ENGINEERING RESEARCH INST ITUTE UNIVERSITY OF MICHIGAN, ANN ARBOR A FOURIER TRANSFORM METHOD FOR THE TREATTMENT OF THE PROBLEM OF THE REFLECTION OF RADIATION FROM IRREGULAR SURFACES William C. M1eecham Office of Naval Research, U.S. Navy Department Contract N60nr-23221, ONR Project NR 385-203 November 1955

A FOURIER TRANSFORM METHOD FOR THE TREATIMENT OF THE PROBLEM OF THE REFLECTION OF RADIATION FROM IRREGULAR SURFACES William C. Meecham ABSTRACT A method is presented which can be used for the calculation of the distribution of energy reflected from irregular surfaces. The formulation is useful for the first boundary value problem and can be used in either two-or three-dimensional problems with any given incident field. The solution is reduced to quadrature with negligible error when the average square of the slope of the reflecting surface is small and when the wave length of the incident radiation is not small compared with the displacement of the surface from its average value, A numerical example is worked, the sinusoidal surface, and is compared with experiment and with 'al method due to Rayleigh. It is found that the Fourier transform method is preferable to previous methods, notably those which can be classified as physical optics (such as Ray'leighfs), since the error in the transform method is of second order in the surface slope whereas the error in previous methods is of first order in the same quantity,

A FOURIER TRANSFORM DIETHOD FOPR THE TREATMENT OF THE PROBLEMI OF THE 'REFLECTION OF RADIATION FROM IRREGULAR SURFACES William C. Meecham.Physics Department University of Michigan, Ann Arbor, Michigan Io INTRODUCTION In the past there has been a considerable amount of work done on the problem of the reflection of radiation from non-plane, or irregular, surfaces. The attention given this problem has increased in recent years partly as a result of the expanded interest in centimeter wave length electromagnetic and acoustic radiation, The approximations which have been used in the past have centered for the most part around perturbation treatments1'2,3 and around the Kirchhoff approximation4 and its adaptations5,6 which may be classed under the broad title of physical optics. I, Lord Rayleigh, Roy. Soco Proc. 79A, 399 (1907), 2, J.W.. Miles, J. Acoust, Soc. Am. 26, 191 (1954) o 30 S.O. Rice, Comm. on Pure and Appl, Math. L, 351 (1951). 4L BoB. Baker and E.T. Copson, The Mathematical Theory of Huygens' Principle (2nd ed,; London:,Oxford University Press, 1T50). 5, LM. Brekhovskikh, Zh. exsper. teor. Fiz. (USSR) 23, 275 (1952). Translated by G.N. Goss, U.S.N. Electronics Lab., San Diego,Cal, 6, C. Eckart, J. Acoust, Soc. Am. 25, 566 (1953),

It is the purpose of this paper to present a new method for the treatment of reflection problems. The method is obtained as follows. From the Helmholtz formula (see below) one can obtain, through the use of the boundary condition applicable to the problem, an integral equation of the first kind whose solution can be used to calculate the field reflected from a given surfaceIt will be shown that under certain conditions (when the square of the slope of the reflecting surface is small and when the wave length of the incident radiation is not small compared with the displacement of the surface from its average value) one can approximate the kernel of the integral equation in such a way that the modified equation can be solved through the use of the Fourier integral transform~ It will be seen that the method is primarily suited to the treatment of the type of boundary value problem in which the field function is assumed to vanish at the reflecting surface (often called the first boundary value problemo. Peculiarly enough it does not seem that there is a parallel formulation for the second boundary value problem, where the normal derivative of the field function is assumed to vanish at the reflecting surface, This is true because of the special form which the kernel of the integral equation must assume in order that the central approximation of the method be useful, The method can be applied to either two- or three-dimensional (scalar) problems.; it will be outlined in detail only for the two-dimensional problem, it being an easy matter to extend the formulation to the analogous three-dimensional problemso The problem to be considered here will now be described in detail0 One is given a half-space of homogeneous material

bounded by the two-dimensional surface; (x); the function 3 (x) is assumed to be continuous, single-valued, bounded, and to possess a piece-wise continuous and bounded first derivative0 It is supposed that radiation is incident upon the surface *(x) from the half-space z < ~(x) (bee Fig 1)) ~i~~. P p Figo lo Diagram used in the description of the Helmholtz formula, This incident radiation may consist either of a plane wave (called the first incident field) or it may be set up by a line source assumed to be located at (coozo) (called the second incident field)0 One must distinguish between these two cases in the general formulation, although the distinction disappears in the special method of solution to be presented below0 If the incident f4unction consists of a plane wave, it is supposed that the propagation vector lies in the x-z plane0 One wishes then to find a function ~(xzst) which satisfies the

wave equation in two dimensions9 e+ M7e (1) where c is the phase velocity, assumed to be constant, In acoustical problems the function f represents the velocity potential, where V(x,zt) is the velocity at (x,z,t); the surface is assumed to be pressure release, For electromagnetic problems, it is supposed that the surface & (x) is perfectly conducting, and that the incident radiation is polarized with the electric vector perpendicular to the x-z plane; then the function is taken to represent the electric field, which9 under the given assumptions, lies entirely in the direction perpendicular to the x-z plane, It will be supposed that the source radiates a single (angular) frequency 9O, so that one can write (x,z,t) = eki-t(xz) 0 (3) Substituting Eq. (3) in Eqo (1) one obtains, _+ 2+ = 0 (4) with k M- the function is to satisfy Eqo (4) throughout the half-space if the incident function consists of a plane wave, and at all points of the half-space except the point (xozo) if the incident field is set up by a line source0 In the latter

case the field near the source is to behave like (x~z) ~ H(i)(kro) t~ 3 (5) where =o ~(x-xo)2 + (Z.Z)2J 1/2 and H1) is the zeroth order Hankel function of the first kind, For three-dimensional problems the Hankel function is replaced bye o o To continue, it is supposed that at the surface (x) ro the function ~ vanishes, 0(x, g(x)) = o; (6) as stated above this is equivalent to supposing that the surface is pressure release or perfectly conducting, for acoustics and electromagnetics respectively~ Finally let 0i(P) represent the value which the field would assume at the point (P) if the sure face 5 (x) were not present, that is if the source system were located in an infinite homogeneous medium with phase velocity c, Then one can write, ~01 - ~! + Or (7) where Or represents the reflected field (by definition)0 The function 4i is, of course, given while the function or represents the unknown. IIo DERIVATION OF THE METHOD It will be shown in this section that when i) (d< )2 l l and

-6.. ii) k % M l l it is possible to reduce the problem to quadrature with negligible error, The symbol indicates 'is of the order of magnitude'' M represents the bound on Z (x), and d< represents the bound on 9d (x) The first of the restrictions i) and ii) is the more important. To proceed with the derivation, the Weber two-dimensional analogue of the Helmholtz formula is needed. By using Green's formula in connection with Eqo (i4) one is able to derive the following result (see Fig 1i)-7 0(p) = i(P) + (2) Hv lH (kr2P) H(1) (kr2p2) 0 ds2 (8) subject to the restrictions that ao The function O(xz) is continuous with continuous first and second order partial derivatives for all points (xz) satisfying the condition z _ 4 (x), with the exception of the source point for problems involving the second type of incident field. In this case O(xz) H(o (kro) as ro -- O0 b. The function $ shall represent outgoing waves at great distances from the surface 5 (x), 7o Baker and Copson, loco cit0 chap~ IIo

The integral in Eqo (8) is to be carried over the entire surface; (x). The symbol, -- Indicates the derivative with respect a 2 to the outward normal direction at the surface point (2)o We have let 0(2) represent the value of the function $ at the point (2), and $(P.) represent the value of the function at a space point P; similar notation will be used for other functions appearing later, Now allowing (P) to approach the surface point (1) and utilizing the boundary condition given by Eqo (6), Eq, (8') becomes, 1 H (kr12) ( ds whe re, rl2 [(xlx2 )2 + (; (a) = (x2))2] 1/2 (10) If Eqo (9) could be solved for the function d its value could be substituted in Eq, (8) to obtain the solution to the problem. One is tempted to let 12 jX1x2j (11) since in such a case Eqo (9) could be solved, Let us examine the error incurred by making this assumption, First let r12 = 1-.... (12) where a is the acute angle between r12 and the x direction (see Fig. 1)o Then define the difference between the exact and the approximate kernel for Eqo (9) as

K(xl, x2) Hl)(o k ~ 1x 21) H (o)(k/xl-x2l) (13) cosa 0 In considering the error made in adopting the assumption represented by Eqo (11), it is convenient to treat the following regions separately: I, k xlX21 << 1; II, k IXl-X21 >) 1; III, all other values of the argument, klxl-x2I o Also suppose in this connection that a << 1 indicates a _ 0ol, and similarly a )) 1 indicates a ~ lOOo Because of the (integrable) singularity of H(l)(y) at y = 0, it is to be expected that I is the most important region of the integrand in Eqo (9). Hence begin by considering ito The Hankel function is defined by, H(1)(y) - Jo(Y) + [r'+An(y) Jo(y) Tl 7r~~~ 7y ~(14) 2i (-l)m(r)m ) 1 02 T 2(ml) 2 where Y = 0o5772o Upon referring to Eqso (14) and (12) it is seen that in I JKJ '" 212n 1 IK o s-. (15) or from the definition of a, assuming d small one has, /KI t d I*I (16) For region II use the asymptotic form for the Hankel function,8 HO(o) i 7Y [1 + O(; )J (17) 80 E. Jahnke and Fo Emde, Tables of Functions (4th ed,; New York: Dover Publications, 1943T-' ---

in Eq. (13) to find K I' x )X2 1/2 (Cosa) exp iklxl-x21( Co _ 1 I Kxe.. x2I1 ~ (o) CI -5 1) t112 (klxo ex21)2 klxl-x o omitting o(l) as a term of higher order, It is easily seen iklx lx:lx2) so that Eq. (18) becomes, K~ e.....x - + 2ei (k-2 (21 Kr ikgIn1orin Iit (kJx1Mx2/)2 x2I ( d tkIxl-x2- H klx lxl 1 1 Finally region ii.-is'considerede Note first' that for a small, K L)(klxx-x2 Ix2k x.x2I o, a (2.) (1) or from the known relation:;HH (y) -H (y), K t p Het ) (knlx -x21 )klx-:21 Os - a ) (22) From the pertinent tables in Jahnke and Emde,9 it is seen that 9. Jahnke and Emde, lbco cit., pp. 157 and 191.

the largest value of i H( (kIxlx2l)k lxl-x21 for 0ol - klxlx21 C 10 occurs at klxl-x21 = 10 and, | 41(10o) ' 1oJ = 2,53 e (23) Again from the definition of a and using Eq. (23) (replacing 2.53 by 3,0), Eq. (22) becomes |K|Z~ U |} (24) To summarize, it is seen that K is negligible in region I if hypothesis i) is fulfilled (cf, Eqo (16)); the same *hypothesis makes K small in region III (cf, Eqo (24)), Finally from Eq. (21) we see that K is negligible in region II if hypothesis ii) is fulfilled. It is noted however that in region II the kernel of Eqo (9) is small so that an error in this region is not so important as errors in the other two, Thus in some problems the approximation may lead to a useful result even though the hypothesis ii) is not fulfilled, Now proceed with a method of solution based upon the assumption that K is small, Rewrite Eqo (9), using Eq. (13), $(l)l) = d) +(25) HO)(kxx2 + K(xx2) ds2. (5 Before continuing it is desireable to make some changes in notation, Let 11) 1 aos(2) (26) F(xl) -4i 0i(l), (27)

and note dx2 ds2 Cos (28) Here Z (x) is the acute angle made by the tangent to the surface at x2 with the x-axiso Then substituting in Eqo (25) we have 00 F(x1) = [H(1)(klx 1X2I) + K(xlx2) s 2xdx 20 (29) -co As the next step consider K(xl9x2) as a perturbation~ Let K(xlx2) = K(xl9x2) 9 (30) (x) = ()(x) + 6 2)(l)(X) + 2(V2)(x) +..(31) where it is assumed that P (x) is an analytic function of 6 for 6 _ lo The solution is obtained by allowing & — lo By substituting Eqso (30) and (31) in Eq. (29), and equating equal powers of & it is evident that: 00 CO S H(l)(klJxl j) @ ()(~.x2dx2 = F(xl) (32) 5 Hol)(k)Xl x21) P (x2)dx2 = g K(xx2). _-oo - (33) & H(l)(k 'Mx2l) (2)(x2) dx2=. ~~K(xl x2) (l)(x2)dx2, There is a well known method due to Levi-Civital1 which may be used for the solution of integral equations of the above 10 T. Levi-Civita, Ro Accademia della Scienze di Torino Atti 31, 25 (1895), See also E.Co Titchmarsh, Introduction to the Theory of Fourier Integrals (Oxford: The..Clareno Press, T 7T,.cap- XI,

=12 -type in which the kernel is a function of the difference of its two variables. The method begins by taking the Fourier transform of the integral equation. Accordingly take the Fourier transform of Eq. (32) and Eqo (33)o We have, (21T)l/2h(t) qy (0)(t) = f(t),(34) (27r1l/2h(t) (l)(t) =- FoTo K(xlx2) (O)(x2 )dx (35) (21)/2h(t) xJ (2)(t) = -FoT I K(Xl, X2), (l)(x2)dx (36)?(t) =1 eitx F(x)dx (37) 2 oO FoTo i F(x) Similarly, h(t) =FeTo, (1) Fx) T W ~)(t) ro= F0T0 ) 1(~(X) } S (38) 1)(t) = FoTo ~ (l)(x) } In order to guarantee that the transforms of Eq. (32) and Eq. (33) will exist and yield Eqo (34) and Eqso (35) and (36), it is sufficient to require that ( (0) O ) (1), and Hol)(k|x|) be absolutely integrable and that F(xl) and SK(XlX2) )(n)(x2)dx2 -00

C=13 have Fourier transforms, If these conditions are not satisfied by P (n) the solution is subject to verification. For the function H(l)(klxl) let k = k' + i6, 6. 0, in order to guarantee absolute integrability, then allow 6 - 0 in the solutiono Now if we add the restriction that P (n)(x) be sectionc ally continuous we can, after solving Eqso (34), (35), and (36) for V (n)(t), take the inverse Fourier transform, It is known that,11 h(t) = (k2t2)/2 (39) We have then from Eqs, (34), (35), and (36): (p)(O) _ 1 2 (kt2)1/2e ixtf(t)dt (40) 2(0 ) (kx) 2 t1/2 1(1/)(X) gy (k1t2a1/22eixtdt 2(2Tr) o { FTo [K(xTVx2) ()(x2)dx2, (41) 00 (2)(x) 1 2 (k2)t2) l/2eixtdt 4 o~ (42) {FT0 L o~oK(XlX2) i (1)(x2)dx2 ( The solution,, (x), is obtained from 4 (X) ~ t(O)(x) + tQ(l)(X) + 00 (43) 11o GoAo Campbell and RoMo Foster, Fourier Integrals for Practical Applications. Bell Telephone System Techo Pub,, 1931; Monograph BD584, No, 918,

14a The symbols in Eqs. (40), (41), and (42) are defined by: ( (x) C=os(x) ' (26) f(t) 1 e itF(x)dx (37) F(x) = 4i Bi(x), (27) K(x1ac2) H()(kr12) H (klx x2 (13) One obtains the solution, A(P), from /(P) =g(P) ~ He ('krlp) ds1; (44) this equation is obtained by applying LEq (6) to Eqo (8), It is seen from Eqs. (41) and (42) that the corrections to the first approximation, 9 (0), are small if K is small enough~ IIIo SPECIALIZATION TO PROE:LEMS IITV0OLVING PLANE 'WAVES INCIDENT (ON PERTIODIC SURFACES In this section attention will be restricted to the class of probleras involving plane waves incident upon periodic surfaces. For such problems the result given by Eqs. (40), (26) and (44) may be simplified somewhato Cor-isider only the zerothl order result, as given in iqo (40); the, development will be formal. Let it be suppoGsed that a plane wa've is incident upon a periodic surface from tihe negative z direction (see Fit., 2);

the propagation vector for the plane wave makes an angle Gi with the z axis. Then, eik(x sin 9i + z cos 9i) P~e~ e(45)ve Figo 20 The figure shows a plane wave incident upon a periodic surface, Upon substituting Eq. (45) in Eq. (37), one finds, f(t) = -/7 5exp {i[tx#+ksinQix'+kcosG x' )]7m dx', (46) or f(t) = j [2 exp i [txksinGix'+kcosei (xt) (2) mao0 C + (t+ksingi)mJ L'C, - (47) where A is the repeat distance of the periodic surface. Now if one makes use of the identity (which is easily established)

-=16 co 00 21 e-imlSA = K 2 &(s-mK ) ((48) nt- -oO co i —to where 6(x) is the Dirac delta function and K = 2 it is found, upon interchanging the summation and integration, that Eqo (47) can be written f(t) = 4iT(t)K 1 6(t mK + k sin Gi) (49) (27r) l (ooK when a-( t) = fexp {i tx + k sin xi xV + k cos @i x (x )J} dx f (50) Now substituting Eqo (49) in Eqo (40) and carrying out the indicated integration, one obtains oO (J(0)(x) () = 2i i e (kK iok c os 9n Ftn(cOS ti) (51) when k cos n = 2 (nK - sin 9i). 1/2 (52) and Ln(cos Gi) = eiixl+ikcosG ( (x5) These latter quantities are, aside from a constant, the complex ik cos 9t t (x) Fourier coefficients of ei By using Sommerfeld's contour integral representation for the Hankel function one may construct the plane wave representation for that function, (1l() i ( -x- k2-s 2 )1/ 2(_Z) Ho (krlp = Y 2 1 ds,(54) (k -s

a17 -valid when Zp-zl 0; for zpiz > O0 one changes the sign of the radical in the exponential of the integrand, Upon substituting the representation given by Eq. (54) for the Hankel function in the integrand of Eqo (44) and remembering the definition of Or (see Eq. (7)), one obtains 0()(P) =. 5 )(0)(x) _-o (55) expis(xp-xl)-i(k2s2)/2(zp- (x1))j d...... 1 2 2) 1/2 (IS dx -~~~ ~ (k )s / where ( ) is defined in Eq. (51). It is supposed that zp <- 5M so that Eq. (54) is valid for all surface points x1. Now noting from Eqo (51) that, _~(0)+) ~= 9(0Q)(x!)ei'Ak sin (56) and breaking the integration into parts as was done in Eqo (47), one finds ~(?) = i ~ ps(0)(xl)eim1/k sin Qi dx1 4rSds (e p)1 (k2 s2) 1/2 the superscript on Sr has been dropped for convenience, Utilizing the identity given by Eq, (48), assuming the validity of the required interchanges of summations and integrations, and substituting the value of g (o) given by Eqo (51), Eqo (57) can be written, 0'rl = r nr(P) = -im ) (xp,zp) Am (58) n3 >|_o8

18~ when cos 0 Am = _~ cos G, Jn(COS B Vi)m n(cos ) (9) n - - mn oo and (~m.) = ~exp [-i(m~-k sin Oi)xp-ik cos GmZp7o (60) The quantities PZ( m) represent plane waves moving in the negative z direction; the waves are homogeneous when (min k sin Gi)2 ~ k2 (61) and inhomogeneous for other values of mi The quantities Am are then recognized as the reflection coefficients for the plane waves (mth order diffracted waves), The quantities cos 'n are defined in Eq. (52); in in Eq. (53). Before proceeding, it is of interest to show that for a given m the series solution for the reflection coefficients given by Eqo (59) always converges, under the restrictions on 4 (x) which have already been given~ We begin, as usual, by observing from Eqo (59) that J Am4 n f I'Ilm n( osGin l)ln co n '- -oo (62) Now from Eqo (52) it is easily seen that cos GnI. Kjnj (63) for n#O0 K' is some positive constant independent of n. Also upon integrating by parts one sees from Eqo (53) that Il n(cos m)If q mn (64)

-193 where the piecewise continuity and the boundedness of d have been utilized (of course the boundedness follows from the assumption of piecewise continuity if attention is restricted to periodic surfaces as above); K"9 is some positive constant dependent on mo Finally since s(x) is continuous with a piecewise continuous first derivative and since E1n (cos Gi) are essentially the Fourier coefficients of eik cos Gi (x) it follows that,l2 I /nt(cos i KFt (65) nt - OC Upon using Eqso (63), (64)9 and (65) it is seen that JAj < e m + cm (66) | cos Gm i which is the desired relation, The positive constant cm arises from the terms n=O and n=mo IV, REFLECTION OF A PLANE WAVE FROM A SINUSOIDAL SURFACE In order to illustrate the foregoing theory, the reflection of radiation fromn a sinusoidal surface will be consideredo This problem was first treated systematically by Rayleigho13 For a normally incident plane wave he reduced the problem to the solution of an infinite system of linear equations, Under the restriction that k <41 he was able to invert this system, 12, See RoV, Churchill, Fourier Series and Boundary Value Problems (lst ed;New York:McGrawISill —Book -Co — 19.1), ppT —85... 13, Lord Rayleigh, Theo of Sound (2nd ed; New York: Dover Publications, 194)5) volo II, p-0i —o

-20 (Actually there is another9 implicit assumption in Rayleigh's development, the effects of which are difficult to evaluate.14 The implicit assumption concerns the question of the representation of the reflected field near the reflecting surface in terms of plane waves; one of the restrictions necessary for the validity of Rayleigh's treatment is the additional requirement that the slope of the reflecting surface be small.) The reflection coefficients obtained by Rayleigh are given by (for the first boundary value problem):15 A(R) = -(i)mjm(2ka cos G.i) (67) The constant a is defined by the assumption that the reflecting surface is (x) a cos Kx (68) The reflected. plane wave, whose coefficient is given by Eqo (67), has a propagation vector whose direction cosine in the z direction is given by Eqo (52) with n replaced by mo Now let the method presented in this paper be considered, Substituting Eqo (68) in Eqo (53), and using a known integral representation for the Bessel function of order n, one finds, 'n(cos si) = (i)nJn(ka cos Gi). (69) Then using Eqo (69) in connection with Eq. (59), one finds for the reflection coefficients 14o BEA. Lippmann, Jo Opt~ Soc. Am,, X, 408 ( 1953) 15l Actually Rayleigh treated only the case of normal incidence, Gi=O, He states the result given by Eq. (62) for m=O; it is not difficult to extend his method to cover other orders, mo o

oO Am J (kacos)J EJn(kacosim)* (70) r-ov o m The series is rapidly convergent so long as ka is not too large. Now, as a check, let the assumption <l1 be adopted, Then neglecting terms of order K (see Eq. (52)), Eq. (70) becomes, A(O) = -im Jn(ka cosGi)Jm n(ka cosgi), (71) 3.-~n or upon using a known addition formula for Bessel functions A(0) =-(i)mJ (2ka cos@ ), (72) m m i which is in agreement with Rayleigh's result given by Eq. (67), V. COMPARISON WITH EXPERIMENT AND WITH THE RAYLEIGH THEORY Experiments have recently been performed by LaCasce and Tamarkin1 which can be used as a check. on the above theory, In these experiments a directional beam of ultrasonic energy originating in water was allowed to impinge upon a sinusoidallyshaped cork surface which floated on the surface of the water, The amplitude of the reflected radiation as a function of angle was then recorded. In Figs. 3, 4, and 5 appear some of the data obtained by LaCasce and Tamarkin, these data being compared with calculations 16, EO. LaCasce, Jr. and PO Tamarkin, Underwater Sound Scattering from a Corrugated Surface, to be published,

m~ce: o.,:....e.ais of the i n t nl. iA o i.L.: 1. daa o v; e.rr.;ted - ' 0 i.:I of. L ' 's.rce i.r t'l.e.S. f'i thcr& i rn ux. idence, o... f 1c3 It.)-a mi-,, 0rll t'Ie cox, r socsncb ril c s.c Luat1i.OnsL is, c on: le Iafylei4;l.a 1fiE ' en 19U. —i aLs )P tv ad c nv Cent:a to 0ctx. - c t - t:.n t e,.,-.~. n': e:'".h tee: n aie. 1 o-..a n z {> -- Q ~initz s c ttt J io o r i '. cc t t o n s t:. tel ", - '" " e~-' i'-a I' t fi s o o;,.s.rol.l T.. te a'l. t Cl. j o' i;:li. i i 1 z:....,,t..,..,:ilveio.)ra-n~ _a C) I lUr~tl-tuIoE.[: fo; i tio;, t b.I ons —:l o. t '" (of i:.-'v one finds the footin5 x lt on, 0053 ecos ela~ ~, -:i' | (7.L4) wlhe;r e tthe sumr rtuns oveOr real valtl.es of cos 'm.. e.or a refloectirig suLrface or fi.nite size, or equivalently for a surfcce radiated by (f) a f'inite bea]m, it is -ossiblo- to obta-in he a''cit:-I'or-, of' the incident ener.y in the mth order for a finite reflectin. surthe face, as follows. Sup-ose that reflecting region of surface is of Wildth L; then the mth order will have a diffraction pattern of angular width O11 G h (75) Suppose that am is the maximum am'litude of the mt. order diffraction pattern, and that ao. is the maximui amplitude of the diffraction pattern (in the specular direction) when the reflectling surf ace os relacsed by a plane Then it follows by

2iy0G = 1+E OA BM GUO17. lCUOPF OUF * TB1LU -ou-[ U J0 UOT 8OXG-[ xo OT)r,4 c 9eputl G;ns 9 r P oT p s'n cU s 'L.:ojJ p 30o 1J -eO s'a:p,;co 'p.o3-oe,,jJT, [;UCoos pu '.sa 'Ti oa:oz etq. u- poaosqo PU-e Pq-3,L,,no rY SOtt4es.OG ~u,4O M EOqslJJ O7;L! C PT '7 TL 0' ~t 5'1 \h.0 MO _~~~ ~ \\ I C) ~'o \/ Q1to / 4Q 'h'OI o O >I b I iC~' i o o Y'~~ "%. ~.~~~~~~~;

inr-....:iy<.. 't; l].e onev;:? conoia:.n.,. --;-. the ml''~ order oye: a ].a:..;. 'e c-l.?ndter, whose a.xs is Iarai l o t..' t i?:':;: C eleen.t or the reflec-tin -:; surface, that, —! t C n S'I -:' l os e i a m (f ) m o s l'i1 ai o Usii —.: apr-;":ai-n t:i.-e cornservw t~o.a of' eneTr:' r;equi.ne:l:L..ent oe sees., CCCO [ O1 = 1O t,:le sum '-:,:.i.n(:'; carried ovcr ' n ie f hor Thi.ch'n co.s g. ' s rPe.l, Ret -- " toa, ti; he relaiseiuni "rw. con,?.id. era.t O t1...on of.....a 'i:a, ti-on (7't),.s usOd to de-l,::".: ie i,. -.e re lative energy.in each order (liaC.-.sce and. '?ila:li{k:in re''o. h. a nt' ttie: e ). It is found -that their resitS i. 'c; 'brleKTIeen 1,0 anrd1 2,.-) for- i1- e left sid'ie of i'Eq. (77); accord'n.:.-, tr'he - irsu.lt;s wer"e d- v:id.ded ib: yr the su ori1 thi e left Jta. -,.si..'.. (7 )o ). he ac. jus d q; anti tes oro t'hle!-iotted exee i:ena..l. -a 3. Jes> i"n Fi':igs4 3, anld 5. lt i s seen f'rom l tl1he fiu:tres tha. t.t' a-ri-,iit between ex'oeriment' x and t'he Fioourier tr;ansforl theor-y is good for values of ka less than 1.50 I —t is to )be empnhasized that for values of ka albove 1.5, order s hif:i hlePr tJhan the second may anpear. Since t'ilese ord(cers wtere not re:-)or ted in It:le experi ments (t- lher are difficult to -lieasure because of the larEge ansles I 'i:ch1 t:hey -a nake with th'le normzal) it is to be expected that t-he true normalized e-peri"mntal avalues are somewhat lower than. those shown in the f'i..:u.res. Furthermore it is seen in the fin- ures thlat the ''ayleigh% method...ives reslv!lts whic-ll are in. err.'oi" by, a.s m. uch as 20~, (referred to 'th.le enery 'in.f L a given ordoer)o inc finls a energmtyJ deficit of

tw;enty-f iie e:1)orcent o. upoD m:n-_ ui.n up the onergy- carried aiwa-y by all real ordelCs as calculatecd usin thle Ciayleigfh noethod, The correspondinfg, deficit for the Fourier transform method is ten percento Finally it is remarlked that the results of the Rayleiegh theory are in many respects almost identicral with those obtained from the methods of }re-khovsbikh and Lckart6 (see LaCasce and Tamarkin6). For the reasons gpiven aboove, then, thle present i method is to be nreferred over these other methods as well. VIO COiQCIrUSlIFO:S AlT ACONO ( LL "PCTiI1S The advantage of the present method over " previous:et-lods applicable to the same class of pro'blems lies in the fact that the error incurred througlh its uLse is of second order in the slop'e of the reflecting surface (see assumption (i) of Section II). The error incurred throu:h the use of lhysi.'eal o-;tics is on the other harl(l o0f first order in thle surface slopel7 (The error in hayeigb' s rnmethod is of the igaie order as that in physical optics, ) It is a pleasure to acknowledge many htelp,:ful discussions with Dr, David?/Nintzer durinri' thle progress of this work. The author wishes also to acknowledge the holi-n of 1lr,Cl Chu rch in the performance of some of the calculations, 17* 'efo -lTeecham, On the Utjse of the Kirclioff A-%proximation for the ~olution of Refloction Probjlems, to be rublished in the JQurial o'f iational Mechanics,

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