BACKSCATTERING OF ULTRASONIC WAVES FROM A ROUGH LAYER by WU-SHI SHUNG B. S., National Taiwan University, 1961 A MASTER'S REPORT submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Electrical Engineering Kansas State University Manhattan, Kansas .1966 Approved by: uj-ah Major Professor ACKNOWLEDGMENT The author wishes to express his deep appreciation of the guidance, encouragement and patience so cheerfully given by Dr. W. W. Koepsel, the Major Professor and Head of the Department of Electrical Engineering. The author also wishes to give his hearty gratitude to Dr. S. H. Durrani of the Department of Electrical Engineering, who was of great help in giving the suggestions and corrections, TABLE OF CONTENTS 5 54,01. INTRODUCTION 1 2. EQUATIONS OF ACOUSTIC WAVE MOTION 2 3. 4. 2.1 Stresses, strains, and elastic constants 2 2.2 The equations of motion 5 2.3 General equations for damped waves 8 REFLECTION OF PLANE WAVE FROM A PLANE INTERFACE 3.1 Boundary conditions. 10 3.2 Reflection of waves at the interface of two isotropic and homogeneous media 10 THE GENERAL KIRCHHOFF SOLUTION FOR SCATTERING FROM ROUGH SURFACES 6. 7. 15 General solution for surface with one-dimensional roughness 15 4.2 Rough surface as random process 22 4.3 Statistical distribution of the field 4.4 The effect of absorption in the medium 26 4.5 Limitations of the general Kirchhoff method applied to acoustic wave scattering 28 4.1 5. 10 ACOUSTIC WAVE SCATTERING FROM LAYER , . . 25 31 5.1 Layer with rough interface in the back 31 5.2 Layer with rough interface in the front 36 5.3 Discussion of the derivation 49 51 EXPERIMENTAL WORK 6.1 Experimental set-up and procedure 51 6.2 Description of layer target 55 6.3 Measurements and results 57 CONCLUSIONS 66 APPENDIX: EVALUATION OF INTEGRALS 67 REFERENCES 69 i LIST OF FIGURES 1. Reflection of plane wave at plane boundary 12 2. Scattering from a rough surface 17 3. Local scattering geometry 17 4. Scattering from a layer with rough side in the back 5. Transmission through a rough interface 6. Scattering from a layer with rough side in the front. 7. Block-Diagram of experimental set-up 8. The plexiglas with a one-dimensional rough surface. 9. Variance of scattering coefficient of layer with rough side in the back 59 Variance of scattering coefficient of the rough surface of the layer 60 10. 11. ... 32 38 . . 42 52 ... 56 Comparison of D{p} from theoretical and experimental results. (a) 9^0° 62 (b) 9^=5° 63 (c) 9^10° 64 ii I . INTRODUCTION The problem of the reflection of an acoustic plane wave from a plane interface separating two isotropic media has been solved by many physicists and geologists. Exact forms of the generated waves are given, related to the incidence angle and the acoustic impedance$.of the media. Brekhovskikh extended the work to deal with a layer having plane interfaces. The nature of wave scatter- ed by a layer with rough interface, however, is generally unknown, The backscattering of acoustic wave from a rough layer is analyzed in this report. The model of the layer considered con- tains a smooth interface in front and a random rough interface in back. The Kirchhoff's approximation for evaluating the scat- tering field of a rough surface (Beckmann 1963) is extended to deal with such a layer; experimental work has also been done for this model at different incidence angles and frequencies. A tentative try is also made on the evaluation of back- scatter from a layer with a very rough surface in the front. Equations are derived for this layer with no experimental support. Owing to the analogies between acoustic and electromagnetic waves, the results of this work can be directly applied to the same problems in electromagnetic waves. The radar cross sec- tion is obtained just by modifying the variance of the scattering coefficient with a scaling factor. Application of the results can be found in the survey of lunar surface and in geological explorations. -1- EQUATIONS OF ACOUSTIC WAVE MOTION II. Stresses, strains 2.1 , and elastic constants In an ideal isotropic homogeneous medium, a wave may propa- When gate without any loss of amplitude due to internal friction. the medium is defomable and undergoes a change in configuration due to the application of forces, the body is said to be strained. It is assumed that a point P (X (Redwood 1960, Ewing 1957). z) is displaced, and the coordinates of the displacement are v, w) . . Y, (u, is displaced by (u+ An adjacent point Q(x+6x,y+6y ,z+6z) 6u,v+5v,w+6w) , By Taylor's theorem, neglecting the higher terms under the assumption of small perturbation, u + 6u = u + {H 6x + |H 6y + |H 6z , v + 6v = v + |X 6x + jX 6y + |Z 6Z , w + 6w . w + |* 6x + »S 6y + || 62 (2.1) . The analysis may be simplified by writing 3u 3x =: xx 1Z -" 3y ' e xy " 3y " 3x 3y — — 3x 3y " co z ' 3w _ '" yy yx yz 3z e 3z ' 3z ' zy zz ' 3x " zx "" xz ' iH-iZso) 3z 3x -2- y ' 3w 8v m "' 3y " 3z | x -3- Then the displacements may be rewritten as u+6u=u+(e xx 6x+ie v+6v=v+(^e yx 6x+e xy yy 6y+ie 6y+|e xz yz 6z) + (u) 6z) + (a) 6z-u> y 6y-u) 2 z x 5y) 6x) (2.3) , w+6w=w+(^e zx 5x+je z 6y+e zz 6z) + (uj x 6y-(o 6x) simple extensions of the medium in the represent e r zz yy vicinity of P (x, y, z) e e xx , ' xy , e e ' , e , yz' the shear strains; and represent * zx o> x , y , w z repre- sent the rotation of the element as a rigid body. To express the displacement in vector form s = where x o . v / o ux„ + wz^ + vy„ o o o and z o (2.4) , are unit vectors in the directions of x-, y-, and 2- axes, respectively. For small perturbation, the shear strain is so small that it has no contribution to the volume change. tion A is defined as A = lim 6x,6y,6z->-0 = e xx + e = div s (6x+e + e yy xx 6x) (6y+e Then the cubic dilata- 6y) (6z+e zz 6z)r6x6y6z yy 6x6y6z zz , and the rotational displacement u becomes {25) -4- •*«*» x o = *• z y-^o j curl s o (2.6) . To discuss the force acting on an element of volume in a medium, nine components of stress, which have the units of force/ Let the stresses be denoted by area, are required. t * . , where the first subscript is associated with the axis normal to the place on which the stress acts, and the second subscript to the In an isotropic homogeneous medium, direction of the stress. there are only two elastic constants, Lame's constants to relate the stress and strain together. fluid and ideal gas and y In the case of ideal The stress and strain relations by 0. u X Lame are as follows: t t t xx =X (e v «x yy zz t xy t yz t zx (e =X (e =t =t =t yx zy xz xx xx xx +e +e +e =pe yy yy yy xy =ue M yz =ue M zx +e +e +e =ye =ue M =ue ^ )+Zye zz' zz )+2ue )->2ue M zz 7 yx , ' zy , ' xx yy zz =XA+2ue xx =XA+2ye H yy =XA+£ue fcH zz / , ' (2.7) xz From (2.7), other elastic constants can be derived as follows, (Lamb 1925) : -5- (a) . Uniform stress and dilatation, x xx =T yy -T 2 . T e , and =(X+|-u)A=kA 3 Shear stress M =t v =t is the compressibility, k=(X+-=-y) (b) zz t xy _ T e xy t as defined by Eq. xy T yz _ e yz (2.7) zx zx is the coefficient of rigidity, (c) . Longitudinal stress, xx m y(3X+2y) X+y v =t zz t m xx xx = ' E is the Young's Modulus. (d) . Poisson' s ratio, a , represents the ratio of lateral contraction to longitudinal extension. T = yy T zz " ° e ' = oe e xx yy " zz In this case, and ' ° = aTI+yT The constants X, y, k can be expressed in terms of Young's Modulus and Poisson 's Radio: X V Lame's constant = = Rigidity = ^ 1+p) k = Compressibility = 2.2 (l+o) (l-2a) (2.8) , ^ E /i_j \ The equation5Qf motion According to Newton's equation of motion, the x-component of the resultant force on the volume element is -6- (p6x.6y 5 z)0 = + < (t T ^x -^* + yx 3t ^« + or where p 3 p ' 2 u + 3x . ap-" , ( 6x - Txx ) 6y62 xx 6 y- T yx l8M « ,6x6 y zx -ar 3T xx )6Z6x / 3t zx yx "ax" + "ay" + ^i~ , t , > For an isotropic homo- is the density of the medium. geneous medium, Equation (2.7) holds, giving + fc (u+2ue X x» 37 " (U 3to 5A V + h <»•«> 3u> Similar arguments apply to the y-, and z- components, so that 2 p 3 s jp- = (X+2y) grad -* A - 2p curl w . (2.9) Equation (2.9) is the equation of motion in an isotropic medium. It is usually rewritten in the form of displacement potentials <J> and % by the following transformations s = - (grad div ty - <J> + curl $) , (2.10) -7- Substituting (2.10) into (2.9), p —-(grad <j)+curli^) = (A+2y )grad(V<()) -ucurl (curlcurl^) , at p— z9 (grad<{i+curl|) = (X4Z;M)grad(V <j>)+ucurl(V J), (2.11) 3t where V 2 . denotes the Laplacian, defined by V 2-*- _ for a scalar <|> and , V $ 2 •* Div. Grad , = Grad Div ijj-curl curl •* ij) $ + , for a vector ij/. In the cartesian coordinate system, v 2 2 2 =i3x + i- 2 3y 2 i- + 2 3z 2 • By taking the divergence and curl on Equation (2.11) the scalar and vector potentials can be separated: —£ 2 9 4> - r C. .. 3t i42 3t 2 7 2 <f> , C. < C s 2 . = A + 2pr Vj , C 2 s In Equation potential $ ,- . = H. (2.13) . p (2.11) and (2.12), the scalar displacement travels with velocity C» and involves no rotation; it is called "longitudinal", or "compressional" or "irrotational" i|» ,_. (2.12) ; p ^ , , or "dilatational" or "P-Wave"; the vector displacement potential travels with velocity C and involves rotation; it is called "shear", or "transverse", or "lateral "rotational", or "distortional" , " , or "equivoluminal" or "S-Wave". , or The names Longi- tudinal and Shear waves shall be used throughout this report. -8- In ideal gases and fluids, the ordinary acoustic wave init- iates a wave motion in which the sign of dilatation changes very fast so that there is no time for sensible transfer of heat be- tween adjacent portions of the medium. The flow of heat hardly sets in from one element to another before its direction is re- versed, and the conditions are practically adiabatic. since u o, no shear The Lame's constant wave propagates in an ideal gas or fluid. X for an ideal gas is found to be X K ,: X : Moreover/ = K , ad = \P n o (2.14) adiabatic compressibility, (specific heat at constant volume) /(specific heat at constant pressure) P : gas pressure. In an ideal fluid, factors. by C, X X = K , is usually influenced by many is related to the longitudinal wave velocity C & /x/p , and an empirical equation giving C, in water is 2 C^ = 141, 000+4. 21t-3.7t +110s+0.018d, C. = 2.3 (2.15) longitudinal wave velocity (cm/sec) t = temperature s = salinity d = depth (C) (1/1000 in weight) (cm) General equation- for damped waves The dissipative forces in acoustic wave motion are propor- tional to the velocity of the particles in the medium for small perturbations. The force of the elastic stress must both accel- erate the medium and overcome the dissipative forces, so the equations of motion, Equations (2.11) and (2.12) are modified (Kinsler and Frey 1950) as follows: -9- 2 ^42 + 7 if- C, V * p 3t ll2 + 9t a || p 3t 3t where n the unit ofc has ,.1. ( = C S 2 (2.16) 4>, V* (2.17) , mass (time) ) (volume) The general solution of Equations (2.16) and (2.17) has the 1 usual form E = Ae reference point. * X • r_U)t) , where r is the distance from certain Substituting into Equations (2.16) or (2.17), the propagation constant X is found to be = K+Kat X where is the wave number, K = o)/C a = n/2pC is the attenuation constant. For water at room temperature, the attenuation constant is about one thousandth of the value of air. For this reason, ultrasonics are very well suited for underwater signal transmisAs for the possible shear sion as opposed to the case in air. waves transmission in viscous gas and liquid, the skin depth is found to be /2£/po> , where K is the shear viscosity. These waves behave very much like electromagnetic waves penetrating a metal. 5.6 x 10~ cm. For water at 1 mc, the skin depth is only In most cases, shear wave propagation can be neglected in gases and liquid. REFLECTION OF PLANE WAVE FROM A PLANE INTERFACE 3. Boundary conditions 3.1 The continuity of normal and tangential components of dis- placements and stresses across the interface give the following boundary conditions (Brekhovskikh 1960) ,-*-* (s ~s )=o -* n \ . 1 ; 2 n x (s ~s 2 )=o )=o ; (t,-t )=o . (t . n x 1 ~t 2 (31) ; 1 n : 2 Taking the z-axis as the normal to the plane, Equation (3.1) can be expressed in terms of displacement potentials and as, <(i n s w v 3x az nxs=-ux +vy o n.T=x ^ zz =-t =a, o =(^ ^ + + T zx^o 3y + ' ^ - \ - 3x3z L»ii "^ SySz*+ 3.2 + ^) x Q + ^ (-j* - + niy + ^ J yQ ; ^Brg^. s zyy 2 2 Z ' ^ —£«.—* 3 ,2. "U $ ^, 3y3z !!i 3y3z 3 Tz 2 *, 7" 3 2 1\, 3x* a2 *x !^x 7~2"~ + _2 a 3z 3y 3 *v o 3x3y / 2 » ».wy 3z3x / o Reflection of waves at the interface of two isotropic and homogeneous media Let A, B, and D represent the amplitudes of the potential of incident, reflected, and transmitted waves, where the subscripts -10- -11- and % denoting longitudinal and shear wave (Fig. 1). s Accord- ing to Huygen's principle, the phase velocities for each wave are equal at the boundary, therefore, C C C 1A . p " sine, ' ls simr, . ' C = ^ls l& sine,' ' slnff m C 2i sine m "' 2 2s sinf 2 It is shown that the longitudinal or vertically polarized wave, ty =$ y is always reflected and transmitted in the modes of , longitudinal and vertically polarized shear waves (Brekhovski Kh, 1960) In other words, the vector displacements potentials in- . volved can always be assumed to be in the direction of the The potentials in media #1 and #2 y-axis and independent of y. are assumed as follows: l =A »* ex P [ iK 1A ( xsine i +zcose i) + Bj»exp J <(>2=D exp [iK 2 (xsine 2 +zcos6 2 ) ] , =D exp [iK 2 (xsinK' +zcosir 2 2 ) ] . 2 iKj (xsinOi - zcos6i)] (xsin^+zcosK^) ly +B s «exp [iK^xsint^-zcosYj) ]y Q o ^!=A »exp [iK ^ [ (3.3) The boundary conditions at z=0 are — 3* 3$ =- 3z + 3x - 3X 3Z * = =%- m —- + 3$ 3* 3z 3x 3X - cY (3.4) |J **' =2- (2 VJ,:5) " . - 3Z 12- PI6.I. Z&Fl&C7\0l\ OP PLKU& WAVk AT rUKJB- &OUU0AZY -13- Is 2 + »u5i? 2s »'(i^ - ^i "^ = 2 + » 2 (^ ^y - Upon substituting Equation (3.3) into Equations (3.4), « • 3 - 7» (3.5), and (3.7), the boundary conditions are expressed in terms (3.6) of potentials as follows: (A rB l )K (A +B )K i t li / +(A s +B s )K ls sinri=D K 2jl cose 2 +D s K 2g siny 2 , (3.8) 8ine -(A -B )K C06f -D K 8ine -D K C08jr l8 1 8 8 1 i 2l 1 2 8 28 J (3.9) u cos6 1 Jl v p l (A + * V l-2-¥*L* 2 *i) +Pi A s~ B s sin2 ^i ( ( ls K P K p I* A £~ B i* " ) 2 2 D (l-2-|^sin K 2s Jl 2 e 2 )+p D 2 8 sin2/2 (3.10) 2 -2-"sin2e 1 "Pi^ A> + %^ cos2,f'i = Is P By setting As , or A^ 2 D £ ^2~-sine 2"" p D COs2 ^2 2 s K (3.11) * 2s equal to zero, corresponding to longitudinal or shear wave incidence, respectively, the amplitude of the generated waves can be expressed in terms of the amplitude of the the incident wave potential. It is to be noted that when the incident angle is small, all the sinusoidal term tends to the limits 1 and 0. Then the boundary condition is further simplified to <W K ir D * K 2l » < 3 ' 12 > -14- - - ( V B )K s ls=- D K s Pl (A t+B t )-p 2 D t "p l ( V p £ 2 At P fs ?2 A " s 2 p C zr p l C l* C C 2£ +P 1 C 2s-?l ls U s 2s 13) ' +p C l ls' 2p C kt p D C £ _s As 2£ +p C l l* 2 2A 2 * "' p ' 15) (3.14), and (3.15) (3.13), f D C C (3 • s s Solving Equations (3.12) B ' (3.14) ; B )= - p D s (3 ; 2s p 2 C l p 2 ^1 2s +p C 2 C 2s l ls Equations (3.16) and (3.17) imply p p (3.16) (3.17) 2 that when the angle of incidence is small, no change of mode occurs at the interface of acoustic wave motion. This is a very important limitation in the acoustic simulation problem. J The angle of incidence has to be very small if change of mode is to be avoided in the experiment. 4. 4.1 THE GENERAL KIRCHHOFF SOLUTION FOR SCATTERING FROM ROUGH SURFACES General solution for surface with one dimensional roughness Beckmann (Beckmann, 1963) has derived the solution for the mean scattered field, power, and the statistical distribution of those quantities by the Kirchhoff approximation method. The principal limit of the approximation is that the surface must not contain a large amount of sharp edges, sharp points or other irregularities with small radii of curvature. The criterion for the validity of the approximation is given as 4Kr cos6>>\ c where r , ' is the radius of curvature, incidence, and A 9 is the local angle of is the wavelength of the wave. The rough surface is given by the function 5 - 5(x) (4.1) with mean level coinciding with the plane z = (4.2) . The medium in the space z>£(x) is assumed to be isotropic in which a monochromatic plane longitudinal wave El =e ik ,r - ia)t l (4.3) is transmitted, where K =il 1 A . Kl (4.4) K1 is the wave number of the incident wave, which is assumed to lie in the xz-plane (Fig. 2) , and r - xx +5(x)z o o -15- is the radius vector f m (4.5) -16- The angle of incidence is denoted by B^, the scattering angle by 6 2 , where the magnitude of k 2 and k 1 are equal: [* 2 - I^J = K - X1 I (4.6) • iL also lies in the xz-plane for a one-dimensional rough scat- tering surface. In order to deal with plane scattered waves , the observa- tion point P is removed to the Fraunhofer zone of diffraction, R'-k», where R' is the distance from P to a point B(x,£(x)) on In other words, the rough surface (Fig. 2). K R'=K R -K -r 2 o 2 where R (4.7) , is the distance of P from the origin. The scattered potential E_ at P is given by the Helmholtz integral where * E and 3E -2— = e iKR' . * -IP- e + iKR -iK„ -r z o R^ . .. (4 ' . Q 9) are the potential and its normal derivative on the rough surface S. The values of those two quantities are approxi- mated in the Kirchhoff method by the value that would exist on the tangent plane at that point, i. e., (E) s Os = (1+R^ , " i(1 - R)E lV"/ (4.10) (4 - ll} n is the normal to the surface at the considered point B (Fig. 3) and R is the longitudinal wave reflection coefficient of a smooth plane. -17- PI6. Z . €>CATTSZlU(j FZOth A Z01J6H 6UEFACE §(x; R6. 3. LOCAL eCATTEKIWS 6£0M£TCY -18- Then = 6,-8 = 9, -arc tan?1x) (4.10), and (4.11) Substituting Equations (4.9), E. (P) = J7)j (4.12) . in (4.8), (l+RjE^-i K 2 nl-ifll-RlE^-n = J.JT e i(V^2>'^ R(^^ 2 )-(^ 2 ) ds -S ds, or E (P) = (E^)e iv ie^)J * r ^ds (4.13) s where iL = K(sin6,x -cos6,z iL = K(sin6 x^+cose 2 n r J = -sinBx +cos8z^ o o = xx +£ (x) =V 2 2 z ) ) . ' z. o / 2 = K(sin8,-sin6 )x -K(cos6,+cose ) z^ / z o 1 o l ? K(sin6,+sine z )x o +K(cos6 z -cos8, i l, ds secSdx, ta.ft3=S'(x) ) z\ o . For a one dimensional rough surface extending from x=-L to L, Equation (4.13) may be rewritten in the scalar form. « E M 2--nR^-j. ik^ k L (as-b)a L iv x+i Vdx (W) -19- where a= (1-R)sine +(1+R)sin6 2 , (4.15) b = (l+R)cos6 -(l-R)cose 1 . (4.16) 1 2 The scattering coefficient p is defined as p-!U E (4.17) . 20 Where E 2Q is the potential reflected in the direction of specular direction (65=6,) by a smooth plane medium-air interface In this case of the same dimension. Vv x =o, J' 5 Oi R = -1, e ikRo f = 8 1 2 so that E 20 " L ikR .. f ^fef— " ike bdX = L ^o~~ „ 2RC ° Se dX i " ' or ike^Lcosei 20 1 Hence, from Equations rR . (4-18) Q (4.14), L 1 f 4Lcos6, 1 (4.17), and (4.18), (ar-b)e iVxX+iVz5 dx (4.19) . /-L For a smooth surface, p * = — 1 _ 4Lcos6, e 1 2 J . _ (1+R) cose-- (1-R) cose, \ lVx X dx -L (l+R)cose2-(l-R)cosei 2cos6, where sine V L x" sin Vx L V..L . S i n cv L , (4.20) -20- As a special case, if R=-l, then p = sine V L x o in Equation (4.20) becomes p (4.21) . For a rough surface with constant reflection coefficient R, a and b are constant; from Equation (4.19), "- 5T P = e 4Lcos = - , 1 AV ^dx- —e 2R+2Rcos(9 1 +9 2 ) cos9 +cos9 1 . 4Lcos9 fl 1 1 L f I> T U z -L - ,-*-* - e^dx-e (L) . 2 The second termacounts for the edge effect tends to zero when L>>\. Thus, ignoring the edge effect, p L.f = 2L L e^dx (4.22) , J-L where F = - Rsec . 1+cob(9i+9 2 ) COS9,+COS9 2 = 1 v-r" - ^|-[(sin9 -sin9 )x-(cose +cos9 ;L 4»2 (4#23) ' 1 2 2 ) 5(x)] (4.24) Rough surface as random process The surface height assuming values £ (x) is assumed to be a random variable with a probability density w(z) z value, denoted by angular bracket <> <Ux)> = , , the mean is , and the mean value of the integral for a stationary random surface is (4.25) -21- e =e ivr,dx -L ivx vx e iv~C z e iv *x , dx -L e = X(v z x dx e ) dx (4.26) -L x(v z ) z w(z)e \ is the characteristic function associ- dz ated with the distribution w(z) . From Equation (4.26) the mean scattering coefficient <P> - jg x(v z )sinc v x L (4.27) The variance of the scattering coefficient D{p} corresponding to the mean of the normalized scattered power is defined as E <pp*> H .2 2 > = D{p} + <pxp*> (4.28) 20 The asterisk * denotes the complex conjugate, and from Equation (4.22), rh -L .iVxCxi-x^) e iv z ( 5l -5 2 ) dxidx2 = pp* 41/ Denoting e iv zU 1 ~K 2 ) '-l &Y -l ; X2 t v z ~ v z )' then from Equation (4.28), L /-L pp * - X2Cv Z r-v z 41/ l-L -L ) e ivx (x x l~ 2 ) dx dx 1 2 -22- "h "e x,(v,-v '2 NV z z v . , iv e " «i-«i> iT LL •t'i.-a>- - Ul " la> -i-a and C,= S(x,) X £a 9 (v Z ,-v Z ) , e,*?(x,) • is the two-dimensional characteristic function of the distribution w( z z , 1 2 ) . The random rough surface is assumed to have a Gaussian distribution 1 = w(2) 2 . /T^ 2 e /0 2 -z /2o^ < 4 - 30) and the two-dimensional distribution is z, w(z,,z,,t) = 1 2 a ss c(t)= <z < z 2iro^/l-c2( T ) 2 2 2a 2 (l-c 2 2 (4.31) 2 (t)J is the variance of the rough surface, > z l 2> is tne autocorrelation function of the rough * < -2c(t)z,z +z -exp g ^ 2 z, > surface, t=x,-x 2 is the separation factor between two points x, and x~« The autocorrelation function usually assumes the following two forms: (i) . c(x)=e 2 . — x 2 /T , — (ii) . c(x)=e I ' T "/T I gaussian correlated, (4.32) exponentially correlated, (4.33) T is called the correlation distance, which is much less than L to assure a random surface. The one- and two-dimensional characteristic functions are -23- evaluated as below (Appendix) 1_. e" 22 / 20 ^e 1xv z" dz 2tto 1 2 ¥ = e" 2 Vz (4.34) where 2 g = a v2 2 (4.35) 5 - X 2 (V Z'- V 2 w(z,,z 2 t)e = ) — 0L» I x (4.36) . (4.28), and (4.29), the variance (4.26), From Equations (4.22), p 2 —00 - exp -g(l-c(x)) of -„ iv.,(z,-z ), z> 1 2 dz^z is D{p} = <pp*>-<pxp*> „ r L r L e 4l/ iv (x 1 -x 2 ) x X2 (v z ,-v z )-x(v z ) x *(v z ) d Xl dx 2 '-L >-L (4.37) Using the relative coordinate = X -X T x already defined as x (4.38) , 2 and introducing the center-of-mass coordinate = *o 7 (X 1 +X 2 (4.39) ) the equation (4.37) can be rewritten as C1J [ D{p}=^ dx 4l/i-L l °;-2L o r 2L 2 -£-.,. 2L. 4L [e ^ * m iv * T x2 X z (t)-xx*] 1-2L VxT 2T)_ 2 f L*2< T jjzhj-^ 1 -1/2 1 +1/2 dx (T)-x.X*l-det 2L = 3(x,,x,) >-XX*>T • , -24- From Equations (4.34) and (4.36), D{p} can be expressed in terms of c(t) as 2L D{p}= F!e-^ e iv x T (e* cCT) -l)dT . (4.40) 2L / -2L Two kinds of correlation functions, Gaussian and exponential, are considered for g<<l (i) and g>>l ? . Gaussian correlated surface, . (A) If g<l, e gc ^ T 'can be expanded by a uniformly convergent series. 2L -2 «/ i f « lv x t f --g\ 2L J-2L » m m m,_v ULa , m g c Z. X * m m (T )dT, since L>>T 2 iv x x £ g c = F 6 -g\ _® 2L m=l ml J 2 _ /ttTF 2L e -g ^ _g__ e—j4m 2 . (4.41) m= i m!/5T If g<<l, 2 2 / D{p} = l|F!g. exp ^- Vx T /4 ^ ; (4.42) g*l _^2 e (B) - V 2 vx T /4 < D{p} 2 < £**! (4.43) . g>>l, ,2 2 r2L zr 2L . ? /-,_.-^ /T , tmm 2 iv x T -g(l-e £\ 2L e |!r ivxT -g(i-e- /t -t / 2 » J t l 2p 2 e -g sinc2v L e J-2L e e .1 dx T2 ^ 2 ; dt -25- Because g is much greater than unity, the integrand is neglible if t/T is not in the close vicinity of zero; the integral is therefore simplified as D(p% 2 2 ,ivx^-g[i-(i-^ /T )]dT = 21" = |_ e ir in /T 2 dx 2 (4.44) ex P 2iT? . 2 e^ T 2 TF (ii) ivx T 4g J Exponentially correlated surface: the same procedure can be followed. (i) (A) g<l, .2 ^?* D{p} e 2 -g m*l _m SL^ ml" -„2 /St (4.45) "XT m +v x T if g<<l, D{p {p } = (B) —t— L * J— 1+v (4.46) x^ 2 g>>l, Dtp) = 4.3 • F — TTX T g*+v x ciT • 2 (4.47) ' Statistical distribution of the field The mean scattered field and power is (4.2); in Equation the probability distribution of those quantities can be found by looking at the random variable (3.22), show/? p . From Equation -26- 3 -L j-l 1 = re * A = £M L < . is a constant, and $ is a random quantity. . 4 - 48 > Beckmann D has shown that for a very rough random surface (g>>l,<p>=o) with zero mean <p>=o, the distribution of the resultant phase \|> is uniform wU) = i- , -»<<», (4.49) and the distribution of the amplitude r is Rayleigh distributed 2 w(r) = •"* /° {p} §fP } (4.50) . The variance of pp* is found to be 2 4 D{pp*}=\ r p(r)dr-D {p} '0 r-) o 2r 5 2 e -r /D<p> 2 dr _ D {p} dTp"} . D 2 {p} (4.51) The standard deviation of pp* /D{pp*} = D{p} (4.52) The normalized scattered power pp* is found having its mean value and standard deviation equal to D{p} 4.4 The effect of absorption in the medium The presence of absorption of the space in which the wave . -27- is transmitted has a great influence on the mean scattered powers. The damped wave equation has a general solution ik«r (l-ia/k)-iwt „ — hi e t which implies a change in the value of V" and x V" of Equation z (4.13) Let V v x = V v x V z " Vz x{vn Z = - v i^ V X K x " *t Vz , ' ' 2 .J_( e tt) 2., 2 2 -z /2a Q AVV i(v z -iaVz ) e *F 2 r Z dz 2 2 = e 10 v z a / k . e -|5_|z_(i-°L_j (4.53) Hence, * x(v 2 )x*(v z / \ i A ~g (l-a = e yv \ ) ' (4.54) ) The two-dimensional characteristic function X(vW) ^ - B x„ iv",-v') is r.^«<W-Vk)Cl-o)- i .i-^di, 2o' (Zj-CZj) 6 " 2a^(l-c 2 ) 00 e iv z (l-ia/k)(z r« 2 ) /2tto2(1-c2) ig°A - |(l-o 2« A 2 \ e d( 1 c 2 , i(l-c)v z (l-ia/k)z 2 ^2\o2 , . e igocA >e -g(i- a2 A 2 )(i-c) # e" (Z2/2a) dz. (4.55) . -28- D{p} . |!\ 2 e iv x T e igac/k 2 2 2 2 -exp [-g (l-a /k )]dx .exp -g(l-a /k )(l-c) L )-2L For a very rough surface (g>>l) c(t) , is significant only for very small value of x/T, D{ P V )=f vx .exp [-gd-aV^ld-OJe^^dT iv x x 2 exp [. g( i. a / k )(i- c )] dT , or D{p} = |_ e 2 (4.56) . From the analogy between (4.56) and (4.40), the variance of p, e. i. D{p} is (i) Gaussian correlated surface: . 2 Dtp} - -j L (ii) . 5 T~ g(l- a 2 /k 2 ) ' Vx 'l ex P 2 — 2 2~ (4.57) 4g(l-aVk' Exponentially correlated surface: n/ D{p}\ = Equations (4.17), p2 L 2 g(i-* /k 2 ) " 2 2„22 2 T (l-a ) g Z ^ (4.57), and will reduce the value of <EE*> A (4.58) , vV (4.58) shows that the abosrption and high-frequency wave causes more attenuation than a low-frequency wave. 4.5 Limitations of the general Kirchhoff method applied to acoustic wave scattering The acoustic wave reflection coefficient is a function of the local angle of incidence, or alternatively, is a function of 6, , and Ctx) -29- R = R(e) = R^ - arctan'U) (4.12) . and <b> in Equation (4.14), which The approximations of <a5'> are always used for varying reflection coefficient case, are much more complicated in the acoustic wave scattering problems. Owing to the elastic property of the medium when 6^ increases, the generated waves change mode from one to another and the reflection and transmission coefficients change very rapidly. If the rough surface is chosen to be a gently rolling surface (T>>a) , (i) the probability distributions of the slope w(C') are . Gaussian correlated surface: 2,12 w(S') (ii) . -=== /4tto^ (4.59) ex P 4a exponentially correlated surface: (4.60) wU') in Equations (4.59) and (4.60) imply a more dense distri- buiton in the neighborhood of zero slope. If 8, is also chosen to be small angle, the value of R can be averaged over as R(e ,5') 1 " RO^ R(C'=<£>)= , since <£,'> = o . (4.61) For longitudinal wave incidence with 9,=0, from Equations (3.17) and (3.16), the reflection and transmission coefficients are defined as Reflection coefficient R 12 (6 1 ) = Z 2" 2 Z 2 +Z 1 (4.62) , l Transmission coefficient D,,(e,) = ;r%-& Z >Z 2 1 ?' I > (4.63) -30- where ±1L cose P z . 2 !l2li lfX . cosSj^ , ' 2 . (4.64) If the #2 medium is free space or air and the #1 medium is liquid or solid RlB^ = -1 , RO^ = . (4.65) Under the previous assumptions, a and b in Equation (4.14) are considered to be constant and the general Kirchhoff approximation can be applied to the acoustic wave problems. ACOUSTIC WAVE SCATTERING FROM LAYER 5. 5.1 Layer with rough interface in the back The evaluation of the scattering field from a very thick layer with roughness in the back is a direct extension of the Thickness d (Fig. scattering of a rough surface. is assumed 4) to be d >> L, so that the smooth boundary of the layer is at the far field of the random rough boundary. The layer to be discussed is a layer without absorption, so that the wave can transmit The smooth interface has the following char- out attenuation. acteristics, at |E |E 2 = d: z = D | 3 = D I in it with- 12 21 | El |E 4 (5.1) | , | , where E., E,, E-, and E 4 are shown in Fig. 4, and D 12 (5.2) , D 21 are the transmission coefficients. 2Z 2 1 " m z +z 12 2 1 2Z 21 Z Z 2 1 z 2+Zl « " m D„, " P = P l 2 1 C C (5.3) 2^ 1« n 2 m = p 1 The rough surface in the back has the same statistical properties as before. Then, -Tl- -32- BZ (P) lunniu/iiiii V^' f&. FIG. 4-. 6CATTE£IU6 F£OA\ A LAYBK. WITH KOUGH S\P£ IU BACK x -33- E x E 3 dsGce - e ik l* E+ik 3 3 = D 12 e iJt 3- (5.4) , ?+ik dsec9 3 3 (5.5) , The mean scattered field <E 4 > at a distance R^ from the origin is .. _ lk-Le — <E„> = 4 +ik-dsec8, 13 iic^r", 4 -a/2 D-.cos e,e y/ 3 = 12 irR, P • J sine v L, «f x lit (5.6) where g and v v is related to k- and jL = k 3 (sin 6 3 x Q - cos 9 3 z Q) , 2P n ^ 12 = c, lp = c, 2p sin sin 6- 6, —sin = i. 8,3 — sin 8. 4 is mean scattered power for a very rough surface (g>>l) 2 k <=4 E 4* > = L 2 C°S R, it TF 2 , T7 The is the Snell's refraction index. -=- 9 D 3*2lT 2 12 rr~ / 9 v ' eXp[ - 2 x T 2 "Tg" ] (5 ' ' 8) After passing through the smooth interface, the mean scattered ik,Le <V " (5.8), are (5.6), and field and power, from Equations (5.2), ikR +ik 3 d(8pce 3 +8pce a ) o D WRo + 2dsec^) J ^ D 12 21 COse 3'2L* _a e 9 «^V (5.9) 2 k <E 2 E *> = -| 2 2 2 tt 2 2 2 2 COS 8. TF 2 D, L D. 21 12 i-S- (R +2dsec8.) o J 2 2L v ^r— /-| / g exp [- V -f— 4g ] .(5.10) ' ; -34- Since ikLe E lkROcose = 2o 1 ' inr o 2 2 k L cos ^20=20** = w e, TT2 R o >> 2dsec6 R so that, for 2 <E~> 2 <P> = "^20 = , 3 cose. COS6- ~, „ ^,. j, i2k,dsec6-, . ,-, rv Vl2 ^ D 21 e 3 3( 3 ScTe^ _ _ »F.-g/2 .__,, TL sincv e x 2T /- ) (5.11) Using Equation (4.37), the variance of p can be calculated for different statistical properties of the rough interface. Gaussian correlated surface: (i) 9 z 2 2„ ~r i D{p} = n 12 D 12 2„ D 2 21 r [ C OS 6, cos 9 3 „2 m TFf 2~ ], 2L, ,— /g /it avrs 6Xp 2 2 v„t* V X * , [ 5g~ , ] ' 1 (5.12) (ii) Exponentially correlated surface . 2 n r x = n D{p} 2n 12 D 2, 2n D 12 21 cos ae, "3 ( cos „2 gT P* - ).—.—^T r e ,- X ,,* (5.13) . g +v x T 1 For the back-scattering from a layer against air © 2 F v vz -RSpc9 H = k 3 ^4 = "^3 ' ~®i i+cos(e +e 4 « 3 — 3 cose.+cose. r (sine ~sine 3 r 4 ) -k (vne +coseJ 3 3 R = -1 ' ) 2 = sec 8. 3 = 2n ilksin6 = 2n l2 = 2ksin6 3 kcose 3 = -2k/n 1 12 , ^-sin iJ e 1 ,(5.14) — -35g = v 112 2 2 o " C = 4k 2 2 a (n 2 12 -sin 2 6.^) 2* The quantities in Equations and (5.13) show the (5.12) (5.11), backscattering from the rough surface in the back of the layer. The backscattered field from the smooth interface in the front of the layer is negligible except for the normal incidence case, which is equivalent to the reflection in the specular direction. If the pulsed signal is used to approximate a monochromatic plane wave, the backscattering from the front surface can be easily re- jected by adjusting the gate position in the experimental measurement. In the actual case, all layers are more or less absorptive. For a layer constructed with a material which has an attenuation factor a, <p> and D{p} should be modified as follow: <p> = n 3*2 _ _ i2n io kdsec0,3-e iy2/k, , D D e 12 ^(ScTe^ x 12 12 21 2 --exp[-2dasece 3 - ^(1-^-j) sincv x L |! . ] k (i) . (5.15) 3 Gaussian correlated surface: D{p} = n 2 12 D 2 12 D cose 2 21 ( 3 2 q) 15551 2 Tp -2L" v • axt> [-4Adsec6, 3 j — —— /gd-d^A^) 2 T z 2 = 5-] 4g(l-dT/k 3 ^) ; (5.16) -36(ii) D{P Exponentially correlated surface: . =n 12 > 222 D 12 D 21 ^ cose 3 2 2 gd-q exp[-4adse6 3 The quantity 2 a A A 2 3 ) (5.17) . ] 2 2 3 2 .-.^—f-^—^. f \s 2 usually very small, so that (1-a /k 3 )**1# but exp[-4adsee,] is very important, it is nearly unity at low frequency, and decreases very fast as frequency increases. For a layer with a very rough interface in the back <p> - o (5.18) . It is observed that the wave backscattered from an absorptive layer with rough side in the back, the attenuation by the absorp- tion increases as frequency increases. Aside from the attenuation by the layer, the wave backscattered from the rough side of the layer has the same characteristics as from a rough surface of the same statistical property, except for a changing in magnitude. 5.2 Layer with rough interface in the front The acoustic wave passes through the rough interface into the layer, and comes back through it after being reflected by the smooth interface. The mean scattered field is negligible if the rough side is very rough, but the mean scattered power, owing to the complicated phase relationship, is difficult to evaluate. A tentative try without experimental support is made here to look at some aspects of the nature of the backscattering from such a layer. If it is proved to be successful, the same -37- method can be extended to the layer with both sides rough. As a first step, the field transmitted into the layer after the wave hits the rough front surface is found by considering the field at Q(Fig. 5), v°> - JK t? *f - > as (5 - i9) • where g ik-R -ik-«r 3 3 g R = (E) s (5.21) (1+RJEj^ g |£ 3n (5.20) o = i(l-R)E.k\.r 1 1 < • 5 - 22 > From the boundary conditions of the acoustic wave reflection, for small incidence angle k^l-RjE-j^ = k DE 3 P 1 (1+R)E (1+R) = —D p (1 _ R) = = 1 2 DE 1 , i ' 1 (5.23) = mD D = l Zifi. D C = np (||) = mDEj^ . (5.24) 2p Substituting Equations (5.23), (E) , l 3 K P 6 into (5.21) and (5.22) (5.25) , rtDE^.J (5.24) . (5.26) -38- /"""X. 7 FI6.^. TKAU5Ml55)Oki THROUGH A KOUGH lUTEEFACB z -39- Substituting Equations (5.10), E 3 (Q) «n) [mDE.f-iJiic " 47 (5.25), - (5.26) ^(inDE 1 ic 1 into (5.19), «n) ]ds where n =-sinBx r = xx Q o + cosgz^ o + C(x)z , Q cose k\ = k(sin9,x jc, = nk(sin8-x„ - cose^z ) 3 o 3~o - 1 z 1 ) ; w = kl " k3 = k(sine 1 - n sine 3 )x Q Moos^ - n cos6 3 ) = W X + w z^ z o X o ds = secBdx tanB = , £ (x) so that, mke ik-R 3 o f L , E 3 (Q) (ac'-bje 4lrR~ 1* r dx (5.27) -L where a = D(sin6, b + m sin8 3 ) -D(cos6, + m cose 3 For smooth interface , C = C , ) = (5.28) ) ) , -40_ E inkR . .-. (Q) 3 = inke o be 4Tr lw x x dx -L or inkROT - D(cose, + m cos9J sinew x L 3 1 2irR_ . . inKe E (Q) 3 For the direction 3 , . E (Q) 3 e inke = - = arc sin 1_. -sine^^ ( inkR_ °l D(cose + 1 5TR~ to/j_ E.(Q) = ) _ 1 n o (5.30) sin 2 . ) e (5.31) i otherwise. , For a rough interface, . . t, E 3 /«^ (Q) inke = o 4TR- inke w inkR_ [-(b+a^ w z e ) J lw,r dx] -L inkR, 1 °D l-nm+ (m-n) cos (e^-6i cose, - n cos 8, . T?R . e^dx -L or inkR E ; 3 (Q) R o e^dx Q 2L (5.32) -L where „ inkD G = - 2ir l-mn+(m-n)cos , # * (6-,-e, 2 1 cose, - n cose 3 w«r = K [(sine^n sine 3 x-tcose^n cose 3 H] ) • (5.33) -41- Equations (5.32) and (5.33) are analogues of the equations for The mean field and mean scattering from a rough surface. scattered power can be written down in similar form. <E > = e inkRo 6 g— -^ 3 T e sinewy If it is a very rough interface <E (5.34) (h = w 2 a 2 >>1) > 3 and, (i) Gaussian correlated interface^ 2 w T X TG *> = *exp <E E 3 (ii) 3 / ZLR ' o / ir (5.35) ih E Exponentially correlated interface: hT <E E *> = 3 3 ~2z h +W LR (5.36) T~~2 V Equations (5.32) to (5.36), which are derived under the assumption that the incidence angle to be very small, show the pro- perties of the transmission through a very rough interface. The next step is to find the effect of the layer's second interface, thus obtaining the overall effect of the layer with rough side in front. This layer is essentially the same as the previous one in section 5.1, except it has been turned over with rough side facing the incident wave (Fig. lower plane interface at z = 6) . The field E 3 strikes the and produces a reflected field E 4 in the specular direction only (see Fig. as the distance from the point Oj 6). If R 1 is defined to the point of observation Q -42- rcp FI6.&. OCATT&ZIU& FZDtA A UAT&Z. WITH ZOUOH <~>\V& 114 F£OWT -43- below the rough interface, then E " e inkR, 1 R 3 x G, 1 *2L e iw, «r, Id x l 1 -L and the reflected field E 4 is related to E 3 by the following boundary condition: (E 4 ) z— d = {R 23 E 3 } z=-d Under this boundary condition, the field E 4 incident on the rough surface from below is i2k3dsec03 (E 4 ) = G e 1Wl R z=Ux) U(d-5)sece 3 23'2T ' r ' ldxi J-e lK ,r 4 2 (5.37) -L The transmitted field E 2 (P)e 3 caused by the incidence of (EJ 6- . can be derived from Equation (5.32), here the subscript denotes that the field E 2 (P)& is contributed only by the com- ponent of E. in the direction of k 4 E 2 (P)e =e 3 i2k3dsec03 Gi G ikRo 2R (d-5)sece . * 23 Then, 2 e '~^? i( Vl +^2 ,? 2 ) dx dx 1 2 3 -L (5.38) where inkLD G " l 2 cose, - ncqs8. 2.r, iKLD G l-mn+(m-n)cos (e«-e 1 12 21 2tt 1 . + C08(fl 2- e 3 -Si f5"^) 00383-1^ cos6 n 2 ) ) 44w, = k. = k_ - Msine. - n - k ? r i 2 Q - McosOj^ - n cos6 )z 3 ( (5.39) , 2 - k(n sine Vo 3 zl o x% o w2 = k 4 sine )x + W 3 - sine z2*o 1 )X Q + k(n cose - 3 cose^S. ' - x iK + c(x i )? o = x x 2 Q + C(s+x-,)? 2"*o It should be noted here that r^ and r 2 are refered to origins 0^ And more -over, since the maximum value of and 0,, respectively. £(X) V is much less then the thickness of the layer, so that, i p) e, i2k 3 dsece 3 e ikR 2R dsece, GjGj . o 3 23 " e 2" i(w .r +w 2 .r 2 1 1 ) dXidX2 ! 4IT -L -L (5.40) In the backscattering direction, inkLD. G, = - ikLD 2ir 21 =-9 2 1 , cos6,-n cos9 3 1 -E+ (^) COS(9 3! e i' cos6,-n cose. $ 1 = K(sin6 1 -n sine 3 )x Q -k (cose^n cose 3 )z Q $2 - then l-mn+fm-njcosteij-ej^) 2 5fir G, = - 6 MsinBj+n sine^x^-Mcose^n cose 3 )z o (5.41) , -45- From Equations (5.40) and (5.41), the mean value of E 2 (P)9 3 can be found, j <E 2 (P)6 > > = i2kdsec9 3 2R dsec( 3 G G ikR \ 2 "^'7? 3 r oo » oo W(Z 1/ Z 2ls)dz 1 dz 2 , L rL , (5.42 -L -L where and £ s in 2 Equation (5.42) is the separation factor between ' s = (2d+C +5 2 )tan6 3 1 iw since s ^ * <<d (5.43 2.dtane. Hence, from Equation (5.42) <E~(P)6-> = A-exp[-h(l+c(s)] 'sinctkLfsinej^-n sin6 3 )] •sinc[kL(sin6.+n sin6 3 )] (5.44) where 2 W„ a h = ,,2 z W = -K(cose ~n cos6 ) 3 (5.45) 1 ik^dsece-i 3 A = e2dR^sec6, o 3 -j »e ikR_ o From Equation (5.44), it is seen that if the front interface is very rough (h >> 1) 9 e -46<E 2 > (P) 3 and, tt/2 <E 2 (P)> = < <e 2 (p) >de e 3 (5.46) = 3 -ir/2 is very much involved, approxi- The evaluation of <E 2 E 2 *> From equation mations have to be used throughout the derivation. (5.40), L L L L (E E 2* ) 2 iw ss e ***" l* (r l* * .-* r )+iW 2' (r 2" r 2 ) dx dx dx' dx l 1 2 1 2 (5.47) 3 -L-L-L-L where the subscripts 2 1 and are referred to the origions 0^ and 2 respectively (see Fig. 6). , The mean value of (E 2 E 2 *) 3 over the rough interface is L L L L )+iW iWz(5 -C z 1 1 , <(E E *)6 2 2 > 3 <e - AA* (5 1 -C 2 , > > -L-L-L-L .e and from Mood ( iw xl tx r X l )+iW Mood 1963 tion associated with W 2 is ) , x2. (x 2" x 2 ) dx 1 dx 2 dx 1 'dx 2 « (5.48) the 4th order characteristic func- J -47- (s,T> S <e iW* (5 r«i )+iW z U 2-«2 > l 00 / 00 / /> 00 00 W(z — OO —00 —00 1 Z2 ,z£,z-, , ,s)e T i ^ U l-«i' +iW^2-«2 ) — 00 / dz.dz 2 dz£dz2 = exp[- |(2+2c(s)-2c(t )-c(s+t )-c(s-t ) 1 1 1 - |(2+2c(s)-2c(t 2 )-c(s+t 2 )-c(s-t z )3 (5.49) , where t,=x 1 -x^ , and t 2 =x 2 -x (5.50) 2 For a Gaussian correlated random surface, , , c(t) = e 2 2 -t7T , and as seen from Equation (5.49), the 4th-order characteristic function X4 (s,t ,t ;l 2 ) is equal to zero except when s<<T, t,<<T, and t 2 (5.51) <<T In the case shown in Equation (5.51), 2 2 (s^^t^^e-^dx -^ )/^]. (5.52) Introducing new coordinates, yl = l^l^V and y = 2 ' 2 (x 2 +x 2* ' and using Equation (5.52) for a very rough layer, Equation (5.48) becomes 2 t, .L r L r L,L (E 2 E*)9 3 =AA*« 2 \\\\ -L-L-L-L 4L AA.. e " 2h + Wx\t ex P 2 2 V ? ^ 2 /£ t I > .« i lT+i,teT *T dT d 1 a yi 21 —SH- 4W ' exp L~5K— , -48- or 2 <(E E*)9>=2L 2 AA**^- ,„2 ,2 2 .exp (5.53) l 8h where h=k 2 a 2 (cos8.-ncos8 2 3 ) , (5.54) W 2 +W 2 2 2 2 = 2K (sin 9,+n sin 9.J Substituting Equation (5.54) into Equation (5.53), / ,-r, r,*x« <(.E E*)6 3 \ > 2 3 • 2irT = -5-5 2 2 L AA* 5 (cos8 -ncos6 K 1 3 ) ex;p [-T (sin 2 6 +n 2 sin 2 e )/4o 2 (cose -ncos6 1 1 3 3 ) J (5.55) In Equation (5.55), and 9 3 ^(E^jjpe^ is significant only when 9 1 are very small angles, i. e. sine, =6, , cos9,=l sine 3 = 8 3 , cos9 3 = : l (5.56) . Then, the limit in the integration for obtaining <E 2 Ei> can be extended to infinity, so that ^ E 2 E 2>=\ «\ 2L 2 AA*--^ K<r(l-n) J.. R AA * << E 2 E 2 )9 3> d6 3 2 23 4d"R*» J6L" y z .exp f-T 2 2 (6 4 4 2 2 nKLP D 12 21 2 16ir L 2 J +e )/4a 2 (l-n) 2 de, J J (i-mn^m-n)^ (1-^.(1-1)) (1-n) -49- Rewriting Equation (5.55), 2 R23D <E 2 E*>= 2 2. 2 D L 12 21 2 256* E*=^ 2Q p #k 2 # 2 I 2 exp ^ejT /4a (l-n) 2 ] (5.57) 2 d R m |l-n| 2 i R * coefficient 2 2 2 Since E 4 !*_ 2- and<E 2 >=o, , the variance of the scattering o becomes D{p} = E E < ^ *2 2> E R__D, D , (1+m = 23 12 21 2 256 Trm |l-n| E 20 20 4 -, ) x —Vad From Equations (5.46), .exp (5.55) 2 2 2 2 . -e£r/4a (l-np (5.58) and (5.58), it is seen that the backscatter from a layer with rough side in front is much smaller than that from a layer with rough side in back. Moreover, the parameters of the media has more influence on the backscattered power as compared with the previous one in Section 5.1, and as the incidence angle e, increases, the backscattered power drop off very rapidly. 5.3 Discussion of the derivation (A) Rough side in back: The fields E 3 and E 4 in the layer have been assumed to be caused by a plane wave and the amplitudes are independent of the thickness "d" of the layer. If the thickness d is much larger then the illuminated length 2L, the amplitudes of E^ and E. will depend on the value of d. From Equation (5.3), introducing the dispersion caused by the thickness of the layer, the value of E 3 given by Equation (5.5) will be changed to in E, m 3 ik3-ri +ik 3 dsece 3 ke s— —— ZifR-. LD, -(cose,+mcose,) J -L* J- . (5.5a) -50- The A similar modification should be made on E 4 and E 2 also. results of these changes in E 2 , E 3 and E , 4 will modify of D{/>} Equations (5-12) and (5-13) to the form n D(p}= fi99 D D 12 12 21 4432r— _kLTF h 008^63(0086^1000863) t~2 74" 2 cos^ 32* m 2 V _ x exp 2 4g~ Jg ^ y d T (5.12a) (ii) exponentially correlated surface: 4 2 cos 63(cQ seL+mcose 3 n D D D{p}= 12 12 21 2 " cos e le/m 2 1 Rough side in front: (£) 2 2 nl K R _, T 3 2 4 (l+m D|^l+m 2 2 D D 2 1 3 5 2,, ) : 256* m |l-n| D{p} in Equations > d V gT _ 2 4 g +vV (5 . 13a) . Under the same condition, D{p} should also be modified to the form in Equation (5.58) D{o}= 4 k L ) ! (5.12a), . TL 2 2 74" . 2 eX p|'-g T /4o 2 (l-n) L 2 ]. (5.58a) J od (5.13a), and (5.58a) are derived under the condition that the layer is very thick. However, in most cases, d is not so much larger than the illuminated length, and the Equations (5.12), (5.13), and estimate on the value of D{p}. (5.58) give us a good 6. 6.1 EXPERIMENTAL WORK Experimental set-up and procedure The experimental measurement of the variance of the scatter- ing coefficient has been done in the Underwater Acoustic Laboratory, Electrical Engineering Department, Kansas State University. Details of the equipment can be found in the report by Toliver (Toliver 1965) shown in Fig. The block diagram of the experimental set-up is . 7. In the experiment, pulsed signals generated by the pulsed oscillator are sent out and collected by one pair of transducers in the water tank. For each single pulse sent out, the received signal will contain a train of pulses. A gating circuit, which is synchronized by the delayed trigger output from the pulsed oscillator, is used to select the portion of the pulse train for feeding into the detector and boxcar circuit. The boxcar cir- cuit has the function of converting the discrete pulse into an analog signal so that it can be recorded by the graphic level recorder. The distance S from the transducers to the front side of the target is determined by S where 6. 0°, 5°, and 10°, Dcose, (6.1) and D is the distance travelled by the radiated signal before it hits the target. It is desired to have D as large as possible so that the illuminated area will be much larger than the correlation distance of the random interface. The choice of 32 inches is made to give an average -51- 2^ 2 < hhL g £ ^ s < C^ ^ VJ O £> Z> \ YMA X N -53- value of L = 0.75 inches for different pairs of transducers. The pulse recurrence frequency, PRF, is also determined by D. It is selected to be the largest value without causing any overlapp- ing of the first few returns with the following pulse sent out from the transmitter. And the pulse length, which should be as larger as possible to simulate a monochromatic wave, is limited by the thickness of the layer. The pulse length must be less than the time that is necessary for the wave to make a round trip in the layer. A safety choice of PRF and pulse length for the model target constructed and the distance D specified is as follows: pulse length = 20 *isec, PRF = 250 pps. The operating frequencies are chosen to be 0.72, 1.0, 1.28, 1.6, 1.9, 2.25, 3.0, and 3.5 mc. Before the measurements of D{p} start, the folliwng procedures are conducted at (a) 8, - 0°, S = D = 32 inches, f - 1 mc. The target suspension is carefully checked by the returned pulse position on CRO to make sure that D is equal to 32 inches for all possible positions of transducers. (b) Transducers focusing is done by adjusting the mounting of transducers for maximum return from a smooth plane target 32 inches apart. After checking on the mounting of target and transducers, measurements proceed as follows: (c) Setting the operating frequency with the help of test oscillator and CRO. -54- (d) The pulse length and PRF is set at pulse length 20psec, PRF = 250 pps. (e) The RF output level and the transducer compensator is adjusted to yield the best possible undistorted pulse. Procedures (c) , (d) and , (e) are conducted iteratively to fit all the figures required. (f) Measurement |e 2Q : | the water-air interface is used as target, because it acts as perfect reflector; and D = 32 inches, 6., = 0° are carefully checked. Then the magnitude recorded by the recorder gives M 1 = K 1 AB|E 2() (6.2) | where A is the magnitude of the output pulse, B is the gain of the transducers, and K 1 is the overall gain of receiver and recorder. (g) Varying 6, to the desired angle (0 , 5 , 10 ) and setting S according to the relation shown in Equation (6.1), then I EJ , which is a function of the horizontal position of the transducers, is measured by scanning the transducers through the target. The gate is ad- justed in the way that only the return from the back side of the layer is detected and recorded. The sample magnitude of a point on the recording sheet gives M 2 = K 2 AB |E 2 | , (6.3) where K 2 is the overall gain in this measurement. The ratio of K 2 to K. can be read from the settings. Since <p>=0, so that the variance of the scattering coef- ficient is E . -55- .(P>-<|P| 2 E., 2 2 M 2 ' ,K, >.<J^> «<».(£) M E 20 l or .2 n/ P D{ . = } < M 2> —Y~ k . * 10n Ml = (K~/K, ,, where K (Id I 1 ) db) WlT ( / in db is the ratio of the overall receiver gain between the measurement of \E-\ and | 2Q \ Attention must . be paid not to overdrive any stage in the steps (f ) and (g) K~ is adjusted to give the maximum possible recording without causing saturation in any one of the amplifiers and in the recording devices. Description of layer target 6.2 The target is a block of plexiglas with a one-dimensional roughness on one side. (Fig. 8) The length of the plexiglas . is limited by the dimension of the water tank; to 6 the width is equal inches, which is much greater then the illuminated area of any pairs of transducers; and the thickness, which should be as thick as possible, is limtied to able at the time of construction. 2 inches by the material avail- The profile of the rough sur- face on the layer is desired to have a Gaussian probability distribution in height with standard deviation o = 0.05 inches, and to have a gaussian autocorrelation with respect to the horizontal position, where the correlation distance is T = 0.15 inches. The profile is calculated by trial and error with the help of an IBM 1620 computer. It is noted that the heights be- tween -3a and +3o were considered in the calculation, and the sample heights obtained from the truncated normal distribution are so arranged that there is no sudden or periodical variations. -SI- Ml D O <: 12 O zz \& IZ o < o X si e>- N oO -57- The profile obtained was then cut by a shaping machine. A sample measurement at intervals of 0.05 inches was then made on the finished target and the distribution and correlation functions were calculated. The measured standard deviation and correlation distance are 0.0465 inches and 0.15 inches, respectively. Moreover, the measured correlation function lies between Gaussian and exponential, and consists of certain periodical variation when the separation factor is greater than the correlation distance T. Moreover, the distribution of slope is found to be different from normal and to have a pair of extra peaks at £'=tan(10 ). This might be caused by the shape of the cutter of the shaping machine. Measurement and results 6.3 The quantity M, of Equation (6.1), owing to the drift in the electronic circuit and the disturbance in water tank, is usually a function of time. After the disturbance in the water has died away, the fluctuation of M, with time is nearly zero. 2 The quantity M 2 is a varying positive quantity, the mean of M 2 is < M 2 >" K gj"2i (6 ' and M 2i denotes the ith sample of M 2 « - 5) The variance of pp* is as follows 2 2 D{pp*} = <p p* > - <pp*> < M 2*> . 10 ( K db/5)-<^10 Kdb/ 5 (6 ' 6) Ml where <"a>"iJl"Si- <6 - 7) -58- The samples {M„.} are taken at equal intervals from the recording Since part of target is not seen at bigger sheet. of samples taken is equal to 96, 90, and for e x = the number , 5 , and D{p} calculated from the experimental data is 10° respectively. shown in Fig. 84, 6^ As a comparison, the value D{p} for backscat- 9. tering from a rough surface is also obtained and shown in Fig. 10. 6.4 Discussion of results To compare the experimental results with the theoretical solution of the backscattering from layer, D{p} is calculated for 6, = 0°, 5°, and 10°, from Equations (5.12) and (5.13). The parameters of water, air and plexiglas are (A). Water: p =1.0 c = 15 x 10 cm/sec, gm/cm*, 4- z (B) . w www = p c 4 = 15 x 10 gm/cm -sec . Air: p a =1.29*10 gm/cm, 4. = 3.4 x 10 cm/sec, c z (C) . 4 a 2 = 0.0041 x 10 gm/cm -sec. 3 Plexiglas: =1.2 p gm/cm, = 27,8 x 10 cm/sec c P 4 2. = 33.4 x 10 gm/cm -sec. z P The reflection and transmission coefficients at the plane boundary of two isotropic medium as given by Equation (5.4) are calculated for the following cases 59- -40,1b- _ -60Jb - -ao<ib-~ -lOOdb— ^^ FIG.*?. VAfclAUCE OP 6CATT2KIWG CO£PFICl£UT OP LAY££ WITH E0U6H 5lp£ IkJ TH£ BACK -60- p« -40db *sV^ -feOJb -30<ib H00«ib - 0.3 PI6.IO. 0.6 a<? 1.2 VAKIAUCE OP 6CATTEKIWG COBFF1CIZU7 OP TH£ ZOUCsH SUZ.FAC& OP 7H& LAY£K *"^*/k A -61- D D R 12 21 z p +z E— w x -£ = 1.15, p ^p 2z = y z p » +z p w x -E. p z -z " 22 z +z* a p 0.744 w * _1 * Other parameters are given as follows n, ~ = 12 CW = — cp ca n 0.54 . The effective illuminated area represented by L is a rather complicated quantity, it is assumed here to be equal to the illuminated area by one pair of beam-limited transducers; mean value is taken for five pairs of different transducers. L = 0.75 inch. The theoretical solutions Equation (5.12), and (5.13) are calculated and shown in Fig. 11(a), 1Kb), 11(c). The experi- mental results are also shown in these figures. The relations among frequency, wavelength, and ratio a/ related to plexiglas are calculated and tabulated below: Frequency (mc) 0.72 1.00 1.28 1.60 1.90 2.25 3.00 3.5 A (mm) 3.86 2.78 2.17 1.74 1.465 1.235 0.927 0.794 Ratio a/A(tf»1.18 mm) 0.306 0.425 0.544 0.679 0.806 0.955 1.27 1.49 From Fig. 11a, lib, lie, it is seen that the measurements of D{p} deviate considerably from the theoretical solutions from a layer having either gaussian or exponential correlated rough inter- -62- m -<+0db e,*o ( - caj-e"^ -60 d b -804 b \00&h - -J 0.3 PI6.II. o-fe COMPASieiOU OF P{f} £XP£KIM£MTAL o.q FKOM "TH&OZ&riCAL AUP jz^euurd 1 3 . > -63- p»i 6,-5- —fOdb - -&o<ib-- -80<*b- -100 J b 4- 03 PiG.IICb), SAME A5 O.S F16.li<a), 0.*=} BUT WITH 1.2 6, = 5° '• 5 * -64- ©, = -80db - -lOOdb - ^0 PIG. 0.3 1 1 CO,- 5AME A5 0.6 FIG.IICa), a<? 10° if 3UT WITH 9,~I0 -65- face. This can be explained by the existence of an attenuation factor a in the layer. If a is a non-decreasing function of frequency, the Equations (5.12) and (5.13) should be modified to the form of Equations The modified results (5.16), and (5.17). are such that D{p} is reduced at low frequencies and decreases faster as frequency increases. For such a choice of a, the experimental results are then expected to lie between the modified theoretical solutions. Since the target constructed is such 2 that the correlation of the thickness lies between e - e I ' T "/T I (5.17) , the experimental work proves that Equations ~r '/T 2 (5.16) and and are a good prediction of the backscattering of an acoustic wave from a layer with the rough side in the back. -66- 7. CONCLUSIONS The backscattering from a layer depends on the thickness, acoustic impedance, and the statistical parameters of the rough side. It also depends on the angle of incidence and operating frequency. The thickness of the layer enables the separation of the second pulse from the first one. In case of an absorp- tive layer, the wave propagated through the layer is attenuated. The attenuation may strengthen the frequency dependence of D{p}, if the attenuation factor a is a function of frequency. D{p} obtained from a layer with rough surface in the back has substantially the same form as that froa a rough surface, on the condition that the layer is constructed with non-absorptive material. If the random side is very rough, the value of D{p} decreases very fast as the ratio o/X increases where o is the standard deviation of the rough interface; and for small angle of incidence, D{p} increases as o/X increases. If the layer is turned over with rough side facing the incident wave, the evaluation of D{p} involves an integral which contains a four-dimensional characteristic function associated with w . Approximation is made by assuming evaluating the integral. e 3 very small in The theoretical solution for this model has a very strong dependence on the thickness d, even if the layer is non-absorptive. -67- APPENDIX: EVALUATION OF INTEGRALS The integration of the improper integrals 2 e -x /2a O 2 e ibx dX/ *5 -x /2a cosbxdx, and rx 2 /2a 2 e i(b-ic)x dx Choose the contour as shown involves contour integration. *-X 2 e -z'/2a' 2 dz=lim e ^ R+00 -x /2a a b 2 dx+lin 2 e , -(R+iy) /2a R+OO -R -R R-»-oo R^oo j 2 R = a b . or, <~ (fee 2 -2 2/o /2a 2 [ dz =\ e" s o, /0 x /2a 2 dx - -6c+ia e b)/2a ' - dx 2 dy — -68- or 2 (x+ia b)/2a 2 e (^ e j ' 2 -x /2a »*/'*» 2 2 2 + e -(x-ia b)/2a dx > -^/2a 2 e -ibx +e ibx dj£ = e" \ x /2a f^ 2 dx . o 2 w ^ = cosbxdx /IT ^-a ^| v j ae 2 2^22 b /2j' (1) Since 2 e -x A ?2a ''" .e ibx "dx dx = e -x /2a 2 cosbxdx +\ e "X /2a sinbxdx 2 f°° = 2] e" x /2a cosbxdx+O , /o So that rx 2 /2a 2 >e ibx 2 dx = /2ira2 . e 2 -a b /2 (2) # And also 2 00 2 r x /2a 2 >e i(b-ic)x dx \ e -(^2+cx) 2 —— * c*) -(,x-a 2 e 2a >e 2 .. .e .e = /27aT. e ia2bc ibx dx , ib(x-a ia2bc .exp 2 . c) d(x-a 2 2 a_c .e—52 2 c) [-^(b 2 -c 2 )] (3) -69- REFERENCES 1. Beckmann, Petr and Spizzichino, Andre The scattering of EMW from rough surfaces. New York, 1963. MacMillan Co., 2. Brekhovskikh, Leonid M. and Lieberraan, David Waves in layered media. Academic Press, New York, 1960. 3. Carlin, Benson Ultrasonics. McGraw-Hill Company, New York and London, 1960. 4. 5. Jardetzky, W. S. and Press, Frank Ewing, W. M. Elastic waves in layered media. McGraw-Hill Company, New York, Toronto and London, 1957. , Kerr, Donald E. Propagation of short radio waves. Publisher, 1964. 6. 7. Kinsler, Lawrence E. and Frey, Austin R. Foundamentals of acoustics. John Wiley Co., 1950. Lamb, Horace Dynamic theory of sound. 1925. Edward Arnold Co., London, Alexander and Graybill, Franklin A. Introduction to the theory of statistics. 2nd Edition McGraw-Hill Co., New York, San Francisco, Toronto and London, 1963. 8. Mood, M. 9. Moon, Parry and Spencer, D. E. Field theory for engineers. 10. 11. Boston Technology , Redwood, Martin Mechanical wave guide. D. VanNostrand, 1961. MacMillan Co., New York, 1960. Ridenour, Radar system engineering. Boston Technology Publisher, 1964. 12. Stewart, G. W. (et.al) Acoustics. D. Van Nostrand Company, 1930. 13. Toliver, Willard J., Kaufman, Dale E. and Durrani, S. H. The KSU acoustic simulator for radar studies. Technical report EE-TR-1. Department of Electrical Engineering, November, 1965. BACKSCATTERING OF ULTRASONIC WAVES FROM A ROUGH LAYER By WU-SHI SHUNG B. S., NATIONAL TAIWAN UNIVERSITY, 1961 AN ABSTRACT OF A MASTER'S REPORT submitted in partial fulfillment of the requirements for the degree MASTER OF SCIENCE Department of Electrical Engineering KANSAS STATE UNIVERSITY Manhattan, Kansas 1966 ABSTRACT The backscatter from a rough surface is usually calculated by using Kirchhoff 's In this report, the Kirch- approximation. hoff's approximation is extended to the backscatter'ing of an acoustic wave from a rough layer. the layer is assumed to have The random rough interface of one-dimensional Gaussian distri- buted surface heights. Gaussian and exponential autocorrelation functions are used to represent the correlation of the heights at two different points. Expressions for the variance of the scattering coefficient are derived in the case that the rough side is very rough. Experimental investigations were conducted at ultrasonic frequencies on a target designed to have Gaussian distribution and correlation. The measured variance of the scattering coef- ficient p, D{p}, has the following properties: (A) D{p} is highly frequency dependent; it decreases as the frequency increases. (B) D{p} decreases more rapidly than that obtained from a rough surface. (C) The dependence of D{p} on the incidence angle such that it increases as 9, increases, if 6, 8, is is small. The experimental results lie between the theoretical results calculated for two kinds of correlation. This agrees with the measured value of the correlation function of the target.

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