing layer,” J. Fluid Mechanics 69, part 3, pp 465–473.

Pierce, A. D. (1989), “Acoustics,” Acoust. Soc. Am., Woodbury, NY.

Wolfram, S. (1991), MATHEMATICA: A System for Doing Mathematics by Computer,

Addison-Wesley, Redwood City, CA, p 572.
15

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NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)

Prescribed by ANSI Std. Z39-18

298-102

Shadow Zone Boundary Limitation of the Effective

Acoustical Turbulence Scattering Volume Using the Turbule

Ensemble Model

September 2000 FY94 to FY95

The Turbule Ensemble Model (TEM) was developed to handle acoustical scattering from

anisotropic inhomogeneous turbulence. A turbule is a localized atmospheric inhomogeneity.

TEM, then, represents a turbulent region by a collection of turbules of different sizes. Since

acoustic sources and sensors are omnidirectional, the scattering volume of the TEM region is ill

defined. Scattering properties of individual turbules in TEM show that the majority of scattering

originates from a constricted volume, called the effective scattering volume. This is true for a

region of homogeneous turbulence and is anticipated to be true for a region of inhomogeneous

turbulence, although no calculations have been made. Estimates have been given of the size and

shape of the effective scattering volume as a function of turbule size for an experiment conducted

in homogeneous turbulence and a uniform atmosphere. In this report, homogeneous turbulence

is retained, but an upwardly refracting atmosphere is assumed, resulting in a shadow zone.

Inclusion of the shadow zone boundary further limits turbule sizes and locations from which

significant signals reach the detector. The finding is that large turbules are less effective scatterers

and that effective scattering volumes are large enough for the number of small turbules to be

large. Implications of this finding are discussed.

Turbulence, scattering volume, acoustic shadow zone

Unclassified

ARL-TR-2234

0FEJ60

61110253A11

B53A

61102A

ARL PR:

AMS code:

DA PR:

PE:

Approved for public release;

distribution unlimited.

23

Unclassified Unclassified UL

2800 Powder Mill Road

Adelphi, MD 20783-1197

U.S. Army Research Laboratory

Attn: AMSRL-CI-EP auvermann@arl.army.mil

U.S. Army Research Laboratory

19

Harry J. Auvermann (ARL), George H. Goedecke

(Department of Physics, New Mexico State University)

email:

2800 Powder Mill Road

Adelphi, MD 20783-1197