ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN, ANN ARBOR A VARIATIONAL -METHOD FOR THE CALCULATION OF THE DISTRIBUTION OF ENERGY REFLECTED FROM A PERIODIC SURFACE William C. Meecham Office of Naval Research, U.S. Navy Department Contract N6onr-23221, ONR- Project NR 385-203 November 1955

A VARIATIONAL METHOD FOR THE CALCULATION OF THE DISTRIBUTION OF ENERGY REFLECTED FROM A PERIODIC SURFACE W.C. Meecham ABSTRACT A variational method is presented which is used to calculate the energy appearing in the various diffracted orders set up when a plane wave is incident upon a periodic reflecting surface. Either the first or the second boundary condition can be so treated. A sample problem is worked showing that if the average absolute slope of the reflecting surface is small (segments of surface with large slope may be included) and if the displacement of the surface is not large compared with the wave length of the incident radiation, the formulation gives results correct to within a few per cent. The calculation shows clearly the existence of Wood anomalies; these are discussed in some detail.

A VARIATI ONAL METHOD FOR THE CALCULATION OF THE DISTRIBUTION OF ENERGY REFLECTED FROM A PERIODIC SURFACE *t W.C. Meecham I. INTRODUCT ION The problem of the reflection of radiation from non-plane surfaces has in the past received the attention of many people employing various approximations in its treatment.(l-5) It is the purpose of this paper to present a variational method for the treatment of such problems, in particular those in which the surface involved has a displacement which is of the order of magnitude of the radiation wave length and in which the surface has portions of moderately large slope, though the average absolute slope should be small. This class of problems is one which is not amenable to treatment by the other methods at present available. * This work was supported in part by the Office of Naval Research. t The method described here was presented in a paper before the thirty-eighth Annual Meeting of the Optical Society of America. 1. B.B. Baker and EoT. Copson, The Mathematical Theory of Huygens' Principle (2nd ed.; London: Oxford University Press, 1950), chap. II. 2. Lord Rayleigh, Roy. Soc. Proc. 79A, 399 (1907). 3. V. Twersky, J. Acoust. Soco Am. 22, 539 (1950). 4. C. Eckart, J. Acousto Soc. Am. 25, 566 (1953). 5. L.M. Brekhovskikh, Zh. exsper. teor. Fiz. (USSR) 23, 275 (1952). Translated by G.N. Goss, U.S. Navy Electronics Laboratory, San Diego, California.

-2 -The method may be described as follows. Following Trefftz, (6) a linear combination of known solutions to the wave equation is chosen to represent the reflected field. The coefficients will be chosen here so that they minimize the square of the error in the boundary condition. (Trefftz chose them so as to minimize the Rayleigh quotient.) This process of minimization is equivalent to orthogonalizing the set of functions formed by evaluating the trial functions on the boundary. Once this set is orthogonalized one can easily construct the estimates of the reflection coefficients for the surface involved. The class of problems to be considered will now be described. It is desired to find a solution 0 of the two-dimensional, time-independent wave equation, 2[' + - + k2 O(x,z) = (1) in a half-space bounded by a periodic cylindrical surface W (x) (see Fig. 1). In Eq. (1) k = W/c when u) is the angular frequency of the radiation source and c is the phase velocity in the homogeneous medium bounded by 5 (x). The solution of the time-dependent wave equation is then given by Oe-i t Fig. 1. Sketch showing the definition of the symbols used in the solution of the reflection problem. 6. E. Trefftz, Math. Ann. 100, 503 (1928).

-3 -Using the method described herein, one may treat either the first or the second boundary value problem. Thus one may require either O(X 5 (x)) = 0 (2) or, a(x,z) o0. (3) I Z= (x) Here represents the derivative normal to the surface. It is supposed that the incident radiation consists of a plane wave making an angle 0i with the +z direction; then one can write the total field as the sum of two components, 0= i + 0r (4) where, ik [x sin Qi + z cos Gi] =i e (5) The boundary conditions given by Eqs. (2) and (3) are frequently encounted in the treatment of problems involving acoustic and electromagnetic radiation. For acoustic problems, the function 0 may be taken to represent the (time-independent) velocity potential, with 0 defined by, v = -V (6) where v is the particle velocity at an arbitrary field point (x,z). Then the first boundary value problem, represented by Eq. (2), corresponds to a physical problem in which; (x) is a pressure release surface. Furthermore from Eq. (6) it is evident that the second boundary value problem corresponds to the physical problem in which. (x) is a rigid surface. For problems 7. One could use the same method to treat problems where the boundary condition is of the form EAOA + B n nz ) = O where A and B may be functions of x or in fact the more general problem where one is given two different media separated by a periodic surface and is asked to find the reflected and the transmitted fields.

-4 -involving electromagnetic radiation on the other hand, (x) is assumed to be a perfectly conducting surface. Then for an incident plane wave which has its propagation vector lying in the x-z plane and which is polarized so that the electric vector is perpendicular to the x-z plane, one chooses the boundary condition given by Eq. (2) where it is supposed that the electric field, which has but a single cartesian component, is given by the function 0. Finally for incident radiation polarized so that the electric field lies in the x-z plane one lets ~ represent the (single cartesian component) magnetic field and chooses the boundary condition given by Eq. (3).

II. REPRESENTATION OF TE PREFLECTED FIEfLD In order to make progress toward a solution of the above class of problems, Rayleigh(8) and others(9'10) have chosen to represent the reflected field by an infinite set of plane wave solutions of the wave equation. In addition to homogeneous, one must choose inhomogeneous waves. The waves must be chosen in such a way that they are respectively either outgoing or exponentially damped as z - -o Furthermore the fact that the boundary is periodic implies that one needs only a discrete set of such waves. Thus one is lead to expect that the reflected field or can be represented by the following type of sum, I -ik sin Q, x - ik cos G, z where sin ~O = Yk - sin Gi cos A, = [1 - sine g 1/2 (8) where K = 2t/AA (see Fig. 1), and where the coefficients, A,, are to be determined through the use of the boundary condition. The angles QO of the various reflected orders are just those obtained from the ordinary grating equation. 8. Lord Rayleigh, Theory of Sound (2nd ed.; New York- Dover Publications, 1945), vol. II, p. 89. 9. U. Fano, Phys. Rev. 51, 288 (1937). 10. K. Artmann, Z. Physik 119, 529 (1942).o

-6 -Lippmann(11) has questioned the validity of the representation given by the expression (7) in the region?< z < (x), at the same time confirming its validity in the region z < 5M (see Fig. l)o The details of Lippman's argument are presented in an unpublished report. It will prove convenient to reproduce here the substance of his argument, though in a somewhat altered form. One may begin by representing the reflected field through the use of the two-dimensional Helmholtz formula (12) = - () (krlp) a (l)dsl. (9) The integral is to be carried over the entire surface 4(x); the symbol a represents the derivative with respect to the outward-drawn normal, (see Fig. 2). A, / a g) o (9) Fig. 2. The figure shows the symbols used in the Helmholtz formula. 11. B.A. Lippmann, J. Opt. Soc.o Am. 43, 408 (1955)o 12. Baker and Copson, loc. cit., chaps. I and IIT.

-7 -For the present purpose attention has been restricted to the first boundary value problem. It is not difficult to verify that, ~(xi + (x +A)) eikisinQi a (X, (xl)) (10) Using Eq. (10) one can rewrite Eqo (9). -4i H(l )(kr ) eikn knAsinGi dsl (11) where r~ [<xp n ~ xl)2 2] 1/2 rn = (xp - nL - ) +(zp - zi) 1/2 (12) By using Sommerfeld's contour integral representation for the Hankel function one may construct the plane wave representation for that function, ik(xp-xl) + i(k2- k2)/2(zp-zi) H 1)(krlp) _, O (kr1P) n>(k2 _ - 2)1/2, (1) -00 where the minus sign applies if zp-zl < 0 and the plus sign if zp-zl> 0. Then by substituting Eq. (13) in Eq. (11), by assuming the validity of interchanging the summation and the integration, and by utilizing the result e-n(= K g (k -nK) (14) where 5 (k -nK) is the Dirac delta function, one arrives at the following representation for the reflected field~ -ik sin xp-ik cos zp (15) =r(P) = Ay e, (15) when zp- 0, and where A = K ) 1 dsl ac(1) i sin Q> kxl+ik cos (xl)

-8 -Further r(P) A- (zp) eik sin @ xp - ik cos zp + A+( ik) e sin ~; Xp + ik cos @ Zp (17) when M<zp < (x), where, A+~ (z) _ Ck I 0 \L + dsl e1 i sin kx + ik cos 00 (xl) v (Z <k cos 4 dxL 41 dx ~~Ct~~~~~~~~ ~(18) and with C-(zp) defined in Fig. 3o ~~~\ - _. Fig. 3. Diagram defining the contours C-(zp). The plausibility of the assertions made above concerning the reflected field now becomes evident. From Eq. (15) it is seen that when the point (P) is removed from the surface, the reflected field is composed of plane wave solutions which are either proceeding in the negative z direction

-9 -or die out exponentially in that direction. Furthermore when M;< zpr < (xp) one sees from Eq. (17) that the field may be represented in a form which appears to be a combination of waves moving in both the plus and the minus z directions with coefficients dependent upon zp. Of course in the latter case, individual terms of the series are not solutions of the wave equation. For certain problems it may turn out that the representation given in Eq. (17) is merely an alternate (and more complicated) form for the representation of the type given by Eq. (15). Indeed this is the case for one special problem which can be solved exactly. The problem is one in which the field satisfies the boundary condition given by Eq. (3): it is supposed that the incident wave falls normally upon one of the faces of the representative groove form (see Fig. 4); /W1OW Fig. 4. Figure showing a simple reflection problem which can be solved exactly. (in the figure, n is an integer). The reflected field for this problem obviously consists of a single plane wave moving in a direction opposed to that of the incident wave and with amplitude unity; this solution is valid in the entire region z,Co

-10 -To summarize the work in this section, it is evident that for some problems one can represent the reflected field by a sum of plane waves proceeding in the negative z direction even in the region M< z <;(x). However, although to the author's knowledge an exact solution indicating the necessity of using a more complicated representation of the type given by Eq. (17) is lacking, it seems reasonable to suppose that in general the plane wave representation is not sufficient in the region near the reflecting surface.

III. THE VARIATIONAL METHOD It will be convenient to define, 0 0=~,, +,, (19) =r rP rNP () where 0rFP1 Av e -ik sin Q x - ik cos z (20) 0rP = - ~ Ade (20) valid in the region z ~ 4(x), and ~rNP represents that part of the reflected field which cannot be written in that form. For the purposes of the present paper, attention is restricted to those problems for which l rNPI << Irp rP| *(21) Lippman and Oppenheim(l3) have proposed a sufficient condition for the validity of the relation (21); it is:.. <.1, (22) when A and X are of the same order of magnitude. The results given in Section IV make it evident that this condition is too restrictive for the present work, probably because of the minimal formulation of the problem. It is possible, through a detailed consideration of the images contained in the region 0(x) ( z _ 0, to estimate the function urNP using a method essentially the same as that outlined below. Problems for which 13. BoA Lippmann and A. Oppenheim, Technical Research Group, 56 West 45 Street, New York 36, N.Y, Final Report on Contract No. AF18(600)-954 -11 -

-12 -such a treatment is necessary will not be considered in this paper. In order to simplify notation it will be assumed hereafter that 0rNP = 0. Furthermore only the first boundary value problem (the boundary condition is given in Eq. (2)) will be considered in detail. When that work is completed, it will be shown how to alter the formulation for the second boundary value problem. To proceed, upon using Eqs. (2), (4), (5), (19), and (20) one finds the following relation for the determination of the constants A. ik sin ix+cosSi 5 (x)] -ik[sinQg x+cosQ,? (x23 e 2..A V e 0(2) ik sin Qix and upon applying Eq. (8), and dividing by the common factor e this becomes eik cos Qi(x) A? -i)Kx-ik cos Q~, (x) (24) To render the treatment of Eq. (24) more systematic let Fl(x) = exp (0) F2(x) = exp (1) F (x) = exp (-1) F4(x) = exp (2) (25) and Al = A0 A A 2 =1 3 of where in Eqs. (25), -i9Kx - ik cos @O (x) exp (~2) = e (26)

-13 -further let ik cos 0i (x) i= e Then Eq'. (24) becomes, i(x) - 2 AS FK (x) =0 o (27) KN I If the series in Eq. (27) is broken off after the Nth term, as must be done in many problems, the left side of that equation is not in general equal to zero. It is proposed that in such a case, the constants Ak be chosen in such a way that the integral over the surface 5 of the absolute square of the left side of Eq. (27) is minimized. Since all quantities in that equation are periodic with period A, it is sufficient to carry the integral from x=O to x= A o It is easily seen from Eq. (23) that this minimization is equivalent to carrying out the corresponding minimization of the error in the boundary condition. It is not difficult to show that if one chooses the coefficients Ak so that they satisfy the set of equations (with R = 1,2...,N) Ak(FX,Fk) = (FA, Oi) (28) when the inner product of two functions, (g,h), is defined by 1 A (gsh) = g*hdx, (29) then the above indicated minimization is accomplished. Rather than approach the inversion of the set of equations (28) directly, it has proved convenient for the purposes of computation to use an equivalent, though somewhat indirect, method-. To see this method, let Eq. (.27) be considered again. One observes that the problem is equivalent to finding that linear combination of functions Fk which is equal to the given function Xi This task is complicated by the fact that the ~ > -.. J..- -....,...... 14. Setting ~rNP (of Eq. (19)) equal to zero is equivalent to assuming that this is possible.

-.14 -functions Fk are not in general mutually orthogonal. This suggests that one proceed by constructing an orthonormal set of functions from linear combinations of the given set Fk o There is a well-known method for doing this5) Let the desired orthonormal set be Gk; then one can write G = " 7 rf )FL 9 (30) with (AScW) = ~mn when mn is equal to one or zero depending on whether m=n or not and where the coefficients o(') are determined as follows. Let G1 be equal to F1 devided by its norm; the norm of F1 is defined as (F,F1)/2 One then takes that linear combination of F1 and F2 which is orthogonal to G1 divides it by its norm and sets it equal to G2. Upon proceeding in this way the following recursion relations for the coefficients in Eq. (30) are easily obtained: (k) (k) _ _ ~1rQ = /2 (31) N where <2 < k, F r6X9 Fk 9 (32) kwhere = 1 and where Nk.9= <& iIj t A)(FndiFoS, )h) } (33) when the star indicates the complex conjugate. Now let 7iN) (x) = BkGk(x), (34) 15. For a reference concerning representations in terms of systems of functions see Ro Courant and Do Hilbert, Methods of Mathematical Physics (Engl. edo; New York: Interscience Publishers Inc., 1953), vol I, chap. II, secs. 2 and 35

where 0(N) represents the Nth approximation to Oi and where Bk are defined by Bk = (Gk)i) (35) or upon using Eq. (30), B (k)* (F ) 3(6) Then by using Eq. (30) in connection with Eq. (34) one can write, =(N)(x) = - Fk (37) K= I when N At = aB r(a) (38) k That A = Ak (39) where Ak are defined implicitly in Eq. (28) can be seen by observing that the Bk as defined in Eq. (35) are the Fourier coefficients of the function isi with the set Gk. It follows then that the quantity A -(N)j 2 i-xi(]dx (40) is minimized and therefore that the coefficients Al, as obtained from the k coefficients Bk in Eq. (38), must also minimize the expression (40). But E. (28) governing the quantities Ak was obtained by minimizing the quantity (40). Hence Eq. (39) must followo Indeed one can verify Eqo (39) directly by substituting Eq. (38) in Eqo (28). One of the advantages of proceeding as above toward the solution of Eq. (28). is that one obtains an estimate of the error incurred by breaking off the infinite system of equations at N, this error being combined with the error involved in the assumption that /rNP = 0 o The estimate is obtained in a way similar to that by which one ordinarily obtains Bessel's inequality. i5) One finds that ht % 1,i ( 2i = (hi0>) - ~j1k 2, (41) O bW_

-16 -or by observing the definition of 0i given by Eq. (26), MSE. = 1 I Bk 2 (42) where M S.E. stands for the mean square error in the boundary condition (the expression (40)). The error arises both as a result of considering only a finite number of diffracted waves and -as a result of assuming that OrKP is negligible. There is a relation which follows from the conservation of energy which can also be used as a check on the accuracy of the calculation. By considering the energy balance within the region of the x-z plane bounded by:(x), x=0, x= A. and z=-C, where C is large and positive, one obtains the following relation for the exact solution, cos G cos ~;, I A1 2 (43) where the summation is carried over those values of > for which cos %Q is real and where the notation of Eq. (20) is used again, remembering the changes made by Eqs. (25). A third relation which can be used to check the accuracy of the calculation arises from the reciprocity theorem (16) To obtain this expression one treats first the problem of the reflection of radiation from a periodic surface which is finite in extent in the x direction, using the Helmholtz formula. This problem is then compared with the corresponding problem involving an infinite periodic surface, By allowing the finite surface to extend to greater and greater distances in the x direction, one finds the following relation governing the plane wave reflection coefficients, cos Q,? A-? (9i) = cos GiA> (@g ) (44) where Ay(@i) represents, as before, the reflection coefficient of the Ith 16. Lord Rayleigh, Theory of Sound, loc. cit., vol. 2, seco 294.

-17 -order wave but with the incident direction explicitly indicated. From Eqo (44) it is seen that the zeroth order (specular component) should be symmetrical about Qi=O, regardless of whether or not the surface is symmetrical. It will now be shown how to modify the work of this section in order to make it applicable to the second boundary value problem. It will again be assumed that OrNP is negligible. In order to obtain the required result, consider the boundary condition given by:Eq. (3) in connection with Eqs. (4), (5), and (20). One easily finds (essentially by taking -3 of Eq. (23)) the following relation governing the reflection coefficients. ikcosgi (X) ik(n sin@ +n cos) ) e 00 Z ~ Ay (-ik)(nxsinQ' +nzcosQ,) e-iKx-ikcos ~ (x) 0 (45) where nx = - 5/1 + (:)2 -1/2, nz = [1 + (,)2-1/ (46) and ad, dx (47) The Eq. (45) is to be compared with the corresponding Eq. (24) obtained for the first boundary value problem. The quantities nx and nz are of course the components of a unit vector normal to the surface 5 (x) and pointing out of the physical region. Now it is not difficult to see that the approximate solution of Eq. (45) is given by Eqs. (38), (36), and (31), (32), and (33.) upon replacing (FnFm)by ( m)2; and (Fn,)i) by (any, i)2 where 1 A g*hdx, (48) (g~h)2 = i) g*hdx (49) i = ik(nxsingi+nzcosgi) Xi (49)

-18 -and 1 = -ik (nx sin g0 + nz cos Q0) exp (0) (50) O2 = -ik (nx sin 01 + nz cos @1) exp (1) 3 =.-ik (nx sinul_+ nz cos O-o) exp (-1) It is noted from Eq. (48) that the definition of the inner product has been changed. in order to keep it dimensionless for the present functions. Furthermore for the second boundary value problem the Eq. (42) must be modified. By analogy with Eqo (41) one can write 1 A i an (N), 2 k2A Ko d-x - (~ i, q i)2 - t Bkl KW I1 (31) where the Bk are those appropriate to the present problem, and where the functions and appearing in the integral are to be evaluated at the surface. Finally then Eq. (51) can be written, M.SE. - -S (52) where M.S.E. is again the mean square error in the boundary condition (now with dimensions of reciprocal length squared). To sum up the results of this section, one uses Eq. (38) to obtain estimates of the reflection coefficients of the various diffracted waves. The quantities Bk are defined in EqO (36); the quantities (k) which are also needed are defined in Eqs. (31), (32), and (33). Finally if the second boundary value problem is being considered, Eqs. (36), (31), (32), and (33) are altered as indicated in Eqs. (48), (49), and (50).

IV. 'RESULTS OF CALCULATIONS AND CONCLUSIONS Calculations based upon the method presented above have, to the present date, been carried out only for surfaces of the class shown in Fig. 5, and for the first boundary value problem. Hence, since the field function vanishes at the surface in such a case, the solutions are appropriate for acoustic problems involving free surfaces or for electromagnetic problems involving incident energy polarized with the electric vector parallel to the generating element of the (conducting) surface (perpendicular to the page in Fig. 5). -e\f\~~q0 W\l.ve Fig. 5. Figure showing the type of surface for which computations have been made. In the figure three representative reflected wave directions are shown, although they have not been chosen to fit any particular case. -19 -

-20 -The surface was chosen for calculation for two reasons. First, it is of some physical interest; the sea surface assumes a shape reminiscent of that shown in the figure under conditions of high wind, so that the treatment of the problem may be helpful in attaining an understanding of the distribu — tion of acoustic or electromagnetic energy reflected from such a surface. Furthermore, the surface is of the type known as an echelette grating which is sometimes used in optical and infra-red spectral work. The second reason for choosing the indicated type of surface for the calculations is that the calculations are somewhat simplified; the integrals shown in Eq. (29) can be evaluated in terms of exponentials when the surface is composed of straight line elements. For the sake of completeness the formulas for the inner products involved in the calculation of the distribution of energy reflected froam a suxface of the. type shown in Fig. 5 will be given here. One finds that (FmFFn) =(Fm9Fn)a + (FmFn)b (53) where (Fm Fn)a = 2kci (mk (cosn) -cos * -l t'ex i ~ o ~ ('m- 'kn) -K C~ (cos*Ou2 -cos3 v - (~4) and _a2 r 2 - (FmF n)b 2i-i L - (vm-m n) -K (c ms*Q mcosQ) ar iemx n) -m C' ()cos*v-cos n Also (Fm ~i) = (Fm $i)a + (Fm i)b, (56)

-21 -where - k (Fi) = +LV ___ml - * (cosOm +C~os i)] * ~exp i'qL m - K (cos *@ m+C~SQi) -1I (57) and (Fm'=i)b i - [ m L K'Y (cos*Qk +cosi +cOS *exp i - K C +cosi) -1}, (58) when 2it ' to = 1+ Cr (59). = tan y, (60) and 1 = o, 2 = 1 3 = -19 =2.... The entire problem, starting with the calculation of the inner products (FkFm) and. (Fk),i), through the calculation of the quantities, and including the calculation of the estimates of the reflection coefficients Ak (or A ') has been programmed for MIDAC, the University of Michigan digital computer. The recursive and cyclical form of the central part of the calculation, the central part being the computation of the quantities rQ given by Eqs. (31), (32), and (33), renders the formulation easily adaptable to a digital machine. The capacity of the machine limited the calculation to ten diffracted waves (N=10). For the calculation presented in Fig. 6 a surface of the type shown in Fig. 5 was chosen with ) = 100 and with A= 1.155X. This ratio of A to x implies that for a given incident angle, at most three diffracted orders appear. The plus second diffracted order never exceeds 0o538 and is too small to show on the grapho It is noted from the figure that the energy deficit of the calculation (as computed from the right side of Eq. i(43))

-22 -CIO3, -30' _ 0 _o* k-4 rNCIDENT ANGLE @c Fig. 6a. The solid curve shows 1 - M.S.E. (where M.S.E. is the mean square error in the boundary condition); the dashed curve shows the ratio of the total calculated, reflected energy to the incident energy. 1.0 z OfI o.N. j 0L LI LI LL Qo o 0o 53 /-30 r00- -60 INCIDENT ANGLE GL Fig. 6b. The curves show the fraction of the total incident energy which is contained within a given diffracted order when a plane wave is incident upon the periodic surface with an angle (measured from the normal) as indicated by the abcissae. The surface is described by k) =100 and ]A =1.155 X.

-23 -averages about 2.5% and never exceeds 5%; furthermore the M.S-.E. is less than 0.025, From these two checks, it seems reasonable to expect that the error for a given order is less than 5% (of its value). As suggested in Section III, the reciprocity relation can also be used to check the accuracy of the calculationo It was deduced from Eq. (44) that the reflection coefficient for the zeroth order should be symmetrical about the normal; it follows then from the energy relation given by Eq. (43) that the percentage of the total incident energy in the zeroth order should also be symmetrical. It is seen from Figo 6 that this order is symmetrical within a few per cent, the assigned error. All other reciprocity checks carried out also agree within a few per cent. For instance one should have, cos 60~ A1(0~) = cos 0~ A1(60Q), (61) and cos 9~8' Al(45~) = cos 45~ A1(90~8) o (62) Actually the numbers from the calculation are -0.1485 +0.1449i and -0.1562 +0.1525i for the left and right sides of Eqo (61) respectively; -0.1787 +0.2312i and -0o1735 +0.2242i for the left and right sides of Eq. (62) respectively. From Fig. 6 it is seen that the main part of the reflected energy is carried by the zeroth, or specular order; this component never drops below 80%. Discontinuities of the type shown in the zeroth order at OQ +80 are known as Wood anomalies(17) (after their discoverer) and have been observed many times experimentally. Rayleigh (18) showed that the positions of 17. R.W Wood, Phil. Mag. 4, 396 (1902). 18. Lord Rayleigh, Phil.o Mago 14, 60 (1907).o

-24 -the anomalies were connected with those angles at which diffracted orders appear (or disappear). Both Wood and Rayleigh concluded that the anomalies appeared only for (electromagnetic) radiation incident with electric vector perpendicular to the generating element of the reflecting surface (the second boundary value problem) and that for parallel-polarized radiation no such anomalies occurred, Recent work by Palmer(l9) has shown that the anomalies can occur for parallel-polarized radiation as well; Palmer concluded however that in this case the anomalies would not appear for shallow surfaces (where the angle k is small). The present calculation shows that they are to be expected even for such surfaces, although the effect here is not large, about 5%o It is to be remarked that the problem of calculating the shape of the anomalies has proved intractable by previous methods. It is seen in Fig. 6 that the anomaly shows a sharp edge on the side where a new order first appears, as is often observed experimentallyo(19) Existence of this edge is connected with the fact that the energy contained within an order falls off rapidly as the angle of the order approaches 90~ (as the order disappears). In fact it can be shown through the use of a perturbation treatment such as Rayleigh's(2) (the treatment being useful for near grazing incidence) in conjunction with an application of the reciprocity theorem() that it is to be expected that the slope of the curve is infinite at this point. Indeed it is just this discontinuity, and its attendant effect upon the other orders through the conservation of energy requirement, that gives rise to the Wood anomalies. It is of interest to compare the results of the present calculation with those obtained for the same problem using other methods. One might 19o CoHo Palmer, Jro, J opto Soco Amo 42, 268 (1952)o

first consider Rayleigh's perturbation treatment. However it turns out that the method, at least in first order, is not applicable, since one requirement for its validity is that k M ~< 1 whereas here kM A 1. Secondly one might consider using Kirchhoff's approximation;(1) the results obtained using this approximation are essentially the same as those obtained from Eckart's(4) and Brekhovskikh's(5) formulations. Kirchhoff's method gives results which are considerably in error; for example the complex reflection coefficient using Kirchhoff's approximation is compared with the corresponding results using the present formulation in Table I. The surface chosen is the same as that used in Fig. 6. TABLE I. A comparison of the results of the Kirchhoff formulation with those obtained using the variational method for normal incidence. Theory Zeroth Plus first Minus first Total energy Kirchhoff -0. 248+0. 720i 01900-0.313i -0.1397+0.1546i 67% Variational -0.1875+0.879i 0.297 -0 290i -0. 0916+0.329i 95% Finally the question of the rate of convergence of the calculation is taken up. In Fig. 7 are shown the successive approximations to the values of the reflection coefficients, as each new diffracted wave is introduced in the calculation. It seems to be reasonable to deduce from the results shown in Fig. 7 that including more terms (more diffracted inhomogeneous plane waves) in the calculation is not likely to significantly improve the result. One then concludes that the residual error arises from the incomplete form of the representation of the reflected field (as explained in Section II). Hence further improvement can be expected only through the introduction of images in the region 5< z / O. However it seems that for surfaces whose average

-26 -absolute slope is small (sections of surface of large slope may be included) and whose maximum displacement is not too many wave lengths, the formulation as presented is accurate to a few per cent. Li L 0,4 Ir 0, 2! E+, U < F10 INUMBER OFTERMS IN APPRoOXIMA'rlON Fig. 7. The graph shows the successive approximations to the energy contained in the various orders appearing when a plane wave is normally incident upon a surface with 4 =100 and with A =1.23 x. The total calculated energy is also shown.

V. ACKNOWLEDGEME]NTS The author wiahes to take this opportunity to thank Mr. Gordon Grant for performing some of the early hand calculations. He also wishes to acknowledge the patient help of Mr. J.H. Brown who offered many helpful suggestions and criticisms concerning the programming of the problem for MIDAC. Finally it is a pleasure to thank Professor CoW. Peters for many helpful discussions, in particular with regard to pertinent experimental results. -27 -

1936 November, 1955 DISTRIBUTION LIST Chief of Naval Research Commanding Officer and Director Undersea Warfare Branch (Code 466) U S. Navy Electronics Laboratory Washington 25, D.C. (1) San Diego 52, California (1) Director Superintendent Naval Research Laboratory U.S. Navy Postgraduate School Technical Information Officer Physics Department Washington 25, D.C. (6) Monterey, California (1) Attn: Prof. L.E. Kinsler Commanding Off icer Office of Naval Research Branch Office Commander Box 39, Navy 100, c/o Fleet Post Office Naval Air Development Center New York, New York (2) Johnsville, Pennsylvania (1) Commanding Off ic er Hudson Laboratories Office' of -Taval Research Branch Office Columbia University 10th Floor 145 Palisades Street The John Crerar Library Building Dobbs Ferry, New York (1) 86 East Randolph Street Attn: Dr. I. Tolstoy Chicago 1, Illinois (1) Massachusetts Institute of Technology Chief, Bureau of Ships Acoustics Laboratory Code 847, Navy Department Cambridge 39, Massachusetts (1) Washington 25, D.C. (1) Lamont Geological Observatory Office of Naval Research Columbia University Acoustics Branch (Code 411) Torre Cliffs Washington 25, D.C. (2) Palisades, New York (1) Director NRC Undersea Warfare Committee Woods Hole Oceanographic Institution Navy Department Woods Hole, Massachusetts (1) Washington 25, D.C. (1) Attn~ Dr. C.B. Officer Catholic University of America Director Department of Physics Marine Physical Laboratory Washington 17, D.C. (1) University of California U.S. Navy Electronics Laboratory Harvard University San Diego 52, California (1) Acoustics Laboratory Division of Applied Science Commanding Officer and Director Cambridge 38, Massachusetts (1) U S. Navy Underwater Sound Laboratory Fort Trumbull Office of Technical Services New London, Connecticut (1) Department of Commerce Attn: Dr. H.W. Marsh Washington, D.C. (1) Direct orJ Yale University Naval Ordnance Laboratory Edwards Street Laboratory White Oak' Maryland New Haven, Connecticut (1) At tn: Sound Division

DISTRIBUTION LIST (cont.) Commanding Officer U.S. Naval Academy Office of Naval Research Branch Office Physics Department 1030 E. Green Street Annapolis, Maryland (1) Pasadena 1, California (1) Dr. R. Mittra Commanding Officer Antenna Laboratory Office of Naval Research Branch Office University of Toronto 346 Broadway Toronto 5, Canada (1) New York 13, New York (1) Dr. C.L. Pekeris Armed Services Technical Information Agency Weizmann Institute Document Service Center Rehovst, Isreal. (1) Knott Building Dayton 2, Ohio (5) Physics Department Brown University Director, Ordnance Research Laboratory Providence 12, Rhode Island (1) Post Office Box 30 Pennsy1vania State University State C'olege, Pennsylvania (1) Attnno Mr. R.J. Urick University of Texas Box 1, Defense Research Laboratory Austin, Texas (1) David Taylor Model Basin Washington 25, D.C. (1) Director Scripps Oceanographic Institute La Jolla, California (1) Attn: Dr. C. Eckart Applied Physics Laboratory Johns Hopkins University Baltimore, Maryland (1) Director Underwater Sound Refer Qnce Laboratory Orlando, Florida (1) Bureau of Standards Washingtbn 25, D C. (1) Attn~ Mr. Cook Marine Laboratory University of Miami Coral Gables, Florida (1) Operations Evaluation Group (Op-374) Navy Department Washington 25, D.C.