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Full text of "Coherent and incoherent components of sound scattered at a time dependent rough surface"

LIBRARY 



NAVAL paSTGRADUATE SC 
MONTEREY, CALIFORNIA 9 



NPS-61MD72121A 



NAVAL POSTGRADUATE SCHOOL 



It 



Monterey, California 




31 December 1972 



NPS-61Md72121A 



Coherent and Incoherent Components of Sound 
Scattered at a Time Dependent Rough Surface 



C.S. Clay 



Physics & Chemistry 
Department 



Approved for public release; distribution unlimited, 



FEDDOCS 

D 208.14/2 

NPS-61MD72121A 



DUDLEY KNOX LIBRARY 
NAVAL POSTGRA.DUATE SCHO' 
MONTEREY CA 93943-5101 



NAVAL POSTGRADUATE SCHOOL 
Monterey, California 



Rear Admiral M.B. Freeman M.U. Clauser 

Superintendent Provost 

TITLE: Coherent and Incoherent Components of Sound Scattered at a 
Time Dependent Rough Surface 

AUTHOR: C.S. Clay* 

ABSTRACT : 

Theoretical expressions are derived for the sound scattered 
at a time-dependent rough surface. The calculations are made for 
a Gaussian shaded source transducer and point receiver. The 
Helmholtz theorem and Fresnel approximation are used. The rough 
surface is assumed to be a traveling wave and to have a traveling 
wave packet type of correlation function. The coherent component 
of the signal is the product of the Fourier transformation of the 
surface distribution function and the smooth surface reflection 
signal. Comparison of theory and experiment shows the coherent 
component to be sensitive to the non-Gaussian character of the 
wind-blown water waves. The incoherent components and the temporal 
correlation function of the scattered sound are given. For the 
special case of a traveling cosine wave type of rough surface, the 
spectrum of the scattered sound includes components which are mul- 
tiples of the frequency of the surface wave. For surfaces describ- 
able by a bivariate Gaussian distribution function, the temporal 
correlation is a function of, but not the same as, the time de- 
pendence of the rough surface. The scattered sound is insensitive 
to the spatial correlation function of the surface at distances 
larger than the dimensions of the transducer divided by the cosine 
of the incident angle. The final expressions are complex error 
integrals and can be used for all values of roughness. This task 
was supported by Naval Ship Systems Command (Code PMS 388) . 



TABLE OF CONTENTS 

I. EXPERIMENT AND THE INVERSE PROBLEM 4 

II. SCATTERING FUNCTION FROM THE HELMHOLTZ 
EQUATION 6 

III. EVALUATION OF THE SCATTERING INTEGRAL 18 

IV. COHERENTLY SCATTERED SIGNAL 24 

V. COSINE CORRUGATED SURFACE 28 

VI. TOTAL SIGNAL SCATTERED AT A NON-GAUSSIAN 
SURFACE 31 

REFERENCES 33 

INITIAL DISTRIBUTION LIST 34 

FORM DD 1473 38 



I. EXPERIMENT AND THE INVERSE PROBLEM 

Sound scattered at a time dependent surface takes on some of the 
time dependence of the surface. For example, the upward or downward motion 
of the surface causes the frequency of the reflected signal to be Doppler 
shifted. It is easy to demonstrate the effects in laboratory experiments 
(Fig. 1.1). Small waves on the water cause the phases of the signals to 
fluctuate. Experiments have shown the reflected signals are (crudely 
stated) modulated by the surface and often the spectrum 
of the surface can be identified as a component of the spectrum of the 
reflected signal. With our growing technology in remote sensing, the 
importance of being able to go from scattering measurements back to a 
description of the time dependent surface is extremely important. It 
may be more important to know when and how far one can go for a given 
technique. 

Doing the inverse problem requires a complete knowledge of the relation- 
ship of the measurement to the quantity being sensed. Too often, the 
result of having this knowledge is "I can't measure what I want by doing 
the experiment I am doing." For example, in the plane wave approximation, 
Eckart (1953) showed that mean square scattered signals are simply related 
to the correlation function of the surface at small ifO 
(f = k(cos 9,+ cos9„)/2, k is 2it/wavelength, a is rms roughness, and 

and 9 are the incident and reflected angles). Unfortunately the 

2 2 
scattered signal is proportional to y a and correspondingly very small. 

If the source and receiver are omnidirectional or broad beamed, the 

scattered sound is usually identified as being a fluctuation of the signal 

and is often inseparable from the noise. 

4 



In this paper I add another limitation (which is derived in the 
Fresnel approximation. The scattered sound is insensitive to spatial 
correlation function of the rough surface at correlation distances larger 
than the dimensions of the transducer/cos9- . 

I believe a bit of discussion of the theoretical problem and 
approximations is needed. In seeking the source of a discrepancy between 
my theory (1971) and some of our laboratory measurements, G. A. Sandness 
identified the calculation of the incident sound at the surface as being 
the difficulty for two reasons, (private communication). First, in using 
the Helmholtz equation, the incident sound signal should satisfy the wave 
equation. The combination of a point source and directional function (or 
illumination function on the surface) does not satisfy the wave equation. 
Second, the limitation of the expansion to second order terms may be a 
very poor approximation for a diverging wave at the surface. 
Another way of thinking about this is to regard the phase and amplitude 
of the incident signal as being the hologram of the source. The hologram 
of a finite object is different from that of a point source. The reader 
may wish to read Melton and Horton (1970). 

Having said that there are difficulties with the Fresnel approximation, 
I have chosen to use it. Also, I use a Gaussian shaded source transducer 
and a point receiver. This shading fits the measured response of our 
transducers and facilitates repeated integrations. I have obtained ex- 
pressions that can be used over a wide range of surface roughnesses and 
correlation functions without resorting to separate expansions for the high 
and low frequency limits. The end result is the mean square scattered sound 
and its temporal correlation function. 



II. SCATTERING FUNCTION FROM THE HELMHOLTZ EQUATION 

The development of a theoretical relationship of a scattering function 
to the properties of the surface requires analysis of the scattering of 
acoustic signals by rough surfaces. This has been the subject of a number 
of papers and is treated in several books, notably Beckmann and Spizzichino 
(1963), Tolstoy and Clay (1966), Ol'Shevskii (1967), Fortuin (1970), and 
Horton (1971). Although we will not discuss it here, in the electronic 
and radio engineering journals an extensive literature exists on the 
scattering of electromagnetic waves by various types of irregular and rough 
surfaces. We are obliged to present the theory in some detail because the 
relevant underwater acoustics theory has not been applied to these types 
of problems. Most of the studies have been empirical. 

The derivation of the scattering equation from the Helmholtz theorem 
is usually based upon the assumption of local plane reflections. It is 
suggested that the reader who is interested in the details refer to the 
development given by Tolstoy and Clay (1966), who base their derivations 
on those of Eckart (1953) and Beckmann and Spizzichino (1963). The sound 
absorption will be ignored in the derivation. 

The general assumptions are: 1) The source and receiver are far from 
the illuminated area. 2) The dimensions of the source are small compared 
to R-. and R . 3) The source is a Gaussian shaded transducer and is directed 
along R^ , (Fig. 2.1). Also, the individual elements are in phase and 
incident sound pressure at (x,y,^) is the integral over ds ' . 4) No shadows 
are present. 




SOURCE 



Fig. 2.1 Geometry 

ds is at the position x, y, ^ relative to the origin. The 
plane of the source is perpendicular to R . 



Because the surface is rough, we can expect that at grazing angles 
some of the surface will be in shadow. The proportion of the area in 
shadow to the total area scanned depends upon the shape of the surface 
and on the grazing angle. Wagner (1967) discussed this for rays and a 
randomly rough surfaces. In any case, the assumption of no shadowing 
is violated. It is also likely that the radii of curvature of some 
features on the surface are small compared to the acoustical wave- 
length. In that case, the sound does not have a local plane reflection 
and in order to describe the scattered sound we mist use higher order 
expansions. 

Eckart (1953) remarked that the boundary conditions are troublesome. 
For example, one can set 

P = Sp^ (2.1) 

where p^ and p are the incident and reflected signals on the local surface 
and ^/ha is the normal derivative. Using ER = -1 for a free surface, Eckart 
suggested that in the smoother areas, the second condition might hold. In 
deep shadows, he thought that 



h,.b. 



Sn = ^^ (2-« 



might be reasonable. Horton and Muir (1967) assumed an average boundary 
condition and combined the two equations involving the normal derivatives 
to obta in 

* P2C2COS9 - P C COS0 

^ " P,c rose, + p^c^cose'' » reflection coefficient for plane waves, 
2 2 1 11 r 

here 1 and 2 refer to medium 1 and 2. is refracted direction 
m medium 2 . 

8 



hp /^ = or hp/hn = (2.4) 

They also used Eckart's small slope approximation and replaced the 
normal derivative by b/bz. Horton et al. (1967) compared theoretical 
computations and the experimental scattered sound and found the 
agreement to be quite good. The slope correction can be included 
by performing an integration by parts [Tolstoy and Clay, p. 196-199 
(1966)]. In the specular direction, the result is the same as 
obtained with h/bn. ~ ^/^z. The result is different for backscattered 
s ound . 

The most direct way to demonstrate the approximations is to 
start with the integral expression for the scattered acoustical 
pressure. With the aid of the Helmholtz theorem [Born and Wolf, 
p. 375 (1965)] it is 



r- /gMs^" ""t \P) 



P = 47- / hi ^- U^ I, 1 ds (2.5) 



i(kR - a)t^)A 



where U = e^^'^" ""^Z'/r (2.6) 



i(kR - cot )/ • 
Pi - B e 1/R ^2.7) 

B^ = npc {2iO'^ 
n = source power 



I 



and the normal is drawn toward the receiver. R and R are the source 
and receiver distances to ds (at x, y, z) . In applying the Helmholtz 
theorem, the surface is closed by a hemisphere at infinity. There 
are no other sources and all waves are outgoing from our source. 
Hence the contribution to the integral for the surface at infinity can 



be ignored and the integral over the illuminated interface is used. 
We choose to use the less conventional form of the normal toward 
the receiver because that is the direction of the wave propagation 
and it does not matter where the source is placed. The substitution 
of (2.1) and (2.2) into (2.5) gives 



^ ^' /. 



"^ %r (Pi") I ds (2.8) 



or the Hdrton-Muir condition, 2.4 gives 

ds (2.9) 



P = 4^ ^^ Pi 






After expansion of the integrands for a moderately directive source, 
the integrals will be the same and the kind of boundary condition 
depends upon a constant factor in front of the integral. 

There are several ways of writing the algebra for the expansion 

and they are all messy. Since the transducer plane is perpendicular 

'2 2 
to R , R and R are 

1 2 ' 2 ' 2 ' 

R = (-R sin + X cos 9 - K) + (y - y) + (R cos0- - x sinS.-^' 

R = (R sine^cose„ - x)^ -f (R sinS^sine - y)^ + (R^cosQ. - 

(2.10) 
With the aid of the binomial expansion and retaining ^ and second 
order terms, R and R are 

'2, '2 x^cos^e,+ y^ x X cose, 

R R I X + v 1 I . I_^ 

- 1 2R 2R R R 

+ X sin0 - ^ cos© + . . . 



2 1 \ v^ r 2 2^ 

1 - sin cos e ) + -J^ (^1 - sin ^2^^'^ ^3/ 



2 . . , ^ 2 

- X sin0^cos6»^ - y sine^sin©^ - ^ cos0^ + . . . (2.11) 

10 



For small slopes, the normal derivative is approximately o/oZ (or ^/^ 
here). On making this approximation and also including the integral over 
the transducer ds', one obtains 



P ~ 



ik BOT e 



-icut 



2n R^ R^ 



r , , r ik (R • + R) 
jgds je 



ds 



(2.12) 



(cos© + cos9 )/2, boundary cond. (2.1 and 2), ^/^ ~ ^/hz 



F = 



1 + cos9i + cosS^ - sinS, sin0- cos9„ , , /« -.^x 
1 2 1 2 3 , slope correction (2.13) 



cos9 + cos0„ 



cos0_/2, boundary cond. (2.4) 



(Tolstoy and Clay, 1966) 



,2 



S = 1^ ^^P 



W 



(2.14) 



Integration over the source yields (after algebraic manipulation) 

f-iQt -ik(R^ + S^l 

ik B9F e -' 

P ~ - T 7- 



2n R^R^d - id^)^ (1 - id^) 



// exp -ax -ay +2 icnx + 2 ipy + 2 iyM dydx 

ik cos^e^R Fl - i (1 - R-/R ) d, 1 
i x L 2 X L J 



a = 

X 



2 R^R. 



1 - id. 



a = - ^ 

y 2 R^R2 



ik R [l - i (1 - R^/R )d^] 



1 - id 



w 



(2.15) 



R s R^ + R^ (1 - sin^e2 sin^e^) 
d^ = kL^/(2R^) d^ = kW^/(2R^) 



(2.16) 



2a = k(sin0 - sin0 cosS ) 
2p = -k(sine sine ) 
2y = -k(cos9 + cos0 ) 



11 



Integration of 2.15 for ^ = yields the image solution. Although 
the size of the transducer is included and the expressions are dependent 
on L and W, I don't think the expressions are accurate for near 
field computations because the higher powers of x and y were dropped 
in 2.11. 

Proceeding, the covariance of the sound pressure is 



/p (t + T) p* (t)\ = 



, 2„2_2_2 -iooT 
= k B gj F e 



2 2 2 2 i: 2 J- 
hT^W (1 + dL^)^(l + d^ )^ 



(il/r^^'i'^x^-V" - V^- V 



2 * ,2 2 * ,2 
-ax'-^-ay -"' 
XX y 



+ 2ia(x-x') + 2iP(y-y') + 2i^'(^ -^') } 

dy dx dy' dx'y (2.17) 

I assume the surface is random and ^ has a zero mean. The correlation 

function of t is X X. T 

^ 2 2 2 

X Y T 
"2 2 2 

where in functional notation ^ could be written as i^(|,ri, t,). 

Assuming a Gaussian surface, the bivariate Gaussian probability density is 

2-1 2-3- 
W - (2na ) (1-t ) 2 



exp 



/ - [ 2 a-^^)o^] ~^ [ t^ + V'^ - 2^^'^3 J (2.19) 



Since the only random quantities are t, and ^', the average in (2.17) 
operates on them as follows: 



•k 

For a given roughness wave length A, the shorter wave length roughness 
in an area having the dimensions of several A has the same statistical pro- 
perties as all other similar size areas. 



12 



(e^irC^ -r)> = //we2i^(^r),^,^. 



= S 



/ \ = exp [ - ^/a^a-^)] (2.20)* 

The following changes of variable permit integration over x" and y", 

X = x" + 1/2 y - y"+ T,/2 

x' = x" - 1/2 y'- y" - T,/2 

iie substitution of 2.20 and 2.21 into 2.17 and evaluation of the 

integral yields, after the usual algebra, 

2„2 2 -iooT 



(2.21 ) 



/p(t + t) p*(t)\ ^ B-F-g?-e 



2nR-^ LW cose 



r r expV- a^l^ -a t]^ -f- 2iQ ^ + 2ipTi - 4r^cr^ (l-i|r)l dl dTi (2.22] 

-00 

2 2 r~ 2 li 

R COS0T 1 + d^ (1-R2/Ry) 

where a, = ^^ 3_J= L ^ 4 

5.22 
2 R2 L 

R ^ Tl + d ^(1-Ro/Rv)^] 

a = y ^ , -^ 

1 2 2 

2^2 " (2.23) 

If '^ can be expressed as a sum of first and second order polynomials 

in 5 , T\, and t, (2.22) can be integrated directly for all values 

of rev. 

The intensity of scattered sound is often expressed with the 
aid of a scattering function S as follows: 



g2 
(p^) = — T S (2.24 ) 



* r 2 2 1 

In Section 6, I replace exp [_- 4y a (l-i|f)J by the characteristic function 
^2 and remove the restriction that the distribution be Gaussian. 

13 



where A = illuminated area 



To change (2.22) into this form I need to express A in terms of the 
transducer dimensions and R . Since transducers are often described 
by their Fraunhofer "beam width", I do so here. For plane waves, 
defining and X as shown on Figure 2.2, and ignoring time dependence, 

,2 .2 

/ fexp I - ^ 

nLW 



|p| = -L- Tfexp I - ^- - K- - ikx'sin0 - iky'sinxj dx' dy' (2.25) 



L2 2 2 2 2 2 I 

- k L sin 0/4 - k W sin X/4J 

The response is shown on Figure 2,3. I choose to use |p|= e to define 
A0 and AX, as follows 



at X = 0, (kLsin0)/2 = 1 
and at 0=0, (kWsinX)/2 = 1 

Thus sin0 ~ A0 = 2/ (kL) 

sinX ~ AX = 2/ (kW) (2.26) 

The substitution of (2.26) into (2.22) and casting into the form 
(2.24) yields 



/p (t + t) p*(t)\ = —^ — f- S (2.27) 

S = k^F ^v^e"^ ^ r Texp -a^^^ - a T]^ + 2 10:^ + 2 i^i\ 

- 4/a^ d-!')] d^ dT] (2.28) 
A = R^ A0 AX/ (cos0^) (2.29) 

Eq. (2.22) is equivalent to Eq. (2.27) and (2.28). We can proceed in several 
ways, direct evaluation at 2.28, by numerical integration, polynomial ex- 
pression of ^ and integration, and special functions for ^ and integration. 



14 




Fig. 2.2 Transducer. The transducer is Gaussian shaded. L and W 
indicate the dimensions for e amplitude shading. 



15 




Fig. 2.3 Response of the Gaussian shaded transducer. The far field 
(Fraunhofer) response of the Gaussian shaded transducer, 
g = exp [-x'2/l2] 



16 



Before we cast off, sail into the confused sea, and expand the 

correlation functions, it will pay us to look at the convergence of 

(2.22). The contributions to the integral are small for values of 
i and T] larger than 

^f > ^^ ^f ^ \ (2.30) 



and for R^ = R ^1 " ^2 



where ^l^ - L/cos0^ 



a ^ ~ W 
Ti - 



Since the contributions are small, we need not worry about the shape 

of ^ at distances larger than i and t] in evaluating the integral. 

Or, the dimensions of the transducer and 0, determine the sensitivity 

of the scattered sound to ^. On the other hand^ accurate fits at small I and 

T] are extremely important at large xo. 

1 would like to close this section by giving my thoughts on the 
physical significance of the I, t] integral for S. This Integral 
ought not be confused with the first integral over the illuminated 
area because the ^ , t) integral relates the phase of the scattered 
signal at any point on the surface relative to a nearby point at 
a displacement i and t]. The contributions to S are small for 
i and i^ greater than i and t) . Paraphased, the integral is 
in correlation space. For a given roughness, the constancy of the 
phases of the scattered components of signal depends upon the dimen- 
sions of the transducer. 



17 



m. EVALUATION OF THE SCATTERING INTEGRAL 

Experimental measurements of the i, t dependence of wind blown 
water waves have shown that ^ has the form of traveling wave packet; 
Fig. 3.1. The envelope moves at the group velocity and the phases 




35-- 



30-- 



25-- 



20-- 



V* 



., _L » 10 ISTSEC 




Fig. 3.1 i a, t) 

On the graphs R(t) is our t 



of the damped oscillation move at the phase velocity. Near the origin 
(i > I , etc), the dependence on I and t can be approximated as 

^ ~ ^if(^ - vt) (3.1) 

for waves traveling in the + x direction. i|f is S3nnetric about 
1=0, and T = 

^ (t) = ^(-e) 

^ (t) = ^(-t) (3.2) 



18 



Near ^ = 0, ^ can be approximated as a polynomial 

t~l-ax -b|xl 
or \1^ - 1 - a^^ + 2 ax&r - b|| - "D'tJ - a^v'^r^ (3.3) 
and the expansion has cross terms ^t. Correspondingly, waves moving 
in the y direction have rix terms and waves moving in an arbitrary 
direction have both. This doesn't make the analysis more difficult 
because t is a parameter. 

In an earlier paper, I used polynomial fits to the correlation 
function for a random surface (Clay, 1971). The procedure was to 
divide the surface into sub areas and to integrate the scattering 
function for each of the sub areas. I will do the ssime here except 
that the t dimension is added to the problem. The I, -q and t 
nomenclatures are shown on Figure 3.2. A polynominal expansion for the 
ijkth sub region is 

- f ...T] - g'. ., T^ - h' ... T - m'..,|T 

i-jk ' ° ijk ijk ijk 



,2 2 2 2 

- Tl . ., T]T - r '. ., I T - S . ., Tl T 

' ijk ' ijk ijk ' 



(3.5) 



It isn't necessary to expand the t dependence as a polynomial in 
some problems. For example, the correlation function can be expanded 
as the product of polynomials in ^ and functions of t such as cos pr 
and sin pr. If I had the polynomial form programed, I wouldn't bother. 
The coefficients of the variables i and t) can be combined 



19 




§ L J '^^^ 




Fig. 3.2 sub areas 

The I, -n map is at constant t. The I, t map is at constant t\ 
^1/ is assumed to be for regions beyond i^^^, r\ ^^ and t 



max 'max max 



20 



2 

C — c' - c' T - h ' T 

ijk ijk ^ ijk ijk 

2 
a, ., = a' . ,, + r' . ., T 
ijk ijk ijk 



xjk = b' , ., + m . ,, T 
ijk xjk 

• , 2 
e. ., = e '. ,, + s '. ., T 
ijk ijk ijk 



ijk " ^'ijk "^ '"'ijk^ (3.6) 



2 
t. ., = c. .. - a. .,1 - b, ., I 
ijk ijk ijk xjk 



2 
" ^ijk^l " ^iik"! (3.7) 



On designating S for the kth t region, we can write 



E 



-ijk ^ 8n ^ h""^ L~ H' - V 



S..,. =k:C£:^-' ll^^p [•_,2 __2 



^-1' ^J-1 
+ 2ia| + 2ipT] - 4r^a^(l - ^^.j^) I dUr] (3.8) 

The integral has the form of the product of two complex 
error integrals. To use the tabulations, we transform the variables. 
Since the integrals on |£ and t\ have the same form, we use an 
integral on x and let it stand for either. Note that erf (z) 
and erfc(z) are defined by Abramowitz and Stegun(1965) as follows: 

erf(z) s — — / e dt 



21 



erfc(z) = _2 



; /"•" 



dt 



-z 



w(z) = e erfc( - iz) 



(3.9) 



We now cast S. into the form of (3,9) by defining the constant C 

ijk ^ ^ ijk 

and functions U , V ag follows 
ijk ijk 



2 2 2 



U V 

ijk ijk ijk 



ijk 32 

=ijk = exp[-4rV(l -c.j^)] 



2 2 

^i 



ijk 



exp 



U-1 
where with the aid of (2.23) and (3.7) 

1 



- B . .1 
~7~ XX J k 



xijk 



(3.10) 

(3.11) 

d^ (3.12) 



2 2 
al + 4r cr a, ., 
ijk 



A 2. ., 
X ijk 



xijk 



2 2 
iO; + 4 r a b. ., 
ijk 



(3.13) 



Similar expressions for the j) dependence are 



ijk 






exp 



- -JL 



^j-1 L ^ijk 



yijk' 



drj (3.14) 



yijk 



2 2 

a + 4r cr e. ., 
T] ijk 



2 2 



B . ., = - ip + 4 r cj f. ., 
yxjk ijk 



(3.15) 



Eq . (3.13) and (3.14) have the same form so we drop subscripts and 
cast it into the form of (3.9) by completing the square and changing 
variables as follows: 



22 



t = X , 

'- + s 



s = AB/2 

g(i-l) =-1^1 + s 



A (3.16) 



I 
g(i) = -1 + s 



A 

2 ^_/..x .2 



A s /-gCi) -t ,^ 
U = Ae / e dt 



/ 

g(i-l) (3.17) 

2 
U = A c^ erfc[g (i-1)] - erfc [ g(i) ] (3.18) 

If g is real, (3.18) can be used for the evaluation of U and V. 

This would be expected for the specular direction when 0! and P 

are zero. For complex g, we change the form of erfc(g) to w(z) 

in (3.9). 

Let z = i g (i-1) 
2 
then e^ erfc [g (i-1)] = exp [s^ - g^(i-l)] w[ig(i-l)] (3.19) 



Eq. 3.18 with the aid of (3.19) is 

2 



U = A e^ ^ e 



2 2 "I 

"^ ^^"^^ w[ig(i-l)] - e ' S ^^^w[ig(i)] J 

(3.20) 

V has the same form as U. 

The substitution of A , ., , B , ., , etc. into (3.16) is simple and 

xijk xijk 

the evaluation (3.20) for U. ,, and V... follows directly. I don't 

ijk ijk 

see much reason to write the expression because there is a chain of 
substitutions through to (3.5) for the t dependence. The analytic 
evaluation of (3.8) is complete. From all of this mess, I would 
not expect the time dependence of the scattered signals to have a 

simple relationship to the time dependence of the surface. 



23 



IV. COHERENTLY SCATTERED SIGNAL 
For my estimate of the coherently scattered signal, I will use the 
Eckart procedure and calculate the ensemble average of p. This method is 
very general and is applicable to a much wider range of conditions than is 
superfically apparent. By considering the more general problem, it is 
possible to use the dependence of the coherently reflected signal on y 
to determine the probability density function of the surface W. 

I begin by assuming the coherent signal is an ensemble average of 
many transmissions \P/ . The sound pressure is given by Helmholtz 
integral and the expansions of R and R' , Kq. (2.11 and 2.12). For a 
moderately directive source and ^ very small relative to R and R' , I can 
write 

R + R' ~ R(^ = 0) + R' (^ = 0) + 2r^ (4.1) 

I use [space] to represent all of the non-random part of x, y, and t 
dependence of the Helmholtz integral, 

p = /[space] e ^ds (4.2) 

>-'-00 

and r.00 



p = / [space] ds 



o /_ 

for ^ = 



(4.3) 



The latter expression is exact for all configurations and (4.2) is only 
restricted by the approximation, (4.1). The ensemble average of (4.2) is 

<^p^ = ^' [space] ds (e^^O ^^'^^ 



24 



The average in (4.4) is given by 

00 

(e^^^; =/w^|i^^d^ (4.5) 

-00 

and this is the characteristic function of W . Hence, we can write the 
following relations 

[<»/Po] = /^W^e^^^d^ (4.6) 

"^ —00 
00 

-00 

Since \pVp is a function of positive yo, Eq. 4.7 should be regarded 
as the sine and cosine integrals from to oo. The normalized coherent 
reflection is the Fourier transformation of the pdf of the surface. 

For a test, I use the experimental data of Mayo, Wright, and Medwin(70) 

[T 9 9 9 

/p\ /p versus ^y a = g, 

where g is often referred to as the roughness parameter. If W is Gaussian 

then 

k'pN/p ~ exp(-g) for Gaussian W (4.8) 

howevei^ their data did not fit (4.8). The data agreed quite well with 
(4.8) for g less than 4 (or ya < 1). At larger yo, the coherent com- 
ponent is orders of magnitude larger than predicted by (4.8). All aspects 
of the measurements were painstakingly re-examined, including finite 
illuminated area, but no "mistakes" were identified. The non-fit became 
the motivation of this research. For simplicity in a numerical test, I 
approximated their measured distribution function by the linear segments 
shown on Figure 4.1. The substitution and evaluation of (4.6) for the 
piece-wise linear function is routine and omitted here. A comparison of my 
calculation of \P//p and their data are shown on Figure 4.2. The phase 
of the coherently reflected signal probably changes as ya increases. The 
phase is needed in the evaluation of Eq. 4.7. Since the phase is needed 

for the inverse transform, I have not attempted to do it. 

25 



The procedure we have described, measure the reflected signal at a = 
and then the coherently reflected signal for the roughened surface is easy 
to do in the laboratory. Doing this for a rough sea surface would require 
highly accurate measurements and calculations. I suggest that an alter- 
native procedure might be to measure the cross correlation of signals 
at a pair of separated hydrophones. 




Fig. 4.1. Experimental probability density function of the model sea surface 
a = .45 cm. The solid line is the piece-wise linear 
approximation to the distribution function. Data furnished by 
Mayo, Wright and Medwin. 



26 




Fig. 4.2. Coherently reflected signal. The data points are from Mayo, 

Wright and Medwin (1970). The solid line is calculated for 

the probability density function shown on Figure 4.1. The data 

are for angle of incidence =45° and a = .45 cm, 

27 



V. COSINE CORRUGATED SURFACE 
The surface of water waves often looks like a cosine corrugated surface 
for relatively long times and over fairly large areas. If one watches 
carefully the phases and amplitudes of the wave change randomly. It is easy 
to make a cosine surface random analytically by letting the phase change 
randomly between each observation (or signal transmission). In addition, 
an ensemble can be formed of waves having different amplitudes. I assume 
the m'-" wave surface is given 



f = f cos K (x - vt - X ) 
^ ^m m 



(5.1) 



Where v is the wave velocity, t is the amplitude, and x is the phase. 

^m m 

The procedure for calculating the scattered sound signal is to substitute 
(5.1) into (2.16) and to expand the result with the aid of the following 
expansion in Bessel functions 



^ia cos bx ^ J ^^ _^ 2 \ (i)"j (a) cos (nbx) 

L 

1 



(5.2) 



The result of integration and manipulation is 



2 r 2 

p = p e"P /^y )j (2r^ )e"" /^x +V(i)'' J (2rC ) exp 
o ^ o ^m / n ^m 



( 2a + nK) . ,^, , , 

^ — ; - inK(vt + x ) 

4a m 

X 



+ ia)\(2nj exp 

1 



( 2a - nK) , . ^, ^ , X 

— ; + inK(vt + X ) 

4a m 

X 



(5.3) 



where 



A S B e 
_ _q 

*o ~ R^ + R^ 



i[cDt - k(R3^ + R2)] 



(5.4) 



28 



A = 
o 



' '^^'l 



- 1/2 



1 - ^Vw 

y.' _l 



- 1/2 



(5.5) 



Before I wipe out most of the terms in (5.3) I should discuss them. The 
traveling water wave introduced an infinite series of time-dependent terms 
having frequencies nKv. These terms contribute a modulation of 
the reflected signal. The higher harmonics are more important at larger 
roughness, i.e. y"^ • If the reflected signal were processed by means of a 
spectrum analyzer, the spectrum would be like that shown on Figure 5.1. 
The powers in the components depend on the bistatic geometry. 



L 



9 e 9 



1 



frequency 
Fig. 5.1. Spectrum of the signal reflected at a traveling water wave. 

O) is the frequency of the sound signal and Kv is the frequency 
of the water wave. 



29 



The simplest ensemble average is obtained by letting all phases of the 
traveling wave be equally likely. An average over Kx from to 2it eliminates 
all of the terms in the summation in Eq. (5.3). In the specular direction 
(5.3) then reduces to 

The distribution function of a cosine wave is 

f = f cos K X 
^ ^m 



(5.7) 



W = C > t 
c ^ ^m 



The circle can be completed by averaging (5.6) over a random set of 
amplitudes. For example, the envelope of a Gaussian random function has 
a Rayleigh distribution function: 

W^ = o'^ ^ exp [- ^^/(2a^)j (5.8) 

r J^ (2ri ) Wj^d^ = exp [-2r^a^J (5.9) 

"^ o 

This would be expected on the basis of the central limit theorem. I suggest 
that fair approximations to non-Gaussian functions can be made by averaging 
(5.6) for several values of ^ . 



30 



VI. TOTAL SIGNAL SCATTERED AT A NON-GAUSSIAN SURFACE 

This is my last section and lest the reader has forgotten, my purpose 
is to do the inverse problem. The basic formulation of the scattering 
problem is given in Sections 2 and 3. There, the emphasis was on going 
from a known surface to calculated scattered signal. This analysis 
started with a bivarate Gaussian surface and its correlation function. 
I could have used the characteristic function and removed the Gaussian 
restriction as follows: 



S 



.<e^^^^^-?'*> (6.1) 

2 2 1 
The only change is to replace exp^ -4y a (1-^1^ )j by C in Eq. (2.22). 



Following my procedure in Section 4, I evaluate p and express Eq, (2.22) 
as follows: 

/ A / 2 - x% -1 -icoT 

//d exp [2iQ^ + 2i3Ti] C^d^d (6.2) 

L2 2 "I 

- a.^ - a T) J (6.3) 

Since a. and a are functions of k, 9, , 9^, and 9., they can also be 

expressed as a function of o; and 3. I make a Fourier transform operation 

-1 , 

on assuming that D and D are slowly varying functions of s and t]. 

Integration of CC and P yields 6(^ - ^') and 5(t] - t) ' ) type functions and 

the expression for C„ is the following 

00 

1 rrP/ ^-k-l vpp*) icjOT r -I 

2n^jjJ ^^\ 2 ^ ^''P [-^(^'^ + 20;^ + 23ti)J dadpdw' = C^ (6.4) 

-00 Pq 



31 



The q: and 3 dependences of a, and a are greatly simplified by letting 
9 = -Q-^> Qo ~ 0> ^'^d holding R^ and R constant for a set of measurements, 
Obviously, one's ability to do the inverse transformation depends upon 
having accurate values of ^pp*^ . I assume the user would do the trans- 
forms numerically. Although the integral has infinite limits, actual 
measurements of /pp'=^ give both upper and lower limits. There is another 
consideration. Since D operates as a spatial high pass filter in (6.2), 
I doubt if C would have much accuracy at I and t] much larger than i and 



32 



References 

Abramowitz, M. , and I. A. Stegun, Handbook of Mathematical Functions, 
N.B.S. Applied Mathematics Series 55, U.S. Government Printing 
Office, Washington, D.C. (1964). 

Beckmann, P., and A. Spizzichino, The Scattering of Electromagnetic Waves 
from Rough Surfaces , The MacMillon Co., New York, (1963). 

Born, M. , and E. Wolf, Principles of Optics , Pergamon Press, (1965). 

Clay, C. S., Notes on Ocean Acoustics, Univ. of Wis. Geophysical and 
Polar Research Center Research Report, No. 71-2, August 1971. 

Eckart, C, J. Acoust. Soc. Amer. 25 , p. 195 (1953)- 

Fortuin, L. , J. Acoust. Soc. Amer. 47 , p. 1209-28 (1970). 

Horton, C. W. , Sr., J. Acoust. Soc. Amer. 51 , p. 1049-61 (1972). 

Horton, C. W., Sr., and T. G. Muir, J. Acoust. Soc. Amer. 41 , p. 627-34 (1967). 

Mayo, N., H. Medwin, and W. M. Wright, J. Acoust. Soc . Amer. 47, p. 112(A), 
(1970). 



Mel 



ton, D. R. , and C. W. Horton, Sr., J. Acoust. Soc. Amer. 47 , p. 290-98 (1970) 



Ol'shevskii, V. V., Characteristics of Sea Reverberation , Consultants 
Bureau, New York (1967). 

Tolstoy, I., and C. S. Clay, Ocean Acoustics: T heory and Experiment in 
Underwater Sound , McGraw-Hill Book Co., New York, (1966). 

Wagner, R. J., Shadowing of randomly rough surfaces, J. Acoust. Soc. Amer. 41, 
p. 138-47 (1967). 



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Unclassified 



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Naval Postgraduate School 
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3 REPORT TITLE 

Coherent and Incoherent Components of Sound Scattered at a Time 
Dependent Rough Surface 



4 DESCRIPTIVE NO T E5 (Type of report and, inclus i ve dales) 

Technical Report, NPS-61Md72121A 31 December 1972 



5 AUTHOR(S) (First name, middle initial, last name) 

Clarence S. Clay 



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December 1972 



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Naval Postgraduate School 
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13 ABSTRACT 

Theoretical expressions are derived for the sound scattered at a time- 
dependent rough surface. The calculations are made for a Gaussian 
shaded source transducer and point receiver. The Helmholtz theorem and 
Fresnel approximation are used. The rough surface is assumed to be a 
traveling wave and to have a traveling wave packet type of correlation 
function. The coherent component of the signal is the product of the 
Fourier trans foarmation of the surface distribution function and the 
smooth surface reflection signal. Comparison of theory and experiment 
shows the coherent component to be sensitive to the non-Gaussian char- 
acter of the wind-blown water waves. The incoherent components and the 
temporal correlation function of the scattered sound are given. For the 
special case of a traveling cosine wave type of rough surface, spectrum 
of the scattered sound includes components which are multiples of the 
frequency of the surface wave. For surfaces describable by a bivariate 
Gaussian distribution function, the temporal correlation is a function 
of, but not the same as, the time dependence of the rough surface. The 
scattered sound is insensitive to the spatial correlation function of 
the surface at distances larger than the dimensions of the transducer 
divided by the cosine of the incident angle. The final expressions are 
complex error integrals and can be used for all values of roughness. 
This task was supported by Naval Ship Systems Command (Code PMS 388) . 



DD 



""" 1473 

I NO V fiS I ^ / *J 

S/N 01 01 -807-681 1 



(PAGE 1 ) 



38 



Unclassified 



Security Classification 



A-3t408 



Unclassified 



Security Classification 



KEY wo R DS 



Rough-surface transmission 
Sound scattering 
Coherent reflection 



DD ,rr..1473 BACK) 



ROLE W T 



39 



Unclassified 



S/N OIOt-807-6821 



Security Classification 



A- 3 I 409 



DUDLEY KNOX LIBRARY 




3 2768 00391402 9