ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN, ANN ARBOR ON THE USE OF THE KIRCHHOFF APPROXIMATION FOR THE SOLUTION OF REFLECTION PROBLEMS William C. Meecham Office of Naval Research, U.S. Navy Department Contract N6onr-23221, ONR Project NR 385-203 November 1955

ON THE USE OF THE KIRCHHOFF APPROXIMATION FOR THE SOLUTION OF REFLECTION PROBLEMS William C. Meecham Physics Department University of Michigan, Ann Arbor, Michigan I. INTRODUCTION In the past there has been a considerable amount of work done on the problem of the reflection of radiation from non-plane (frequently termed 'irregular) surfaces. Of the various approximations used in approaching this problem, perhaps the most popular is the one due to Kirchhoff [i]. (For work using this approximation see references [2], [3], and [6J.) It is assumed in this approximation that the field near every region of the surface is essentially what it would have been if the surface had been flat with a slope equal to that of the irregular surface at the point in question. The approximation has been assumed to be useful when the surface variations are in some sense large compared with the wave length of the incident radiation. Using this assumption concerning the value of the field at the bounding surface in conjunction with the Helmholtz formula (see below) it is possible to obtain an estimate of the field in regions removed from the reflecting surface. Despite the prevalence of the Kirchhoff approximation in the treatment of the problem of the reflection of radiation from an irregular surface there does not exist, to the author' s knowledge, a systematic derivation of the approximation which demonstrates the way in which it may be corrected, nor for that matter, which shows conclusively the size of the errors incurred through its use. For this reason it is hoped that the derivation in the present paper may prove of some interest.

-2 -For simplicity consideration will be restricted to two dimensional problems in which the field function is assumed to vanish at the irregular surface (in acoustical work, a pressure release surface). It is a simple matter to extend the arguments to cover the case where the normal derivative of the field vanishes at the surface as well as to extend them to cover analogous three dimensional (scalar) problems. One wishes to find then a function ( (x,z,t) which satisfies the wave equation in two dimensions, 2 + - 2 2_ (1.1) x2 - z2 c2 t2 at all points except the source point (xO,z0) of the field; c is the phase velocity,and is assumed to be constant. In acoustics, the function ~ represents the velocity potential, v -= - V (1.2) where v(x,z,t) is the velocity at (x,z,t). It will be supposed that the source radiates a single (angular) frequency ), so that one can write ~(x,z,t)= e-it(xz). (1.3) Substituting Eq. (1.3) in Eq. (1.1) one obtains, a2+ 2 + k2j (x,z) =, (1.4) L ~ with k = A; the function 0 is to satisfy Eq. (1.4) at all points except at (x0,zO). Near this point the field is to behave like 0 (x,zZ) - (kr)() where r = [(x - x0)2 + (z - z0)2 1/2

-3 -ikr For three dimensional problems the Hankel function is replaced by r To continue, it is supposed that at the surface 5(x), the function 0 vanishes,,$(x, 4(x)) =. (1.6) The function W (x) is assumed to be continuous, single-valued, and bounded with continuous first and second derivatives; the function is defined for all values of x. Finally let 0i be the value which the field would assume if 5 were not present; in our case,i = H(l)(kr). The Kirchhoff approximation consists of supposing that ' 2A on the boundary, where is the outward-directed derivative normal to C(x). It is supposed in this work that k is complex, that is that the medium is slightly absorptive. Comment will be made upon this assumption at the end of Section 2 (following Eq. (2.35)). We now proceed with the derivation of the approximation. II. DERIVATION OF TKE KIRCHHOFF APPROXIMATION In the treatment to be presented here we shall attempt to develop the Kirchhoff approximation in a systematic way. The Helmholtz formula is considered in connection with the determination of the normal derivative of the field function on the boundary; an integral equation is obtained. The solution by iteration of this integral equation will then be considered. It will appear that using the first term is equivalent to adopting the Kirchhoff approximation. We then investigate the next term of the iteration solution, in particular observing that it has the order property, )II C M' 1-xI + M" 1klRm (2.1) Here I is the first term neglected in the solution; Rm is the minimum radius of curvature and d;M is the maximum slope of 5(x); M' and M" are dx

constants independent of surface properties. On the basis of this property, it seems reasonable to suppose that the Kirchhoff approximation is valid if -- i) -dM << and ii) lkJRm >> 1 We proceed with the outlined program. First the Weber two-dimensional analogue of the Helmholtz formula is needed. By using Green's formula in connection with Eq. (1.4), one is able to derive the following result (see Fig. 1) O(P) = Oi(P) + E 5 7 (l) H )(krlP) - H( )(krip) 0a ~(l)) ds1, (2.2) subject to the restrictions that -- a. The function 0(x,z) possesses a singularity of the type (xz) — H1 (krp0), as rp — 0; (XO,Zo) is the source point and rp = [(xx0)2 + (z_z0)2] 1/2 b. The function O(x,z) is continuous with continuous first and second order partial derivatives for all points (x,z) satisfying the condition z _ 5(x), with the exception of the source point. c. The function 0(x,z) shall be of order e-at as r -- O, where r = (x2 + z2)1/2. The constant oC is the imaginary part of k defined by k = k + i, C > 0. (2.3) The integral in Eq. (2.2) is to be carried over the entire surface (x). The symbol e indicates the derivative with respect to the outward normal direction at the surface point (1). We have let 0(P) stand for the value of the function ~ at some space point P and similarly for other functions appearing in Eq. (2.2) and later. We now use the boundary condition given by Eq. (1.6); Eq. (2.2) becomes

0(P) = i(P) H( )(krlp) ds. (2.4) If we allow P to approach the surface at the point (2), we see =16(1) 0i(2) - 4i H (kr12) (1) ds (2.5) 0 12/ Fig. 1. Diagram used in the discussion of the Helmholtz formula. Once the function B.1 is known, the solution to the problem can be obtained from Eq. (2.4). In order to obtain an estimate for this function we set up an integral equation of the second kind and propose a solution by iteration. We take the derivative in some direction, of Eq. (2.4); the point (P) is then allowed to approach the surface along the normal at the point (2); at the same time the derivative approaches. One obtains 1. The integral equation to be developed here (Eq. (2.16)) has been obtained previously by Maue [81 for both boundary conditions for the three-dimensional problem. It is repeated here in order to show the detailed development for the two-dimensional case. For the use of integral equations in the treatment of diffraction problems one can also refer to references [4] and [7].

-6 -= f(2 ) 1 Lim a 0(1) ds (2.6) Y? 7i (P) —(2) - (krEp) -? ds1. (2.6) 2 26 That it is legitimate to differentiate under the integral sign for rp2 ' _ > 0 may be seen in the following way. From the asymptotic formula for the Hankel function one has, H(1)(y).(129) / i(y) y >. (2.7) Using this representation together with the radiation condition, hypothesis c) above, it is evident that the integrand of Eq. (2.6) is less in absolute value than Me2(Xl-X2l for some M so long as rP2 O. It follows that the integral is uniformly convergent; furthermore O H( lP is continuous for rp2 a g, as is ~ by hypothesis for z 2; (x). Hence the interchange of differentiation and integration is Justified. H(0) The quantity a in the integrand represents a distribution of dipoles. It is a well known result of potential theory that limiting processes such as the one in Eq. (2.6) introduce integrable singularities in the integrand. In order to evaluate the limit, the integral of Eq. (2.6) is split up as follows: 1 Lim H ol)(krlp) 30(1) ds 4i (P)-(2) Rep 1 7 1 Lim C Ho (krlp) 0(l) dsl 1 Lim (krlp) d (28) 4i (P)-+(2) 2 de a a~p u1 The limit in the first term on the right side of Eq. (2.8) can be moved inside the integral sign. Then we are left with the task of evaluating the second

-7 -term on the right side of Eq. (2.8). For convenience we write 1 m H(l) (krlp) a(l ds 4i (P)-(2) 1 = 1 Lim HO (krlp) 3 (2) 1 Lim C Ho1)(krlp) /a(1l) - ~2) dsl. (2.9) 4i (P)-+(2) "a )a The Hankel function is defined by, o (Y) = JO(Y) + a [ + n ) ] Jo(Y) 2i j (- ){+++ (2.10) 2 1 2 m or H(y) ( 2i in(y) Jo(y) + G(y) (2.11) where G(y) is analytic at y = 0, and 1 is Euler's constant, 0.5772... Using the assumption that; (x) has a continuous derivative, together with the expression for the Hankel function for small values of its argument, one obtains the following result: Lim {_1 Lim, bHo (krlP) 6 0(2)d 1 B ~(2) seD 4i (P)-(2) 28 2 a' (2~12) rt/ P 2 2 To see this, 5 o is chosen small enough so that the region of surface (r12 _; ) is essentially flat. Then using Eq. (2.11), Eq. (2.12) follows directly. Also since by hypothesis -- is continuous, we see that

-8 -Lim C 1 Lim K aHW (krlp) ( 'l) - _ (2)\ TI 2 4 i (P) -(2) 0 )V - 4rzP( aa<t jy 1 2 (2.13) Now allowing 0- 0 in Eq. (2.8) and substituting in Eq. (2.6) we find, (2 H _?ld,(g) r (1) __,_ - Xi7(2 _ 1 1 0( 32Ho (kr10 12) 0(1 dsi (2.14) ge2 8) 2 -,2 4Y2 1 Equivalently Eq. (2.14) may be written, =( 2 i(2) 1 H eHl)(krl2) O ds1 aa an, (2.15) 2 2 2 1 Now we consider the solution of Eq. (2.15) by iteration. The procedure is as follows [ 6]: one substitutes the right side of Eq. (2.15) for the quantity a appearing in the integrand. This again gives the quantity X appearing on the right side, now under a double integral. By repeating the substitution process indefinitely, one generates an infinite series of terms in which each term involves, in general, a multiple integral with the known function appearing in the integrand. This process then yields the series, i(2) i ( H(l)(krl2) i() =2 - - 2 dsl 2i3 2 2 2 1.1 f H.l r H<1) (kr12) 1 i aHl (k(1) [ a Ii3) d7 d 2i 2D1 7 Go 0y1 2) 3 1 L2 j ' 2 ds. ds1I, i; 5 (2.16) Under the assumptions on;(x) and using Eq. (2.10) it is not difficult to show that o(2 is continuous in x and x2 with the exception of a removable singularity at xl = x2. Then from Eqs. (2.3) and (2.7) it is seen

-9 -that each integral in each term of the series given in Eq. (2.16) is absolutely and uniformly convergent and moreover that each term is continuous. Finally it will be seen below that for I | sufficiently small and for jklRM sufficiently large, the series which forms the iteration solution is absolutely and uniformly convergent. Then one sees that Eq. (2.16) is a solution to Eq. (2.15). The series can be integrated term by term since each term is continuous and the series is uniformly convergent. It also follows that under the given conditions the series representS a continuous function of x2. Furthermore, it is not difficult to show that the iteration solution when it converges is the only continuous solution of Eq. (2.15). We now retain only the first term of Eq. (2.16) assuming for the moment that the remainder of the series is negligible in comparison to it; this gives, -a (2 0 (2) ' b 2 (2.17) 2?2 As a check on the result, it is not difficult to see that for 4(x) = 0 (a flat surface), Eq. (2.17) is exact. To complete the solution, Eq. (2.17) is substituted in Eq. (2.4) to obtain 0( =) 0(P) - H)(krlp) 2 i ds (2.18) when 0(l)(p) represents the first approximation to the field 0. We shall now consider the first term neglected in the approximation represented by Eq. (2.17). We make use of the relation d (1) (1) dy H0 (y) = H (y), (2.19) and also 9 sin t1 (2.20) '1

-10 -with ' 1 defined as the angle between the tangent to the surface at the point (1) and the radius vector connecting (1) and (2) (see Fig. 1). Then we have for the second term of Eq. (2.16) a i H(1)(P12) F aei( ) 5 H(1),ka 0ii(l)..1...2i 2 f ds1 = S Hi (kr12)k sinZ2.) dsl L~(0)kr2 I i()(2.21) We shall label this quantity I; it must be small if Eq. (2.17) is to be valid. We need the expression for the function H(1) corresponding to Eq. (2.10): (1) i 2 HE (Y) = J1(y) y + 2 ln J1(y) (-)ly) 1 1 1 1 1 1 (2.22)... +-' + +... - + -+- + -.+ -- + m! (m+l)' 1 2 m 1 2 l+m From Eqo (2.22) it follows that for kr12 1, 1H1l)(y)lt H (2.23) Using the asymptotic expansion for the function H1) 2l1/2 i(yH(1)( ) -2 e 1 + o(i j1 (2.24) one finds that for krl2 2 1, H1 (krl2 1 Me*2 (2.25) Finally from the definition of 0i; it follows that, i C |k1Mi. (2.26) In these expressions mH, MH, and Mi are all of order unity.

-11 -With these definitions in mind, we proceed with the consideration of the function I. We examine separately the regions of the integrand of Eq. (2.21) where (klr12 C 1 and where (klrl2 > 1. We have I = Il + I2, (2.27) where I1 = i H1 (krl2)sin 2 2 - kdsl (2.28) and I2 = (kr12)sin 2 kdsl (2.29) I - Ir '', ' kds1 Let I. be considered first. To begin we note that, M..... _ (2.30) X2 - X1 l dx (230) since the quantity on the left is Just the average slope of the segment of surface between the points (1) and (2). By examining Fig. 1, and using the result given by the relation (2.30), it is easily seen that ~sin Z 2 |: 2 {dx | (2 31) Upon using Eqs. (2.25), (2.31), and (2.26) one then finds without difficulty that I-2 ' (k Mi 4Mf l +(d 2 j d | (2.32) We now proceed by considering Eq. (2.28). The following relation is easily established (cf. Fig. 1): i r12 \sin jg - | = 1 12(2.33) when Rm is the minimum radius of curvature. One then finds upon using Eqs. (2.23), (2.33), and (2.26) that, I1 (kt MimH [1 + d(2

-12 -Hence from Eq. (2.27) it follows that, M\2Th1/~1 kI ~mH d +.djkldxM i m I (2:~M C k!M 1 + ix.35) l+:d1 v( dx ko R It is noted here that as one considers media of smaller attenuation (o), one is also forced to restrict consideration to surfaces with smaller slope in order to guarantee a given bound on the error using the present development. In order to remove this difficulty, one must deal with conditionally convergent integrals and in particular consider the question of whether or not they are uniformly convergent. To continue, one sees upon examining Eq. (2.16) that, | 2 k. 1 + 2 + (2.36) 2 where 5 (l + 1 d )l/2k1 4MH + (237) the series in Eq. (2.36) converges for s < 1 d M This proves the assertion made above that for sufficiently small and for!klRm sufficiently large, the series in Eq. (2.16) converges absolutely and uniformly. It is worth commenting that for lkLRm << 1 the terms neglected in the approximation represented by Eq. (2.17) are of the same order as the terms retained for regions of the surface where the radius of curvature is small. III. THE RECIPROCITY THEOREM We now show that the approximate result, expressed by Eq. (2.18), satisfies the reciprocity theorem if there is but one source, i.e., i(P) = H(1)(krp0). (3.1)

-13 -A general transformation of Eq. (2.18) is first shown. We consider a system having sources giving a field 0i without the bounding surface C (x), so that the sourceSradiate in an infinite homogeneous medium. Then applying Eq. (2 2) (1 11 (( ~0i) =a~ _( + 4i~~(i(1) - o Ho (krlp) d1( 7- H oHl)(krip) ds 1 (1) d s1. r5 ~1 1r (3.3) Now Eq. (3.3) is used in Eq. (2.18) to obtain, (P) = Oi)P) - (H ) (kr 0(1) 1 (34) It is evident that if we specialize the problem by applying Eq. (3.1), Eq. (3.4) is unchanged by the interchange of (0) and (P), for we have, - H PO1(kr ) 4 l ) (HO (krlp)H(1) (kr10)) dsl (535) Thus when a single source excites the field, Eq. (2.18) satisfies the reciprocity theorem. We now examine the value of the approximate result, given by Eq. (2.18), on the boundary. From the boundary condition, Eq. (1,6), the function 0(l)(2) should vanish. Subtracting Eq. (2.18) from Eq. (2.4) we see, O(p) - 0(1)(p) -= _ j H 0 - 2 O dsl. (3.6) When (P) lies on the boundary at the point (2), Eq. (3.6) becomes, -(l)(2) = - H'( l -2 ] ds (.7)

-14 -Hence if the error in the determination of the value of a on the boundary is made small by hypotheses i) and ii), one may expect the value of the approximation given by Eq. (2.18) to be small there, and indeed to be of the same order. IV. CONCLUS IONS AND ACKNOWLEDGEMENT It is shown that the Kirchhoff approximation is really the first term in the solution by iteration of an integral equation governing the value of the normal derivative of the field function at the surface. The neglect of dSM the remaining terms of the solution is seen to be justified for dx sufficiently small and for IkIRm sufficiently large. Correction terms for the approximation are given. Finally it is shown that the approximation to the solution obtained from the Kirchhoff assumption satisfies the reciprocity theorem. There is one further point of interest in connection with the present approximation. It is easily seen from Eqs. (2.21) and (2.31) that the first term neglected in the approximation contains a term of first order in the surface slope. It seems reasonable to suppose that the effect of this error would be greatest in regions where the reflected radiation is far removed from the specular direction. Brekhovskikh, cited in reference [2], finds such deviations in comparing calculations using the present approximation with an exact formulation due to Rayleigh. It is a pleasure to acknowledge the helpful criticism of Dr. David Mintzer and of Dr. C.L. Dolph on several points concerning this work.

1. Baker, B.B., and Copson, E.T., The Mathematical Theory of Huygens' Principle, (2nd ed.; London: Oxford University Press, 1950), Chap. II. 2. Brekhovskikh, L.M., Zh. Exsper, Teor. Fiz. (USSR) 23, 275, and 289 (1952) translated by G.N. Goss, U.S. Navy Electronics Laboratory, San Diego, California. 3. Eckart, C., J. Acoust. Soc. Am. 25, 566 (1953). 4. Franz, W., Proc. Phys. Soc. Lond. A, 63, 925 (1950). 5. Lovitt, W.V., Linear Integral Equations (New York: McGraw-Hill Book Company, 1924), Chap. II. 6. Rowland, H.A., Phil. Mag. 35, 397 (1893). 7. Dolph, C.L., Memorandum, University of Michigan, EMF-26, Willow Run Research Center (1951). 8. Maue, A-W., Z. Phys. 126, 601 (1949).

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