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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). 33 INITIAL DISTRIBUTION LIST No. of Copies 1. Conunander, Naval Ship Systems Command 2 Attn: Code PMS 388 Navy Department Washington, D.C. 20350 2. Commander, Naval Ship Systems Command 1 Attn: Code PMS 302 Navy Department Washington, D.C. 20360 3. Office of Naval Research (Code 468) 2 800 N. Quincy Street Arlington, Virginia 22217 4. Director, Naval Research Laboratory 1 Technical Information Division Department of the Navy Washington, D.C. 20390 5. Director, Advanced Res Projects Agency 1 Technical Library, The Pentagon Washington, D.C. 20301 6. Defense Documentation Center 12 Cameron Station Alexandria, Virginia 22314 7. Commanding Officer 1 Office of Naval Research Branch Office 219 S. Dearborn Street Chicago, Illinois 60604 8. Commanding Officer 1 Office of Naval Research Branch Office 1030 East Green Street Pasadena, California 91101 9. Commanding Officer 1 Office of Naval Research Branch Office 495 Summer Street Boston, Massachusetts 02210 10. San Francisco Area Office 1 Office of Naval Research 1076 Mission Street San Francisco, California 94109 34 11. New York Area Office Office of Naval Research 207 W. 24th Street New York, New York 10011 12. Director Naval Research Laboratory Library, Code 2029 (ONRL) Washington, D.C. 20390 13. Commander Naval Ordnance Laboratory Acoustics Division V?hite Oak, Silver Spring, Maryland 20910 14. Commander Navy Undersea Res. & Develop. 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(Code 480) Department of the Navy- Arlington, Virginia 22217 35. Dr. Paul A. Crowther Naval Division Marconi Chobham Road Frimley, Camberly, Surrey England 36. Dr. H. Medwin, Code 61 Md Naval Postgraduate School Monterey, California 93940 37. Mr. R. Uranicar (Code 370E) Naval Air Systems Command Department of the Navy Washington, D.C. 20360 38. Sonar Directorate (Ships 901) Naval Ship Systems Command Department of the Navy Washington, D.C. 20360 39. Librarian Applied Research Lab University of Texas Austin, Texas 78712 40. Prof. C.S. Clay Dept. of Geology & Geophysics Geophysical & Polar Research Center University of Wisconsin Middleton, Wisconsin 53562 41. Dean of Research Naval Postgraduate School Monterey, California 93940 42. Dr. Jerald W. Caruthers Department of Oceanography Texas A & M University College Station, Texas 77843 43. Dr. R.G. Williams NOAA Rockville, Maryland 20852 37 Unclassified security Classification DOCUMENT CONTROL DATA -R&D iSiTuniy classification ol tlllu, body of ahslraci and indexing annolaliun mu.sl he entered when the overall report is classified) 1 OHIGINATING ACTIVITY (Corporate aultior) Naval Postgraduate School Monterey, California 93940 2a. REPORT SECURITY CLASSIFICATION Unclassified 26, GROUP 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 6 REPOR T DATE December 1972 Sa. CONTRACT OR GRANT NO. b. PROJEC T NO. 7a. TOTAL NO OF PAGES 39 76. NO OF RE FS 13 9a. ORIGINATOR'S REPORT NUMBERIS) 9b. OTHER REPORT NO(S) (Any other numbers that may be assigned this report) 10. DISTRIBUTION STATEMENT Approved for public release; distribution unlimited, 11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Naval Postgraduate School Monterey, California 93940 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