THE UNIVERSITY OF MICHIGAN 8525-3-T BSD-TR-67-143 DIFFRACTION BY THE CONCAVE SURFACE OF THE PARABOLOID OF REVOLUTION Technical Report No. 3 F04-694-67 C-0055 Lt. J. Wheatley, Contract Monitor by S.E. Stone February 1967 8525-3-T = RL-2184 Prepared for Ballistic Systems Division, AFSC Deputy for Ballistic Missile Re-entry Systems Norton Air Force Base, California 92409 This document is subject to special export controls and each transmittal to foreign governments or foreign nationals may be made only with prior approval of Ballistic Systems Division (BSOMS), Norton AFB, California 92409.

THE UNIVERSITY OF MICHIGAN 8525-3-T FOREWORD This report was prepared by the Radiation Laboratory of the Department of Electrical Engineering of The University of Michigan under the direction of Dr. Raymond F. Goodrich, Principal Investigator, and Burton A. Harrison, Contract Manager. The work was performed under Contract F 04-694-67 C-0055 "Investigation of Re-entry Vehicle Surface Fields (SURF)". This work was administered under the direction of the Air Force Ballistic Systems Division, Norton Air Force Base, California 92409, by Lieutenant J. Wheatley BSYDF and was monitored by Mr. H.J. Katzman of the Aerospace Corporation. This document is subject to special export controls and each transmittal to foreign governments or foreign nationals may be made only with prior approval of Ballistic Systems Division (BSOMS), Norton AFB, California. Information in this report is embargoed under the Department of State International Traffic in Arms Regulations. This report may be released to foreign governments by departments or agencies of the U. S. Government subject to approval of Ballistic Systems Division (BSOMS), Norton AFB, California, or higher authority within the Department of the Air Force. Private individuals or firms require a Department of State export license. Publication of this report does not constitute Air Force approval of the report's findings or conclusions. It is published only for the exchange and stimulation of ideas. BSD Approving Authority William J. Schlerf BSYDR Contracting Officer iii

THE UNIVERSITY OF MICHIGAN 8525-3-T ABSTRACT Let r7 denote the focal length of a paraboloid of revolution, and let D be O the closure of the domain bounded by its concave surface. Then for a point source, with wave number k, located in D and on the axis of the paraboloid, the diffraction by the boundary of D is considered not only if kr >> 1 but also if k] << 1. If kr >> 1, an asymptotic representation of the total field on the boundary of D is derived for the Neumann boundary condition provided the source is far (with respect to wavelength) from the focus and the field point is far from the tip of the paraboloid. This representation is interpreted in terms of geometric optics. If kr7 << 1, an asynptotic representation of the total field anywhere in D is derived for both Dirichlet and Neumann boundary conditions and for the source (field) point in the near field and field (source) point in the far field as well as for both source and field points in the near field. The near field result is compared with the solution of the corresponding potential problem. A necessary and sufficient condition for the existence of a solution to the corresponding Neumann potential problem is also derived. iv

THE UNIVERSITY OF MICHIGAN 8525-3-T TABLE OF CONTENTS I. Introduction 1 1.1 Preliminary Discussion 1 1.2 Mathematical Statement of the Problem 2 1.3 Coordinates of the Paraboloid of Revolution 4 1.4 Integral Representations of the Solution 7 1.5 Results of the Asymptotic Analysis of the Integral Representations 21 II. Low Frequency (Thin Paraboloid) Diffraction 28 2.1 Low Frequency (Thin Paraboloid) Poles 30 2.2 Residue Series for the Near Field 35 2.3 Dirichlet Potential Problem 45 2.4 Neumann Potential Problem 50 2.5 Residue Series for the Far Field 57 2.6 Interpretation of Far Field Results 63 III. High Frequency (Fat Paraboloid) Diffraction 66 3.1 High Frequency Zeros 67 3.2 Convergence of Residue Series 79 3.3 Equivalent Integral Representations 89 3.4 Saddle Point Analysis 142 3.5 Whispering Gallery Waves 169 APPENDIX A: Normalization (Point Source Normalization) 179 APPENDIX B: Analyticity of Resolvent Green's Function Rk 181 APPENDIX C: Closing the Contour (Convergence of Residue Series) 182 APPENDIX D: Uniform Asymptotic Representations 207 -ikR APPENDIX E: Asymptotic Representation of -e /R 224 APPENDIX F: Comparison of Cone with Paraboloid of Revolution 269 BIBLIOGRAPHY 271 v

8525-3-T I INTRODUCTION 1.1 Preliminary Discussion For the most part, the solutions of diffraction problems have been confined to convex surfaces; relatively little has been done in the case of concave surfaces. For high frequency waves, diffraction by convex surfaces gives rise to single reflections and creeping waves while diffraction by concave surfaces gives rise to multiple reflections, caustic surfaces which are the envelopes of a family of multiply reflected waves and, in addition, whispering gallery waves (a form of traveling waves) if the source of radiation is near (with respect to wavelength) the concave surface. Some of the early considerations of these short wavelength effects are found in a paper by Rayleigh (1910) and also in his book (1896, Chapter 14). A recent investigation of the short wavelength diffraction by concave surfaces is illustrated by the papers of CKlkr (1961a, b), which treat the circular cylinder and sphere, respectively. In this report, we shall study the diffraction by the concave surface of the paraboloid of revolution not only of high frequency waves, but also of low frequency waves and in the long wavelength limit. In the high frequency case, we shall consider a point source located on the axis of the paraboloid and, in addition, far (with respect to wavelength) from the focus. For waves of low frequency, the point source is considered to be anywhere on the axis of the paraboloid. The restriction that the point source be far from the focus is the result of a fundamental difference between the short wavelength diffraction by the concave surface of the paraboloid of revolution and that of either the circular cylinder or sphere. This difference arises from the fact that while the paraboloid behaves as a typical concave surface for the point source far from the focus, the point source at the focus is a configuration whereby in the short wavelength limit, diffraction by a concave surface gives rise to a single reflection. The ca'e of a dipole, with moment perpendicular to the axis, at the focal point of a perfectly conducting 1

8525-3-T paraboloid of revolution has been investigated by Fock (1957) and Skalskaya (1955). Pinney (1946, 1947) considered the moment both perpendicular and parallel to the axis. Although there is a double reflection, it is natural to consider the plane wave problem in this category. The diffraction of a short wavelength plane wave by the concave surface of a parabolic cylinder (Dirichlet boundary condition) was studied by Lamb (1906), who indicated that the method could be extended to the paraboloid of revolution. Perhaps the best starting point for a study of diffraction problems pertain.ing to the paraboloid of revolution is a book by Buchholz (1953) which includes a complete bibliography. Worthy of separate mention is an earlier written paper by Buchholz (1942/3) in which he considers a point source at the focus (Neumann boundary condition). 1.2 Mathematical Statement of the Problem A paraboloid of revolution (with focal length rr) divides three dimensional space into two domains. The domain whose boundary is the concave surface of the paraboloid shall be called the interior of the paraboloid of revolution. Let D be the closure of this domain and let p( r) (r is the usual position vector) dente a point source distribution in D. The precise form of p(r) depends on the definition of the point source; it will be specified later. Then the solution of the initialboundary value problem for the inhomogeneous wave equation (with source distribution p(r)) in the domain D is a function u(r, t) defined in the space time domain Dx(t > 0), which is in C2(D) for each fixed positive t, which is in C2(t > 0), and which satisfies 2 1 a2u iwt V u- 2 2 - p(r) e u = 0 on the boundary of D (Dirichlet boundary condition), or (1.P1) au = 0 on the boundary of D (Neumann boundary condition), some prescribed initial conditions. some prescribed initial conditions. 2

8525-3-T The solution of the time independent problem corresponding to (1. P1) is the function v( r) which is in C (D), and which satisfies V2v+k v =p(r) (k=/c), v = 0 on the boundary of D, or (1.P2) 3v - = 0 on the boundary of D, an as rI -- oo, v(r) corresponds to an outgoing wave and the flux across a confocal paraboloid is approaching a fixed value (radiation condition). In this report, we shall find the following functions: (i) v(r), if kr7 is sufficiently small, the point source is located anywhere on the axis of the paraboloid, and r is the position vector to any point in D (except the source) or on the boundary of D, and (ii) v(r), if kro is sufficiently large, the point source is located on the axis of the paraboloid at a distance d from the focal point where d = O(r ), and r is the position vector to any point on the boundary of D. In recent years, there has been considerable effort devoted to obtaining, for various domains D' and distributions p'( r), a relation between the solution u'(r,t) of (1.P 1) (with D=D', p(r) =p'(r)) and the solution v'(r) of (1.P 2) (with D=D', p(r)=p'(r)) in the form u'(r,t) = v'(r) e +u(r,t) where lim u'r,t) = 0. For a certain class of unbounded domains, such a result t —oo has been obtained by Zachmanoglou (1963). However, this class of unbounded domains does not contain the interior of the paraboloid of revolution, and with the exception of this observation we do not consider the question of existence of this type of relation for the interior of the paraboloid of revolution. We conclude this section by considering, for a moment, problem (1. P2) defined on a domain D" representing the closed, three dimensional exterior to a smooth, bounded, convex body. Then the radiation condition of (1. P 2) becomes the three dimensional Sommerfeld radiation condition 3

8525-3-T lim r( +ikv) = 0. r — oo Moreover, if p"(r) is a distribution which has compact support in D", then Ritt and Kazarinoff (1959, 1960) show that the solution v"(r) to (1. P2) (with D= D", p = p"(r)) is equal to the limit lim+ v"(r,s), where v"(r, s) is a function defined in the domain D" x(B > 0 and which satisfies V2v 2 p"(r) Q =(Wl) V + 7V = pr) (7 = - (W- is), v" = 0 on the boundary of D", or (1.P3) av" an - 0 on the boundary of D" On S v"(r, s) 2dV < o (dV = unit volume in D") JD" We do not attempt to prove that this is true for a class of domains containing D and distributions containing p(r). However, we show that it is true for D and p(r) by deriving an integral representation from the formulation (1. P3) (with + D"= D, p"(r) = p(r)), taking the limit as s — 0, and then comparing the result with the corresponding integral representation for v(r) derived from the formulation (1. P2). This is significant since for large r ], D is approaching a cylinder which implies that, for large lr 1, the behavior of v(r) is no longer three but two dimensional in character (see Buchholz (1953, Chapter 18) for the form of v(r) as Ir| — co). The shape of D also affects the potential (see Section 2.4). 1. 3 Coordinates of the Paraboloid of Revolution A natural system of coordinates, i. e. a system for which the wave equation separates and the boundary of D is a level surface, may be defined in the following manner: two families of confocal paraboloids of revolution ( = Ia, rl ). with focal point at the origin, given by the equations 4

8525-3-T 2 2 22 2 p = 4(i- z) (p =x +y ), and p =4r(+z), together with the usual azimuth angle f, where 0,, r7 < oo and 0 < 2 < 2r. This system of coordinates is called the coordinates of the paraboloid of revolution and is illustrated in Fig. (1-1). If the point source is considered to be located as shown in Fig. (1-2) (either possibility), then the domain D defined above is described by 0,, r7 r, 0, < oo, and 0,< < 27, while the boundary of D is given by r) =. Moreover, since p is the radial coordinate of theusual cylindrical coordinate system, the coordinates of the paraboloid of revolution are related to the rectangular (x, y, z), cylindrical (p, 0, z), and spherical (r, 9, p) coordinates by the following equations: x = p os = rsinecos) = 2 j cos, y = psin = rsin0sino = 2 7 sini, and z = z = rcose = -r7. Throughout this report, we shall perform calculations in the coordinates of the paraboloid of revolution. These calculations are greatly facilitated by the use of the general relations of curvilinear coordinates developed in Chapters 1 and 5 of Morse and Feshbach (1953). To use their relations, we observe that if we set 1 = V 2 = 27 and 3 = cos0, their generalized curvilinear coordinates (e,,2' -3) will be defined in terms of the coordinates of the paraboloid of revolution given above. Hence, the scale factors hl, h2, and h are: 2 2 2 1 6all) 1 h2 = e + g 2+ ( 2 = 2 2 2 5

8525-3-T x z FIG. 1-1: COORDINATES OF THE PARABOLOID OF REVOLUTION x (0, z y FIG. 1-2: LOCATION OF POINT SOURCE 6

8525-3-T and aand x29a ( 2 az 2 h = \a + + in3 a3 aA3 3 sin3 Using the above two sets of equations, we can directly apply the relations of Chapters 1 and 5 of Morse and Feshbach (1953). 1.4 Integral Representations of the Solution An integral representation of the solutions can be derived from both formulation (1. P2) and (1. P3) of Section 1.2. We first show how the method in Ritt and Kazarinoff (1959, 1960) can be applied to derive an integral representation from the formulation (1. P3). Thus, we begin with the inhomogeneous wave equation in which the wave number has an imaginary part: V2 + vy p(r) (= (W- is). (1.1) In the coordinates of the paraboloid of revolution, V v has the representation given by (Morse and Feshbach, 1953, Chapter 5, together with Section 1.3) V 2( l a) L ( a) an 22t a J But since p( r) represents a point source on the axis of the paraboloid of revolution, the problems have axial symmetry; hence, the 0 dependence can be removed. Therefore, equation (1 1) becomes a (av) + Av a+ y2(e+ 1) = (Q+A)P(Sr). (1.2) We consider first the point source at (_, 0). In this case, (~ +r)p(Q, ra) = C 6(- - -)6(r), where C is a constant which depends on the precise form of p(r ), i.e. on the definition of a point source. We now make the stipulation (or normalization) that our definition of a point source is such that C = 1. (We show in Appendix A that this implies that p(r) = 47r6(r- r ), where r is the position vector of the point source.) Substitution of this choice in (2) yields position vector of the point source.) Substitution of this choice in (1.2) yields 7

which can be written as -L v- Lv = 6(- -)6(r), (1.3) where LxY= d (p(x) -) + q(x)y with p(x) = x, q(x) = -y x. In order to proceed, we need to make a study of the operators L and L defined by L yd= — d -72, OT < < and Ly (Sd) -y2~y 0~<co Lfy = - 'd dI ~ ) < since these operators do not correspond exactly with the ones studied in Ritt and Kazarinoff (1959, 1960). As they do, we shall use the papers by Sims (1957) and Phillips (1952) as the basis of this study. Before beginning, we note 2 2ws p(x) = x, q(x) = -7y x which implies Imq(x) = 2 x > some q. c Thus, although the operators L and L do not correspond exactly with the ones studied in Ritt and Kazarinoff (1959,1960), the conditions on p and Imq do. For L, the homogeneous differential equation to be studied is L y- Xy = 0. It can be written as d2 + + (+ )y = O. (1.4) -1/2 The substitution y = ur]1/2 results in the equation 8

8525-3-T d, (2 X2 1 + (2 + )u = 0O dr 4r or u + 1+ X + 1 d.2 d( 2iy1)2 ( 4 21 f 2i) 4( 2i) (t.4a) which is Whittaker's equation. It has the two Whittaker functions M+ X2iy, 0(+2iyr), W+ X/2iy, 0( ~ 2iyr) as linearly independent solutions. A complete discussion of this equation together with these functions is found in Buchholz (1953). The solutions M+ X/2i, 0( 2i1) are regular at n= 0 and lie in (0,r1 ), where (a, b) is the class of all square integrable functions on (a, b). Except for certain values of X, the solutions W~ /2i, (2) are not regular at zero, but are in 24(0, %) for all values of k. To see which pair of linearly independent solutions are most natural to use in the definition of the me solvent Green's function, we make use of the fact that the solutions M_ /2i, 0(t 2i17) are linearly dependent while the solutions W+ X/2i7 0( + 2iTr~ ) are linearly independent. Then since yl(.7.) = r M1/2 M (2i d) and Y2(l. X) = 172 W.1 i (2iy) are two M,2i7 0iWk 2 X/2iy, 0o linearly independent solutions of (1.4) such that (1) Y1(r, X) is regular at r =O, (2) y2(n, X) is not regular at r = 0, except for the value 2- n+ - or 2i'y 2 X = iy(2n+l), they are the natural solutions to use in the definition of the resolvent Green's function. Furthermore, the properties of L, O0.< ri g, are: (1) r = 0 is a regular singular point of L y-ky = 0, p(0) = 0 (2) p(r1) 0. 9

8525-3-T (3) for ImX <q = 0 (hence A cannot equal i(2n+ 1)), the homogeneous equation L y - y = 0 has exactly one linearly independent solution regular at r = 0. To find the resolvent Green's function of the operator L, 0 i r7 < ro, we need a solution 0l(r, X) of L y - Xy = 0 which satisfies the boundary condition at,i together with a solution 2(n, A) of L y-Xy = 0 which is regular at n=0. If we consider the Neumann problem, this is accomplished by the choice /dy1(ri,)\ /dy2(r., X) (nA, -) = y2(n. X) ( dr - YI(n, X) di =\ =r)=7r0i and 02(no X) = Y1(nr,.. (Hereafter, we shall denote the derivative (, drn= 7 \, dr) = by F'( rlo ).) With these definitions, the Wronskian W[0(rl. X), 02(nl, X) of l(nr, X) and 2(n, X) becomes W[0l(t. X), 02(Te, t = (2liy)yI(no. X) w[2(n. X), Yl(n. X which reduces to (Buchholz, 1953) w[01(n, X), 02(m, X = (2iy)y(Io, X) ( - A)2iy It should be noted that 0l no X) = 21y W[y2(T0. X), y1(no, )], 2i2y i- oK2 Thus, the resolvent Green's function GN(rY, rl',A) can be GN1r( X ) f= (o)2(' ) GN(1, n * ) - (2iy)yI(,70 X) 0 (n',x)0 (n ) 1 0 1 2 written as n >n' n <n' 0 10

8525-3-T while the resolvent operator has the representation (N) R y S GN(T, ' X)y(r')dn 0 For the Dirichlet problem, a solution l(rl, X) of L y - y = 0 which t1 satisfies the boundary condition at r7, together with a solution 22(nr X) which is regular at ra = 0 is given by 1(rL7 X) = Y2(r1, X)yl(lo0, ) - Y1(f X)y 2(o0 X) ) and ~22(n, A.) = yl(r, A) Then W [(l1 X)\), 2(. l = (2iT)y1(0., X) W[y2(r), A)(, Y1l(. which reduces to (Buchholz, 1953) (2iOy)yl(no, ) Thus, the resolvent Green's function can be written as F -( 21~ GD ) = (2iTy)y1(o, X) l(,.MO2(n. ) n < r' while the resolvent operator has the same representation as above (with N replaced by D). The properties of the operator L can be written down immediately. They are: (1) e=0 is a regular singular point for L y-Xy = 0, p(0) = 0 11

8525-3-T (2) for ImX<q = 0, the homogeneous equation L y- y = 0 has exactly one linearly independent solution y l ( M X) -1/22 MX/2i0(2i) v which is regular at e =0 and lies in 4(0. o) (O < o < co), plus exactly one linearly independent solution v2(f, X) 1 -1/2 W 2 o(2iY) which is regular at infinity and lies in <(Qo' o0). Hence, the resolvent Green's function G(Q, ', X) can be written as r-1 X- l ( )y ) 2(I) k< G(,',) = 2i< y1( ', )y2(~, A) e > e' -" while the resolvent operator Rk has the representation R y = G(f, E', )y(Q')dC' It should be observed that in this case RA is not only analytic in ImX < q = 0, but also in the larger half-plane Im < k. We can now proceed with the method in Ritt and Kazarinoff to find the integral representations for the solutions. Consider the Neumann problem, and let r be a path in the complex X-plane defined by thestraight line running from -o - ia to oo- ic, 0 < a < k. (We shall always assume that the X-plane is cut at X = -37r/4 and thus, argX satisfies -37r/4 < argX ~ 57r/4.) Then, applying the " See Appendix B. 12

8525-3-T (N) "1 resolvents R x 4RX successively to equation (1..3), using the resolvent rela.tion R (L)Mx - Xy) = y, and integrating along rP noting that the singular points of (N) R lie above Prwhile those of R_ lie below ',we find the integral representation VN. n#71,E,Os19) =2iri JG~~) N(?OXd where v g, T,7, 0, s) denotes the solution of the Neumann problem. Substituting for GZ-X) and GN(?,ODX), we obtain (0) t 2 'i 1 1' where (0)(~ a;:; 0)=lm 0, 5), g = min(g.."-), N ~s —#01 ' 1 < g2 and 7> 0. But (0. X) = (27y) 1/2 1' and 0 (r& X) = Y (t~jA)Y l(flo, X) Y 1(riA)y I(rj 0 X) Therefore (0) _ Vjj~ ( 7 ~: Q) Am lim S -*ol 0 (2iy) 3/2 rK& - + * I X~(ti X) Y1 ( 1D X X)y2( Q 2j (1,5) If fl7 =n 0 (nItX) = 21~y 1 0 7 1 0PQ 13

8525-3-T Thus, the field on the surface is given by the simpler formula; (0), - i (2iT) 1/2 v.,In, s- O" 2= lim s -0 ' o Sr Y{(Oo k) Yl(l -X)Y2(2Q -x). 1 ~ ~(1.6) (0) The above procedure may be repeat6dtoobtain v (), r, -, 0), the solution D to the Dirichlet problem for the point source at (0, 0). Let r be the path in the complex X-plane defined above; then i(0) = (21) -3/2 dX vD,,-,,. o)= lim+ 2 - d s —* e yl(n.o ) YorX)Y 22' -) [Y2(, )Yl(no, X)-yl(. )Y2(no, ). (1.7) The field on the surface for the Dirichlet problem is given by the normal derivative Using the relation (Morse and Feshbach, 1953, Chapter 1, together with Section 1.3) av () 1/2 av() an 1/2 an we observe that the equation for the field on the surface is av(o) r..o) D(, n,-, 0) an 1lim + 1/2 r2+)-1 - - o dX i'i.+ l( [n ) 1/-2 2vri yl(n, X) n=,ro P 0 r ' Yl(Jl, -)y2(E2, -x), (1.. 8) 14

8525-3- T where the condition 1< ~ is understood not only for (I..8) but also for (t1. 5) through (1. 7). In addition,, we note the condition rj > 0 for (1,.5) and (I17)." In the case of the point source at (0, H), ( + Ti)p(9,, ri) =C 6(9)6(T-q-H). Then the choice of C = 1 implies that equation (1. 3) is replaced by -L 77v-Lv=6(6(I-H Repeating the above procedure for the Neumann problem, we find VN (9. Ti,0 (, W P)Y =2j'ji G(g, 0, XA)GN(TiD H, XL)dX while the solution of teDirichiet problem has the same representation with N replaced by D. Substituting for the resolvent Green's functions,, we obti VN(., iO,,,H) = l m ~ 27r dX N5s-40 + 2f r r 1 - x 14 + x 2 2 ky 2iy.. -X)y X). y I (ri 4, X) Y2 Q, 1(771.p 1 0 [ Y2i17 X-)y 1(n. X) -'y (VI X)y I(n0 Ii A.)." (0) = lime (21ri vN,(, ri, 0 H) l0 SdX y I(0,qX) y2 (9,-XA)y1(HISX)) r l (1I. 10O) D s5. — 0+ (2-y312 ir I jdX y1(Li0r x Y2 (90 -A.)y1(nit X). [ Y2 ti 2D X)y1(VI0, -X) - y1(nl2# X)y2(770I X] (111 15

8525-3-T (a, Hi -1/2_........ + x n 1/2 2ri 3 y(1( a,X) y2(,-X)y(H, X) (1.12) where the definitions il S min(n, H), q2 = max(Y, H), and the condition ~ > 0 are understood for (1.9) through (. 12), while the condition 771 < 2 is also understood for (. 9) ad (1. 11). Consider now equations (1.5) through (U. 12). We note that there is an essential difference between these equations and the similar equations in Ritt and Kazariubff (1959, 1960). For these equations r is independent of s; thus, it may be possible to take the limit as s -* 0 inside the integral. According to Buchholz (1953), the fnctions y1(x, X) and y2(x, X) are entire functions of X/2i-y while the path P is defined so that the functions Pr( + -) and r (-. + ) are analytic functions of X/2iy on r. Therefore, we can take the limit as s - 0 inside the integral and equations (. 5) through (1.12) are valid without the lilm condition if the parameter 'y is replaced by the paramter k. s-&0 Buchholz (1942/3, 1953) derives an integral representation for v(r) from the formulation (1. P2). For completeness we reproduce this derivation. We first consider the point source at (.-, 0). For p(r) = 4w6(r - r ) (r is the vector to the - -o0 -ikR point source at (F, 0)), the free space Green's function has the form -e /R (R = |r- r, time dependence e ). Thus, he first derives an integral representation for -e /R. It is -IkR -a, +loo e - 21k 11 (0)(, (0) H - 3 |ds r(s+ ) P(-s+) 2 )m (2lk)w (2ik2)w (2ikii) -o'-t -a where |a'| < 1/2, - n >0, m (2ikx) = (21kx)- /2M 0(2ikcx) and w(0)(2ikx) = (2Akx) -/2W (21kx). Then he assumes v(9, n,, 0) has the form a B,,0 16

8525-3-T v(,,ri -, O) = v'(I, n,E, ) + e-ikR with i 1 1 (0) (0) (0) sr-c i 2 2 s o v'(~, n,-,O) = 2 i ds r(s+2) F(-s+ Im s kSw ike^A m, -ato- o where again [a' <1/2 and A is an unknown function of s. Hence, v(,, r,, 0) formally satisfies the inhomogeneous wave equation while the boundary condition may be satisfied by a suitable choice of A. For s a = 0 on the boundary, On while for (0)' (N) w (2ik) A = 2ik -- I m (2ikO ) w(~)(2 W7 ) (D) -s 0 ~' S 0 v = 0 on the boundary, where again F (2ik ) = (- F(2ikl)) 0 0 We continue by considering explicitly the Neumann problem. The obvious modifications can be made for the Dirichlet problem. Substituting for A ) we 8 obtain 2ik 0)8(2ikf2) 21Jr 2 1m (2ik ) -aN(',, 0) - 2r d' r(s+ -)r( -s+ -ao' - i 0a -8 (2ko w(2k (0) )- m(o )()2ik -oW(2 )m (2ikr) - m (2 ikn )w( 2ikr) -8 -S O -8 (1.13) 17

8525-3-T as a formal solution to the inhomogeneous wave equation which satisfies the boundary condition. If as e -- oo we compare the behavior of the integrand of (1.13) -ikR with the one in the representation for -e /R, we note that vN(,, 7 0) satisfieb the radiation condition. Thus, it remains to show that the integral exists. Using the previous y1(x, X), y2(x, X) notation in equation (1.13), and substituting -s for s, we find (d-12 '"'at i OD y -^ t s _.-) 1 (k1 yl(,I-2iks)y2(2,-2iks) (2ik) 1/2 __'-_i__ _ _ _2_ _2 _ vN( = a.- 21 0)\ dsr(s+ )r(-s-+ 2)( 2ri 2 2 y2(o, 2iks) a'+ioo [Y(r, 2iks)y (r 0, 2iks) - y2(r, 2iks)y(n, 2iks)j. (1t14) It is shown in Appendix C or Buchholz (1942/3, 1953) that r(-s + 1) [y1(r, 2iks)y2( I 2iks) - y2(n, 2iks)yi(rn 2iks) is analytic in the complex s-plane, y(n r, 2iks) has a countable infinity of simple zeros which lie along the imaginary axis, and the integrand of equation (1.14) is exponentially decreasing on a large semi-circle in the righthalf s-plane. Hence for 0 < a' < 1/2, the integral represents the zero solution which can be omitted by the further restriction -1/2 < a' < 0. Moreover, for a' = 0, the integral is not defined, and so we finally obtain the restriction -1/2 < a' < 0. In addition, it is seen in Appendix C that along this path the integral converges. Thus, vj. r, -r, 0) given by equation (1.14) together with the restriction -1/2 < a' < 0 is a solution of the Neumann problem defined by the formulation (1. P). We write it as (2k)/2 1 1 y1 -2iks)y2( 2, -2iks) N O) (2ik 1 /2 dsP(s+i)r (-s+4) ( 2iks ia'+ iOD (-1/2< a'< 0) y* y2(n 2iks)yl(r0, 2iks) - yl(n). 2iks)y(, 2iks). (l. 15) 18

852&-3-T If in (1.15) the substitution s = X/2ik is made, this equation becomes vN() = (2ik)32 vN(,,t' 0) = 2w oo + 2ika' ^'.,eik+ 2 r- 2ik -) - Y-y(n - A) -oo+2ika' 1 ~ *Y )y i(, A) - y(n. )yI o, A with -1/2 < o' < 0 which implies 0) -(2ik)3/2 VN(g?":' 0)' 2ri - d'ik - +i + ) y (11 Y2, 4) -oo - ia (0 < < k) [y2(1. A)y(\. A) - y(r A)Yn, 2X) o (n, (1. 16) As asserted, the integral in equation (2.16) agrees with the one in equation (2.5). Substituting 7r = rl in (1. 16) yields the equation for the field on the surface (21k) /2 OD0 i - VN ( o2ik)r = 0 ( (2+i) 1( 2 1 iar2. )22 N 0 = r(27r i) yYI(n, X) (0 <- k) (O <o < k), (t,17) where the integral in (1.17) agrees with the one in (. 6). Similarly, for the Dirichlet problem vD(,-20) = (2 ik)-3/2 S iaX (O < a < k) 2(, X)y1( -, ) - y, X)y2(, (. 18) 2 ' 2~ i0 while for the field on the surface 19

8525-3-T (,, O)n \ an =tl 0~ -1/2 1oo - i -1 (2ik)1/2 - 1/o( 0 2 2ri dX( (O < a < k) 'k 2) ~+D Yl({1, X) "yl(O,o, x).... (1.19) where the definitions = min(,ZT), f2 = max(g, ), and the condition f$ - are understood not only for (1.19), but also (1.16) through (1.18). In addition, the condition rl > 0 is understood for (1. 16) and (a. 18). For the point source at (0, H), we begin with the following integral representation for -e /R: -ikR e-i 2ik R 27ri 1 1 (0) (0) (0) ds P(s+ 1)r(-s+ )w ()(2ikj)m ()(2ikr )w )(2iki2), 28 -8 I -8 2 where |'| < 1/2, > 0, tl= min(r, H), 72 = max(rl, H), and rl H. Proceeding as above, we would then obtain 2ik)3/2 Go- Ya(' -X)Yl(nitl X) vN( r.O. H)= ( \ 2ik+)-2/) 2ik+2' y 1\) 27ri '%m 2 k 2 ( -O -ia (0 <a < k) [y2(r2 )y;n(rl, ) - y1(n2, k)y (, x)., (1. 20) H (2ik)-1/2 VN(, r0o, 0, H) = Tlo(2 i) 0 - dXr__ _ )Y2('-)Y(H) ry. ik -nlo X) -oo- icr (0 <a < k) (1.21) 20

8525-3-T vD(V7I, 0, H) = 2i r(21ki + ) r 2 + y2 X)y(n -co -ia (0 <a < k) * [2(n. )yl(no. A) - yl(n2, A)y2(no, X ( 2 8 1 0 1 2 2 o0 (1.22) __________.1 (2ik)-'^ dxr +) An -n= Oco-ia (0 <a<k) Y2(, -X)yl(H, X) -, (1.23) Yl(77o, x) with the above conditions on r1, r2, and ~. 1.5 Results of the Asymptotic Analysis of the Integral Representations In Section 1.4 we derived two equivalent sets of integral representations, each containing one representation for the solution of the various problems formulated. We first discuss some general results pertaining to each representation. We do this by considering explicitly equation (1.16), and noting that the results pertaining to (1. 16) either pertain directly or with a slight modification to the remaining integral representations. In Appendix C, we prove that the integral representation (1. 16) may, for all positions of the source and field points, be replaced by a convergent residue series expansion obtained by summing the residues at the zeros of the function y'(r, X) (poles of GN( riY', )). In Section 3.2, we prove that this series conerges quite slowly for short wavelengths or high frequencies, i.e. for wave numbers k (k = w/c = 27r/X) and focal lengths r7 such that kr ~>> 1, Hence if krv >> 1, this residue series behaves like the Mie type series found in the diffraction by smooth, bounded, convex bodies (Ritt and Kazarinoff, 1959; 1960). The above behavior for kri~ >> 1 leads us to investigate the residue series for long wavelengths or low frequencies, i.e. for wave numbers k:and focal lengths 21

8525-3-T r7 such that kr7<< 1. If this series is to be truly analogous to the Mie type series found in the diffraction by smooth, bounded, convex bodies, it should readily yield the first term in the asymptotic expansion for vN(,, ) if kr7<< 1. We show this to be true in Sections 2.2 and 2.5. The explicit results established in Chapter 2 are summarized below in Theorems 1 through 4. The possibility of further analogy with the diffraction by smooth, bounded, convex bodies suggests that in order to find an asymptotic representation for vN(g, 7- 0) if kr >> 1, we should examine the residue series obtained by summing the residues at the poles of the r-function r( - +2) (poles of G,,-)). We show in Appendix C, that this residue series is convergent if and only if the source and field points are subject to the restriction 1 + 2< fr'. Thus, any asymptotic representation of vN(, r,, 0) derived from this residue series would be subject to the same restriction and could not be used to describe VN(S, r,7, 0) throughout the interior of the paraboloid. We show in Sections 3.3 and 3.4 that if kr >> 1, an asymptotic representation of the integral in equation (1.17) which is valid for all positions of the source and field points can be derived by considering the saddle point contributions to the integral. The explicit results obtained in Chapter 3 are summarized below in Theorems 5 through 8. The results of this report differ from those of Buchholz (1942/3, 1953) in that his analysis is not directly concerned with large or small values of kri. He is primarily concerned with analyzing the total field by considering the separate terms (or modes) in each of the two residue series, particularly the one obtained by summing the residues at the zeros of yl(n, X). However he does obtain, by different methods, an asymptotic representation of the zeros of Yl(no, X) (1953, Chapter 17, pg. 189, eq. 14) and y (r 0X) (1942/3, pg. 432, eq. 3.10) if kno<< 1. Comparing his results with those of Section 2.1, we see that the first terms agree and the error term in these equations (Section 211, equations (2.15a) and (2. 13a), respectively) has the same order as his explicit second term. 22

8525-3-T 1.5.1 Low Frequency Theorems For kr~ << 1 we study two distinct cases. The first considers both the source and field points to be near (with respect to wavelength) the focal points; it is called the near field and is defined by the mathematical conditions kg << 1, k= << 1. One reason for interest in this case is the investigation of a relationship between the first term in the asymptotic expansion of the total near field and the solution of the corresponding potential problem. For the Dirichlet boundary condition, we obtain (Sections 2.2 and 2.3) the following result: Theorem 1: If krl << 1, then for the Dirichlet boundary condition the first term o in the asymptotic expansion of the total near field (of a point source on the axis of the interior of a paraboloid of revolution) is equal to the solution of the corresponding Dirichlet potential problem. In the attempt to prove a theorem for the Neumann boundary condition corresponding to Theorem 1, we find two essential differences between the Neumann and Dirichlet problems. The first is that the asymptotic representation of the total near field contains a term which has a logarithmic dependence on kg. The second is that the Neumann potential cannot be regular at infinity in the usual sense. Instead, we find (Section 2.4) the following condition at infinity: Theorem 2: A necessary and sufficient(condition that p be a solution to the Neumann potential problem (for a point charge on the axis of the interior of the paraboloid of revolution) is that lim 0 This result is most likely a characteristic of the domain defined by the interior of a paraboloid of revolution, since the corresponding configuration for a cone does not contradict the usual regularity condition. In addition, it implies that the solution of the Neumann potential problem contains a term which has a logarithmic dependtoe on f. Furthermore, if we call the other terms in this solution the

8525-3-T regular part of the Neumann potential, the results of Sections 2.2 and 2.4 also yield: Theorem 3: If kr << 1, then for the Neumahnn boundary condition the term, 0 which is independent of k, in the asymptotic representation of the total near field (of a point source on the axis of the interior of a paraboloid of revolution) is equal (to within some constant) to the regular part of the solution of the corresponding Neumann potential problem. The second case studied if kr ~<< 1 is the point source (field point) in the near field and field point(point source) in the far field. This case is defined by the mathematical conditions 1A 0= 0(1), kg 1, and corresponds to a configuration of some physical interest. We obtain (Section 2.5) the following result: Theorem 4: If kr ~<< 1, then for a point source (on the axis of the interior of a 0 paraboloid of revolution) in the near field and field point in the far field, the solution to the Dirichlet problem is exponentially small, while the solution to the Neumann problem contains a single propagating term. This result can be interpreted in terms of waveguide theory. At a great distance from the source, we can assume that the field is approximately governed by the homogeneous wave equation. In addition, the paraboloid locally resembles a cylinder. Thus the local far field satisfies the boundary value problem: - r -) + 2 + k v = 0 within an infinite circular cylinder, r Or a r^ az2 v = 0 or -v = 0 on the boundary of the cylinder. an The solutions of this axially symmetric cylindrical waveguide problem for the Neumann boundary condition have a propagating term for all values of k (r) fixed), while for k small enough the solutions for the Dirichlet boundary condition have no propagating terms. 24

8525-3-T 1.5.2 High Frequency Theorems If kro > 1, we again investigate two distinct cases. The first considers the source to be far (with respect to wavelength) from the focus and far (with respect to the transition region of y (nr, X)) from the tip of the paraboloid. In this case we explicitly study the field on the surface if the point source is at (-, 0) and the Neumann boundary condition holds (equation (1.17)). We obtain (Sections 3.3 and 3.4) the following results: Theorem 5: If kn >> 1, k >> 1, k >> 1, then vN(S, n10,, 0) can be asymptotically represented as a finite sum of integrals, where the sum always has at least one term and the total number of terms depends only on r 0: and f. This first term is approximately equal to -2e i/R, while the remaining terms (if they exist) can be estimated by the saddle point method. Theorem 6: Let rl, and e be such that there exist N terms (N > 1) in the sum of Theorem 5, and let n be an integer such that 1, n, N- 1. Then a sufficient condition that the saddle point equations of the (n+ 1l)th integral have an approximate solution is 1 (Q /A2)2(2n+1).< Moreover, if, 2 =, and 1 (-,/)2(2n+) << the (n +1)th integral is approximately equal to 1 n -2 2(2n+1) -i". i0 o n) i2 where ) 25

8525-3-T Theorem 7: Let r, and e be such that there exists N terms (N > 1) in the sum of Theorem 5, and let - and e be such that (E/) /<< 1. Then the geometric path traveled by a ray that leaves the source at (1, 0) and arrives at (,?n ) via one reflection is approximately equal to -I-+r? +2n = - Z,,.o,1). We do not extend Theorem 7 to values of n > 1. However, in Section 3.4.3 we do show that if a ray leaves the source at (, 0) and arrives at (, r1 ) via n reflections where the path length of the initial ray and each subsequent reflected ray is a priori large, then the total path length traveled by such a ray is approximately equal to (E-,, no n). This relation together with Theorems 5, 6 and 7 gives strong impetus to interpret the results obtained in terms of the theory of geometric optics. This interpretation is discussed fully in Section 3.4.3. We also show how similar results can be obtained if the source is at (0, H) and -k H is outside the transition region of yl(n 0 k). The second case studied if krn ~ 1 is the point source at a point (0, H) 2 o near the tip of the paraboloid (-k H is inside the transition region of Yi(n, X)). In this case the sum in Theorem 5 depends on krl. Hence, we consider directly the residue series of equation (1.21) obtained from summing the residues at the zeros of the function yl(nI, X). We express our results (Section 3.5) in the following theorem: Theorem 8: If kr >> 1, k >> 1, and -k H is in the transition region of yi(ro X), then the residue series for vN(, r 0, H) (residues at the zeros of y{(ro, k)) has two distinct sets of nonexponentially small residue terms. Let E denote the set whose terms become exponentially small if -k H w 2 is allowed outside the transition region of yj(rI, X). Then if -k H lies inside the transition region of y(n., X), the approximate distance-dependent 26

8525-3-T phase of each of the terms of w is equal to the arc length from the tip of the paraboloid to the field point (g, n7) under consideration. Furthermore, for field points ( rn ) such that g/n >> 1, the approximate distance-dependent amplitude of each of the terms of E is equal to 1/p, where p is the radius of the cylindrical cross section of the paraboloid of revolution. Therefore, in this case we observe the existence of a set of residues (previously exponentially small) that can be interpreted as cylindrical waves traveling along the surface of the paraboloid of revolution. These waves are called whispering gallery waves. 27

8525-3-T II LOW FREQUENCY (THIN PARABOLOID) DIFFRACTION As indicated in Section 1.3 a paraboloid of revolution may be characterized by a focal length ro. For a given wave number k, the mathematical condition ki ~1 corresponds either to small values of k (low frequency diffraction) or small values of r~ (thin paraboloid). We wish to investigate the integral representations in this case. For convenience these representations are now given a slightly different form. Let us define (the m (z), w, (z) notation being used by x x Buchholz, 1953) vl(x,X) = m/2ik(2ikx) = (2ikx) /2 M (2ikx) = (2ik)1/2 y1(x X) V2(xi,) = W (2ikx) = (2ikx 0(2ikx) = (2ik)-/2k) 2k)/y(x, X), 2(and ) = ik and v i(xo, X) =(d2x i(X X) i 0 k) - (This notation for the derivative is different from that of Chapter 1 and Appendix C where the primes simply refer to differentiation with respect to x.) Then equations (1. 18) through (1. 19) become A.v 1 v(o, X)v( X) v ( -0) = + r + 1) (o<<k). [v2(n, X)v(.) - v(, X)v( (n j (2.1) 1O~-iv VN('. o 0"-" ) = 2r i(2ik-oo) - (O <a + 1 v1(91S, -)'v2('2o, 4. ) dxl\?k+2 1 0o (2.2) 28

8525-3-T V jO + /\ v 1(91 -X) V2(~2 9 -X) (9,n,''.O ar + r + 1 1De (0 <cr c k) (2. 3) L2n X~1(n0. X) - v6,X) V2 (Y~, X] ( 9. 77 ~a0 cl0 -1 1 (~~~2f +712 Ir (O <a Xr( + 1* V 1(91" X)L)v2( 2,p - X) la v1(r0.),- < k)(2.4) whee gan =ml(~ 2 = max(g# Z) and 9 i r > 0. Siml~arly, equations (I1-20) through (1. 23) become 2iI (0 < la d4r X +1 10 - LU ".a < k) (2,.5) X n A) VI (VI XA) - v1(rp2 A.) VI x}"A. I vN (9 p Yi,poH) = 1 2ff i(2iki7). +) 7 2(9.XF 1(,, (2.6) /-cO-ia (0 <a<k) C- Ia 1\v1 y(9. -X,)y (n la X) --,.- -~2CD21k2 (0<a<k) *[2(n2. x)-v1i(no X) - IV1(r 2p.)'V2(n0i ) x] (2.7) 29

8525-3-T n (e, 0 H) - 1.1, -A V 2E.-x)vX(HX) K [ ] 2 \ Vw( +.X) \ ) (O<a<k) (2.8) where as previously, n11;min(n, H), n2 = max(dr, H) and n 4 Hi, f > 0. 2.1 Low Frequency (Thin Paraboloid) Poles The "low frequency poles" are those poles of the integrands of equations (2.1) through (2. 8) corresponding to the zeros of the functions vl(nr, A) and vi(r, A) if kr << 1. In order to analyze these zeros we consider vl(r, X) for kr<< 1. We first assume O |X/2k|: 0(1). Then the power series expansion for v (r, A) (or m (z) to keep the equatlns brief) is (Buchholz, 1953, Chapter 2, equation 7) OD 1 r (0) = -z/2 2 X)r which for z -0 can be written as o r z zr 1. (,x)(-x+1) 2 m(z) Zl- + + O(z3L + ( x)z+ 12 ) z+oz23] X = 2 4 2z +=4 Z or 1 1 (0)z 1 2 21 2 3 mI ( ) 1 -K X+ -1) — 4 --- z + 0(z3 We set X = X/2ik. z = 2ikr to obtain n)z) ) - 1 - +O + ((kr) )2 implying that v 1(r0, A) has no zeros for 0 IX/2k | 0(1). We also see that for z>0 for z -- O 30

8525-3-T d (0) X3 z 2 2 dz m (z) = -X+ z+x ]+O(z2) This implies vr(., 0) = O(k7), V vi(r, ) = -2ik + O(k) (X O), or that X = 0 is asymptotically a zero of v1(o, X), and there are no other zeros of vI(,X.) for 0X |X/2k(0(1). To investigate the zeros for I/2kj ~ 1 or IX/k > 1 we can use the theory developed in Appendix D. In the Whittaker equation d2\ 1 X 1 d(2 ( 2ik)2 (2ikl)2 we identify from Appendix D the parameters I = k/2ik, z = 2ikri and thus 8= 2ikr x 4 -2ik This implies that for -37r/4 < argX: 57r/4 (recall that this is the convention adopted for the X-plane), -r/4 < arg s < 7r/4. In particular Is I= knr k/ << 1, s = kr << 1, and we can apply equations (D. 35) to MA/2ik 0(2ikrl) for any value of argX. Therefore MX/2ikO(2ik7) = (2ikn)l1/2 J( )[+O(k1) )] if lN.s MX/2ik (2ikn) = (2ikn)1/2J (?) [1+O(k/)j if I>N, where C is given by equation (D. 28a). Then we can write 2x1/2n/ + ( +0 k) (2.9) 6 XVR ''X _ 31

8525-3-T It immediately follows that v (rn,) = J() + ((k )] (I ), (2. 10a) v1 ( X) = J( ) [ + (k/X ( i N), (2.1 Ob) from which Vi 1"' X) = Jot() M l+O(6ki7) l+Jo(0)(k/X) ([ N) 1 vl0=o0 d(2ikn) L + \X/+Jo)O ) The second term for IC |>N is present since the order term 0 ((kr) ) also occurs in equation (2. lOb) but is not written explicitly since it is of lower order than 0(k/X). Using equation (2. 9) we obtain vi = 1/2 ) E+ o ((k)D) + J () Ok/) (1jIN) (2. lla) 1/2 v( ) = 1/2 J ) [l+0 (k/) +J ()0O(k/X) (1I>N). (2. 1b) 2ikn Let r' r = 1, 2, 3..., r = -1, -2, -3,.. denote respectively the positive and negative zeros of J' (C). Then the zeros of the function v (X, X) can be repre0 sented by C = C + A. In order to find A, we use the Taylor expansion of J' (?) r o about a zero r. Thus both (2. lla) and (2. lib) imply that for a zero of V1(ri, X) 1/2 2ik/2 0(1) = 0(k/X), and hence the zeros of the function v l(7, X) are given by the equation ( 1/2 k3/2( r= +O~ ) 3/2' (2.12) r,.'X22/ 32

8525-3-T for arbitrary C. Since we are explicitly interested in the zeros X in the complex X-plane, we must solve equation (2.12) for A. Substituting for C we obtain r the equation 2X1/21/2 f+ o(kr )-) = 1r+ 0kn)1/2 k3/2) which has the solution 2 X = rE+o(kF). (2.13) r But since rr= - rwe need only consider r = 1 2, 3,.... Then kr) < 1 implies X /k > 1, since 1- 3.832 implies X 2 2 r r -1 - - — a >> 1 k 4kn? 4kn This demonstrates consistency with the assumption of IX/kl >> 1 and therefore Vt (7o, X) has real positive zeros given by the equation 2 k )2 = r 1+0 0 (r = 123,...). (2.13a) r 4o 2 1 r To find the zeros of v (,1 X) we need, in addition to equation (2. 10), the estimate 1(./2 2 k2 2 of l(1, AX) at a zero of J (r) (Appendix D). Thus if (r=1,2,3,..., r=-1, -2, -3,..) denote respectively the positive and negative zeros of J (C), the zeros of v,(rn X) are given by the equation r= r +(kr)/2(kl)2 ) (2.14) r \ X2 33

8525-3-T for arbitrary C. Substituting for C we obtain the equation 1/2 [1 o((k).) = +O (L)1/2(k)2 k2 which has the solution 2 kX r +O kn 2,) (2.15) r Again since 3r = -1,we need only consider r = 1, 2, 3,.... Also kr << 1 implies X /k 1, since 31 - 2.405 implies 32 2 r r 1 k 4kr > 4kr~ Therefore we again have consistency with the assumption of X/k >> 1 and vl('0, X) has real positive zeros given by 12 F7(kno) 2] - r | +0 (r= 1,2,3,...) (2.15a) r 4o L r We conclude this section with an observation about the zeros given by equation (2.13a). Since cos(C- 4) governs the behavior of J (M) for large C, 4 o we see that the limiting behavior of these zeros for large r is described by r equation (C. 4a). That this limiting relationship exists follows from the fact that for fixed kr) the value of 2 s = -- -kr(k/Xr) r can be made small by taking k/k small enough (X /k large enough). Then although kr k — k. 34

8525-3-T is not small, the asymptotic representations given by equations (D. 34) and expansions given by equations (D. 28) through (D. 30) still apply. A similar observation may be made for the zeros given by equation (2.15a) 2.2 Residue Series for the Near Field Let X = 0 and X, r = 1, 2,3,... denote the positive zeros of v1(rIoX ). Then we prove. In Appendix C that the integral representation (2.1) can be replaced by a convergent residue series expansion. The result obtained is OD= A., 0 v_.\ -)v (.-X) N _ 2 r) *V f [V ~2 k 1 + -( V.r.)... r=0 X=X r *[ V 2- X rvj,Xrvi r o' )[ I ] 22( r, 0 rr'-v'~ 1 rv 2,0o. r ~ which upon separating the r 0 term reduces to r(1/2)r(1/2)vl(E1, o)v2(2 0)) N(5..-. S0) = v — -- ^) |V2(nO)v;(n 0)-v (n,0)v;(1 7o W f0) d.... 2...... 00 r We make this separation since although vi(rl, O) GO, v2(n, O) and v'(no, O),oo and thus the ratio 1d I.o [v2(rl' O) - Vl(r, )v(7, 0) must be carefully evaluated. In order tonaalmo the derivative in the r = 0 term,we employ the power series expansion for m((z) given in Section 2.1. From the expansion of d m (z) we obtain 35

8525-3-T d d (0) (z) dx dz X Using this form with X = 2 &ik = -1+Xz+O(z2) and z = 2ikr1, we find d 1v((rl, X) =- +on+ o and dA vn X) 1d o ~ =O -2ik -~kO The other members of the r = term are given by (Buchholz, 1953, Appendix I) v1(O1 O) = J (k,) v2(F2o0) = - i 1/2 (2) v(, O) 2 Jokn) v1(r, 0) = ~o(kn), kir0 V,(ri, ) = - i v1 o0 4i [1 + o(Qk) 0 v2(r1 O) = 2 H()(k2 ) = 1- k o0 - 1/2 2 2(1) 2 +0(1)+ (+ ) 4~rl/2 ^n + ky7)2) vI(o o0) = - 1./2 21k /2 0 kri + 0 4ir 1/2 kro 2 + O(k0o) Therefore equation (2.16) becomes 36

8525-3-T V Ng' rlp = ir J (kg ) H(2)(kg2 )L1+O(krI] + 2n0 2 OZD k ~'rv(D-*X) lk r14 + - ~ % )~ r V (T " x) *(2. 17) V1( r~ To estimate the fuctions depending on i7 in the rt (r >, 1) term, we use the asymptotic representations obtained in Section 2. 1. From equations (2. 10) and (2. 11) we find V1~ r ~(r)[+O 2L)] r - v (77,x ) = (C (r) )E+o~~) r (r(r)J><N). Vi(n,) = ~1/2 Jl~(% )[1+O(kr ))] +J (Q7 )O0(k/X)) 0 0?)00 VI(X >) = (12 )[1 + ONk + J (C )o0(k/X) 10 2ikvj 1/2J'70 on7 00 where cJ% (KJ (<,.N). >N) D n7 02- E' k0 )] (r = 2Xr '7I1+o((knhj-)I n7 -Moreover.. by substituting for X the value given in equation (2. 13a), we observe that 37

8525-3-T C(r) = Cr + ~ r Therefore, taking the derivatives with respect to X, setting X = A, and expanding the functions 0J (r+ ~ o) ( ^ V) in a Taylor's series, we: obtain Vd t o- = 2 Jo(r) 1 + r1 2ik, r (Crr N) d 'I 2ik ok r rl o 2+: r (Cr > N). Since r is a positive zero of J' (C), Besselfs differential equation, r o J"() + J',()+ (J) = 0 O implies that J"(r) -J (r) Hence, the above derivatives become r d 21 1 ~ =X 2i r 1 r___ -ik (r)1+ / 2k o r Ir (CrS N), (2.18a) L J (r ) k o r 1+0- ) +0r ( > N), (2.18b) r 38

8525-3-T In order to further estimate (2.17) we use the mathematical conditions defining the near field. Since the near field corresponds to the physical problem of both the source and field points near (with respect to wavelength) the origin, these conditions are k << 1, kg < 1. Then for these values of k, 51' 2 J (k1)= l+O(k1)2, (2.19a) H +(kg2) —+) I + 2n (2.19b) o 2 i 2 i 2 where y is the Euler constant. We now determine the behavior of vl(rl -A ) and v2(2, - ) for r >1, k << 1, j=1,2. It suffices to obtain M_ X/21k o(2ikc1) and W (2ik g). These functions are solutions of the equation u d2 r 1 + + u = 0 (j = 1, 2) 2 2ik(2 ikg ) 72 d(21ik.)2 2i 2ikj) 4(2ik.) and upon identifying from Appendix D the parameters I = -Xr/21k z = 2ikCj, we obtain 2ikg 2 Then we see from equation (2.13a) that sj is real valued and positive. We also note that for -3ir/4 < argX 5ir/4, the s-plane defined by 2ilS s - X is described by -/4 ar 7/4. In addition, since is described by -X /4 <: args < 77X/4. In addition, since 39

8525-3-T 29 s = 9j Ar 4(k~ ) 2 V0)< as well as x = Ik-s <(1 Jti we can apply equations (D. 35) and (D. 38) to obtain M-A /2ik, 0 (2ikf1) = (2 ikg1)1 /2 ( ( r))L+o(2 r 0 C (I (r) K~lN)0 M-X /2ik., 0 (2ik%1) = 12 0 r (1 (r)1 > (2ig1) i0(9 )1\I c1>N 1/2 F1/2 (1) rI FL 12ik(2ik~2) H0 ( c +02. r 2.. cr _0 where FXr2ik =exp4x log — r and- from equation (D. 28a) (r) = ((1/2 )(kn )N r He~unce 11' r (r (r) -1<. N)0 1 (2.20a) 40

8525-3-T (1'-X) =J (r) O 0 L r (4r) >N) 91~ (2.20b) (2.21) ) - /2 H(1)((r))2 +0 2 a) v2( 2 r 21/2 F2r) 772 F-Xr/2ik oo2 r Now a/ 2 8ki1 >> 1. and since arg X /2ik = -r/2, Stirling's formula (Erdelyi et al, 1953) gives r(r + )= exP-{ log 2e} +O /]. Substituting this and equations (2.10), (2.18), (2.19), (2.20) and (2.21) into (2.17), we obtain VN(fn,.o 0) = 11 OD J ( r)) H ) ( r) )i (C ) 1 i7r iO.1 0 >2 0VI - In kg - --- 2 ~o -' — - 2 o k2 70 2~o o r o r=1 LJ(ri 1+0 (2.22) r where only the terms of greatest order have been retained. Expanding J (( ) o 1 and H(1)(C(r)) about (r)) = 2iXl/2(0).l/2 j = 1,2, where () = 24 o S i r j r (r (r) =2 1 '2 o expanding J (C(r)) about C(r)(0) = 2 l/(0) 1/2, and substituting the results into equation (222), we find into equation (2.22), we find 41

a525-3-T V (e,.-, 0) = 10 2 1 i - in k~2 7+ — + o ~o 2tlo OD f{ (r) (0) H(1) (r)' 0 (r) (0 J(1 (2 (O\J 7o r=l [Jor) +<r r (2.23) where again only the term of greatest order has been retained. The residue series for the point source at (0, H) may be similarly obtained. Summing the residues of the integrat representation (2.5) yields V (SI, o, H)= a + 1 2lky o rz ril/2)r(1/2),2(e, o) v l(l. o), ~o ( Id X),) ) [ )vj, 1 ) 0l( oO) Vl(f12 0)v 2(7o0 + ~ r + 2 2 xr)V l(t r) Vl(r2,x r 2., r.i +.)*.v( q (d C ~l(77o Xr) r 24) which is analogous to equation (2.16). This can be written as N(, r 0, H) = - H -(k + 2ikro r=i +( V2 (' Xr) l(1 xr). vl(n2 r) dX i ~ = or).r Vl(n 2x ) X=X r (2.25) Then, since the near field is defined by k << 1, substituting the corresponding asymptotic representations gives 42

8525-3-T VN( o.H)= cnk)((+)Jo(- _r)Jo((r) ) V i7.1 Op H) In kS + 1 1Y 1~ - o o 2o ' o o r=l Jo ()r]2 i +(E I ) a (2.26) r where (r) 21/2 rl/2 F t )(k0~2 ) - r (r) = 2l/2/ +Okg) r 2 r and only the term of greatest order has been retained. Expanding about (r)) = 2l/ 2(0)n/2 'rj = 1 2 () 2 /2 () 1/2n r r.1 1 i 7 VN(~o r77 O, H) In kg + = + - 7 2ro r=l2 r where again only the term of greatest order has been retained. The solutions for the Dirichlet problem are found in the same manner as above. Consider the point source at (-, 0). If Al.2.. '... now denote the positive zeros of V1(nio X) (given by equation (2. 15a)) then the integral representation (2.3) becomes 43

8525-3-T VD( n.-. 0) = +r( _ V+ 1)r. -Xr) 2( - r) ZIOr + r + r and on using the Wronskian relation this reduces to VD(Ti n..o 0) v,1 (nAr) - 1 - E v{(.o xZ ) & (2.28) Then in order to evaluate (2. 28) asymptotically, it remains only to find (dA Vl(no, X)) From equations (2. 10) 1 ~ 4 r 1/2 d -l(noA.) =,I(C ) ) +o ky) d= J)0 o o Xi/21 711/2 d V,(nr X) = JI(t ) 72 + (k/X dX1 0 0 o l1 if 1l 1 N a if C l> N. " Mo0 Thus (d 271 - y(7oX)) = o() r 1+0 O ( 3rkIlN), r (2.29a) (2. 29b) Vl(rn. )), =- Jo(p r ) +(k (jBr1 >N). Substituting equations (2.29) and the other corresponding asymptotic representations into (2.28), we obtain, upon expanding as before and retaining only the terms of greatest order, 44

8525-3-T vD (o;)HOa() =l -) ( (....)) (r( () o r=1 I- L 9l 2 o r or (2.30) For the point source at (0, H) the integral representation (2.7) gives VD, 1o r Vr L v-r)VlT r)r V'1 (n2r) rl O — # -aH) -'I X )LJ 0 r=f1 V (77, X) 1 o =1 = (2.31) r Then substituting the corresponding asymptotic representations into (2.31), expanding as before and retaining only the terms of greatest order, we find c() H (J (r) (O Jo 2 )( ir _ 0771r- 0 2k v D(g, 1, O, H) = - (i(t0 r 77o r= o r] r (2.32) 2.3 Dirichlet Potential Problem We consider first the Dirichlet potential problem corresponding to the wave problem for a point source at (, 0). In order to represent the solution to this potential problem we use an integral representation of 1/R given by Morse and Feshbach (1953, Chapter 10). They define a paraboloid of revolution coordinate system by 1 22 1 2 x = Xocoso, y = Xsin, z = (Ax-w ) (r = (X2+2)) 2 6 in which, for a point source at (X = X, w = 0), the integral representation of 1/R is OD - ir\J (t)J (tX )H(1)(itw)tdt (W >0) R 0 0 0o o "0 45

8525-3-T Then if the substitutions X = /2, o = A2, X = \2Z are made, the above coordinate system reduces to the one which we are considering, and the above integral representation for 1/R becomes = itr 0 (t2 )J (t )H()(it 2)tdt (rl >0), (2.33) X\ ^ 0 1 0 ' J0 where again 1 = min(g,Z), 2 = max(g,-). Therefore we can assume that the potential OD(, rl,-, 0) has the form D(, n,, 0) = -+ 0S(go r,-, 0), where (S rnIo 0) = ir (t\ J 1T )Jo(t 2g2)AtJo(it /2)tdt, with At an unknown function of t. The boundary condition implies H(1)(it oFi~) A- -. t J (it{2n~) ' 0 and hence ~ j (t2rl)Jo(t 22) D( t 0) = -r iftdt J (it,/I ) H )(it /)Jo(it2o) - H()(it 2ro0)Jo(ltt/^) is an integral representation of the potential. In order to compare the potential with the term in equation (2.30) which is independent of k, we obtain a series representation for the potential. In order that this series repress ntation be more readily comparable, we first substitute t = E which yields 46

8525.-3-T = 21fSvvJ0(2vvt97)J 0(2vf92) [H'(2i) p) 0(iv) - H '(2iv[noF)J (2ivF (2.34) Then we consider the fuction JP(V = H 1) 2iv )i(2i /T) - H (2ivjTI)JO(2iv ITWj Jtv J(2 ivfF) H0~(i / )J2iv (1) whereupon we note Ov e-"=')2 (2w ) (2iv(-) + H(2)(2ivf/Wlb J (2iv/~) -4H(1) (2iv 1F+H (2) (2iv yro)7 J0(2ivVW] which reduces to ON (e - HJl(2iiv)J 2ivno - H (2iv ftF)Jo(2ivri]) or Therefore if we write equation (2. 34) as OD (go 7',Q) = -1-ff vdvJ (2v -FF)H ' (2v g!F);pkv) - (2) - ~rvdvJ (2v g7)H 0(v'2Ov 0D 47

8525-3-T -?i we can consider the substitution v = we in the second integrand to obtain co -(1) (~ -0) - - v VdvJ (2v/i )H (w)(2vw) "V + +0 + iir \ wdw J (2w )H( )(2w )O(w) This reduces to OD(?, ni 0) = -ir vdv J (2v ) H 2v v) -0o and thus the equation for the potential becomes OD J (2VN o (2v r[( D(g. r0 _ 0) = -i.r vdv ~ 0 --- H (2iv(2)J(iv o ) - D 3j (2iveFo) L 0 O - H(1)(2iv i )Jo(2iv ]. (2.35) Let 3r (r = 1 2, 3... -1, -2,-3,...) again denote the zeros of J (). Thus v = i3r/2yf~ (r = 1, 2, 3,...) denote the zeros of J (2iv j/F) along the positive imaginary axis and hence also the poles of the integrand of equation (2.35) in the upper half plane. Now for large Iv in the upper half plane, we can use the asymptotic representations (Erdelyi et al, 1953) of J (2v 1), H (1)(2v J), H )(2ivr) and J (2iv ri') to show that the integrand of (2.35) is exponentially O o small provided n > 0, 1< g2' Therefore the residue theorem implies i..,0) = 2 i(-r) () EHi'w2 ivjr)J (2ivf.] r 48

8525-3-T which upon using the Wronskian relation for the Bessel functions Jo(2iv jV), (1) ~ ~ H (2ivFj7), and substituting vr= i3r/2 5;T, reduces to i7 o i Jo(ir/o)H(1)r9 %/; D(t n- o)= i- 2 J"o( i (2. 36) This series is identical to the one in equation (2.30) repriseznting the term independent of k. A similar argument can be employed for the Dirichlet potential problem corresponding to the wave problem for a point source at (0, H). In this case the integral representation for 1/R becomes = i o(tV2)J (it /)H(l)(it?r)tdt where as previously rl = min(rp, H), 12 = max(rn, H). Then the potential has the integral representation D ({ 0. H) = -iDr 0tdt J(t ~ [H (it 2)J (it2 ) - - H( )(it4rl)Jo(it 2 )which by the same analysis as before reduces to D( r 0 H) = - vdv J (2 i1(2 i )J (2 i i - -00i - H( )(2iv/)J (2ivVf;]. ~~o V' (2.37) For g > 0, r1 < 12 the integrand of (2.37) deorea ees Ianentially as $vls - o in the upper half plane. Thus as above 49

8525-3-T tD(',O.H)= LK E '/'qo) olprE7 )Jo(PrV 2T ~ (2.38) -. o......... the series being identical to the one in equation (2.32) for the term which is independent of k. 2.4 Neumann Potential Problem We consider first the Neumann potential problem corresponding to the wave problem for a point source at (Z, 0). If we attempt to represent the solution to this problem in the same manner as we did for the Dirichlet potential, we find that substituting the value of A implied by the boundary condition leads to a divergent integral. Therefore we seek another method of representing the solution to the Neumann potential problem together with some insight into why the previous technique fails. We recall that for interior Neumann potential problems the condition that a solution exist implies that the boundary condition cannot be specified arbitrarily (Koshlyakov et al, 1964, Chapter 18). Consequently we seek an analogous con-o dition for the domain defined by the interior of a paraboloid of revolution. Let us consider the bounded volume Vi, defined by the intersection of the paraboloid of revolution n7 =?I with the paraboloid of revolution e = g' (Fig. 2-1). If S represents the boundary of Vt,, we define the surface areas S1 and S2 by s1= sN(1= ro) s2 = snh( = r,). Then we can apply Gauss' theorem (Morse and Feshbach, 1953, Chapter 1) to the volume V-, and any suitably defined vector A to obtain AndS+5 A.ndS2 V AdV, 1 2, 50

8525-3-T S S2 S1 S F GSS FIG. 2-1: VOLUME AND SURFACE AREAS FOR GAUSS' THEOREM. 51

8525-3-T where n is the unit outward normal to V{,. We substitute for dSl, dS2, dV, (Morse and Feshbach, 1953, Chapter 1 together with Section 1.3), and taking the limit as I' - ax of both sides of the equation, we find 2 \ A-ofo + (n 0+)dd] + lim A - An2 T/i +r 1ndo 0 0 O I \ \ \ 0 V *A[2(g +n) ddndd. (2.39) Jo J0 J0 Let p represent a possible solution to the Neumann potential problem under consideration. Then 0 must satisfy the boundary condition (an = 0 as well as the condition =11o Or 5 r v2 [2(c+rl)ddrdp =4 4ir Jo0 0 J0 which follows from the Poisson equation. But an allowable choice of A in equation (2.39) is A = V0 (Morse and Feshbach, 1953, Chapter 1). Therefore (2.39) becomes lim V *n [2' '('+1T) drnd]= 4r j' —O oD 0 0 Now since A'n =V n =On' we find, upon using (Morse and Feshbach, 1953, Chapter 1, together with Section 1.3) a (1) 1/2 (an:, 8a (g.+)1 2 changing the dummy variable from g' to ~, and carrying out the 0-integration, 52

8525-3-T Umrn =n 1 (2.40) This equation thus represents a necessary condition at infinity for the existence of a solution to the Neumann potential problem being considered. It is also the reason why we cannot represent the solution to the Neumann potential problem as we did for the Dirichlet potential problem; assuming a solution in this form implies a behavior at infinity which is contrary to the validity of equation (2. 40). This result is characteristic of the domain defined by the interior of the parabolbid of revolution. The same is not true for the domain defined by the interior of o cone (half cone angle less than v/2) where the application of Gauss' theorem does not lead to any oontradlicton. This is shown in Appendix F. We can show that condition (2.40) is also sufficient for existence by producing a solution to the potential problem V2 = 4r6( r-r ) (r the vector to the point (0)), 0 0 - = 0 on the boundary = r (2.P) 9 — - OD by the method of separation of variables (Morse and Feshbach, 1953, Chapter 7). We write the inhomogeneous potential equation in the coordinates of the paraboloid of revolution a +an),(().)* (2.41) Then if r r =1, 2, 3,... again denotes the positive zeros of J' () = 0, we ob(r) (r) serve that the functions J (y ), y = Jr, and the function 1 are solutions of the homogeneous potential equation: satisfying the boundary condition. Thus we assume a solution of the inhomogeneous equation to be of the form 53

8525-3-T 00 (r) PN( r- 0) = A (9)+ A (g)J 0(r) (2.42) In addition, since the functions 1, J (y () form a complete set in the space of C functions, by the theory of distributions, we can find the expansion OD ((r))..6 (17) +7(2.43) 6(n) = I + I E ~ '2 * (2.43) Substituting (2. 42) and (2.43) into (2. 41), we obtain the following ordinary differential equations d dA (9)> d \ dJ = 6(9-) (2.44), dA () 1 6 -d)) _ (O)A r() 1 - 22 er=1,2,3,.., (2.45) r" o r J[Jo(r)2 where again X (0) = 2/4. We solve equation (2. 45) by constructing the Green's function from linearly independent solutions of the homogeneous equation (Friedman, 1956, Chapter 3). The pertinent solutions are J (y(r)) (7Yr) i which are regular at =0 and lie in 2(, ) (0 < < oo), and (1) (r) o ) The H(1) (r) Ho (^Y ), which are reuar at infDtty and lie nto '2(o,). The H(o ()) solutions are required in order that condition. (2.40) not be violated. Thus since j((r). H(1)( (r) d (r) (1), (r)) = i o" d{"o r~ ) - d,.oWY o u~ y= 110 Green's function for equation (2.45), written as d XdA i 6(-a) d- dX (0)A (9) = -.r r r 54

8525-3-T is G(ts-. (0)) =- r/i [. (8x (r),), (r), - Hence the solution to equation (2.45) is given by Ar(C) = - G(,.-XA(O)) -7 [o1 ' (-s 2] or ((r)H (1), (r) J (( )Ho (y) ~-'1 ' 2 Ar() / -' r 1 2 3,... (2.46) where (r) 2o 1 2 1 2 We now show that condition (2.40) is sufficient also for the existence of a solution to equation (2.44). To construct the Green's function we use 1 for the solution of the homogeneous equation which is regular at 9 = O0 and lies in ( (0 o) 2 0 and log 5 for the linearly independent solution which has the 9 dependence at infinity required by (2.40). Since the Wronskian W(logf, 1) = -1/g, the eolution to equation (2.44), written as / dA ( ) dT ( d ) 1(f -7) st given by < 1 { A () 0 o tlogs > - (2.47) H(C). no 55

8525-3-T Substituting (2.47) and (2.46) into (2.42), we find o () ) H )(y(r) )Jo(7rlr) _ __ Jo 1..2 0 i1 H(1(2.48) ON 0) = 7 H(9o(r) -2 0 o r=1 as a solution to the potential problem (2. P).. It should be noted that this solution plus any constant is also a solution to (2. P). Examining the series in equation (2. 48), we see that it agrees (to within some constant) with the term, which is independent of k, in the field (equation (2.23)) for the corresponding Neumann problem. This comparison completes our investigation of potential problems. We do not consider any relation between uniqueness and the behavior at infinity. There is little difference for the Neumann potential problem corresponding to the wave problem for the point source at (0, H). The first part of the previous discussion does not change at all; equation (2.40) remains the same. The second part of the discussion changes only in that we are now considering the equation 2 V 2 = 4r 6( r- r ) (r the vector to the point (0, H)), which when written in the coordinates of the paraboloid of revolution becomes +) (E )+a )=6(e) -).- (2.49) The assumption of a solution to (2.49) is then of the form 0O N(e n. 0, H) = B0( g H) + Br(g, H)Jor) (2.50) r=l while the expansion of the 6-function is now 6(o r=1 (Jo(7 ) r1 F r12 56

8525-3-T with -rH = Ir Hi. This leads to the ordinary differential equations d dBo ( 7 (), d9 9 i0 I (2.52) / dB (g) ( Jr (() d ( I-r )X(0O)Br() = 6() H r = 12,... (2.53) dwith doltr on with solutions Bo(f) - nlog 0 0 (f > 0) j, (r) (1), (r) B (e) = i/i (. H o(.)a %or] (f > 0), r=1,2,..., (2.55) and thus N(, t7, O. H) = o00 o-1 1 1 T i o o rN= (1) (r) (r) (r) H()('Y ')J(o'ay )J(o ) [0~r1 (2.56) plus any constant. Examining the series in equation (2.56),we find that it agrees (to within some constant) with the term, which is independent of k, in the field (equation (2.27)) for the corresponding Neumann wave problem. 2.5 Residue gries for the Far Field We consider first equation (2.17) without any assumptions on k, j 1, 2. Then using the formulas developed in Section 2.2 we can write V N( no- 0), - Jo (k 1 )H((2) - O 2 2~0 OD -o o r=1 exp exp j lIg } vl( 1, — Xr)v2(2, -.r), (2.57) 57

8525-3-T where the error terms have been omitted for the sake of simplicity. We will continue to omit them throughout most of the section. The far field is defined by the mathematical condition k2 >> 1; the case of physical interest is =1/ro 0(1). For these conditions the behavior of v1(1, -X ) is at once determined by equations (2.20). It also immediately follows that (2) 2 1/2 -ik2 11r/4 HJ(kg2) - 1e (2) Substituting for Ho2)(k2) and vl(.1, -r). we write equation (2.57) as 2 _ 2 11/2 -ik -S - J ( r o ) 0 (4o2 exp k * (2.58 ) so as to retain the dependence on kl in the first term. To estimate v2(F2-, X ) we need to determine the order of magnitide of 2r Jkg2 k 4ke o O X (kJ2) -, 2(k~2). 2 r r 2 This depends on the relation between kro and ke 2 as well as the value of Cr; o 2 r the latter dependsnce is a function of r in addition. Since k2 1 there are three different possibilities for its order of magnitude; they are kg2 >> /kro, 42 = O(l/k7o) and k2 << 1/kro. For k2 >> l/kno we can write the series in equation (2.58) as 58

8525-3-T M J (r (0) ) (r)) 0) -- -- exp log r=M+l L.....ik 2 (r (2.59) where M is such that r(M implies r=O(1) (s >~1), N is such that r>N r 2 implies r O(k2) (s2<< 1), and re[M+1, N] implies ~2= O(kn)(kg2)) (s -o(1). Wewill refertothethreesumsasZi Z. 3 Thu2 sthe problem of estimating v2(~2, -X ) is reduced to estimating it in each of the three r sums 3 In order to evaluate v2(h2, -Xr) in and Z2 we use the results of Appendix D. We can write = -2Ui -2i( ) Xr/k - r Then in Zi.,? is large and a is large; thus expansions (D. 9) and (D. 10) inserted in equations (D. 15) yield v2(2,- r) (2ik2) 1/2 exp - Alog(2ik2 e. (2.60) Hence 1 becomes 59

8525-3- T >1 1 I r70 lk 2 r11 exp ik log 2k.0 AO oexp A2 ik log(2ikg e 2 Evaluating the two exponentials we obtain Z1 1 I?o 2 m i (r) (O i(r)(0 11 o 91 r=l (C 2 0 r exp 8knie c2 c2 -exp 8y0log 16(kg 2Xkfio~e} (2.61) For Z2.0 a9=1~)D and equations (D. 15) give '' ^. 2k )-1/2 (X k)16ep — jj log r.(I* (2) 1/3 2 (2.62) with s (0) = 2(kg2) 2K2 and -(r) (O c9 2 x r a 2 Therefore z2 becomes 01%,#0.- IT 2 1o ik92 N 2 7/6 ri (c n 1 [~(Cr 2 c2 - r 1. exp 8kyj 0, 2 2 2/' (2.63) 60

8525-3-T From equations (2.61) and (2.63) we see that both 1 and 2 are exponentially small. In order to evaluate v2(2, -X ) for Z we need only use equation (2.21). This gives o J ( O 1.0 o)2) 3 o r=N+1 [Jo( )2 Since C((O) =,(/ )l/2 we may write as rj o 3 31 r (C 2 r r [ (2.64) We write the product of the exponentials as exp r E /no) O I (92/rro)1/2 22 But since <l /ri o 0(1), and k2 > l/kyr implies 2/1 l /k 2o, this product 1 0 2 0 2 0 0 is much less than exp4 r[io< xp1} and Z3 is also exponentially small. This implies that the contribution from the residue series in equation (2.57) is exponentially small; thus it is much smaller than the error term of the r ter. Therefore we may write equation (2.57) as (2) vN(, n 3,0) s2 Jo(ke Ol y (k). Tl+e0(k )at. (2.65) while upon using kl << 1 equation (2.58) becomes 61

8525-3-T v )N(,O)1/2 i O( (2.66) since ko>> l/k 2. In the case of kg2 = 0(1/ko) we can write the series in equation (2.58) as o r - E o ~ 1 exp Bli 2 7or=1 [Jlr)] lo r=M+1 ()-2 2k 2 2 r 07(,r, [2....... where M is such that r < M implies 2r = 0(1) (E O(, and for r >M >> 1 (s << 1). Then the discussion above for 2 applies to the first sum 2 2 here, a similar discussion to the one above for 3 applies to the second sum. The results are the same as above; equation (2.65) is valid, while we add the error term O(l/kC2) to equation (2.66). Finally, for the case k2 << 1/krt we see that the series in equation (2.58) does not need to be divided into separate sums. The condition sa ~ 1 is true for all terms; a similar argument to the one for the third sum above s2hows the series is exponentially small. Equation (2.65) is valid and in equation (2.66) we replace O(krn) by 0(1/kg2). For the corresponding Dirichlet problem we first consider equation (2.28) without any restrictions on kj, j = 1, 2. Using the formulas developed in Section 2.2, we obtain ftOD, ( (r)(0)) - rX X VDOr),2 Eexp; 10 og 2ik - Vl(' r 2 -2 (2O - 67 (2.67) 62

8525-3-T However, the preceding arguments for the Neumann problem show that this residue series is exponentially small; thus we have the result that v j.,-,o) ~ 0. vD(g3n -O) 0. If we now consider the point source at (0, H), it is easily seen that the arguments do not differ from those above. We begin with equations (2.25) and (2.31) without any restriction on kg. Substituting the formulas of Section 2.2, we find vN(, O0. H) = _ H (kg) (2..69) equation2( (2refer to the p8) arguments. o r=- (C 2.6 Interpretation of Far Field Results o rr (2.69) Then, since the f field is defined by kg >> 1, comparison of these equation with equation (2.58) shows that we can refer to the previots arguments. 2.6 Interpretation of Far Field Results The results of the previous section can be simply interpreted. Let us consider a source in the near field (at (0, 0) or (0, H), it makes no difference), and investigate the field uN(C, ri) or uD(9, a) in a neighborhood of the point (, n) in the far field. We make the following two observations: (i) The field far away from the source is approximately governed by the homogeneous wave equation V2u+k2u = 0. 63

8525-3-T (li) The radius of the circular cross section of the paraboloid is p = 2 Ji. Hence at (fy tf) d '. = / ~1 d -I<<1; thus the paraboloid looks like a cylinder in a neighborhood of (f, of). Therefore the local behavior of the field at the point (Sf, rf) is governed by the axialy symmetric cylindrical waveguide problem a Z au ara2 2 u = 0 or a = 0 at p = pf 2/i. This problem can be solved by the method of separation of variables (Morse and Feshbach, 1953, Chapter 11). For the Dirichlet boundary condition we write the solution as uD(PZ) k2-X) A e r +B re 1, (2.70) r=l where p, z are the associated cylindrical coordinates (Section 1.3), A and Br depend only on r, X is a solution of the equation 22 2 k pf - 2 r (2.71) Pf and Cr (r = 1, 2, 3,...) are again the positive zeros of JO () = 0. From (2.71) we see that there are values of k (considering rl fixed) for which all X are 22 2 negative (k Pf - 12 < 0). For these values of k the solution becomes, upon defining iA = A, Ar >0, r r r 64

8525-3-T a0 UD(P. ) E Jo(Pk2 r-AX r r=l which is exponentially small. Therefore as k- 0 we find that for k less than some k there are no more propagating solutions ("low frequency cutoff") and the field is exponentially small. This corresponds to the result observed in Section 2.5. The solution to the Neumann problem can be written as 5i-2 -i zz: uN(P.z) = N Jp/k - F Ce r + D e r (2.72) where C and D depend only on r, Xr is a solution of the equation 22 2 k Pf- r - - A (2.73) 2 r Pf where 0 = 0 and the 3 (r = 1, 2,3,...) are again the positive zeros of J () =0. 0 2 /0r~f-\ 0 Since =, J (pk - )= 1, and choosing the solution corresponding to a time dependence of e we obtain It11\ iIz i gr~2 uN(pz)=D e- + J(pY ) e De D r. (2.74) rN(P) o r The series behaves as in the Dirichlet problem. Thus as k- 0 it exhibits "low frequency cutoff". However, the r =0 term remains; there exists a propagating term for the Neumann problem. This again corresponds to the results in Section 2.5. 65

CHAPTER III HIGH FREQUENCY (FAT PARABOLOID) DIFFRACTION In this chapter we investigate the integral representations for the total field if ki >> 1. This condition corresponds either to large values of k (high frequency diffraction) or large values of ro (a fat paraboloid). We noted previously (Sections 1.4, I.5. Appendix C) that Buchholz (1953) shows that vl(nl, X), vX(rlX k) have a countable infinity of simple zeros which lie along the real axis. Moreover, the proof that all the zeros are real valued does not use any asymptotic representations (Buchholz, 1953, Chapter 17). Let ' \' 3...; A1 A2' A3... denote the zeros of Vl(r,X) and vI(r., ) respeotively. It is proved in Appendix C that the integral representations (2.) through (2.8) can be replaced by convergent residue series expansions. For equations (2.1) and (2.5) we find r 1> v -A)v2(.. -) vl(. r) VN(,, l, t, 0)= 2ik v' 2ik r=- 2ik 2 d vi( )) v 1 ) l r~l - V(I o, r \dX 1 ~ A r (3.1) X=A r (3.2) the other representations yield similarly behaving residue series. We first want to determine whether equations (3.1) and (3.2) can be used to obtain an asymptotic representation of the field. That is to say we sebk to learn whether or not the residue series are converging rapidly. It will be shown in Section 3.2 that they are not rapidly convergent. This demonstration is based upon the detailed discussion of the zeros of v1(rl, X) and v(ril, X) that is given in Section 3.1. The positive results we have obtained are given in Sections 3.3, 3.4, 66

and 3.5. These results stem from the application of the saddle point method to derive an asymptotic representation for vN(, rio -0) if kro>> 1, k ~ 1, and k >> 1. In either case (convergence or saddle point method), the discussion of the Dirichlet problem is quite similar to that of the Neumann problem and is not explicitly considered. 3.1 High Frequency Zeros We first note that we need only to consider vl(nr, A) for krl >) 1 and X real (see above). We shall study the zeros of v1 in three pairs of regions of X, namely the regions corresponding to the conditions 0: IX/2k << kur, | X/k\ = O(kyr), and I X/k | kir. These conditions correspond to regions of validity of the various asymptotic representations of v1 derived in Appendix D. If kr) is large, these regions of validity overlap and cover the real line. However, we do not use this fact and do not attempt to prove it. Roughly speaking, what we do show is that there exist sets of zeros in the range of X where I X/k I = O(krn), sets of large cardinality for which the corresponding residue terms have a nonnegligible sum. If we make the same identification of parameters as in Section 2. 1, we observe that the condition IX/k~ >>kr implies that s [. -= knr7k/ << 1 (recall that 21kn s8 4=/2ik ) Therefore, this case has already been considered in Chapter 2 (as indicated in the discussion following equation (2.15a); see also Appendix C). The zeros in this range (see equation (C.4a)) do not affect the rapidity of convergence of the residue series. The residues they contribute are exponentially small, involving negative exponents that monotonically decrease. Therefore, these zeros are not considered further in this section. The situation as regards the zeros in the range where 0 [k/2k << kr is rather more complicated. If in addition to the condition k >> 1 (kg ~ 1), we stipulate that k1 >> 1 and k2 > 1 (kll > 1 and kr2 > 1), the slow convergence of the residue series (3.1) ((3.2)) may be demonstrated without considering these zeros. However, if k ~ 1 (k< 1 << 1), these zeros generate the dominant residue contribution and o, are discussed further. Since |2ikrl[ > j/2kj, then (Buchholz, 1953, Chapter 7) 67

-, X/2ik ik MX/2k, o(2 = i e [1r */1 X+* 2 2ik 2 2i from which we obtain = (2k)-1/2 (2ikn)- X/21k v1( X) = (21kn) 1^ X \2 " 2ik/ 2~k A-k / k,/ r + J + (2 12i e e-7 i/keX/^ l[ /k )] 2ik (3.3) Hence, the zeros of v1(1 o, X) for k ~>> 1 and 0 (X/2k << ki? are asymptotic to the solutions of the equation -Xk/21ik o 0 A./ X/2ik -iko2 (21ko) e ( -2ik e e /2k)i/2 (34) (2 2k) p(l + ) e e (3.4) r +X ( 2 2ik 2ik That such zeros exist for some values of krlo can be demonstrated by setting X=0. Then (3.4) becomes i(klo - ) -i(kr- ) o 4 'lo- 4 e +e 0; thus if krn is a solution of cos(krl- -)= 0, a corresponding zero X of vl(v, X) exists and is symptotic to 0. 68

A similar result follows for vI(n0, k). By differentiating in equation (3.3) we find -1/2 (2ikre)k/2tk e-Wi - irX/2k ~ri/ X(n A) = (2ikn) X e2ik)/ e e e eiJ *[1+ 0(/kr1). (3.5) Therefore the zeros of v (ri, X) for kil~ > 1 and 0 X [ /2k | << kr are asymptotic to the solutions of the equation ( ) e (2ikn e -/2k ri/2 (2 2ik 2 + 2ik/ 7r * This implies that if kro is a solution of sin(kr - 7) = 05 a corresponding zero A of vI(nrI. X) exists and is asymptotic to 0. The determination of all the zeros in this range depends on an explicit solution of (3.6) (or (3.4) for the zeros of vl(non,)). We have not obtained such a solution. The important region is that where |I /k I = O(krl). It contains the zeros which govern the rapidity of convergence of the residue series (3.1) ((3.2)) if kn1>1, k1 ~1, 12 (kg > 1, krl 1, ky2 > 1). We observe that AX/ki = O(kn) implies s | = 0(1). In this range we can thus use the Airy function representation for v (r, X) if args = 0 and s > 1, or if s < 1 but 1-s =O(1/(k T)2/3. If args =0, s <1, 1-s l>>/(krk), and, i2/3 n %1 t s >> ~ /(k) /, we can use either the Airy or Bessel function representations for v (rX, X); while if arg s 3= r, we may use only the Bessel function representation. 1 T? These cases will be presented in the order determined by beginning with negative values of A satisfying the inequality X <-k 27, proceeding to negative values of X satisfying the relation X >-k 2r, and finally ending with positive values of X. 69

Case I. X < -k r; 2X/k = O(kn) For negative values of X (argX = 7r, args = 0) satisfying the inequality 2 k X< -k ir we observe that s < 1. Thus for values of X such that nq (k)2 X 2/3 (k~l) (-s >>_ 7 1 ) r7 (kq)2/3 (k?7)"/ equations (D. 34) may be used to obtain 1/2,8(s )p1/2 M X/2k, 0(ik i) = (X/ik) Lks )T 1/2 / )1 /2 MX/2ik, 0 (2i) = (X/ik)1/2 1 "1~140'~;'1" ia,;(s ) c5(1) J (k7-)9/2 ( Kk N) (3.7a) J (? )+ o n 1/2 Oir -ir 8(s / 2[e ) + e 0(1) (kn)3/2 (r I > N) (3.7b) with j2(s )= 1(1- ), Ti 5 Ti where by (D. 23) <s ) = S 7(-s) ds (3.8) )= kik (3.9) In order to avoid repetition of long equations, we observe that the representation -1/2 of v l(rX) = (2k17) / M/2 ik 0(2ikn) follows immediately from equations (3.7). By differentiation we find 70

vI(n. ) = (2ik1)-l/2 [M k O(2ik )+O(llb/k l (3. 10) where d M/2zik, 0(2 d(2"ikil) MX/2ik, 0(2ik") Therefore, since ds d(2ikrL) = O(1/kr) d(2ikrj) 2 and = 1 ( T d(2ktkl) = 2 8s equations (3.7) imply 1/2 1/2 ()/2 fl s 11/2 Mliik~k0(ik) = - J J~ ') r ' +O(1/k) (3.11) V2 ik,, 0 ik L.L Oin)1 0 2 s a / for all allowed i. We can now assert that vl(), X) and v(l(n, A) have no zeros in the range of x where equations (3.7) through (3.11) are valid. This follows from the fact that equation (3.9) shows that arg D = r/2, and there are no cornplex zeros of J (r) or J' () (Erdelyi et al, 1953). 0 0 2 We continue to consider negative values of X satisfying X < -k 2.; in addition, we consider only those values of X for which 2 X>> 2/3 (e>> 2) (kl)23 (k) 2 T?1 1+ - ( 23) (1.. -8 ( (-i) *) W 2) \ 7 k7 \ 2/3// 71

Then equations (B. 21) yield X/2iko 0 ( kl TO1/ -vkrkX S. + 0(1/kv7) c17 (3. 12a) X2k,0(2 iky) = c(ik LEMI + -.. ff (3. 12b) with 2( Ti 17 (x2 -(31/6,11/2 (X)1/6 D TI = (2 TI D and, from equations (D. 3), (D. 4) and (D. 6)v ~(s ) = TI dos doa (3.13) TI k5 TI S / (3.14) — tI 11.I]6 [ -TI1/2 (3. 15) 72

Representations for v1(ri, X) follow immediately from equations (3.12). In order to obtain a representation for vI(nX), we find one for M k 0(2ikr). Since X/2ik, 0(2ikrl). Since ds d( ) 2=1k = Q(l/ky~) (recall that X/k| = O(kr)), d(-a) 3 1/ d? d = /1 -s d(2ikr) -2 d(2ikr1 ) and _ _ 1= ( i d(2ikrl) 2 Ti differentiating in equation (3.12) gives MC, 2 0(2ikr) = )eiri/4 -7 X/2k -[s 7 3 j1/3 Mi2o 0(2ikr) e e sr{2 s) } ' i1-) 1s1/2 Afi(-a ) 2 -81 + 0l/kr) (3.16) Ti for all allowed r. We can now assert that vl(T7, A) and vi(r), X) have no zeros 2 Ti 1 if X <-k r). This follows from what has been shown in the previous paragraph, together with equation (3.14) which implies that arg = 3ir/2 and arga = n (-a is positive, and there are only negative zeros of Ai(a) or Ai't) (Abramowitz and Stegun, 1964, Chapter 10)). Case II. -k2 T X < O; X/ki = O(krt) We proceed to consider negative values of X satisfying the relation X -k2rl. In this case s > 1, thus equations (3.12) and (3.16) are valid for M/2k 0(2ikr) and Md/2ik 0(2ikTr), respectively. However, equations (D. 3) now yield -X/kl) ds (3.17) Jl 73

For s =1, this implies? = 0, which in turn implies both a = 0 and -a = 0. Consequently, the zeros X (4A ) of v (r,X) (vI,X)) satisfy the inequality 2 2 n n 1 12 A > -k 2 (A >-k rt). We consider two possibilities for values of X >-k A}. The first assumes values of A in a neighborhood of -k Ti defined by the relation -2-1 = o(7=) ( 1 (2 0))' while the second treats those values of X for which A.,2/3 2kri (ki/) t7 (ki) In either case we note that arg 7= 0. This implies arga = 0, or that -a is negative; thus v 1(7, A) or vI (r, X) may have zeros in these ranges of X. To see whether such zeros exist, we note that for the first possibility equations (D. 7) and (D. 8) give -1 2 (s )= ( )l 1+ k 2)23$ '(3.18) ) 3/2 1 From these equations together with (3.13), (3.15) and (3.17), we find + (- 1)3/2+O ~ (3.20) 3 Ti (kT) 23/).s )= (2/3)l/6 E+ (/. (3.21) In addition, a ( /3(s -1) + (3.22) T7 k r L- (ki) '23 74

If we now substitute (3. 21) into (3. 12a) (in this range of X = 0(1)) and use the definition of vl(rj, X), we obtain /6 ^r,1/4 -,l/2k 1 1 v1(,X) = ( 2i (2/3)1/6 C(2Xk) ei/ e / Ai(-a )+(O(f) (3.23) with a given by equation (3.22). Then if -a (p = 1, 2, 3,...) is a zero of Ti P Ai(-a) = 0 (ap>0, ap+ > Op), we can perform a Taylor expansion of Ai -a +O 1 P 1 P about -a to show that in this range of X there exist zeros Xp of vl(r, }) which P P 1 are given by +O 1 /3 (P) 1 ia +) P (S e- 1) ' i L 0 / j (3.24) p k 2/3 '2 where s(P) =-k rt/X. This equation may be solved by noting that -7 P P -1 = ~(k T3) implies X = k2 E (k)+ (3.25) where A is independent of kr) and Al < (krt)2/3. Substituting equation (3. 25) into (3. 24) and solving for A yields X -k2 a P a te)2/3 ev z4/e as the equation governing the zeros <of v ) in the given range of X. 26) as the equation governing the zeros of vl(rq0 \) in the given range of ~. 75

A similar equation exists for the zeros of v (v, X). Let - p (p= 1,2,3,...) denote the zeros of At'(-) (p> O p+l > ) Then using equation (3.16) for M/2ik 0(2ikrl). we can proceed as above to derive 2/3 +. (kru) (ku) // k2 L k (( o4). (P=1,2,3. (k~)2/3 (3 27) as the equation governing the zeros of v (rl, X) in the given range of X. The second possibility for X > -k r (4n-1 )1~ \ 2,(kfl) 2^) will be treated somewhat differently. We show that zeros of vl(rl, ), (vi(ri, X)) exist, but do not represent them explicitly. This is permissible since we never need such a representation in our discussion of the diffraction problem. From equation (3.17) we note that in this case O = O(kr); thus we see from equation (3.12b) that there exist zeros of v l(r, X) if there exist solutions Xp of the equation r 1/2 2/3 op+O(1/kr) = -2k I ( ) d (p >some p). (3.28) Since a are the zeros of Ai(-a) = 0, such X exist if P P f(X)) - 2k 1.1) /2 is an increasing function of X. But -f'() 3X 3/k n J)2 k f '(L) -3/2k ( d) - (k a2 1 7 7 X/~ 76

and we can use equation (D. 5) to show fi'() - 3/2k [(/ - 1) - log T1 + + a B t) or f'() = log( -T +. ) Since s > 1, f'(X) > 0, which implies that f(X) is an increasing function of X. Therefore, there exist solutions X of equation (3.28); these solutions are the zeros of v 1(r, X) In this range of X. A similar argument can be made for the zeros of vl(X, X). We can show as above that there exist zeros AS of v(i(n, X); these zeros are solutions of the equation E 3r 1/2 ]23 + 0(1/k) ds (p > some p. (3.29) Case m. X > 0; JX/k0 = O(kr) Last we consider positive values of X (arg = 0, args = 7r). In this case equations (3.7) and equation (3.11) are valid for M 2ik, 0(2 ik) and M/ 2ik, (2ikn), respectively. Moreover, by (D.25), 4(s ) =-i (s — ) ds (3.30) 0o and as a result X/k ( ) ds (3.31) If we let s = -x in (3.31), we find 77

Is ()1/2 T=X/k ( ) dx; (3.32) Jo thus arg 7 = argX = 0. and r = O(kri). Therefore, we observe from equation (3. 7b) that there exist zeros of v1(Ti, X) in this range of X if there exist solutions X of the equation r 1s.1 1/2 r+ O(1/ki) = /k= (./ - } dx (r>some r) (3.33) J O where the 3r are large positive zeros of J o() (r = 0(k'y)). Such X exist if o r r 2 0 k { | ( )L /2 g(X) = X/k +1) x+-t or Thus ) > which implies that g) is an increasing function of and that g,(X) = is or 1 ' x+ 1 Integrating by parts, we find l".l7~, 1/2 Is dII Thus g'(X) > 0, which implies that g(X) is an increasing function of X and that 78

there exist solutions X of (3.33). These solutions are the zeros of v1(n, X) r in the given range of X. An explicit representation of these zeros is shown by Buchholz (1953, Chapter 17). An analogous argument applies for the zeros of vi(q, X). There exist zeros Ar of v1 (r, X) which are solutions of the equation + 0(1/k) = ( )) dx (r > some r ) (3.34) where the Cr are large positive zeros of J (0 (r = O(kry)). 0 o 3.2 Convergence of Residue Series We can now investigate whether or not the residue series (3.1) or (3.2) can be used to obtain an asymptotic representation of the field if k)o >> 1. We also assume kr >> 1 in (3.1) and k >> 1 in (3.2). These mathematical conditions imply that the field point is far (with respect to wavelength) from the focus, Thus there are two possible choices of source and field points for each series. They correspond to the conditions l >> 1, kg2 >> 1 or kfg << 1, k2 >>1~l in equation (3.1), kI ~>> 1, kr2 >> 1 or k1 << ~1, k72 >> 1 in equation (3.2). The first possibility for (3.1) corresponds to the source far from the focus as well as the field point far from the axis. The second possibility demands either the source to be near (with respect to wavelength) the focus or the field point to be near the axis. If we set rg = rn to obtain the field on the surface, the axis becomes the tip in the above possibilities. Similarly, the first possibility for (3.2) corresponds to the source far from the focus as well as the field point far from the axis. The second possibility demands either the source to be near the focus or the field point to be near the axis. If we set ra = r0~ in (3. 2), the first possibility then corresponds to the source far from the focus while the second possibility corresponds to the source near the focus. 79

We shall show that the first possibility for each series implies the existence of a large number of terms which must be summed in order to calculate the field. This slow convergence prohibits these series from being of practical computational value. It also negates the possibility of dobtaing any asymptotic representations. The second alternative for each series involves a difficulty of an entirely different nature. Instead of a large number of terms, we encounter a degree of uncertainty as to how many terms are needed. The nature of this uncertainty will be discussed later in this section. We consider first equation (3.1) if kl > 1, k 2 > 1. To demonstrate what was indicated above, we study the terms originating from zeros in the range argX=o(args = ). X/k = O(kr ), <k2g1, kg1/X-l >>l/(kn )2/3. Inths range we note from equation (3.34) that the separation of two adjacent zeros is defined by k k k - (1) Thus there are O(kro ) zeros in this range of X. It remains to evaluate the order of each of the residue terms. Hence, we find asymptotic representations for the Whittaker functions that appear in these terms. We observe that ds rl ikn - 1 O(1/krl) dX l/i (-) k ~ and by (3.32) dC Is 1/2 1+1 1/2 n i= 1)1/t dx+ X( i kIo)1/21) Thus differentiation with respect to X in equation (3.11) yields 80

127 2 1 /12s 1/2 d dV (rl, A) = (2ikr) -1/2 [ [ - J 1 ( )1/2 d i ~(sn). L J(S ) ' is- dX + (k 2) (3.35) k 1k7) 9. If we substitute ir = r7, set X = I, observe that (3.34) implies ( )A= A= Cr+ O(1/kTlo) no r perform a Taylor expansion of J" ( +O(1/kno)) about {r. and use Bessel's o r 0 r differential equation, then (3.35) becomes (d v( ox) =(2iko LA 3:1 /2o r 1 s(r) 1/2 '(dA9) k+ ) (3.36) The asymptotic representations for vl(r,2r) and v/l(T, A) follow from no 1 o r equation (3.7b). However, to find representations for vl(1, - A ) and v2(2, -A ) we need some additional analysis. We again identify the parameters of Appendix D as in Section 2.2. Thus arg Ar= 0 implies arg 0(r) 0 j (r) 2ik arg = j = 1, 4(- A/2ik)/ 2 () (r) (r)>s >1. Also while Ar <k22 < k 22 implies s Also 2 1 "2 s1 81

- 1 A r >> r1 (k.7 o2/3 k22 implies - -1 >> 1 I or (r) 1 s1 1 (kI )2/3 0 implies -(r) 1 >> 1 2 (kn )2/3 0 Finally, equation (D. 3) yields -(r) Ar v k (r) 1 1)1/2 ds a / (3.37) the above inequalities imply that (r)>> 1. Therefore, equations (D. 15b) and (D. 18b) yield 3 v2(2-Ar) (2ik-1/2 exp{Ar -(r) A -ic 9 -2 e gike aC((r) 1 /Z ' 92 (3.38) and A /2k (2ikl )-1/2 e_ v Q(j -Ar) (3.39) respectively, where p 2(, rl) j 1 (r)8 1)) S j =-3 s(r) 9j Now into each residue term 82

2o 1 Vl(n ' r ) 2ik o2 (d VI( v1(n, Ar x=A r we substitute for v l(r, Ar) and vl(ro, Ar) the asymptotic representations obtained from equation (3.7b), together with the expressions obtained from equations (3.36), (3.38), and (3.39) for the remaining Whittaker functions. This enables us to observe, upon using Stirling's formula for 1( + 2) that these residue terms have order )1/2 O((k)/2) Hence, each residue term is small (not exponentially small). But since there are O(krlo) terms, their sum is not necessarily small. Therefore, to obtain the field we must calculate the sum of a large number of terms. We consider next equation (3.2) if kr1 >> 1, kv2 > 1. The large number of terms which contribute tothe field now originate from the range argX = t (args = 0), 2 2 2/3 Ti kI/kl = O(kno), X >-k '2, -k 0 / - 1 >> l/(krlo)/3. To show this, we first observe from equation (3.29) that, in the given range of X, the separation of two adjacent zeros is given by k k = 0(1) This follows since for large p = -O2/3) (Abramowitz and Stegun, 1964). 83

Thus there are O(.Ir ) zeros in this range of X. It again remains to 0 evaluate the order of each of the residue terms by finding the asymptotic representations of the Whittaker functions that appear in them. As above, we note that ds,/dX = 0(1/krl) and by (3.17) that dC 1 = - dX k 1.i(l / kl) Hence, differentiation with respect to X in equation (3.16) yields d v(n,) = 1/2 f,~i/4 -erA/2k i(, { (3 1/3 2 lV v(nX) = - - C(_ e e i 4 2) - 2 1 * Ail"(-a ). TI F2 d L )dX (3.40) If we substitute rl = Tr, set X = A, observe that (3.29) implies (-a );A= - + 0(1/k1o) O p P perform a Taylor expansionof Ai"(- p+ 0(l/kro)) about -.1 and use Airy's differential equation, then (3.40) becomes (2ik 1o2 A - (i rA/2k VI ( C (P ='i ce/4e p P(s) kik =2T0 21k e. Ai(-} 2 p 1+oU(l/ ki] (3.41) The asymptotic representations for v (H, A ) and vl(r, A ) follow from 2 and since by definition equation (3.12). Since we assumed X> -k Ti1 and since by definition T1= min(Ti, H), 84

both sH > 1 and s > 1. Thus, the Airy functions in the representations for n7 (P) vl(H, A ) and vl(r, A ) are oscillatory. Finally, argX = r implies argsp- =r. Here s(P) _ 2ikg ' 4(- A /2ik) p Hence (D. 25) becomes _(P) -(d (3.42) 0o which shows P)>> 1. Then from (D. 42b) we find v( p) (2ik2) -/2 1/2 exp 2- log 2 33 with 2 (>) 1 (P) ( p)) Now into each residue term 1 An( V2( -A - )vl(H, A) V(1 A 2ikno i k 2 (dv( ) vl(o' P we substitute for v (H, A ) and v (7, A ) the asymptotic representations obP P tained from equation (3.12), together with the expressions obtained from equations (3.41) and (3.43) for the remaining Whittaker functions. This enables us to observe, upon using Stirling's formula for 85

r A^> \2ik 2) that these residue terms have order _ 1. 0..... 1 But since there are O(k7r) terms, their sum is not necessarily small. Therefore, to obtain the field we must again calculate the sum of a large number of terms. We now investigate the residue series (3.1) if k l<<1, kg2 1. We begin by noting that, independently of the condition on k1l, the terms of this series have exponentially decreasing order in the range argX = r, IX/k[ = O(ko ), X >-k2 This is shown by first substituting for v1(r, A ) and Vl (rA ) from (3.12), p p using (3.40) for d. (r ) ) (3.43) for v2(t2, -A ), and observing that equations (D. 34) imply is BP)) 1/2 P[,p) ) A li p ) 1/2 A)1/2 L1/ 0 L 1 s (3.44) with (P) given by (3.42). Then, by Stirling's formula for ( + we see 1 7A /2k k that each residue term has order O(e. Thus, since arg A = ir, they p are exponentially small with negative exponents that decrease monotonically. We consider next the range (argX = 0, etc) previously discussed, but with the condition k1 << 1 replacing k1 > 1. All asymptotic representations remain k the same with the exception of (3.39) for vl(1, -A-). Since sB = k1 << 1 if k 1, eq. 3) k ~ << 1, equations (D. 34) yield 86

1/2 A r/ l'Ar) (2ik l)1/2 ( ((r)) (3 45) vr I 1 (345) where (by (D. 23)) (r) B() 8(rr 1( 1)1 d/ (3.45a) But < 1 implies (r) 2 r (8(r))1/2 m1 1 k 1 This in turn implies (r) << r/.2k. Hence, each residue term has order -i A /2k f1 O pe r O(e ), and is exponentially small. Tierefore, all residue terms in the range I X/k = O(kr) are exponentially small. The same is true for terms in the range |X/k >> O(ko) (Appendix C); thus only terms in the range jX/kj << ~ need be considered. That there now exists a degree of uncertainty is reflected in the discussion of this range of X. We did not derive an asymptotic representation for the zeros of vI(r, A). In all probability, equation (3.6) can only be solved unme rically. Hence, the problem of extracting any information from the residue series (3.1) if k1 << 1, k2 ~ 1 remains open, since we cannot say how many terms are needed to obtain the field. If we recall the possibilities for source and field points, we see that the above discussion applies to a source near (with respect to wavelength) the focus and field point far (with respect to wavelength) from the axis. The source exactly at the focus is governed by the condition C1 = 0. This implies v1(91, X) = 1 for all values of X. I we use v1(1,X) = 1 instead of (3.44) or (3.45) in the above discussion, the results do not change. Thus for the source at the focus and field point far from the axis, the conclusions of the previous paragraph apply. 87

We conclude this section with a similar discussion to the one above for the residue series (3.2) if krl1<< 1, kr2 >> 1. We first observe that, independently of the condition on krll, the terms of this series have exponentially decreasing order in the range argX - 0, etc. This follows by substituting for vl(H, r), vl(A, Ar) and vl(ro, Ar) from equation (3.7b), and using (3. 36) for (d vl li o) as well as (3.38) for v2(, - A ). Then, on using Stirling's formula for 2 r -r-A /2k / +,we see that each residue term has order O(e ) and is 22 2 exponentially small. We consider finally the range (argX = nr, > -k n ) where the residues were not exponentially small, but with the condition krl > 1 replaced by kr << 1. All asymptotic representations remain the same with the exception k of the one for v1 (r1, A) (71 = min(l, H)). Since s = -k 1 f 1 1 1 ~ 1 X lI 1 "- 1 equations (D. 34) yield v1 Ap11/2 1/2 0171 v(n.i 11p (2ik1)-1/2()2 Jo( (P ) (3.46) L )1 -fl where (by (D. 23)) (P) (p) = | (S) 1-1/2r (3. 46a) Ti1 ik But () <<1 implies C(P) 2 P ( )1/2 ti I1i I k "1 88

This in turn implies 7r(-A ) l(p)I << 2k 171 2k irA /2k Hence, each residue term has order O(e ) and, since argA = xr, is P exponentially small. Therefore, all residue terms in the range I /k| = O(kr o) are exponentially small, and the above conclusions apply to the residue series (3.2) if klI<< 1, k2 >> 1. 3.3 Equivalent Integral Representations The discussion of Section 3.2 shows that the behavior of the residue series (3.1) ((3.2)) if k1 >> 1, k2 >> 1 (kr1 >> 1, kr2>> 1) is analogous to the behavior of the Mie type residue series found in the scattering by smooth, bounded, convex bdiies (Ritt and Kazarinoff, 1959; 1960). Since we wish to derive an asymptotic representation of the field, the possibility of a further analogy suggests that we study the residue series obtained by closing the contour of integration around the alternate or r-function poles of the integrand of the representation (2.1) ((*.5)>). We prove in Appendix C that we can close the contour and thus replace the integral representation (<2 1) by such a convergent residue series expansion only provided the inequality 1 + / < F;, or 1 + B2( < 2 - T Iif we write (2.1)as -ikR /O \ o rl - -r( + ' rV 9,' 7s M 0) R 27r i dx r 2ik. r 2ik -oo-iC (O < a < k) vl(1,Q -X)v2()2, -X) v(o) - (t7, X)v'(7),X) is satisfied. In the same manner it can be shown that for the integral representation (2.S))the relevant inequality is + /j; < /i ( f+ i < 2 '-;2). This latter inequality can be satisfied only in a limited region of the interior of 89

the paraboloid of revolution. The same is true for the former inequality, which also has the property that for source distances - > nl (. > 4r0) it cannot be satisfied in any portion of the interior. Thus, any asymptotic representation of the field derived directly from the above residue series is subject to the same restriction and cannot be used to describe the field throughout the interior. This situation is somewhat analogous to that of the integral representation for -e /R. However, in Appendix E we show that an asymptotic representation of -e k/R can be derived, without using the (-function poles of the integrand, directly from the integral representation by evaluating a saddle point contribution. This method can be extended to derive an asymptotic representation of the total field (scattered plus incident). As in the analysis of Appendix E, the extnsion depends only on the evaluation of saddle point contributions to the integral representation and is independent of the r-function p6les of the integrand. We now note that instead of considering the integral representation (2o,1) ((2. 5)) for the field anywhere in the interior of the paraboloid of revolution, we shall study the integral representation (2.2) ((2,0)) which governs the special case of the field on the surface of the paraboloid. The behavior of the field on the surface is itself important. Moreover, the relation between the surface field and the field anywhere in the interior is well known (Morse and Feshbach, 1953, Chapter 7). However, the main reason for investigating (22) ((2,S)) is that the derivation of the asymptbtic representation may be demonstrated with much less detail. It is then not difficult to see how the method may be extended to the more general representation (2.1) ((2.5)),. Although we consider only the Neumann problem, the Dirichlet problem is susceptible to similar treatement, the details of which are illuminated by the discussion of the Neumann problem. 3.3.1 Equivalent Integral Representation for the Source at (-, 0) We begin the proof of Theorem 5 by starting with equation(2.2) and proceeding quite formally at first. A discussion of the motivation will appear later. If we let 90

All# 12' o' A)= 2i(2i o) r + I(') v(-X)v2(I2' -) ) 0 2w (2ikt0) (3.47) anddenote the path -ao-la to -ia4 (0 < a < k) by C, then (2.2) beeoms F((Ia t2,3 o k. X).(C.-.o_~,)-' dA 1 (.o) ~A).. C 1 o (3.48) According to Buchholz (1953, Chapter 2, equation 20a), v1(ro, X) has the following decomposition: -tX/2k, -X*. 0 v2 (ne X) (no ) = r-/2o - v,,,.~,.,,,/(i '~:h' -rX/2ke il/2 (n ) - 0 v2%oI X +. (3.49) If we recall that v(no ) = d(21k7) Vl(, X)) 0=o then (3.49) leads to the relation -f./2k, e-I i A.) a- v 2(nro ^. vI,0 o3.i/2 r, - A.) 21k e-ir/2k ei i/2 v.,(y,X) r+Q + A).\2 2,k (3.50) This can be written as vI (vo, ) = lo -irA/2k / - riri A, e v(n e -) 16 7 -2o r ( -2l V Ik 2(not A) (3.51) which upon defining 91

e-7rX/2kv, (l -efi 4) e / V(r< e,-A) o 3ri/2 1 x\ e - 2 X)rk X(o k,'X) = - 1,4) r; e ) eiik ( V '-.) f becomes v;.(t X) = g(o, k, X) - X(n o k,;)]. (3.52) Substituting (3.52) into (3.48) yields VN( Sd F(~ 1,2' o' k, X) vN O g(o, pk, A)1-X(r1, k, A)] C (3.53) or more briefly = F VN(., no,, 0) = \ dX( - Jc (3.53a) This form of equation (2 2) simplifies the proof of Lemma 1 below (the convergence of the integrals appearing in Lemma 1 is shown in Appendix C). Lemma 1: For all integral values of M >1, M-1 VN(E 'o'.-" o)- d X &.-. xn = n=O S dX F xM g(l - X) c (3.54) Proof: From equation (3.53) we obtain 92

or VN(DT1 I) 5 F 4ft.CX VN(-* TI*1D 0 S g[l -x) "I Sc g(1 -RiX) Thus the lemma is true for M = 1. Let us assume it holds for M-1L. Then VN = 0~ 0) - M-2 n=o 9d g(l. X) g4 Therefore M-1 VN5 IR01 1: 0) " N 0 n=0 or VN(co 17f OD 0) - gW M-X Consequently the lemma follows by mathematical induction. We continue the formal procedure by now considering,, on the Interval 0 < z< the two equations (fT)n n ~2 ~ + T2 -Z(3.55) (r -)n+ 1 nzz (3.56) 93

without regard as to how they arise. In Appendix E, in connection with equation (E. 54), we prove that on the given interval w(z) is a decreasing function of z and u (z) is an increasing function of z. Thus u (z) is also an increasing function of z. Since u (0 ) — 0, equation (3.55) has exactly one real solution in n the given interval if and only if w(g1) < un( ) or 2IE;l+ 2 1 /^ & ' (j)2n We write this inequality as y< (j2-i+ j2)' ~ ~- ~n+l (< )n+ (3.57) But I+ + < /< + t1. Hence 0 - /j1 < F or '<1 Thus for fixed g1, 2 the inequality (3.57) will not hold for n large enough. Therefore, there exists an N (depending on 91 and 2) such that equation (3.55) does not have any solution in the given interval for n >,N. In addition, we can choose N such that n +l /31 4 2 for n a N. Since u (z) is increasing for all real z, we observe that for n > N and 5j satisfying 1- ((k 7.. ).1+ o() 1 k,7) r(kn) 3 94

(rO~g - r (- r_ ) an+1 2n+1 2 T2 Then for these values of n and / (rl- - ro n+1 3"I 1 2 < 2/ ~ 2/ or /( n+i.- n+1 ~, 2 (0 + (1 n3 (3.57a) In Appendix E we also prove, in connection with equation (E. 64), that g (z) is an increasing function of z on 0 < z < C1 However, g (z) is not so simple. Differentiating with respect to z, we obtain g(z): " = 1 n+ q2n+z [(2n+3) / ( y- j -(2n+1)z 2 +n z 7 z - 2n +1). Thus gn(z) has an extremum at the solutions of nr - _ I)2n+ 1.oZ Tz -f 2n+3 z or + 2o+ z) (3. 58) Squaring both sides, we see that g (z) has an extremum at the solutions of 4 2 (1 - 2n) 2 z + (2n+3) rloZ = 0 (2n+ 3) 95

(2n- 1)(2n+ 3) or at z =, z= (n - )( r. The extremum at z =0 is outside the given interval. For n large enough, - (- >; hence the extremum at (2n- 1)(2n+ 3) 1 Z nO is outside the given interval. We assume the latter to be true for n n1. For z small and positive Ir+ + O = 0 - 0no implies (2n+3) iz (JiTFz - Ji)-(2n+1)z = (+0-n)z[l+O(z Thus for n > nl gn(z) < 0 on 0 < z < (1, and g(z) is a decreasing function of < n o n z on this interval. In this case the minimum value of g (z) is i 2n+1 gn(fl) g (ro+ n+ We observed above that rro Consequently for fixed l and n,nl, gn(1l) is increasing with n. But f(z) is obviously a decreasing function of z on 0 < z < 1 with maximum value less than 4 '12. In addition, there exists an n2 such that for n ) n2 nl and fixed 1, ~2, gn(fl) > 4 1 2. This negates the possibility of a solution of f(z) = gn(z) for n >n2. Therefore, there exists an N (depending on 51 and 2', N = max(n2, N)), such that for n >N equations (3.55) and (3.56) do not 2 o 2 o have any solution in 0 < z < f1, and the inequality (3.57a) is valid for 1' satisfyi g 1 kfy x x ^ 'yo 96

We conclude the formal procedure by combining the results of the previous two paragraphs together with Lemma 1. We wish to approximate vN(5 no. 0), which is described by equation (3.53). Let N (e) denote the smallest possible N defined in the previous paragraph (with f1 =min( ),. 2 = max(W,^)). Then according to Lemma 1, (3.53) can be replaced by 6N ( vn \ dX l-x. (3.59) n=0 C C 0 SI g- X-X) C C We shall show that for klo >> 1, kZ >> 1, kg D 1, the integral defined by F N0 R(f, o, 0) = P dX g(FX) X o (3.60) C satisfies lim R(T, o 0) =0. (kro) D -~o This implies N (R)-1 We call equation (3. 61) the equivalent integral representation for VN(., r, 7, 0) if 1y >> 1, k- ~ 1, kg ~ 1; the finite sum is the one in Theorem 5. Before considering equation (3.60), we shall discuss the motiviation behind the above procedure. We do this by presenting a heuristic argument that begins with the integral representation (2.2, and leads to a series of contour integrals which have the same asymptotic representation as those of equation (3.61). This heuristic argument leads one to develop a rigorous derivation of equation (3.61).

Proceeding heuristically, we observe that one could try to analyze the original representation (2.2) by estimating, as in Appendix E, the integrand along various portions of the path C. Upon doing this, we find that the only contribution to the integral arises from that part of C defined by the condition IX/k[ = 0(kn ), argA=-68(). with 6 > 0 and 6 << 1; in this range arg s = r+6 args = 6. Then according to (D.34), v(rn, X) is governed by (3.11), and according to (D. 21) and (D.. 20) respectively, vl(S1, -x) is governed by (E. 34) while v2(2, -X) is governed by (E. 35). Thus for |X/k| > k1 and X outside the transition region of v (1, -X), the integrand is exponentially small. For [A/k| < k1 and X outside the transition region of v1(91, -A), equations (E. 35) and (E.36) apply to vl(f1, -A) and v2(92, -A) respectively. Since the Bessel function in (3. 11) is a sum of exponentials, factoring out the exponential which is larger than one on C, and expanding the other factor into a geometric series leads to integrals of the type that can be evaluated by the method of steepest descent. However, to do this directly involves many computational problems. To avoid these, the above procedure was developed. Equation (3.51) is factorization of vI(rn, A.) corresponding to the factorization of the Bessel exponentials. Lemma 1 represents the expansion in a geometric series. As one may have recognized, equations (3.55) and (3.56) are the saddle point equations. The choice of N (F) insures that there are no saddle point contributions in the evaluation of R(g, ro -, 0). Finally, we observe that g(rl,k, X) has no zeros in the region |ImXI < k (Klante, 1959). This avoids the consideration of poles when evaluating the saddle point contributions to the integrals of equation (3.61). The remainder of this section is devoted to a proof of (3.61), namely a proof that R(, r,, 0),is small. In order to estimate R(, rl,_, 0), we first subo o stitute equation (3.52) into (3.60) which gives C F(1, f2, io, kA ) N (e) R( rlo,,0) = d (. to, ) * X o (o, k, k). (3.62) Since 0 C < k represents the condition that C lies between the zeros of vI ( O, X) and the poles of Yx +,. we lose no generality in the estimation of v(lo, ) and the poles of 98

R( nr0,-, 0) if we assume a/k = 01). Moreover, we note that for n =0 the (2n- 1)(2n+ 3) extremum of go(z) at a= - 4 Io is negative. Thus, it cannot be greater than 1' Therefore, N (Q) is at least one, and R(E, r '-, 0) differs from zero for all values of r o., f. We shall estimate R(, ro,, 0) if,/rlo 0(1), 52/ 1. This configuration corresponds to that considered in Section 3.4.2, where we obtain approximate solutions of the saddle point equations. We write R(g, rn0>, 0) as N (Q) F o BR(,,O) dX X (3.63) j=l ( 0X) j where for some Mi > 0 (i= 1, 2,... 11) with M>> 1 and M>> 1 0C1= - oi - a< r<-Mlk22<0J C2 = - ia-( Ml +M2)k22< xr<-M3k2toj i C3 r a(M3+M)k r 5 C4 = - ia - (Ms+ M6)k < Xr < (M +M7 )k C= X -iaIM k<X <(M +M ) k2(T C 6 { r-ia | Mgk 70 < Ar < (Mlo M}l l) 1 2 6 r-o 2 r < 1011 < O C7= r 1 and we examine each integral separately. The estimation of R(,, rn,, 0) if /rlo = 0(1) and f2/e1 = O(1) differs mainly in the definition of the intervals covering C. This difference has been discussed in Appendix E, where we first 99

considered 1/n1 = 0(1) and 2/91 = O(1). and then investigated the resulting changes for 1/n o 0(1) and 2/1>> ~1. We immediately observe that in Appendix C the integrand of (3.63) over C and C7 is shown to be exponentially small. Consequently, the integrals over C and C7 do not contribute materially to the sum. To estimate the integral over C4, we use some formulas from Appendix E and two results from Buchholz (1953, Chapter 7): v ) (ik 1/2 (2ik )-/21k aik (2ikn)X/2ike-ff/2ker i/2e-ik v 1(nX) (2ikn) X +( 2i 2) ( ik 2)(3.64) 2ik 22ik 2X (3.64) and v2 e -X) ~(2kl (2ik ei)/2i e -ik (3.65) (3.65) Substituting (E.8) and (E. 9) into the definition of F(1, f2, rnO k, X), we obtain F(e ' 2o'.' kkX)+ F-(1 2' + o' kX), (3.66) where + C(k,,lo 2) ike, -Ik2 X/2ik2 )/2ik F e(Cis e2 1 ea k) 2e (2ikro e (e (2/ 2 ik 0 (3.67) CF(k, S1 2 -ikl 'l 2e x2 r'k +-i \ 2i *(21kl ) -/2ik(2Ik )-X/2ik eX/2k ei/2 (3.68) 1 2 with Ck., 2 =......1 NC(oehoe0 ( ) i/2 'g e 2 22r i(2ik )l/2 (2ik )/2 Note that over C v, ) governed bE. 10). Note that over C4, v2(rn~ X) is governed by (S. 10). 100

Since ImX = -a < 0 and a/k = 0(1), equation (3.64) yields 2ok 2 (2ikr1 ) / (2iko)-A/ ike ~oBy (3.66) and (3.69) -ikr1 F, 2e r ( 2iko)Al/2ik r F++, v1( X (2iko)_ /2r2k2/ [ +j or F(e1De2Dn,kX) + vl(r.) - 2X nLX( 1 2 ro)+ I(A, {1 2' o (the full expressions for I.(XA, 1D f2 ro) and I(A, 1, 2. ro ) are given as equations (E.12) and (E.13)). In the range C4 it remains to evaluate X(l, k, A). From (E. 10) and (3.65) we obtain (2i k o)-1/2 -k 2()- - e (3.70) ~. X -(2ikr1 e l) (/2ik )ikne ____ _..... -iX/2ik ilko v( e- k)- - i —....(2ik1 e' e (3.71) Substituting (3.70) and (3. 71) into the definition of X(no, k, X) we find X (k.) - eli/2 x e 2ik X(nkg) + eT i/2 2 (2ikr7) X/ike-rX./2k (3.72) Since ImX = -a < 0 and a/k = 0(1) equation (3. 72) yields the estimate that on 04 101

X(rj.,k,,X) — u- 0(M) 1 0(kn )a/k (3.73) But from the definitions (E. 12), (E.- 13) we see that on 04 Therefore, on 0 4 0(1) (k1a/2k kla/2k (ka4/2kk )a/2k ki0) F VI (n ',X N0(9) 0 OM1 L (k g ) cr/2kk /k+ 1 + IN 0(9) + Since C 4is of length 0(k)., 4 w% kO(1 (kg )1/2 kg ) (knW )1r/2 1 2 0 1 +(K )-a/k 10 1 - or, since K 1> ~1 and a/k >0O, 102

3 NF d( 1 ( )o/a2k (1) 1 0 vIo ) (k )l 2 (k2 )2a/2k 4 - -0(t9 - 1 ' oi7k (3.76) The behavior of the Whittaker functions over C2, C3, C, and C6 is governed by the asymptotic representations of Appendix D. We consider first the intervals C2 and C3. On these intervals argX = r + 6(X), where O < 6() << 1, so that / ikg \ arg s T- 6(WX) i = Thus from (D. 34b) we find v 1(1, -X) is given by (E. 5); and using (D. 42b) we conclude 1/2 X 2) 2 1/2 ex 2k 2ke ( t (3.77) (-X/ik)l/2 ex p where for both v 1(E1 -) and v2(2' -) 1~ 1/2 X ) de (j 12). (3.78) Moreover, f1 s has the following possible order of magnitude relations on C2 and C3: (i) 8 >> 1 with l/kl =0(1) (ii) Js i 1. with 1 <<~X/k << kro (1) |s, =- 0(1). I >> 1. with IX/kl O(kro ) (recall that we assume I [>>I. -,,~ 103

l 0 = O(t70)' 2 >> l) ' (iv) s 1~<<. |s [= 0(1) with |J/k|=0(k2 ). (v) s ~<<1, s1 <<1 with |X/kj> (k2). Possibility (i) can occur over a portion of C3 of length O(k). The relations se >> 1 ( ikg>> |I-X/ik|) and IX/k = 0(1) imply that the equations (E.8) and (E. 9 are also valid for vl(1, -.X) and v2(2, -X), respectively. Furthermore, the relation IX/ki = 0(1) indicates that the formulas corresponding to the interval C4 apply. Therefore, equation (3.76) holds for this case. For the other possibilities, we begin by using (D.26) in (3.78) to obtain = ) -1 - log ( -] (3. 78a) Since arg s = X -6( ), arg (-s ) = -6() and arg(l - s ) 0. This implies arg -s (1-s ) -log i - s - F~0^v for all possible values of se on C2 and C3. Hence on C2 and C3, arg Cg arg(-X) ~ 0; and since I J f >0 (koM0) (again recall that we assume 1 = O(i ) and f2 >>~ 1) to approximate v l(1, -X) we can use the exponential representation of the Bessel function in (E. 5) while equation (3. 77) can be replaced by (E. 6). If in these equations we set 1e =Re e +iIm C and note that Re C (X) -? (r ), we find that for X j j j J j (X -ic) on C2 and C3 r 2 3 2 r1/2 rice ( -Imp x) -ire (A ) Imr (C ) -(2ik l)1 /2 e 1 )e ()+ie (r) e 9(J) 1 i(3.79) 104

. X 1-l E2(r) Im C 2(X) v2(2-) (2ik )-1/2 )1/2 xP 21k g 2ike 2 (3.80) If we now consider possibility (1i), we note that the substitution of (D.32b) into (E. 5) (exponential form) and (E. 6) shows that v I( 1 -X) and v2(e2,-X) are again governed by (E.8) and (E.9), respectively (as well as by (3.79) and (3680). Moreover, equations (E. 8) and (E. 9) may be written as (2ike )-1/2 C (iar) 2kea/2 Le (2kl )a/2k + /2k1 1 -i:;l(X) Elk(Aor(2 k O(1) ~ 2ke i mmJ a _ (3.81) 1 1 xex -x — 1 v2(,-X)-, (2ikg )_1/2(-i)1/2 ep 2ik l 9y2 (2k)off/2k a (1) * j j2ke / e (3.82) x 2ke which is the result one would obtain by substituting the indicated imaginary parts in (3.79) and (3.80) and retaining the terms which are not of 0(1). But in this sub-region 105

k,1 ~ /k1 | IXr/kl; thus - I' IXr/2ke a/2k >> " a.2k (2kz ) 1 a and (3.81) reduces to (2i )-1/ 2 2 - (r) v1, '-X) ~ - F -,1/21 1 1 ^ (s ))2 ((2Cg )a/2k 1 X /2ke |a/2k r^ k 0(1) k e (3.81a) For possibility (iii), we see at once that v2(92, -X) is again governed by equation (3.82). It remains to consider |sI 0(1). In addition, we can conider simultaneously s I 0(1). Thus for a 0 (1), we first note that Im O(6(X)) j(Xr). Then by (3.78), (X() =I X/k 0(1) and since o(6())- O(a/|X ), equations (3.79) and (3.80) become vl(l, -X) ", v2(2, -X), Finally, ing Is ~1<'. (21kg1) /2 (2kU )O1/2 2 0(1 e (3.83) 0(1) k (3. e4) (3.84) -i2 Xr) i ^2 we can complete the study of possibilities (iv) and (v) by considerBut by (z 27), s << 1 implies i 1/ 1.0]f I1/4 < - --— c ) 1 9 < [0 i J 106

In addition by (D. 28a),, (r) = ( |Thus O(( 1/2 1 k Ir since s << 1 implies \| >> k2 2. Therefore, equations (3.83) and (3.84) apply. To find the representations of the Whittaker functions which depend on nj, we first recall that on C2 and C3 argX = ir + 6(X), where 0 < 65() < 1, so that / 3 bn\ arg. -6(X) 0a Then equations (D. 21) show that equations (3.12) are valid. This in turn shows that vi(rlo, ) is governed by (3.16) together with (3. 10). In addition, we see from (D. 15) that v2(r(, X) is given by (E. 7), while from (D. 14) we obtain K io,i)-1/2 1/6 x x v2(ne -X) - (2ikre- l(-A/k) exp - logA ~ erk/2k (1)( 2 (. (3.85) The behavior of the derivatives v(rno, X), v( e -) follows from that of v2(17,X), v2(rje,-X) in the same manner as the behavior of v(rT, X) follows from that of vl(r, X) in Section 3.1. However, the value of? depends strongly on the position of X with respect to the transition region. This must be considered since the width of the transition region, defined by 2 __ _/3_ AXA -k X- =- X o(;b 2 =kO k ) 00 is A = 0o (( )1/3 while the minimum distance from the path C to the turning 107

point -k 0 is A-k = =0(1). Consequently, in this range we need to study three oases, namely s > 1 and away from the transition region >1 > 1 IB s-lI~ --- —---- o (kn,) 2/3 I|S < 1 and away from the transition region, s in the transition region. For 1s j > 1 and away from the transition region, we consider first s 1 >> 1 with jA/k [= 0(1). This can occur over a portion of C3 of length O(k) and has already been discussed (possibility (i)) when studying the Whittaker functions which depend on 1 and 92. We next consider s J > 1 and away from the transition region with either 1 << /k~ < k ( s 5 1), or I|/k| = 0(kOl) ( s = 0(1)). In either case, equations (D.3) show that i is given by (3.17). By (D. 5), r = -X/k l8 ( -1 - log s + (3.86) But args a -6(X) and arg(s -1) 0. The latter relation is obvious if 70o >> 1. If |s = 01). it follows from the definition of 6(X) = a/l |r. since s 1 0(1) implies Xr/k O(kno), or 60() - 0(l/kno) = 0(1/kTo). Hence / k2 -1 _ -1 — 10(1/ko) 0 r and arg(s -1) not close to zero implies k2~ - -l=0(1/k). r 108

This is contrary to [sao being away from the transition region. These estimates for the arguments of s and s -1 show that in the cases under consideration arg s (s -.1) -log n- 1 + JS t0 Consequently arg? ~ o arg(-X) ~ 0, and /3-0 arg(-Z ) + (a ' ) ) 7 Since | [ | =(kni), the asymptotic form of the Airy function given by Erdelyi and Swa n (1957, equation 4.6) can be used In (3.12) to obtain,V - - t o+ie no 2,B,.( /)l2 (3.87) while the asymptotic forms of the Hankel functions given in Section D. 1 can be used in (E. 7) and (3.85) to obtain 0 (3.88) VI(noe -A) (2ikeo/e ) /exp 3 g2le} et /2k()e o D i (3.89) where in all three equations we have substituted (s) for(s -l/1/ If s I >> 1, the substitution of (D. 1Oa) into (3.87), (3.88), and (3.89) shows that v(7,X), vI(no.X), and vI(n e-,-X) are governed by (3.5), (3.70) and (3.71) respectively. Moreover, equations (3.5), (3.70) and (3.71) may be written as 109

(2 iki )l1/2 VI (noA) v"' 0 0-irX >/2k [e + i? 77 (Ar) + ie0 aO ek (-) e 2 70 D (3.90) -1/2 Fx Xl -ic 7( VI'(vi X) -- (2ikn )'M exp - log ----(-)e 0 2 o 0 L2ik 2iJ a O k0i 2 \Y'% /n 0 (3.91) Ti r Y.r VI (70e.1-X) — % (2 ikIQ e f~/exp43 log~jT 2 (2kTI )a/2k eX2k (i e 0 N 2kiera 2 a O In0(1) 0i s /, (3.92) which is; the result one would obtain by setting C = RerC + i~mC 7 in (3.87). (3. 88), and (3. 89), an notin that ReC1 (X ^ — ( Cx)O wh~Ile retainn the terms TI0 r which are not of 0OM in Im. 10 But in this sub-region kT ~> IX/ kI'- x r/kb; thus 0 1 X/2ke 1a/2k I r 2ke] a/2k (2kTI )a/2 0 I and equation (3.90) reduces to (2kT.1/2i () vt X ^.* oW70 -T X/2k Tie 0 xr) v(n.X * e i (2kTi0)a/2k A. Xj2ke ja/2k a O k e 0 110 (3.90Oa)

We now estimate the behavior of the integrand in the case corresponding to formulas (3.90), (3.91), and (3.92). In this sub-region I << /kl<<kro, thus equations (3. 81a) and (3.82) are valid for v l(l -X) and v2( 2, -)), respectively. Substituting these equations into the definition of F(1, 92' r7o. k, X), and using (3.90a) for v(n o,X) yields F (2kg1)a/2k(2kg2)a/2k vi(n' X) 1 2C( k. ) 2 a/2k |k) a/2k 1 2kto * r2Ie r 2ik) +exp iklog 2ike. e-./2k 1_(s2 2 rlo e 1)0(s 2 ))(s% )]1/2 LI ~^ 2k nhilj -irl(X)r) e 1 -itS2(r) O(1) e e But argX = v +6(X) implies arg2 = + 6(), as wellas arg+ (). Hence, by Stirling's formula (Erdelyi et al, 1953) i2k 2 ik 2ik and 4 2ik 2 2 exp k log- 2 Inserting the first form into the above estimate of F/vi(ron X) shows that in the case under consideration IF.4 I(.0, x) 0(1), _ - 1 a 1 a 1 a (k)2 2kk2)2 2k(ko)2 2k 2 o X 2k e snce Xr < 0 and 1 << IX /k I< k7. But a > 0 and 0 < a < k. Consequently, F/vi(ro,X) is exponentially small. 111

To estimate X(r7, k, X), we substitute equations (3.91) and (3. 92) into the definition of X(o. k, X). This gives X A +- exp log 2, ke. lr/2ke l / k+ 2) exp - 2A log,,ke- e/X-/2k (2ki7o')/k -2i ^o ()r) a 0 e 0 k (i) e Using the above forms for the r-functions, we obtain -2ir o(xr) Om Irj/2ke (X/ 2ke V > X ^ ie o e 0 - -: =0 ~ (2k 7 Since Xr/k ~<<kRo, IX I<1, andthus X h<1. Therefore, i/ xr No( ~ is also exponentially small in the case under consideration, namely, s >>1 with 1<<X/k<<~kn. We now consider |s > 1 and away from the transition region, but with |s = 0(1). If in (3.87), (3.88), and (3.89) we set C1t = Re? +ilm1n, and note that Im? =!=(al/Xrl)Ren, e (X) Re(?,r Ck (L) r |/k I (1) 0o 0 0o vIo(no,.) ~ o e-i /2k ' e + ie r ek 2(ik)-, (3.93) 112

-k -0 (1) vro.) (2ikro) e7exxp kl 2ik e e * (-1/2) ((2 )) 1/2 v(r7 e -i) -1/ 2) (8 )a /e k (3.94) (3.95) To estimate the behavior of the integrand in the case corresponding to the above formulas, we first observe that in accordance with the configuration of r)?1 and F2 under consideration the relation js | 0(1) (MX/kJ = O(ko)) implies that the relations sg l 0(1) and s r~ >1 are valid. Thus v1(e,.-X) is governed by equation (3.83i. while v2(92, -2 is governed by equation (3. 82). Substituting these equations into the definition of F91l, 2'1 n, k, A), and using (3.93) for v(I(n,X) yields (2kg )a/2k vi(,o,. 2C(k, 1 r,2' ) I /2ke /2k k 1 x! r + )exp 2log ie - rX/2k L-_I '2 2 e AS, )'s- )p(s' )/ 2 1) -i l -i n * e( () Inserting the above asymptotic form for P(2- + ) into this estimate for F/v (rpo, X) shows that in the case under consideration 10 113

X r _ F 0(1). 2k I) 1 ~a 1 a (kgl)l/2 (k 22k )2 2k since Xr<0 and iA/kl 0(kno). Consequently, P/v.(noA) is exponentially small. To estimate X(ro, k, A), we substitute equations(3.94) and (3.95) into the definition of X(r, k, X), and using the above forms for the r-functions, we obtain -2i (r) k 0(1) X ~- ie o e = 0(1). Therefore X/ XN() Therefore X ~ 0(1). and( /vI(n,X))X is also exponentially small in the case under consideration, namely, s > 1 and away from the transition region but s t(1). For s < 1 and away from the transition region, we consider first the 1 0 case where is is not much smaller than 1, s = 0(1). In this case equations (D. 3) and (D. 6) show that? is given by equation (3.14). If we perform the inteno gration indicated in (3.14), we find Om X I396) = i[ - sin ji - J-T ( s1-. (3.96) But arg s = -6(k), while the previous argument indicates that for s= 0(1l), arg(1-s ),00. Since nO Re _ -sin - (s (1-s ( ) dt 0 (r) (r) - r 114

0<0(1) = 1 (r)] t(rL i)(r) -(r) dt r-sin-s i- (1-8 ) o t o a these estimates for the arguments of s and 1- s show that arg -1 - (1-s8 VO. Then arg " ~ arg() -. 33r/2, and arg(-C ) O 0 O 2 ( 3) Since r r = O (kyr ), the asymptotic form of the Airy function given by Erdelyi and Swanson (1957, equation 4.5) can be used in (3.12) to obtain (2iko) 1/2 v'(n,) - -irX/2k -i o e e (s )),/2 ro 1 (. 3.97) 0 o/ while the asymptotic forms of the Hankel functions given in Section D. 1 together with the continuation formula (Erdelyi et al, 1953) H(2) ( I ri) i/3. (), 1/3 1/3) 1+/3) can be used n (E. 7) and (3.85) to obtain v.(7o0) ~ (21kro) -1/2exp4.log x - ) l e ^(. 1/2 1 2 (3.98) 115

(k 7\ir-1/2 fj \ x 1 v(~e e -- ) (2ikoe O) exp - log 2ik 2 -i ) * (3.99) where we have neglected the term containing e ~ in (3.99), since argl ' 3v/2 and |r |o = O(k. ) implies that this term is exponentially small. 10 We now estimate the behavior of the integrand for is | < 1, so | not much smaller than 1 and s away from the transition region. Using the asympno totic forms of the P-functions together with equation (3.98) and (3.99), we immediately observe X(rn0, k, X) 1. If in equation (3.97) we set?q1 = Re? + Im, o 0 110 andnotetthatjRe? I O(/IxrI) ImV n II Iron | r)I I? (Xr)| = 1xI/k|O(1), this equation becomes '1/2 '.2 ~ "om '2 (2ikn )/ eX /2keio(xr) iOk(1) 1 V X) (/?. e e0 0 (3.100) As above, vl(1,. -A) is governed by equation (3.83) and v2(Q2, -A) is governed by equation (3.82). Substituting these equations into the definition of F(1 2 n I k, AX), using (3.100) for v(qo, X) and the above asymptotic form for ik 2) + 12we find that exp fy d (t ) } F 1M0(1) -10 vk.,X) 1 (kg. 2+ ro.. )l/2(k ~2 2k(kqo)2 2k ~1-2k 116

since X < 0 and | /k | O(knr). But in Appendix E we show (investigation of C ) that the ratio jIx - - 1../2. 2k e is exponentially small in this case. Therefore, F/v(rlo, X) and N () 1 ~ (F/v(i.,X)X 0 are also exponentially small for | < 1, 1 | not much smaller than 1 and s away from the transition region. r)o We complete the study of s < 1 and away from the turning point X =k r by considering s ~<< 1 (sn 1/(k 1 )27/3). But args - -6(), and this together with ]s << 1 implies that arg(l - ) 0. Hence, equations (3.97), (3.98), and (3.99) also apply in this range. We immediately obr X(ro,k, ) ~ 1. For Is 1<<1, s ) 1/2 4. and sincethe rlo T0o relation jI (X r = IXr/k 0(1) remains valid in this range, equation (3.100) is 0 replaced by -1/2 i (X 1(n ~e e e 2 1. (3.101 V( o F e-e / r e k * 1 -475. (3.101) Furthermore, in accordAem with the configuration under consideration 9,1/n = 0(1).,2/1 >> 1. the relation s. << 1 implies that the relation Is] 1 1 s valid, while I ] satisfies either ) r Is = (1) or 1. Thus IEl I2 at2 2 in this case vl( 1, -X) is governed by (3. 83), and v2(92, -X) (either possibility) is governed by (3.84). Substituting these equations aton the definition of F(-1,+ -2n k, ), using (3.101) for v(r, X) and the above asymptotic form of rpGa - + we find that 117

exp k-1jrt (f)"} F 0(1) '10 V) 1/2 11 1/2 ' X V(oA) (kgl) (kg ) (kn 1/ r 1 2 0 k e since X < 0 and /kI~ 0>>(krj ). Then the argument of Appendix E shows that F/vI(r~, X) and consequently (F/v(, X)) () X s exponentially small for |S << 1 (Is >> /(kn)2/3). The case where s 1 i/ in the transition region (jsJ0 \ 1) can be handled with much less detail than the above ones. If s is in the transition region, then o 1/6 equations (3.18) and (3.19) show that?70 = 0(1), 1(s )a~ (2/3). Thus the Airy and Hankel functions in (3.12), (E. 7), and (3.85) are bounded independently of kro0 and equation (3.16) implies v = 0 (l/(kr) 2/). Usng the asymptotic forms of the r-functions, we obtain IX(no,0 k, X) < Ml, for some M1 independent of kno. But j| 1 1 implies Is1 | = 0(1), IsI | 1. Ience substituting (3.83).o th a I E2 for vl(l, -x) and (3.82) for v(e2, -X), the above order of vi(ro0, ), and the asymptotic form of r( +-j into F/vi(tl X). we observe X N (_ 0(1) ' VI F.X 1 - 1_ (kgl)1/2(kg2)2 2k(k o)2 2k N () since X <~ and | k/k |O (k%)Trf. (/ X is exponentially small for |s J 1., 0 * The above range (the range for which s I N 1) exhausts the possible sub-reglons of C2 and C3. In these sub-regions we have shown that the integrand 2 3' 118

is exponentially small, except possibly in a sub-region of C3 where it behaves as in the interval C4. Therefore the integral over C is not contributing materi4l 2 ally to the sum (3. 6), while the integral over C3 is either not contributing materially to the sum or equation (3. 76) also exists for C3. We continue the study of equation (3. 63) by considering the integrals over C5 and C6. In this range argX = -6(X), where 0 < 6(X) ~1, so that args r+ 6(X) Thus equation (D. 34) shows that v (I7. X) is governed by (3. 11) together with (3.10), where rn is given by (3.31). Moreover, equations (D.41) and (D.42) yield v2(r1e X,-) (2ikre-)1/2 1/2 exp 2e t/2kV(l)( (3.102) -1/2 -(- 1/2 F - } v(2) V(7,) (2i) exp log - V ) (3.103! 2k1/2.XP 2 -with ~ again given by (3.31). The behavior of the derivatives v(n e 1, -X), V Iro 2(lo,X) follows from that of v2(7r e, -X), v2(n, X) in the same manner as the behavior of vi(no, X) follows from that of vl(rn, X) in Section 3.1. In addition, by (D. 26) o kL t '1- -log( - lOJ. (3.104) 0 O O 0'I Since args = +6(X), then arg-s = 6(0) and arg(l-s ) -a 0. This implies arg v-s ( 1-8 -alog( -;;0 ' 'o i ro o for all possible values of s in C5 and C6. Hence on C and C we find o 6 119

arg o " argX '0, and since j >| O(kro ) we can use the exponential repreo '0 sentation of the Bessel function in (3.11), the asymptotic forms of the Hankel functions (Section D. 2) in (3.102) and (3. 10.3 co obtain (2ko) -1/2 -i2i'.o i /2] -/ v'(,) -..* - [ - e 7e i/2 (s 1/2 o1 0) 2i L "e 1 lo O0 (3.105) -V,1 -) (2k-1/2 1/2 / e i^eO o * (-i/2)(p 2'))-i/e (3.106) v(rl e,-X) (2ikn e (-i) exp 2klog- 2 0/2i ( q(o/2 2 ~ )-1/2 ) )1/2 )( ) S o behavior of the integrand. We first note that argX = -6(X) implies arg/ik = 6() and arg- 6(), which in turn implies the validity SubsttutWe nowg thestigate together with equations (3.106) and (3.107) into the of the asymptotic forms definitions of X(, k, X) and g(ro, k, X), we find 120

-2iC I i/2 X(#,k,o) * e eoe 2 (3.108) (2 k. o)1/2o 0-i /4 1/2 21o - - *.- - (3.109) In the sub-regions of C5 and C6 there are four possibilities for |sr | namely, s >>1 with |x/k = 0(1), I ~ 1 with 1 ~I|/k| ~kr, i tI 0 and |A/k| - o(1) imply that ln this case v'(r,X) v2( X) and v(1 e 1,-) 10 v 0 O 6 are also governed by (3.nto (3.70), and (3.71), rewpectively. Furthermore, the relation |/k| 0(1) indicates t(at th o c the formulas corrsponding to th rval C apply. Thpreforo, equ ngth O(k). h e relations of where /k| - 1). For js ~> 1 with 1 ~ X/k! < kr, we observe that the substitution of (D. 32b) nto (3.= O105) (3.106) and (3.107) shows thate (,,). v(), ) and Var( e, -X) aragaingoverned by (3.5), (3.70), and (3.71), respectively. Moreover, these equations may be written as (2iko) 1 r) -ir/4 (2k )a/2k O(l) v(X) - - 2i (Xr/2ke) a/2k _..2r 0,or) (-r/2O)a/k ea ~( 0o(1 kL )(3. 110) O (3.110) 121

VI.-K i -7ri -1/2 1/2 A. A. 7rX/2k 2(no 0 0 -X) ~ (2ikio e (-i) exp - 2ik log - 2ike e (Lr/2kefa/2k e -12 VI (n. X) -'o (2ikn.) /(il2expji log - 2j ji) eTO (X /2ke)'7/2k *g 0(1) (kI)'7/2k e.(1/2) (s7))/ which is the result one would obtain by isettin r =Re r'I +ilm C'I in (3.105), o (3.106),, and (3.107), and noting that Rer (x) —r- (Xr) (X= - Ia), while TI r r 0 0 retaining the terms which are not of OM1 in Im~. Thus -2iC (Ali / (Xr/2ke)/k (.0(1) X(TI, kp X) -,.. e nor0v2ek (3.111) 02~o / adsince kn4,~ I>X/ki-oA Xrk, equation (3. 11 0) reduces to (2ikrp )1/2 irT11o(xr) (2kv1) a/2k.2O ~ ge - i r / _ _ _ _ _ _ v io )A - 4yr21 e/2k e 1 (Xr/2ke)a (3. 110Oa) Ifsa7 = 0(1) then in (3.105) and (3.108) we set Cn =Re C7 +limhnCn T0 TI0 T0 and noting that fIm IO I= 0(c/xfRe~ In (X =)Lr*i(p Ix> O)s Re~ C'O (Xr)t 122

X r r k oA)-2= 0(1), we obtain -2iC (X )( X(n, k, ) e i/2ek (3.112) 0 (2ikrlo)-1/2 ie n/4 k 0(1) V (noA " - 2 e e 1-2ilo(}r) li/2 kTl 1/2 -e2 ( r) 0ei/2ek (s )1/2 (3.113) This latter equation contains the function o ro(1) Y(ro,k,,a) = l-e?o r eri/2ek We know (Section 3.1) that there exists values of X in this range for which -21 (kr) r i/2 1-e ~e e / 0. But since a/k = 0(1), no such Xr can exist for Y(ro, k, kr, a). Finally, if s ~<<1 ([ X/k >> k ), the substitution of (D. 28a) into (3.105) and (3. 108) shows that we can use equations (C. 4) and (C.21) for v'(rno X) and 1o X(0, k, X), respectively. In order to find the representations of the Whittaker functions which depend on j,. j =1, 2, we first observe that argX = -6(X) implies ( ikg Ni args =(X) -X/ik Then equations (D. 21) and (D. 20) show that vl( l, -X) and v2(2, -X) are given by (E. 34) and (E. 35), respectively, with C given by 'J 123

-k (x-7)/ de J! However, in this range s depends on the position of X with respect to the turning point X = k2 (. Tils must be considered since the widths of the transition regions (defined by (E. 78)) are again A 0 (ko)1'/), while the minimum distance from the path to the turning points is A- a/k = 0(1). As indicated in 2 Appendix E, we do not allow k 2E to be in the transition region of v1(1, -A.). Consequently, in this range we need to study the following cases: | I| < k21 and away from the transition region of v l(l, -X), X in the transition region of 1(1, )) I, X I > k IX <k 2 and away from the transtion regloof both vl( l,-A) and v2(2,0 -), X in the transition region of v2(E2-X ), LI |>k2e and away from the transition region of v2(2. '.). For IX | < k l and away from the transition region of vl(1, -X), we consider first the case |I/k] = 0(1). This can occur over a portion of C5 of length O(k) and has already been discussed when studying the Whittaker functions which depend on o. We next consider X | <k <1 and away from the transition region of v1(1, -X) wit thther 1 << I/k|<< k1 ( si |>>1, s | 1) or | 4k 1 m t0(kLC = >ki) (Js | - 0(1), s.j | l1, recall that the configuration of El1 12' and ro under consideration assumes =1 O(ro0 ), ~2 >2> ~ ) In either case, equation (D. 5) shows that [/s a-j) -log I + j j But argsf = 6(A) and arg(se - 1), 0. The latter relation is obvious fl If |BJ | = 0(1) (although the relation B.2 | 0(1) need not be considered for this case, we develop, for future use, the estimate of v2( 2,. -) for s1 | - 0(1) J I 2{ 124

along with the estimate of v1( 1, -X) for s I= 0(1)). it follows since arg(s -1) not close to zero is contrary to being away from the transition J j region of vj(C, -4.) (see the previous argument for s and X in the interval C ). These estimates for args and arg(s -1) imply j fj arg [jji) log( C + [i] -. As a result, arg - argX 0. Then since 1?3 |0 (kr1). vl(1,-) and v2(2,' -X) are governed by (E. 38) and (E. 39), respectively. If t l|I/k|<< kl (s1 |>> 1), the substitution of(D.lOa) nto (E. 38) and (E. 39) shows that (E. 8) is valid for v1(1, -4), rolle (. 9) is valid for v2(2, -X). Moreover, these equations can be written as v(,) (2ik2)-1/2 ~- A /2k ea 2k \ 1 -T e 1 a/2k + (2kC1) (2k)k 2 0(1) (X 2ke (3.115) (a/2(/ e ) 21 (2kC )Y/2k O e 'mom 2 i (3.115) 125

which is the result one would obtain by setting f =Ree +iIm in (E.38) j j 1 and (E. 39). and noting that Re F (X) - ( r), while retaining the terms which are not of 0(1) in Im. But since k1 l>X/kj kXr/k, equation(3.114) reduces to -1/2 _. Oa(1) (2ikg) / 2k. 1 e. r V1Q1. e 1 /2k e os (X a2ke) 1 r(3.114a) We now estimate the behavior of the integrand in the case corresponding to equations (3.114) and (3.115). In accordance with the configuration of y1, f2 and ro under consideration, the relation 1 << X/k [<< k7o is also valid. Thus inserting equations (3.114a) and (3.115) into the definition of F(y1, 92P r, k, X), using (3. llOa) for v'(r, X), and the above asymptotic form for (2k + 2) we observe F 227 C(kglA2ri0o ) (2kg1)/ (2kg2)a/ XV It [n(s, )( )k )])(1/2 /2k (X/2ke)a/2k 1 2 0 ) 2 0(1) i-(~ ( with -(x) =- (X ) - (x )- (). 1r 2r r But in this case W k2 2 1 r 2 r 1- (r) - s) In and ito, euti Hence, equation (E.48) shows that Jk (r) = Xr/k 0(s ). In addition, equations r 0 o I 126

(3.104) and(E.9 ) = /k (s( ). Therefore, (3.111) and the above estimate of F/v (o, X) yield F vI(r,AX) 1 o N (~) X 2 27 C(k, 1' 2' no) sr(s )0(s r)0(s dl2 1 2 03 ^ (2kl )U/2k 1( (2kt )/2k 0 (2kg2)a/2k 2 (X/2ke) /2k ) N i- o (X<) 0 r N (S)_ (X /2ke) ~ k. r N (S) (2k7o) k 0 - 0(1) ()1/2 ek n iN () 2 e e (3.116) where N o ~(r) = 0 r Xrk(), (r)) 2 -(2N )+l) X f ~o(s(r)) The estimate given by equation (3.116) enables us to estimate the integral _ F N o ) vl(1X) 0 where C' is the sub-region of C discussed above (1 << A/k| << k). For X = Xkr i, dA = dXr, and this integral becomes 5 d F N ( ) 0 r vi(no'X) X (C )' I ~ 5 r with (C)r = r X -i C5 1 << 5 r r r 5' k << kS 127

Substituting equation (3.116) into this latter integral, we see that it may be estimated by the method of stationary phase. It follows from the computations in NAppendix E ((E. 50) through (E. 53)), that ~(r ) has a stationary point in (C )' o r 5r provided the equation (/ 2N ()+1 ( - o has a solution in 0 < z < 1. But N (g) was chosen so that no such solution exists. 1 o Therefore, the integral can be estimated by integration by parts and we obtain F N(?) F N0() d v. x o -) o' v, r v(no X) 55r 1 1 a 12 i (1o+) ((kn )/ 2 k 2 0 2 )1 ( 2) (3.117) We now consider I It<k 2 1 and away from the transition region of v,(l,-X), but with |/k = (kEl) ( =0(1) |s >> 1). We also derive but do not use, the estimate of v2(e2, -X) if s > 1 and away from the transition region of v2(.2, -X), but s2 = 0(1). Thus in (E.38) and (E.39) we set - = Re + i Im, and if we note that Im = O(a/r )Re j r Re i) r i (X ). (A) O(1), we find r r k (2ik -1/2p e a ( 31 fl'^. - e, / e (3.118) 128

1/2 x- i 2(x ) oa v2(-2, ->)- (2ik2) /exp 2ik log- 2ike 1/2 e 2 (3.119) To estimate the behavior of the integrand for jX < k 21 and away from the transition region of vl(1, -X), but IX/k = (k) ( =1) ), we note that in accordance with the configuration of g1, 2, and rio under consideration, the relations /kl = (ko) ( s = 0(1)) and [/kL[<kg ( |s > 1) are rlo 2 also valid. Then v2(i2, -X) is still governed by (3.115) [with (A /2ke) = O(kEg while v (9, -X) is governed by (3.118), and ven7m X) is governed by (3.113). 1 11 0 Using these equations and the above asymptotic form for r + we find -a/2k o(1) F 2 i7 C(k, g 2' )(2k2) /2k 1/2 k_ () VI-( 7 X) 1/2 a/2k Y(67,k, A.oa) )0(s )(s ) /2(2k) k o r rl 2 ro But in this case the relations (r) () (Xr) sr >1 and >1 r 2 r r 11 2 77 0 together with equation (3.112) gives together with equation (3. 112) gives 129

or/2k F N0(9) 2 27r C(k, 9 1, 92 -P flo )(2kg2) 0 x 0%,f - VI (n OX) - 1/2 a/2k 1 0 (s9 )0(s9 )0(s 77) (2kg 1) 1 2 0 (j)1/2ek Y(t~~,k, X au) r 3 (3.120) vinN (Q) 2 e iI where N N 0 (A.X) A.r (r) 11- 7 (s9 )1 i~2 (N(M+) (r. (r)) k r.ios ) o -X - ((r)) k 0 -r D (0r)(N(+ - - (s )- N g2 i) x r (r) k (is ). 770 The estimate given by equation (3.120) enables us to estimate the integral F N 0(~) 5+06~VI(n,X) where (0 +0 )' is the sub-region of C and C discussed above Lt O.I 56 0(). s 15 1~1 For X =Xr-icv, dX =dXradti integral becomes dX F Nx r v~~f.A (0 +0) 0 wt(C +C6)=Xjx -ioEC5+C2X <k2 1and away from the transition region of v11 (,,but X.k =0(kgij 130

Substituting (3.120) in this latter integral, me see that it becomes two integrals which may be estimated by stationary phase since Y(ro, k, Xr, a) has no zeros in (C + C )'. It follows from the computations in Appendix E ((E. 50) through 5 6r (of thes. 53)), that one integrand has a stationary point in (C +C ' provided either of these equations VT --- 1 z + 1 o +z - T70 v T 2 -z + V72 z + , z + l/= F)a + Z- fl 0( 0) 2N (~) +1 zjf (T) o 9 have a solution in 0 < z < 91. But N (e) was chosen such that these equations 1. o do not have any solution in 0 < z < f1. Therefore, the integrals can be estimated by integration by parts and we obtain F N o() v(.o, x) ) ( (C5+ C6)r F oN0() r v{(,,X) '(C5+ 6)' 56b 1 __' - -(1:+a) (-a) (k3 )1 -) 2 2 k(3.121) 1 2) For A in the transition region of vl(fl -X), s - 1. Thus in accordance with the configuration of 1l, 2', and ro under consideration, s 2> 1 and |s = 0(1). In this case | | = (1). Hence, the Airy function in (E.34) is bounded independent of kr7, while (D. 7) and (D. 8) imply j(s )~ (2/3)1/6. Moreover, equation (3.115) [with (Xr/2ke) =0(kg1)J is again valid for v2(2, -X), 131

while vI( JX) is again governed by (3. 113). Using these equations and the above asymptotic form for (ik +-), we find F I(r10 X) _e a 0(1) pC(k.. ^ )p(2 )/2k (2/k) /6Ai(_-' )e 2 2 ___ 1 rS 7 (s i 12 (2kg )U/2k Y(t7lk,Xo) r'"2 0.o 1 i(k r) * e with r(r) = - (r) -r (Xr) r f2r rlo r 2 o But in this case the relation k2C (r) 2 > 1 92 Xr is again valid. Thus, the equalities of the case equation (3.112) give 1 << /k << kl1 together with F N ({) vl(, OX) F o C(k, 1' l2' ro) (2k2)a/2k [(s )(s )1/2 (2k )/2k C2 r70 o (X/k) / Ai(- ) Y(ro. k, Xr. c) N 0(1) i ~(X ) k o r ~ e. e (3.122) where N o r - -(r t1) (r) It remains to estimate the tegral It remains to estimate the integral F F N (9) \ dX F X ~. ) v,(n AoX) (C5+ C6)I 1 0 6 b 132

where (C5+C6) ' is the sub-region of C5 and C6 discussed above [on it Is V1, s I>>l s, =0(i. ForX = -ia, dX = d, and this rI rr 2 1 integral becomes N (.) d F o r v'(noX) (C + C g)" 5 6 r with (C5+ C6)r = r r is in the transition region of v1(1 -. Examining the estimate of the integrand (equation (3.122)), we observe that the latter integral can be estimated by stationary phase since Ai(-a ) is independent of k7o and not oscillating rapidly in (C +C )". It follows from the computations in Appendix E (E. 50) through (E. 53)) that o ~(Xr) has a stationary point in (C+ C6 )" provided the equation,2N ()+1 5 r6 has a solution in 01_O 1 $z, 51+ O 1 )2/ 1 kr2/3 o 0 But N o() was chosen so that no such solution exists. Therefore the integrand does not have any stationary points in (C5+Cg6)r and the integral can be estimated by integration by parts. This gives 133

(C0 5+C6)r" ( 61 'For Ix > k2C1, IX<k 2r andawayfromthetransitionregionofboth v1(g1 ->) and v2(f2,-X), we first oonsider IX/ki = O(kgl)(ls j = O(1M) and thus in accordance with the configuration of 1 i2 and ri0 under consideration I8 >> 18 I = 0(1). In this case we can again use equation (3.115) for 2 o v2(2 -X). However by (D. 6) 1 /2 = i(-A/k) 2- sin- s (1 - s1) =t —; i Sm ( 1)1/23 Now args = 6(X), and since arg(1- s- ) not close to zero is contrary to s being away from the transition region of v1m(1, -) (see the previous argument for s and inthe interval C, we fi nd arg(1-/ s 0). Then these estimates for arg sc and argw(1 -e s ) show that 1I 1 (see the previous argument for s and X in the interval C3) Consequently in o 134

in this case arg -f v /2 + arg(-X), 3wr/2, and arg(-oa ) - 0 =3 /-2 1 1 a =2 (, ) /. Since |= O(ko), the asymptotic form of the Airy i 2,1 function given by Erdelyi and Swanson (1957, equation 4.5) can be used in (E.34) to derive (2i~-1/2 (2k1l rX/2k ie v ( e ( )/ ii If in this equation we set -= Re C + i Im 1 1 Re 1 = O(a/Xr) Im |O Im / -Xv | equation becomes (2ikg )-l/2 V(,-X) - 1 X/ 2k v 1(9IIV2-7-6 fk C, and note that 1 x i( <r)1l Ki(Xri = r)k0(1), the X 1 1i2 i ( I ) iexp r ( d3.) 1 4) 1/72 ' (3.124) Inserting (3.124) together with (3.115) into the definition of F(g1, 2 r70, k, X), using (3.113) for v'(T,X), (3.112) for X(l, k,X), and the above asymptotic form for P (,p - +, we observe I F N() I v1X 1 - (kti)2 2k(kg )2 2kl(k, )1/2 o(exp f ds- r ((r) since X > 0, A /k = O(kWr ), r) < 1 and not in a neighborhood of 1. lenc the integrand is exponentially small f IX|>k21. AXI < k 2'2 |x/k| = O(k) and away 135

from the transition regions of both v (, -X) and v2(C, -<). Since equation (3.124) remains valid for 1s << 1 ( j>> 1/(kr 0)2/3), this conclusion is also true if |X/k |= O(k 1) is replaced by kg1 << lX/k << kg2. We complete the case lXI > k2 [x I < k 2 and away from both transition regions by considering |X/kl = O(kf2) ( s = 0(1~). Then in accordance with the configuration of 91, 2' and 7o under consideration, the relations e1 << 1 8 |>> 1/(kT7i )2/3 and Js | ~1 are also valid. Thus equations (C.4) and (C. 21) govern the behavior of vI(rjo, X) and X(r, k, X), respectively. Furthermore, equation (3.124) is still valid for v l(l, -A) while v2(f2, -X) is now governed by (3.119). Using these equations and the above asymptotic form for r (2ik +) we find F No(l e 1 weo |(n 0 (kg )1/2(k2)1/ )1/2 -2nn / O(xp - r) ), (3.125) aince Im C? < 0,. Xr /k = O(ki ), s8 < 1 and not in a neighborhood of 1. Hence the integrand is exponentially small if Il >k 2f1 |XI <k f2, X /kf = O(kg2) and away from the transition regions of both v l(1, -X) and v2(2, -X). For X in the transition region of v2(f2, -X), we can again use (3.124) for vl( l,-A), (C.4) for v(ro, X), and (C.21) for X(n, k, X). We also know that in 1 1 1 this case = 0(1). Thus the Airy function in (E. 35) is bounded independent f. il 1/6 of kn, while1(D.?) and (D.8) imply J(s }) (2/3) /. Therefore, equation 136

(3.125) with (k 2) / replaced by (k 2) / describes the behavior of the integrand 2 2 in this case. Consequently, for X in the transition region of v2(2, -X) the integrand is exponentially small. For X J > k2%2 and away from the transition region of v2(2, -X), equations (3.124), (C.4),,, and (C.21) remain valid. Moreover,? is now given by 2 = i(-X/k) [- sin - (1- S) Hence, the previous arguments [I||>k 21, lX/kl= O(k1l) or IJ/k[~>>k show that in this case arg 'a 3tr/2. Since I> O(kr o), the asymptotic forms of 92 -2 ~ the Hankel functions given in Section D. 1 together with the continuation formula (Erdelyi et al, 1953) H(2)z e i) e i/3 H(1). (2),( 1/3 z 1/3z + 1/3, show that (E-35) becomes 1 _ X X 57r/6 v2( -X) (2ik2) exp - log - e exp k-| (r) d8s } sf i O___(1) ("s2 e where we have set =Re C +iIm, noted that IRe2 = 0(/Ar)Im, 2 2 2 2 2' lIm I | I 2 ]r (f ) = X/k A0(), and neglected the term containing 2 2 1 r ^ 1 \1/2 exp ds -- (r) 2Thu Thus 137

N () 4N ()Im 7Xo F o e V (r1i. O (kgl 1/2(kg/2(k 2 e -2 Im (r) Xr(x 2 ds q 1/2 P' I- (r) ince Im < 0, >0 r/k > O(kr) (s <( and not in a neighborhood of (r) r 2 s. Therefore the integrand is exponentially small if I XA > k 2 and away from 2 the transition region of v2(Q2, -X). The above range exhausts the possible sub-regions of C5 and C6, and thus completes the study of the intervals C2 through C6. In these intervals the integrand is either exponentially small and the integral is not contributing materially to the sum in equation (3.63), or the contributions are described by equations (3.76) (for C4, and possibly C3 and C ), (3.117), (3.121) and (3.123). Examining these equations, we finally observe that lim R(g,,-,0) = 0 (kn o)- o as previously asserted. 3.3.2 Equivalent Integral Representation for the Source at (0, H) If in the integral representation (2.6) we let G(, rio, H, k,,) = 2) r( + v2(, -X)v(H, X) * (3.126) 2iri(2k ik 2/ 2 13.126) and again denote the path -co -ia to ao- ia (O < a < k) by C, then (2,86)be~oms G(f, nH, k, X) vN(, n. O, H) = dX V-, k( X) (3.127) C 1 138

Substituting (3.52) into (3.127) yields S G, n,H, k, X) vN(, no 0. H) = IdX g(n0, k, X) - X(, k, (3.128) C or more briefly N(. O,H)' ' )g-X) G (3.128a) C Then the proof of Lemma 1 can be repeated to obtain Lemma 2. (The convergence of the integrals appearing in Lemma 2 is again shown in Appendix C). Lemma 2: For all integral values of M > 1, M-1 vN(f0, no. H)- I dX GXn = dX X (3.128b) -0 g~ -x) We now consider, on the interval 0 < z < H, the two equations Hn-z + ___n n n (3.129): ) =_ z, fz).v () n (ii.+ vHi)Q ) nz+ V Tf 7 i(z) (3.130) together with the assumption that H is bounded away from n7 and satisfies the relation T2 1 1 H (knr)2/3 139

This case can be studied by using the method employed in Section 3.3.1, and will be discussed immediately below. The same is not true when H is in a neighborhood of ro governed by "0 1 s 1 H (ky 3 This configuration is studied in Section 3.5. Since H < r, we know from Appendix E (analysis of w(z), (E. 54)) that on the given interval i(z) is a decreasing function of z. Similarly, we know from Appendix E (analysis of u(z), (E. 54)) that on the given interval i(z) is an increasing function of z. It is obvious that v (z) is a decreasing function of z on this n same interval for all n >1. Thus for n large enough, the product v (z) 'u(z) will be a decreasing function of z with minimum value v(H) *(H) H nn 'H But H >1 thus v (H), and with it v (H) -(H), increases with n. In paticular, we can n n choose n large enough such that v (z) i(z) is a decreasing function of z on 0 < z < H with minimum value v (H)u(H) > 1. On this same interval, the maximum n value of I(z) approaches fH/o < 1. Therefore, there exists an n3 such that for n: n3 equation (3.129) has no solution in 0 < z < H. We also know from Appendix E (analysis of g(z), (E. 64)) that g(z) is an increasing function of z. Again it is obvious that f(z) v (z) is a decreasing function +n of z on 0 < z < H. Since g(0 ) —0, equation (3.130) has one real solution in 0 < z < H if and only if f(H) v (H) < g(H) or n 140

H VT+-H- VT - (3.131) But we can again chose n large enough so that this inequality cannot hold. In addition, we can choose n large enough so that vi ' + )( -H ntis (an is valid for H' satisfying H' >r+H-RA (3.131a) H- ~ H' 4 H+ ~ 2/23 \(kn73 kj)23 Hence, there exists an n4 such that for n n4 equation (3.130) has no solution in 0 < z < H and the inequality (3. 131a) is valid. Then for n N1 = max(n3, n4), the equations (3.129) and (3.130) have no solution in 0 < z < H, and the inequality (3.131a) is valid. Let N1(F) be the smallest possible N1 defined above. Thenmacordig to Lemma 2, equation (3.128) can be replaced by vN(, n1O., H) N (-) -1 n=0 g(l - X) ~C (3.132) We can show (see discussion below) that for kro > 1, kH > 1, k >> 1, 7 12/3 - 1 >> 1/(kqo), the integral defined by Hu o G N I(;) R(9, rO. 0, H) = dX. X) ' X 0 h 1 g(1-X) (3.133) satisfies lim (kiL ) —aO 0 R(, ro,' 0, H) = 0. This implies 141

N1W()-l vN(, nl0, O, H),v \ dX G * n (3.134) n=O We call equation (3.134) the equivalent integral representation for vN(g, rl, 0, H) if ko >> 1, kH >>, k >> l, and 0 1 — 1~ H (kn )2/3 The details of the proof that lima R(Q, no',O, H) = 0 (kWo)-* oo follow quite closely those of the proof that lim R(n0,-,O) =. 0 (kno ) —a) ~ Because of this and the extreme length of the proof we do not present it. The necessary asymptotic forms and arguments have all been developed in Section 3.3.1. The only difference (as one might expect from the difference between the cases of Appendix E) is that now the integrand is exponentially small for the entire range C5+ C6 while the various arguments involving the stationary point equations and transition regions arise in the range C2+ C3. 3.4 Saddle Point Analysis In this section we demonstrate how an asymptotic representation of the total field can be obtained by estimating the integrals occurring in the equivalent integral representations. The details of these estimations do not differ very much from those of Section 3.3 and Appendix E, where all the necessary asymptotic forms have been developed. We will refer to Section 3.3 or Appendix E whenever 142

possible, and discuss only the differences in the estimations. We find that each of the integrals can be estimated by a saddle point integration. We show that the first integral (n =0) is approximately equal to twice the incident wave. For each remaining integral (n =l, 2, 3,... ), we consider the field point to be "far" from the source if the saddle point equations have an approximate solution. This criterion, different for each n, is precisely defined in Section 3.4.2 (for n = 0, see Appendix E, equation (E. 55b)). These approximate solutions yield terms in the asymptotic representation of the total field which, by a comparison of their phase with the geometrical path length, can be identified as the reflected rays of geometric optics. We consider explicitly the source at (Z, 0). The same results can be established for the source at (0, H), provided we redefine the concept of field point "far" from the source (see the end of Appendix E for n = 0). 3.4.1 Saddle Point Integration We complete the proof of Thebrem, 5 by first writing equation (3. 61) as r N()-l vN(,ro) AdXC ~ (3.135) ^c n=1 and considering the first term separately. Since g(rj k, k,) has no zeros in the region I Im x < k (Klante, 1959), and F/g is exponentially decreasing for \xi — oo in the same region (Appendix C), the first term is equal to \ dx-. Using the definitions of F(y1,?2' n o k, X) and g(o70, k, X), we see that 1 6 00 143

(0)( roY0 k) - d 2ri(2ikri) 2ik -00 -00 e3wi/2 Y Q " XUY (9 -X) -X/2k 1 -ie 2 (3.136) e v (n oe We estimate J( )( 1 2' r1o. k) by setting 5 J( (1j \2P70, dEL O (3.136a) J:1 C where the Cj have been defined in connection with equation (E. 4a) (with rI = ),. and examining each integral separately. We have already mentioned that the integrand is exponentially small over C1 and C5. Thus, the integrals over these arcs do not contribute materially to the approximation of J(0) (1 2' r10 k). On C2, we can use equation (E.5) and (E.6) for vl(1,-X) and v2(f2, -X), respectively. The various asymptotic forms of vI(i e- i, X) (depending on the position of X with respect to the turning point X=-kft )are given by equations (3.89) and (3.99). The arguments of Appendix E and Section 3.3.1 then show that the integrand is exponentially small over C2. This implies that the integral over C2 also does not contribute materially to the estimation of J(0)(E0 ri. k). To estimate the integrand over C 3 we substitute (E. 8) for v1(y1, -X), (E.9) for v2(2',-X), and (3.71) for v2(n le,-X). We find F(g1I 92' f'l, k( X)3 +k,X)_ 22 (i > ~ X0;,2 l, el C2,o)+I,.;1';2' ro (3. 137) 0 where Itg(, gI2.,n) and 2I X.,g 2.Tr)) are given by (E.12) and (E. 13), respectively. Hence, the argument of Appendix E implies s - =( ) (3.138) -00 144

Therefore, equation (3.136a) can be written as (0) F J ~( t 2'2 ok) I dXA * (3.139) 4 On C4 Vl(41, -4) and v2(2, -X) are governed by (E. 34) and (E. 35) respectively, While g(ro,k.X) is given by (3.109). Using these equations, we observe that on C4 2C(k,1,' C ik)] e i/4 8s )Ai(-a )e /1 )Ai(-a ) I 1 2, 2 (3.140) ( )/2 or F- 2 Asymptotic Representation of I(k, 1 (3. 140a) where I.(k,,1'2' ) is defined by equation (E. 3). Consequently, the arguments of Appendix E apply and equation (3. 139) becomes -ik + g2+rq )2- 4 1 (0) -2e 1 o 1 2 J (102''(0 k) 2, -. (3.141) 11 2 o 2 provided 4g21 o 2- ( l+.o) >> k)2/3 2a (k )n and 145

k) + j(O)( 1' 2' k) v 2I dXIk,1 2 n1 ) I (3. 141a) (k2~) if 4 k (2 1+ i = ( 12/3) The latter integral calls to mind formula (E. 42) from Appendix E. But for those 1 2 (2 and rn for which (3. 141a) holds, I- (in (E. 42)) has no stationary point on C4, and therefore (E. 42) becomes -ikR. (k2 + - \ d XI(X1', n) (3. 141b) (k2 )Further, according to the definition of R. (Buchholz, 1953, Chapter 4, equation 3), RB= (e+ -+ )2 -4. Therefore equations (3.141a) and (3.141) imply -ikR. J(O )(i1 2'no'k) R ' (3.142) for all values of g and I. We suggest a geometric interpretation of the occurence of the distinct representations (3. 141) and (3. 141a). We do this by showing that if ~ < ', the relation 2 = e l+ri, which now becomes -= +r0, implies that the line joining the two points (-, 0) and (Q, no) is perpendicular to the surface of the paraboloid of revolution. Such a ray path gives rise to a caustic, and in analogy with the analysis of other diffraction problems it is reasonable to conjecture that a caustic should be associated with the condition under which (3.141a) holds. Thus let 2 (i, o) be a point on the surface of the parabola x = 4r ( + z), with x coordinate given by + 2 1i;, and z coordinate given by e - lo (see Section 1.3). Then the slope of the line perpendicular to (9, no) is 146

1 dx If this line intersects the z-axis at (, 0), then necessarily - > g, and 2 -o = - i7 (Hence 2n = -+ r- e, or -= +no. The significance of this geometrical relation is further discussed in Section 3.4.3. If e >, the equality g2 = ~1+ro becomes e = Z+r. This has no special geone tric interpretation; it only reflects the inability of the Green's function to distinguish between the two configurations. We now consider the terms in equation (3.135) which have the form 2 1 k) a dA Xn n =12, 3,.... (3.143) C We can estimate J( )(1' 2' e,0 k) by setting de F no. k) = dX I * X (3.143a) j=l =1 where the C have been defined in connection with equation (3.63), and examining each integral separately. We know from Appendix C that the integrand is exponentially small over C1 and C7. Hence, these integrals do not contribute materially to the approximation. On C4, equation (3.137) together with the argument of Section 3.3.1 yields d F )xn 1 a/2k 0(1) 1 _ l_1 S d. T (2/~1 1/2 ( a/2kL a/k 4 (3.144) Finally, the arguments of Section 3.3.1 show that the integrand is exponentially small over C2 and C3, Therefore, the integrals over C2 and C3 do not sml ove Ca 2 147

contribute materially to the estimation, and using (3.144) we see that (3.143a) can be written as J 1n) (21'r ok) Nv s dX5+ C 5+C 6 (3.145) To estimate the integral over C5+ C6, we first observe that the arguments of Section 3.3.1 can be applied directly to (3. 145) to obtain J (,(kfl2r o k) " dX -- Xn+ g g (C5+C6)" D (3.146) where (C +C) = kX C 1 <</k <kg and away from the transition region of v1(1 -). and in (C5+ C6) 1.1'1 F Xn, g 22J7r C(k,. )( 1)2 e [(s8 )A(s )0(s 1]'/2 -1 92 ro ir X) iie ( 3.47. e +Hie n, (3.147) with +,(.) = - (A) - -2nC (A) n:x 1 = 2 no 0n0 (3.148a) J Cn0 = -C (A)- C (A) - CW-2nC ( (A)i n 1 2 2 o while in the range (C +C )" (defined in Section 3.3.1) F [(eS (/ (Ai(a )(2 2 C ( A-) er i/4z 1s)Ai(_a )(i.i 1/2 (3.148b).1/2 t 2 ir 1) e e -- I ~ II I I I.... (3.149) 148

with Cn(X) = -2 (X)-S (X)-2n? (X). (3.150) 2 0 o For these values of n, the definition of N (e) implies that at least one of the O following three equations has a solution in the indicated interval: (i) (3.55) in <z< 1, (ii) (3.56) in 0<z< l, (ilVT ^n. = ({; - no 2(fl ' in 1-7()3/ ) -z7 g1+( 2/3) * (3.151) (o0 0 Equation (3.151) results from seeking real solutions of d n(X)/da = 0 in the transition region of vl(91, -X). We shall not explicitly consider these solutions, but will discuss their significance in Section 3.4.3. Thus, let us assume that the triplet (i,, 0o) is such that possibility (iii) cannot be true. Then the argument of Section 3.3.1 yields -1 d-)Xn= j -l -- ) /3) 1 2 which, when inserted into (3.146) gives (n)k) dF. (3.152) g 149

In order to analyze equation (3.152), we write it as J (n)(1 20 o0, k) - | (C5+C6) dX F.xn+ g (C5+C6)1 5 6 I dXL.Xn g J (3.152a) where F =Xn F.Xn g n {+(), i- iC (X) 2 C(k, 1' 2' r1 )(-i)l/ e e 1/2 s (B^ )O(s 9 )O(8 TI 1 1/2 7ri2 2 'Ti C(k, 1' ^2' r)(-i)1/ e 2e aj 1 )W(s2 )0(s )17 (3.147a) (3.147b) If equation (3.55) ((3.56)) does not have a solution in 0 < z < g1, then the arguments of Section 3.3.1 imply F+. Xn F l gF 1= 1g 1 1 1 2 (3.153) However if equation (3.55) ((3.56)) does have a solution z (z) in 0 < z < i + t-\ + 2+ - then (X) ((X)) has a saddle point at x+k Zn ( = k z ). Thus, the n n n n n n integral dX F Xn g ( FdXL g g 150

can be estimated by deforming the contour (C5+C6) to a new contour C shown in Fig. 3-1. This is possible because the fact that g(&o, k, X) has no zeros in nmX < k also means that X(nr, k, k) has no poles in the same region. The argu- ments of Section 3.3 apply directly to the nonreal horizontal pieces of C. To estimate the integral over the nonreal vertical pieces, we first note that the estimates of F /g ( F-/g I) and |x I on these vertical pieces are obtainable directly from Section 3.3 by replacing a with a', where 0 a' 4 a. These estimates combined with the fact that the vertical pieces have length equal to 0(k), yields as the estimate of the integral over the nonreal vertical pieces. But since the ^10 + 2 + I real piece of C contains a stationary point An = k zn of - * Xn, we find thn gat that o e x(n]1/2) * />| n, (3.154) provided Cn (A ) 4 0. There may be more than one solution of (3.56). If we call the solutions zn(i), the corresponding result for n d F x t g is 151

X = Imx i k2z+ k2 II X =ReX. I I I I x = -ik (C +0 ),~5 6 C FIG. 3-1: CONTOURS OF INTEGRATION IN THE X-PLANE. 152

Xn = S d Xn 9 b (c+) i signc E 2.2r C(k 1 2 n o)(-i) /2e 2 e n e 4 n n 's~ ) *8 )O(.rlo)1/2 1 2 )0( / ( n (3.155) provided " (n(i)) $ O. The existence of solutions ) (Xn(i)) such that rn (x) = ~ n W = 0) will be discussed in Section 3.4.3. However, such solutions will not be explicitly considered. Thus we assume that the triplet (I,-, ro) is such that 1(n (+) 0, (n(i)) 0, for each i. Then defining n n,) the terms of equations (3.154) and (3.155) as E ( n), and E(n(i)), respectively, and substituting equations (3.153), (3.154), and (3.155) into (3.152a), we observe that J(n)( 21, k) E+(X+), if possibility (ii) is not true, (3. 156a) J(n)( l. 2r o k) ' E- n) if possibility(i) is not true, (3.156b) J(n)( F 2, k) - EX+ I E (ii), if both possibilities are true. i This completes the proof of Theorem 5, (3.156c) 3.4.2 Approximate Solutions of the Saddle Point Equations The exact form of the representations (3.154) and (3.155) depend,,of course, on the exact solutions of (3.55) and (3.56). However, even for n = 1, these equations appear too difficult to solve exactly, and we investigate approximate solutions of the type discussed in Appendix E. Thus we assume that 153

=/r 0(1) and /g << 1 (here the condition implied by the symbols < is understood to be independent of kn o); this represents a configuration of considerable physical interest, the configuration studied in Section 3.3. 1 and subsequently. Let us consider (3.55) and (3.56) for n= 1. They are W__ iii i ii)2 1 (3.157) ( + + 1= (3.158) with,1 =, 52 = e for the given configuration. The condition that (3.157) has a solution ((3.57)) becomes <( -;) (3.159) Examining (3.157) and (3.158), we see that since gl(0 ) — oo, equation (3.158) always has a solution zl for small values of z, while u (0+) - 0O A+ implies that equation (3.157) has a solution zl for small values of z provided (3.159) holds. But for fixed -, (3.159) will eventually hold for all e larger than some fixed o. Since we wish our results to be valid for these values of A, we assume a priori that the ratio f2/A1 is such that (3.159) is true. To define this mathematically, we first write equation (3.159) as 1 < + +1 (3.159a) or since 2/1 >> 1, as 154

1 <2 2 ' +1 - (3. 159b) But rlo/1 = 0(1) implies +1 - =0(1) Thus if we consider (9/.) >> 1 added to the defining conditions, we observe 21/3 1 - a (3.159c) which implies that (3.159b) holds and consequently that the inequality (3.159) is valid. This inequality not only guarantees the existence of a solution to equation (3.157) but also that equation (3.158) has only one solution. This is so because: (1) f(z) is decreasing from 4J1- on 0 < z < f1; (2) g1(z) is either decreasing from wo, or decreasing from oD to a minimum and then increasing, on 0 < z < E1; and (3) (3.159) can be written as f(il) > g(E). In addition, (3.1596) implies that equation (3.151) cannot have a solution for n =1, since the condition for a solution to this equation can now (2 >> 1) be written as o o 1I Finally, thesolutins ^1 and z1 can be approximated by writing (3.157) and (3.158) as 1 (Z)I = (-1- [+O (z1, (3.157a) 4 \ [l+O 2 0(zj z. f) [1+o(zj (3.158a) 155

Therefore Z1 = 4 71/32 / 1+0 e 1/g2)1 ' (3.160) 4 3 2 (/) [+ l/2. (3.161) We now evaluate the result of the stationary phase integration if we use the stationary points given by (3.160) and (3.161). For positive real values of X, equations (3.148a) and (3.148b) become () = ()- 2n (s ), (3.162a) n o "ok 0 W() = o()- 2n (s ), (3. 162b) 0 where.k (X), Ji(X) and 4(s) are given by equations (E. 46). (E.47), and (E.49), respectively. Then the computations in Appendix E (equations (E. 50) through (E. 53)) show that dX 2ikX 0(ai t )c(8o )0(s c) (.1 - 6 4) ds2 2ik \(s )0(8s )O(B )7 [(X =i(- z) - ) -zs +(+n) - ) /v2n 7?I+ Z n n"o(3 Ib 1 2n (3.164a) (3.164b) 156

But =Z1 ~21 A+ T0 ~1-z 1 = 1+O(9 1/g2))4/ AS 1 D 8 ) 9 =Z 1 = 1+Q(9IA/.2) 5 2 =1 2 A- TI A+ ol? iwis )i 77 A-TI7 O = I = 1+o((91/y)2 = 1+0(g1/92 ) p Substitutin these equations into equationsp(3. 163) and (3.164)., we see that =-1 QA )1~ /3 10gA)/ (3. 165a) ( 21- 2+Og 11L =k~ X () X1 2 c1 O1 = -kg 2- kg1- kTi7-k(2Ti)+ kTi 0 (91/2 (3.1 65b) (3. 188a) (3. 166b) 157

Therefore, equations (3.154) and (3.155) yield 1 -2 1/ 117 gi/2 'IX -%# 1 91g2 a which for 91=E,.-y9 becomes i+ 1 216 WEAm++2n1 / dXX.~ 0 0 (C5+06) (3. 1674) and cxF dX xl1 ___ __ __ __1/2 -IIk(2+c 1+ro to 0/ / 9 /2 (91) 92 9 which for 91: = 2= becomes F dX*.X g (1..3/2. 1~-fyo (3. 167b) Insertin equations (3.167) into equation (3. 152a), we obtain or since (Z/9)1/6 ~<1 ek.. +n+2v e i/'2 g13 'ft Lo' k 7 7 -r/al 0 0 (3.168) In order to establish Theorem 8, we now investigate the asymptoic representation of j(n) t7, k) for n > 1. We need only follow the argument for 158

n = 1. The part of that argument essential to the case n = 1 was the appropriate condition for (3. 57) to hold and for equation (3.55) to have a solution, namely that 1/6 (/.) >> 1. As above, we observe that if n > 1, the corresponding sufficient condition is ( )1/2(2n+l) >> 1. Under this condition, as for n = 1, equation (3.56) has only one solution in 0 < z < 1, equation (3.151) does not have any A1* solution, and the two solutions z and z are approximated by n n z=4 - [+/) (3.170)()] ^+o 1 +O l/) ) (3.169) n ' n- 4,o(, 1!2-1 i 1 /2n1 ) 0 (f/f2+l/ 2). (3. 170) Consequently, the stationary phase approximation gives 1 1- -ik(g-+++2nn) ni (n) -2.2(2n+) 0 0 2 J( )(-, f 0o, k) 2 r. f- (-/2 e e. (3.171) This result enables us to obtain an asymptotic representation of the field vN( 0 r,,., 0). We first substitute definitions (3.136) and (3.143) into equation (3.135) and find N (g)-1 0 o v(gr O) (, o k) n=0 Using equation (3.142), we observe that - ikR N (f)-l -2e + (n)( vN(',o", 0) R R- + 3,k). (3.172) - n=l Then if n It) is such that (g/)1/2 1 > 1 for n. n1(), we can substitute equations (3.168) and (3.171) into (3.172) to find 159

_____- 2 1w..L.. go o.,, n f l ) ' n=1 + L J( )(., rok) (3.173) with,, no,n) =e- +ro+ 2no. This completes the proof of Theorem 6. 3.4.3 Interpretation of Results We conclude Section 3.4 with a partial interpretation of equation (3.173). Included in this interpretation is a discussion of the asymptotic representation which arises from a real solution of dr n(X)/dX = 0 (defined by equation (3.151)) in the transition region of vl(1, -X), as well as a discussion of the asymptotic representation which arises from a real solution X- of d{n(X)/dX = 0, with the additional property that (d a ) =~n We need not consider the latter case for C (X) - we shall show that solutions n of this type are not possible. The model for the interpretation is suggested by the first term of equation (3.173). Thus we try to identify the successive terms with the reflected rays of geometric optics. To do this in general is very difficult. If n > n1(C), we cannot even hope for such an identification without additional analysis of the stationary point equations. Such additional analysis is quite complicated even for J( )(2,,ro,k) if i/2 = O(1), i/n = 0(1). and we cannot use the approximate solutions given by equations (3.160) and (3.161). For n < nl(9), the geometric analysis involved in the identification grows rapidly with n. Consequently, we investigate only the n = 1 term. 160

We first prove Theorem 7, i.e. for a triplet (-, rvi) such that /IrO 0(1), (/ 61 <<' 1, the approximate phase of J()(,, r, k) which depends explicitly on distance, (, f, n o 1), is equal to the approximate path length of any ray that travels from (, 0) to (, nrl) via a single reflection. A ray of this type is shown in Fig. 3-2. The point (Q' rlo) tnast be determined in terms of (eli o ) and ( 2' no) If tan 01 denotes the slope of the incident ray from (-, 0) to (I', rn ) tan j2 denotes the slope of the normal to rt = ro at (5', r 0), and tan t3 denotes the slope of the reflected ray from ( ', %o) to (g, ri0), then the condition that the angle of incidence equal the angle of reflection implies t/3 = 2'2'- 1 or 2 2 tan V2- tan /1 + ttan 2tan tan 3 — 2 1-tan2 2,+2tan 2 tan 1 Since tan 01 = x/z- 1 ' tan &2 = -x/27o, tan j3 = x2-x/z2-z, this equation can be written as 3 x x x x- x z - 2 2-x l 4l2o(z -1) 2 --- —- 2 x ' (3.174) Z x X2 zz2 2 U-9 1 r2 ro(Z- l) 0 But 2x2 x2 X2 t-v ', z2 4 vo o 0 0 thus equation (3.174) becomes 161

x 0 0 (z X) z P-A to3 (z 2,- x2 ) FIG. 3-2: A RAY TRAVELING FROM (-:1$0) TO (.rl) VIA ONE REFLECTION.

4ij ~ 4i -1) - 70- x1) X2 - 4I X20 2 oX (x2+ X) -4 11 - x - 4oX + _ 0-owhich reduces to 4 4x +2rox + C1X2x + 4ro(l+r) 0. (3.175) 0 The solution of equation (3.175) gives the coordinate ' (C' i x /4rjo) in terms of 1 and 2' We do not solve equation (3.175) exactly, but instead -s-mte the order of the solution. Since we are seeking a C' > I, the two possibilities for a solution are x =O( ) or x ~1 If x O(1), the terms A - x4/4r, B 2r x2 1 0 0 I C x2x, D 4 o(El+ro), have orders A O( 3), B = 0(g ), D - 0(f), C O(C3/2 1/2) ~/ 1 2 This shows that C dominates A, B, and D, which implies that x- O(e) cannot be a solution. Hence x >~ l, which shows that A > O(C )x, and so dominates B ' O(0x2 and D= o(3 ). Therefore the solution x obeys x /4r0 21 |x2 x (x2 negative), or x 3/4r0; gl x2. Consequently, 3 ( 3/2 2= 2(1 2/2) =, = O(2/ 3 1/3 and '/ ' O)1/3) >>1, I 2/' - 0 /1)2/3) >>1 163

These conditions imply that the distance R(91, ') from ( 1 0) to (i', n ), as well as the distance R(Q', g2) from (', r ) to (e,2'n ) can be written as 2 0 g 0 RH(Sl') - + 1+'+o) 4e2 4 = ' '+qo- 1+noO(1/') ' R(,2 2 +-') 70 2 + ) 2- 2 + 1 0 ((g '/g2 ). Thus, the approximate path length is given by d = f'+ 0 - 1+ (f2- '+ 2no) - 2 f -1 + o+ 2 0 or d -m-+ro+2o, which is equal to 0(-,, rlo, 1). This ends the proof of Theorem 7. The latter part of the above derivation may be extended to (-Z. 9,,o n). Suppose we assume that a ray leaves the source at (-, 0) and travels to a point on the surface (., n ) via n reflections at the points ((n), rno), where the angle of incidence of the initial ray and each subsequent reflected ray is small. This latter condition is equivalent to assuming that the ratios (0)/ (1) (J)/ (j+l) are small for 1 < j n (Q(O) = -, n+l) ). But then the distance R((0 i(19)) from (0), 0) to (1), 0o) as well as the distance R(i() (3 +1)) from ((i0).o) to ((J1 r- ), can be written as 00 Th(0) (1) ) t at p(1th (0) l t(1r) i(s g+nO ( by) R((0) (1); ( 1) + (+ o ( + r- + o( ~ u - + 4xo p+a lg ) c Thus, the approximate path length traveled by such a ray is given by 164

dn=(1 )(0) + n((l)- e(j)+ ) j=l 0 and this "telescopes" to yield d (n+1) -(0)+ n +2nn e -+ r + 2nqo which is equal to 7-,;,, na). To complete the proof that for n K n (a), (, r, r ) is equal to the approximate path length of any ray that travels from (-, 0) to (Q, rn0) via n reflections, we would need to show, as we did for n = 1, that the condition (/) 1/2(2n+l) >> 1 implies that the ratios g(0)/(1) ), j)/(J+l) are small for 1 * j < n. This will not be attempted here. The above equality between path length and phase which depends explicitly on distance is an important step in the identification of the n = 1 term of equation (3.173) with the corresponding reflected rays of geometric optics. To further complete this identification, one ought to show that the amplitude of the n = 1 term can be obtained by applying the optical form of the principle of conservation of energy to a bundle of rays which eminate from the source at (-, 0), strike the surface at (', 1.o), and are reflected to the point (, ri0) (Levy and Keller, 1959). The geometry of the paraboloid of revolution suggests that this is a difficult problem, and its resolution will not be attempted here. Finally, the additional phase factor which appears in the n = 1 term (as well as the other terms for n < n1() must be accounted for. This again is a difficult problem, we simply discuss the nature of the difficulty involved. In most cases when one uses a geometrical interpretation of the asymptotic representation of a total or scattered field, one finds that there exist field points where the ray picture is no longer valid. These occur either as isolated points, curves, or surfaces, called caustics. They are discussed by Levy and Keller (1959) and by Kline and Kay (1965). An example of caustic surfaces in the case of 165

diffraction by a paraboloid of revolution is the envelope of all the rays which are reflected n times. For each n there is one such caustic. A caustic line in the paraboloid of revolution occurs when a ray strikes the surface and is reflected back along its own incident path through the source. Then the caustic is the path this reflected ray travels after passing through the source. Therefore, the portion of the axis from the source to a) is a caustic, while other caustic lines are generated by rays, discussed in Section 3.3.1, perpendicular to the surface. The intersections of the caustic surfaces and lines with the surface of the paraboloid of revolution form caustic circles on this surface. For the source at (0, H), we immediately note, from the failure of the analytic geometry problem of Section 3.3.1 to have a solution intersecting the z-axis at z = -H: no ray leaving the source can be incident perpendicular to the surface. Therefore, the only caustic line in this case is that part of the axis from the source to oo. However, the caustic surfaces formed by the rays reflected n times remain. The behavior of the field at a caustic is characterized by an enhancement of its amplitude by a factor dependent upon ka, where a is the characteristic dimension of the body under consideration, compared to the amplitude of the field given by the geometric rays. Thus for the paraboloid of revolution, we expect the amplitude of the field at a caustic to depend on (k o), with b > 0. This behavior cannot be determined by stationary contributions of the type discussed if n (X+) $ 0 and n (X ) f 0, since equations (3.163) together with equations n n n (3.154) and (3.155) show that these contributions have no dependence on kr. Therefore we investigate equations (3.163a) and (3. 163b) with regard to zeros. The forms below equation (3.163a) indicate that the zeros of this equation correspond to zeros of - 2 / (3. 176) (3.176) 166

But the function (2- Z)/(gf- z) has a derivative equal to 2-1l >0. )2 ( -z) Thus it is increasing on 0 < z < 1 and -1 X - 1 on this interval. Hence 92-z l1-Z f2 {l on 0 < z< 1, and (3.176) has no zeros in 0 <z <. Consequently, the integral \ dX -.xn g1 cannot determine behavior at a caustic. Similarly, the zeros of equation (3.163b) correspond to zeros of Z -z -Z Z -+ + (1,+ 2n),* * or the solutions of 1 + 1 (1+2n) (3.177) 'w 'wi'^ ^ +* in 0 < z < f1 Since the right side is a decreasing function of z on 0 < z < E1, with maximum value (1 + 2n), and the left side is an increasing function with 167

maximum value approaching co, and minimum value approaching 1, on 0 < z < E1. equation (3.177) has exactly one real solution in this interval. Thus the condition that (3. 177) and (3. 56) have a simultaneous solution, would correspond to the relation between the parameters C,Z, rI at the intersection of a caustic with the surface of the paraboloid of revolution. There is another possibility for the description of the behavior at a caustic. For positive real values of X, equation (3.150) becomes ) - ) ) is ro( ) - (3.178) Cn k ~0 92 ok fo^st 0 k o Y0o with ((s. ) and o(s ) given by equations (E.48) and (E. 49), respectively. 2 o The computations in Appendix E (equations (E. 50) through (E. 53)) show that d2 (X) io(sn )+(1+2n)>(s ( n o- 2' (3.179) 2 2iL(s )a(s )] Then from the form of equation (3.149) and the analysis of Section 3.3.1 leading to equation (3.123), we observe that a solution of (3.151), z, with "(X n) 0, n 1/6 a n gives rise to a term with an amplitude which depends on (kr). Hence the condition A; ~ ~ + ' By2 would describe the intersection of a caustic with the surface of the paraboloid of revolution. An additional property of caustics is the appearance of a phase jump of r/2 in the geometric optics field as the field point passes through a caustic. This phase jump has not been explained geometrically, but appears in the asymptotic 168

representation associated with the given problem. This is the origin of the factor e7 i/ which appears in the n < n1() terms of equt ion (3.173). The approximate analysis of Section 3.4.2 is fruitful precisely because, for a given n, it corresponds to field points which are well away from not only the nth caustic line but also from the region of influence of the nth caustic surface. This is not true for an n such that n > nl(g) in equation (3.173). Thus a physical description of the statement (which we have defined mathematically) that a field point (f, r0) is "far" from the source is that it be so far that it is far from the caustic lines and the region of influence of the caustic surfaces corresponding to the integers n < n1(). For other field points, we most likely need a careful numerical study to obtain any information from the saddle point analysis. 3.5 Whispering Gallery Waves In this section we prove Theorem 8, that is we investigate the source at (0, H) where H is in a neighborhood of rit governed by H (kTi )2/ H (k )2/3 0 0 This implies that the N 1() of equation (3.134) is increasing with krlo. Therefore, the use of the representation (3.134) is subject to the question of the convergence of the series therein. We do not consider this question. Instead, we investigate the residue series directly. We show that there exist residues, exponentially small if oi0 1 H 2,/3 (k)o)/ 1/6 which have an amplitude dependence of (krn )/ The phase of these residues which depends on distance is approximately equal to the arc length from the tip of the paraboloid to the field point (g, roi) under consideration. Moreover, for field points (g, 1i) such that /n0 > 1, the distance-dependent amplitude of 169

these residues is approximately proportional to 1/p, where p is the radius of the cylindrical cross section of the paraboloid of revolution. Therefore, these residue terms can be interpreted as cylindrical waves which are traveling along the surface of the paraboloid of revolution. We shall call these waves "whispering gallery waves" after similarly behaving waves of other concave surface diffraction problems (Section 1.1). 3.5.1 Asymptotic Form of Residue Terms If r) = r, then the residue series (3.2) becomes 1 00^ iA. vA(~ -A )v1(H, A v (Q, r) O.H) = i +2) 2 v (1 D) (3.180) N*o yo p= l ' Vik T 7 p residues which are exponentially small. Thus (3.180) can be written as 1 A V2( -A )vl(H A) p for some M > 0. We define the parameter H1 by the following conditions: H has the dimensions of r1, to -k H is not a zero of v( X). Then we further decompose equation (3.181) into 170

VN(^E riO, OH) 2ik~ O N 0 2ikrl 2ik I (dv{(nox v2(,-A)v1-(H, A) P + 12ik 2 r+(f V2(, -A ')v1(H, A) 1 p X AP -k2Hl<A<Mk d =A P In this section we consider only the sum -1 ni < ~ 1 v (2, - 2)v1(H, A '"o' 0o AH<) Hik, kHo 2i p 1 dX o' X=JI (3.182) (3.183) The zeros Ap are given by equation (3.27) (in order to avoid confusion of notation, we replace the Zp of (3.27) by wp). The asymptotic form of v2(-, - ) is governed by equation (3.43) with CP) given by (3.42). The asymptotic forms of the Whittaker functions in equation (3.183) which depend on H and r0 have also been derived in Section 3.1 and 3.2. The function vl(H, X) is governed by equation (3.12) (for = H), with?H given by (3.17) (for r = H). Since o is also given by (3.17) (for rl = o ), we observe that d(2kik) - s i 2X/ik -11/ 21 d(2ikr)) k2 8 ' 2-/Lk - 2 r)'~' 171

and as a result equation (3.41) for ( V (n ))P X can be replaced by \d 0 X =^I (Qi A)),,(10, x ~ (2iko)-/2 - i i/4 2k ) (p) s(P) 1./2 L84) Now by (3.17) and the definition of a rlo d, dC io I \ l~ ~-"-,i/' 1], - 1)1/2._ doI k -l d + 01) (3.185) However in this range of X, s -1 = O /(kr ) ), thus equations (3.18) and (3.19) are valid (for rl = r), and equation (3.185) can be written as d? ri o l ro 1/2] k 8 - 1) (3.185a) dl kL t i 0 0 Moreover, equation (3.27) implies (P)-1 - o (kr )2/3 0 Consequently, (o- 1 d1,0 1 -. (3.186) Then using equation (3.186) together with the fact that equations (3.18) and (3.19) 172

(p) 1/6 imply IX(s7 ) -~ (2/3). we find that (3. 184) becomes no VI 770 X "i 2ik7 ) 1/2 1 X= 0 ______F e 2k(2/3)1/ Qc Ai(w )j -i 2 k~ ki )2/3 0 (3.187) In order to estimates ) (sP) a we write it as 21 A -A p p D a(p)= H 2 H a or 8(p) L 02/3 ['h23 /3i (kit0)(kyl I L (kn~)2/3 ~ o which implies that 0(p) 1 1 - - (kyn) 2/3 -The refore, equation (3. 18) and (3. 19) are valid (for tj = H). This shows that 0() (2/3) 1/6. and H - \~H/ f. (s(p) )3/2)2/ p 173

Using these results in (3.12), and performing a Taylor expansion of the Airy function about w - h. we obtain v (HA) (2WkH)1/2 /4 2k (2/3 Ai( - h). (3.188) Then inserting (3.187), (3.188) and equation (3.43) for v2(,-A ) into equation (3.188), and using the definition of (s P)), we find N o 1 A e A<-k-H Ai(-w +Ih) -1 ((P) * (u )A?(, ) e 5, (3.189) P P or, upon defining the dimensionless parameter WHby k(- H1) - (knr) /3wH HN 0 1 0, v*( n 0 H) (k^ )1/6e-li/4 (' o /2 (. 1/4 0{jo +o <Hi Ai(-w +h) -i+(p) *(-w)A ) e e * (3.189a) P P 3.5.2 Interpretation of Results We first note that the residues of equation (3.183) are exponentially small if H >> 2/3 This follows because now H < H, sH = -k H/X < 1 for X < -k H1, and X away 174

from the transition region of v (H, X) implies that v (H, X) is governed by the decreasing exponential form of the Airy function in equation (3.12). The same conclusion can be drawn if we consider the residue series (3.2) for the field point at (, ri). Unless both H -1 = (() and — 1= o( 3 H 3 nk)2' the residues arising from zeros of equation (3.27) are exponentially small. Thus any physical consequence arising from residues of this type is confined to an r-layer near the surface of the paraboloid of revolution defined by — 1 = 0 2 Finally, from the form of the residue series (3.1) for the source at (-, 0), we see that these residues are exponentially small for all values of -. This is because the asymptotic form of vl(1, -A ) is governed by equation (3.44). We now investigate the distance-dependent phase of the terms in equation (3.189a). Using (D.26), we see that (3.42) becomes (p) ( I >-8(p)) + log(P) (- p). (3.190) k But since (J- 1- ) 1 -t I+ ) 1, args = implies log ( s - s) -log (l- + \ Hence we can write equation (3.190) as - + - ) (<3.190a) r(p) =1 -a 8 + log a ) - a (3.190a) of kV~Vb \1~-' 175

Then using equation (3.27) together with the definition of sP) (p).2ikL ' 4A /2ik P we observe that I(P) kyrl/ (+i; + log (i+ +, for each p, and consequently -iP) v -ik +3T + r log( + + /rf (3.191) 2 for each p. The arc length from the vertex of a paabola x = 4r)z to point (z' - 4n z' ) on the parabola is," -ikL, foreach p. (3.193) ~ o L, 'zI'+7o) +riolOg _ + (See the Smithsonian Instittio n bles, 1922, p. 46, eq. 2.404). In our coordinate system. (Section 1.3), z' = rvo+z, z =- ro; thus z' and L = 7(i+r0) +0olog + ^ + (3.192) which shows that for a field point (9, ro), -i ~, -ikL, for each p (3.193) There is another factor in the right-hand member of equation (3.189a) which can be interpreted geometrically. This is the distance-dependent amplitude of each term in (3.189a) which we write as 176

1 1/4 A(-,-n ) = / (3.194) o. (%) = 1/2 +n If we assume that the field point (0, rl) under consideration is such that /ro >> 1, then (3.194) becomes 0 A(f, 9o)- =i 1+(, /l, or A(Q, o' ~ 72 (3.195) 0 (971 )/2 0 But according to the definitions in Section 1.3, the radius p of the cylindrical cross section of the paraboloid of revolution is given by p = 2 nJ. Thus if /ro >> 1, A(S, io) is approximately proportional to l/p. Equations (3.193) and (3.195) have an important consequence. They show that: (1) each residue termn. has a distance-dependent phase approximately equal to the arc length from the tip of the paraboloid to the field point (Q, v ) under consideration; and (iI) if g/n >> 1 the distance-dependent amplitude of each residue term is approximately proportional to the inverse of the radius of the cylindrical cross section of the paraboloid of revolution. But this is exactly the distancedependent amplitude and phase that one would expect of a cylindrical wave traveling along the surface of the paraboloid of revolution. Therefore, we can interpret the residue terms of equation (3.189a) as cylindrical waves which are traveling along the surface of the paraboloid of revolution. By equation (3.193), we see that each wave is arriving at the field point (g, v7) approximately in phase. Waves of this type were first noticed by Rayleigh (1896, 1910) in connection with the propagation of sound in a room with a concave ceiling. He called them "whispering gallery waves", a name which we keep. Kimber (1961a, b) shows the existence 177

of such waves for the interior of a circular cylinder under line source excitation, and for the interior of a sphere under point source excitation. Keller and Rubinow (1960) discuss not only these domains, but also show the existence of waves of this type in the case of bounded cylindrical convex regions. 178

APPENDIX A NORMALIZATION (POINT SOURCE NORMALIZATION) In the text, we first assumed the point source in question to be represented by p( r). Let J(, rl, 0) denote the volume Jacobian in the coordinates of the paraboloid of revolution. Then for the point source at (-, 0) 1 C p(*, n) = J(, n, ) C 6(-)6(n) = ( ) 6( —)6) and so e i27r oo Oo \ \ \ p(?,r)dV = \2C6( —)6(r)d dydo = 47rC J J 0 0 = 47r, for C=1. Therefore, the choice of a point source such that C = 1 implies p( r)dV = = 47r which in turn implies C" = 47r or p( r) = 47r6(r- r ). Since the free space Green's function for this p( r) is -e /R (R =|r-r |), we demonstrate consistency (indicated by the agreement of the integrals of equations (2. 5) and (2. 16)) by showing that the solution to -L v- Lv = 6({ -)6(7):, (*) |v(,r,s)2| dV < o, all space -ikR has -e /R (R=| r-ro|), for its limit as s- 0+. According to the theory of Section 2.4. the solution to (*) can be represented as 179

v(, n, s) =2ri (, -,-X)G(n, o, X)dX, n ^ * where r is a straight line contour between the poles of G(g,-, -X) and G(r, O, X). But G(6, 0, X) = l(0 X)y2(X, X) (O < ) is analytic in ImX < k, and _yl(_,-__(_,-__) (~<E) G(Y, 2- -)X) 2i > 1(E, -X)y2(f. -k) (? >-) w- is) is analytic in ImX >-k. Therefore if r is a path defined by -o - ia < X < o - ia, I al < k, P v(?2is) = (2ih I 2 ir( ) (2 + )y 1, -XA)Y2('2o -X)Y2(*? X)dX. p Arguing as in Section 1.4, we see that the limit as s —> 0 may be taken inside the integral by replacing the parameter-y with the parameter k. Then using the inte-ikR gral representation for e /R in Buchholz (1953, Chapter 16, equation 9), we observe that -ikR e lim v(, rl,s) = -7 e 8.0+ which demonstrates consistency. See Appendix B. 180

APPENDIX B ANALYTICITY OF RESOLVENT GREEN'S FUNCTION R. The analyticity of RA follows that of (1, '.k) which was represented as r{1 ) Yiy(X)yl(')X)Y2i). <, G(, X)A ) 2i. (')7 (,)y2(o ), s > ' According to Buchholz (1953), the functions y1 and Y2 are entire functions of X. Thus the singularities of G(f, ', X) are those of r - -), which are simple A. 1 21 poles at the points 2i- n+, n=0,1,2,.... G(S, ', ) will be analytic in any 2iy 2' domain which excludes these points. Consider then the expression X/2iy with 1 x = x+iy and = -(w- is). We observe c i -i ' -i(x+iy)( (w+t s) 2i' 2~ 21y12 21712 Thus sx + W ix is y X c c c c 21 2i(12 WX a S and if this is to be real - implying x = y Then C C (x 2 2 s y +wy C WC y 2 2 ( + + W 2iy = 22c- 22wc (s +w ) 2 * 1 22 2|X _ But y12 = r= - (w +s ), and so for real X/2iy we find = 2w 2 2iy 2w c To exclude the poles of the r-function we need, for real /2iy, - < This implies < or < =k 2i7 2 p 2w' 2 c But since X = x+iy, y=ImX. Therefore, the condition for analyticity in a half plane becomes ImX < k. 181

APPENDIX C CLOSING THE CONTOUR (CONVERGENCE OF RESIDUE SERIES) In this appendix, we consider the behavior as Ix | —oo of the integrands appearing in (i) the representations (1. 18) through (. 23), (ii) the equations (3.54) (Lemma 1) and (3.128b) (Lemma 2). Due to the similarity of the various integrands involved, we shall study explicitly only the integrands of the representation (1.16) (of which the integrand of (1. 17) is a factor) and equation (3.54). The other integrands are susceptible to similar treatment, the details of which are illuminated by the analysis below. For convenience, we repeat the representation (1.18) (2ik)-3/2 X +1 1 1,' - )y22 - vN~t o)- \|d2riik 2 2ik i) i(nx) -xo-ia (0 < a < k) [ y2(n, X)y( I(n )-y 1(n X)y2Io i ' and equation (3.54) C r C where C is the path defined by -co- ia to oo- ia (0 < a < k), and F(f1, 52' o,' k, X) is given by equation (3.47), while g(r), k, X) and X(r o, k, X) are defined following equation (3.51) We shall show that as I I —>ao, the integrand of equation (1.,1) is exponentially decreasing in the upper half plane (ImX >-a) for all values of 91, 92' il and to (g2 > Y1' r > 0). We shall also show that in the upper half plane this integrand has poles only at the zeros of the function y'(rn, k). These zeros 182

lie on the real line (Buchholz, 1942/3; 1953). For |X I — oo, we investigate their behavior and find that they lie only on the positive half of the real line. The form of these zeros together with the fact that the above exponential decrease is true for positive real values of X away from the zeros of y\(r X), enables us to replace the integral representation (1.16) by a convergent residue series expansion. The behavior as Ix -oa) of the integrand of equation (1.16) in the lower half plane (ImrX <-a) is different from that above. In this half plane, the integrand has poles at the poles of the r-function r(-~ +1) which lie on the negative imaginary axis. We shall show that for a portion of the lower half plane which includes the negative imaginary axis away from the poles of r(/ k + ) the integrand of equation (1.16) is exponentially decreasing as X 1 - co if and only if \ + T< </T Thus, the integral representation (1.16) can be replaced by a second convergent residue series expansion if and only if A1 + I2f< ~ ' If 1 + gf2> ~/, we observe that the integrand is exponentially increasing over the above portion of the lower half plane. If y1 + v~2 == we find that the integrand is decreasing like some power of x I| as JA I — co, but that the second residue series is not absolutely convergent. Finally, we shall show that as x I -* co on C, the integrands appearing in equation (3. 54) are exponentially decreasing for all values of gf0 f2 and ro. Therefore, the integrals appearing in equation (3.54) are indeed convergent. C. 1 Behavior in the Upper Half Plane We begin by noting that the function Rn. no, k, X) defined by frs, o,k,X ) = + 2 )Y ) - Yl(n' }I)y (rIo,, (C. 1) is analytic in the complex X-plane. Since y1(r, X) and y2(7, X) are analytic functions of X (Buchholz, 1953, Chapter 2), the only possible singularities of flnr, ro, k, X) are poles which can occur at the poles of r(- -2 +-). These o \ & li) ~ 183

poles lie on the positive imaginary axis at X = ik(2n+ 1), n = 0, 1, 2,.... But at these points y(n.X) = y [, ik(2n+ l) = -1 /2 M+, 0(2ik) = (2ik)l/2e-kn L~)(2ikr). (C.2a) Y2(nX) - Y2[. ik(n+ = n-/2 W+/ o(2ikr) - (-1)nn (2ik)1/2 eikL()(21kn) (C.2b) where L )(2ikn) is the corresponding Laguerre polynomial. Thus n Y2 E ik(2A++l) y. ik(2n+1] - Y1 [n. ik(n+lj] y2 [o ik(2n+1l) is equal to (1) n. (2ik) e- 1L( )(2ikn)[d (e- L)(2ik)Y1 - W (0) d -(-1) nW (2ik) ee L 2 iknlL) Of and therefore cancels the simple pole of r - +2) at X = ik(2n+ 1), n= 0,1, 2,... This implies the analyticity of fi(, Xr, k, X). Before proceeding with any more calculations, we consider some conventions to be used in this appendix. We recall that the X-plane is to be cut at X = -3ir/4. Thus, argX satisfies -3w/4 < argX $< 5r/4; this convention does not change in any appendix. Henoe for X -o ao, the upper half plane (ImAX.-a) can be characterized by -6(X) 4 arg ir+ 6(X), where 6(X) is a small positive angle which decreases as I Xl increases. Similarly, for I X -* ao the lower half plane (ImX. -a) is characterized by -3wr/4 <argX< -6(X), xr+6(X):argXA< 5r/4, where 6(X) is as above. We shall use this characterization throughout this appendix. 184

We now consider the behavior of Yi(ro, X) as I X-ao). For -3w/4 <argX, 5r/4, yl(no, X) has the representation (Buchholz, 1953, p. 98, eq. 17a) / 2 1/4 y (( 77k [2 12 - co. (C. 3) which implies.. 4kI2n X1/4 - Yi(nA) -1 -) in [2V i -. (C.4) Therefore, as lxi -bO the zeros of y( ) are the zeros of sin [2 - which obey the relation I + Nr kN - (8-2) N —o, (C.4a) 0 and consequently lie on the positive real axis. For -3r/4 < arg < 0, the representations (C. 8) and (C.4) may be replaoed by / ( 2iVo, >. ) X ~ y(l,<>) ve2i. y(ry;<,X) -ye while for 0 < argX. 5r/4, the representations (C. 3) and (C.4) may be replaced by y -2, yr,-e21/. yr ) on e where we have oonsidered only the exponential fator in X. We do this whenever the nonexponential factors or exponential factors not dependent on X are not essential to our arguments. To demonstrate the exponential decrease of the integrand of equation (1.10), we first examine the factor ( 1't o2' k, X) where the function (f1' 2', k X) is defined by 185

f(E kx) = (2ik)-3/2 Y(X)y2(2,-X ) (C.5) With this definition of f( l, 92 k, X), we note that equation (t. 1)8 can be written as f A91 ' k, 2 ). vN(O) = )X)l f(rL ro, k,kX) C ~ Now? we need to investigate y1l(1, -4) and y2(E2, -). For r/4 < argX < 57r/4, we can use Buchholz (1953, p. 98, eq. 17b) to find that as |lI -aoo Y,(5,.-A ) Qcos [2S e e \i/ 4 + e e. (C.6) while for -3r/4 < argX<: ir/4, the same equation shows that as IX |-o yl(l-X) c O 1 e /2- 4j - e~ -ir/4+ 2 1 1i r/4 y,('-") ~cos|2 e 4 e ~ +e e. (C.7) The function y2(2, -X) is not so simple. In order to examine its behavior and then that of f('1, I2, k, ) as IXI — oo, we divide the upper half plane into the sub-regions -6(X) 4 argX < 0, 0 arg < r/2, argX I r/2, r/2 <argX Tr, 7r < arg X < r + 6(X). On each of these sectors we first obtain the behavior of y2(52, -), and then study the function f(l 2'2, k, X). We begin by noting that we can obtain the behavior of y2(2,- -X) on (i) -6(X) 4 argX < 0 and (i) 0,< argX < r/2 with one calculation. However, on these sectors, we cannot obtain the behavior of y2(2, -X) directly. Instead, we make use of equation (21b), page 19 of Buchholz (1953) which asserts y2(r2, -k) 2 -e/2k e )^k/2k e -2 ei 22 186y2(2 e, 186

Now -6({) < arg A < r/2 - -r/2 - 6(X) arg 2k < 0 > Im k < 0. In addition, arg- k(2ik2e ) = argo(e r) E -6(X), 37r/2) and arg 2(2ik2e- ) = arg(Ae )r [-7r-6(k),-r/2). Therefore, equation (19a), r 2i - 7r i page 99 of Buchholz (1953) applies to both y2(e2e,X) and y2(Q2 e, ). But argXe [-6(X), r/2) implies ReX >0, thus Y2( 2 -X) "r( r( 2i k 2( 2e- ). -Tri Since the above-mentioned equation for y2( 2 e, X) can be written as y2(E2e,X) ~ exp ik log ikeJ e we find y 2(-X) r1 A- 2 r A,) e2i e'X/2k exp { log 2ike- e2 X Y22 2 2 22 ^ ik 2k log2ikeJ~ The range of arg - implies that Stirling's approximation is valid for the r-function r'+ ), thus from Erdelyi et al (1953) we observe r 2 + -^ exp i log 2ik (C.8) Consequently, the above estimate for y2(Q2, -X) can be written as 2 2 ()+ 2ik) ~ r2( -2 krya/2 2 2 J-2 However, from Erdelyi et al (1953) we also observe X (C.9) cos 2ik and using the exponential representation of the cosine, we see that as 1XI - on -6(X) < argX < 7r/2 187

y2(2-X-) v e /2k e. (C.10) Therefore, using (C.4), (C.7). (C.9) and (C. 10), we find that on -6(X) argX < O0 -2Rej^ 2ReI1 f e e YI(no) -21m.since -6(X)/2 < arg\JT < 0 > Re /> 0, Im IT< 0. Similarly, using (C.4), (C.6) or (C.7), (C.9) and (C. 10), we find that on 0 < argX < r/2 and provided X is not a zero of yl(ro,.) -2ReV-2 2Re 51 f e e e since 0 arg/ < /4 m<ReB >O. Im O>,. In either case, since C2>e. f/l"(rno, ) is exponentially decreasing as | | — D away from a zero of yi(r). X). (iii) argX = r/2. For argX = /2, we set X M iIX. Thus -X/2ik = -Ixl/2k. X/2ik = |Ix /2k, and equation (20), page 100 of Buchholz (1953) can be used to give r X l 1x ' X -2/ 2^ cos r /4 y -x) exp log }e. (C.ll) But (C.8) applies for r + ) and using this together with (C.4) (C. 6) and (C. 11), we note -2 [xL cos r/4 2/ co ir/4 f V- e....... e, yi(.o2 A) 2 iX oos r/4 whiis exponetallysmall as sce which is exponentially small as Ix I-*oo 'since 92 > 1' 188

We conclude by observing that we can obtain the behavior of y2(j2, -X) on the sectors (iv) ir/2 < argX < r and (v) r < argX.r +6(X) with one calculation. In these cases, we note that 0 < arg 2 < 2 + 6(X), which in turn implies Im /2ik > 0. But in addition X -ir X 21k e 2ik X and arg- (2ikg2) - argX c (r /2, r + 6(X)] C (-Tr 30r). Hence, equation (20), page 100 Of Buchholz (1953) applies to y2(2., -X), giving Y2(2,- exp ik log 2. (C.12) But equation (C. 8) also applies; thus, using it together with (C. 4), (C. 6) and (C. 12), we find that on r/2 < argA X r -2Re 2Re f e e.. yt'r*X) 2Im# e since r/4 <argTi<r /2 ==>ReT m O, Im ' > 0. This is exponentially small as I|xl-oD since C2 >C1 and Im > 0. For rI < argAX Z r + 6(X), we must argue slightly differently. In this case (C.8), (C.4), (C.6) and(C. 12) give -2Re 'i2 -2Re 9; f e e since r /2 < arg _ 2+ == >Re -<0, Im O>0. But as [X| I- on r < argA ~ yr + 6(A). [Re f j< k while Im fT increases without bound. Therefore, 189

-2Re~ -2Re e\ 2k\ 2k 2ImiW < 2Im /YF e e which is exponentially small since ImJT is increasing without bound. Consequently, f/y\(ro. X) is exponentially decreasing as I|x -ao i n the upper half plane. We now examine the behavior as I X| -- of the function Ifr), ri, k X). We shall find a representation for fnr, n, k, A) which is valid in the entire halfplane -6(X) arg, r+6(X). On this half-plane, yl(rn,) and y(rni, X) are governed by (C. 3) and (C.4) respectively. It remains only to find representations for y2(rl,X) and y2(rn, \). For 86(A) < arg X r +6(X), - '+ 6(a) arg i < + 6(X), and arg -2 (2ikT) - argx f E6(X), +6(X C (-2w, 2w). Hence, equation 19, page 99 of Buchholz (1953) yields 2) 1/2 U k211/4 [2-T k Y2(n. X) (< ( 4k12/2exp log 2A cos2 - + 4 q ik) xJ lkej and by differentiation (2)1/2 1/4 X y(nT ) - (-4 ( _4k t)/ exp2 log sin2 -r 2Nik ' But equation (C.8) for the r-function P - +-) is valid, thus these equations can be written as 1/2 (i2n /4 + Y2(t1 ) 11/2 A(ik 2) 2 4 (C.13) (2)1/2 2 /4 y(, k) VO) sn( 2i 0.14)k 4 (C. 14) 190

Therefore, using (C. 3), (C. 4), (C. 13) and (C. 14), we find - ( A +. 1) fA + 2) o X +008 [2 2; 21 2ik 2/ 'ik2M f rQ: +) r(.) ewx/2k e2i;; 2i + l ] + r( 4 ft + ) (A +i) e X/2ke2lnen2ii+,.2l1 ] (C.15) To examine the behavior as I X| -- of the integrand of equation (2. 1) ( *cf), we again divide -6(X): argX A r +6)X) into the u-rb-zleion -6(X) < argX <0 0 0 argX < r/2, argX r f/2, r/2 < argX:< r, r < argX< r+6(X). and we examine each sub- re ion sep rateb-I.. (1) -6(l) arg A < 0. In thit mb-egaon, ReX >0, and 2 arg lT<0 implies Im T< O, Re jT > 0. Then using (C. 9) and (C. 15) together with the exponential representation of the cosine in (C. 9), we find f e2 since, r n. This together with the previous representation of y-r( ) shows that on -6(X) < argX < 0 0 f -2Ref12I 2Re V 21m 2 XrT y,(n f O' e, e * which Is exponentially decreasing as IX| -- since Tm <0 <, Re >0, and E2 > 1. 191

(1i) 0< argX< r/2 In this sub-region., Re A > 0, and 0 xr arg Xr < /4 implies Im >0, Re \ > 0. Then using (C. 9) and (C. 15) together with the exponential representation of the cosine in (C. 9), we find - 2i Xo o f ~e Ve, since r)<n o This together with the previous representation of f/yl(rl X) shows that on 0< argX < r/2 f -2Re X2 2Re /ij -2Im yl (e ee which is exponentially decreasing as |Xj -oa (provided X is not a zero of y (rn, )) since ImT >,0, Ref >0 and 2 > 1. (ili) argX = r/2 In this b-region ReX > 0, and arg fX= r/4 Implies Im 7= Re/- = VTX cos 7r/4. Using (C.9) and (C.15) as above, we find v e 2 2xIn cos/4 e2 cos r/4 f,re-20 ~1, since rl7, o. This together with the previous estimate of f/yl(no kX) shows that for argX = r/2 f -2 T l cos 7 /4 2 F 0cos r/4 -2 ixj co08 r /4 *f v je e e which is exponentially small as |Xj ->a since 2 > C1 (iv) ir/2 < argX < In this sub.reogton, ReX <0 and r/4 <argC3(,< r/2 Implies Re7 T 0. ImF'T > 0. Using (C.9) and (C. 15) as above, we find 192

f,,e2! * e.-i o since r1 $. Thus, the argument follows closely that of (ii). If Re T = 0. the exponential decrease is implied by Im ~T > 0, rt > 0. (v) ir < arg X < r + 6() In this sub- rgion, Re < 0 and /2 <arg T + implies Re\ < 0, ImJT > O. Using (C. 9) and (C. 15) as above, we find - 2ie 21-2iiX;F since rl, rio. This together with the previous estimate of f/y,(ro, X) shows that on A < argX r + (X) f - -22Re ^ -2Im22 yi f ('ve e e Butas || —co on r <argX 'r6+6(X), r/2<argk 2+ 6). or Re'TLk, while Im ' increases without bound. Therefore, -2Re -2Re 2k 2kdf -2 m e e e 2 e e e which is exponentially decreasing as IXl J — o since rn >0. Consequently, the integrand of equation (1. 18) is exponentially decreasing as IX J.-o in the upper half-plane for all values of y1, 92' r and r1o (2 > ' n >0). Before proceeding further with equation (1.18), we discuss the convergence of the integrals appearing in equation (3.54). This convergence will be demonstrated by showing that the integrands appearing in equation (3.54) are exponentially decreasing as [X -- oo on C. If we compare the definition of F(F1, 2' Io,k, X) (equation (3.47)) with that of f(1, 92' k, X) (equation (C. 5)), we observe that the exponential behavior in X as X — >ooa is the same for both F/vI(no, X) and 193

f/y (rn, X). Thus we have already proved that F/v(rio, X) is exponentially decreasing as I X -w co on C. It remains to consider g(no k, X) and X(7o, k, X). Examining the definitions of these functions given following equation (3.51), we see that in order to derive representations for them as AJ I — oo on C, we need only to derive a representation for y2(ro e, -) as ix| - co on C. We consider first argX = -6(X). Then X ifr T and arg- 2 = arge 2ik 2 6() and arg (- ) (2ikre ) ~-= arg (e 2. 2ikeW) = argX - -6(X) Thus, we can use equation (19), page 99 of Buchholz (1953) to assert that as y2 (re,-X) - exp 2k log- 2ik cos22 I+ + 4 and by differentiation, y(,e -X) exp ~ 2ik log - 2ike sin +2 - +2ik +I (C. 16) Moreover, for arg-2ik = 2 - 6(X), Stirling's formula is applicable to the r1-function (- -2i + (Erdelyi et al, 1953). This gives r- A+ ) exp- log } (C.17) Using (C. 17) together with the fact that argX = -6(X) implies ReX > 0, we observe that (C. 16) can be written as y 2(e i-X) ~. x + e e (C.18) Similarly, Re X > 0 implies that (C. 14) can be written as (with n = T ) 194

yx) + e I, (C.19) Substituting (C. 18) into the definition of g(r1o k, X), and (C. 18) together with (C. 19) into the definition of X(rl, k, X), we find that as Ix J- OD on argX = -6(X) g(nr kX) e ~(C.20) and -4i 1/Xr X(n o,k,X) e ~. (C.21) Comparing (C.20) with (C.4) (for argX =-6(X)), we see that F/g is exponentially decreasing on argX = -6(X) as IX) - ao. But argX = -6(X) implies arg ~'= -T, which in turn implies Im r'< 0. Hence, X(no, k, X) is also exponentially decreasing as O|X -a) on argX = -6(X), and with it the integrands appearing in equation (3.54) n and F X' forany n *) We now consider argX - v +6(X). Then X -i X and arg- arg e i. + 6(x), and 21k 2ik 2 arg (- ) (2ikne ) = arg(e' 2X) = -2-r +argXi -r +6(). Thus, we can again use equation (19), page 99, of Buchholz (1953) to assert that as ll-.o y2(e 'lTi P)), 2exp log- cos [2 e + + and by differentiation y:(oe., -) s~ exp log- gsin [2 e- i+ J. (C.22) 195

X i In addition, arg- k - + 6(A) implies that (C. 17) is again valid for -2ik 2 rP- 2-i + 2. This together with the fact that argX = r + 6(X) implies Re < 0, shows that (C. 22) can be written as -W r f x k -1rA./2k 2i~ O oe -) 1k e * (C.23) Similarly, ReX < 0 implies that (C. 14) can be written as (with n = rro) y X) r(^ +2) e X/2k e io (C.24) Substituting (C. 23) into the definition of g(rn, k, X). and (C. 23) together with (C. 24) into the definition of X(r, k, X), we find that as x I -* oa on argX = r +6(X) g(n k,a.) e." e X/k (C.25) 0 and X(r0, k,X) o 1. (C.26) Examining (C.25), we see that the dominant term is e /,, and so Ig(,k, X) is exponentially increasing with j J on argX = ir +6(X). Thus, F/g is exponentially decreasing as IXA -- on argA = +6(X), since in this case we showed that -2Re -2Re; 2k 2 2k g F e e e e Since we have already proved that F/v-(n, A) is exponentially decreasing as XJ -oo on C, then equation (C.26) implies that both F.xn (n>0) and vn (n o) 1 o are also exponentially decreasing as IX — *oo on argX = I +6(X). Consequently, the integrals appearing in equation (3.54) converge. 196

We continue the study of equation (1. 16) by considering the implication of having shown that the integrand is exponentially decreasing as | X -+ o in the upper half plane, and, in particular, for argX = 0 away from a zero of y (r1, ). Let A1 < 2 < X3... denote the zeros of Y{(no1, X) along the real axis. For large | x|, these zeros have the form given by equation (C. 4a). Therefore, by use of the residue theorem * O n Furthermore, the Wronskian relation between y1(no, X) and Y2(v, X) implies n n(C.) But In (C.27), argX 0 for n sufficiently large. Thus, the previously developed Henasymptotic forms, 0) can be wrused to investigate the consergce of the res in,1..~ -/2^ \ Xy y2(,is,_ i)2(2 lutely convergent provided 2 > >. Moreover, thse are the conditions assumed when derIving (1. 18). For the most obvious attempt to represent the series in (C. 27) as the sum of the incident plus scattered fields, we use the integral representation of 197

-e ikR/ (Appendix A) to write equation (1. 1) as oD - i e-ikR (2-k)-3/2 VN(C,,Tip00) = + 2i.... R 27I ik 2 2ik 2 (O <a < k) y1(1-. -X)y2(2. -X) [ y (no, x) - (no, Y1 (0WoO 1 2 Then the previous calculations can be used to show that the integrand of the remaining integral is exponentially decreasing in the upperialf plane ifannd only if 1 - C + / < 0. But in the upper hlf plane this integrand possesses poles not only at the zeros of y\(ro, X), but also at the poles of the r-function o/ \ \ ~ r (- 2ik +2 which lie along the positive imaginary axis at the points X = ik(2n+ 1), n 0, 1, 2,... However, the contribution of these latter poles to the residue series (absolutely convergent if rg - ~ + (] <0) simply can-ikR cels -e /R. This follows immediately from the existence of a residue series (in terms of Laguerre polynomials) for -e /R if f - 2 + F< < and the calculation which shows f(n, n', k, X) analytic in the upper half plane. The final result is the previously obtained residue series. C.2 Behavior in the Lower Ialf Plane In the lower half plane, the integrand of equation (I - 1) has poles at the poles of the r-function r + which lie along the negative imaginary axis at the points X -ik(2n+ 1), n = 0,1, 2,.... To investigate the behavior of the integrand as [X — + o, we consider first the regton 7r +6(X) < arg X 5r/4. As always, Yl(n). ) and yi(r(n, X) are governed by (C. 3) and (C. 4), respectively. In addition, yl(l, -X) is governed by (C.6). It remains to find y2(2, -4), y2(r, ) and y2(ro X). We immediately see that the arguments leading to equattons (C.12), (C.13) and (C. 14) can be repeatedfor r +6(X) 4 argX: 5T/4. Thus as JXj — o in this rgion 198

-2Re 2 -2Re f e e 1y( o' 2Im0iXF e and -2Re S -2Re -21m - f..e. e e But r +6(X) <argk 5wr/4 implies2 2+.~:arg Z<5r/8, which in turn implies Re < O., Im t > 0 and -Re T < Im T. Hence, -2Re -2Re\ V 2 -Im2 2ImX 2Im Y -2 Im e e e < V e e e Therefore on ir + 6(X) argX 5 r/4, the integrand of equation (1.16) is exponentially decreasing as Ixl - o if ~ + < /. We now consider the remainder of the lower half plane which can be characterized by -3ir/4 < argX < -6(X). -I this region,, we first examine the factor f(1' 92' k, X) YI(n,X) 1 0 Since y1(rn,X) is governed by (C.4) and y1(S1, -X) is governed by (C.7), it remains only to find a representation for y2(2, -X). We can derive a representation for y2( 2, -A) which is valid for the entire izgioani -3r/4 <arg A. -6(X). In this iT X ir X case - = e, and so -2 =e 2ik. Then -3r/4 <arg < -6(X) implies,r/4 < arg- 2ik -6(x), while arg (- )(21kU2) = arg(e X) = r +argX e (r/4, -6(X3 C., 2i). Hence, equation (19), page 99 of Buchholz (1953) may be used for y,2(2, -X). This gives 2-X) -exp 2ik log - c2ik [2y + +] 2 2 k i Cos + 2-k 4 199

Since in this case (C. 17) is valid for r (- 2- + 2 this estimate for y2( 2, -) can be written as y -+) +- cos[2i ^ + + (C.28) Then to examine the factor f1, 2D k.X) we divide the regonl -3wr/4 <argX $ -6(X) into the sub-regions -i/2 < argA. -6(X), argA = -r/2, -3;r/4 < argX < -r/2, and consider each subregion separately,. (i) -r/2 <argX -6(X) Since -s/2 <argX.4 -6(X) implies ReX > 0, then (C. 28) can be written as y2(2-X) (r 4A + /2k T (C. 29) Using (C. 4), (C. 7), (C.29), (C.9) and the exponential representation of the cosine in(C.9), we find -2Re ^~ 2Re,1 f ~ e e Y( X* ) -2Im since -r/2 < argA < '6(x) =0 -r/4 < arg/T: X 2 Re 7 > 0, Im ~f < 0. But 2 > 1; thus, f/yI(on 10) is exponentially decreasing as lXI ->oo on - /2 < arg< -6(). (ii) argA = -,/2 In this case X =-i|X L hence (C. 28) can be written as I 2 2 cos/4 Y. - e.30) Then using (C. 4), (C. 7), (C. 9) and (C. 30), we find that away from a pole of ' 2ik 2) 200

2 j[| cos r/4 2 |XI| cos r/4 f e 0 Yl(,X) e2 2 JFcoes/4 0 which is exponentially decreasing as AX -oa if and only if l + J < 7. (ili) -3r/4 < argX < -/2 Since -3ir/4 < argX < -r/2 implies ReX < 0, then (C. 28) can be written as v2 r(2ik+.2) e'Xf2ke2 (C.31) Using (C. 4), (C. 7), (C. 31), (C. 9) and the exponential representation of the cosine in (C.9), we find 2RefW1 2Re I2P f e e yi(yiX)A -2Im y;n e since -3r/4 < arg X < -r/2 = -3r/8 < arg T < -?r/4 =s Re f > 0, Im\ I < 0. But in this case Re f < -Im ', thus 2Re R 2Re 5 -2Im S1 -2Im/f e e V e e -2Im 2 Im -2Im(X ' e e which decreases exponentially as IX| -ao if i/ + < fT But as argf? appraches -7r/4, -ImIT approahes ReV. Hence, f/yi(r )o ) decreases exponentially as X — ac only if V + f < Fo. To examine the behavior as X-| - oo of f(rT, r), k, X), as well as that of the Integrand of equation (1. 16)( y (r,) * f), we again divide the region y1 -o -3r/4 < argX< -6(X) into the sectors -r/2 < argXA -6(X), argX = -r/2, -33r/4 < argX < -r/2, and we examine each sector separately. As always (C.3) and (C.4) are valid for y1(r, X) and yi(70, X), respectively. It remains only to find a representation for y2(r), ) and y(Y, X). 2 2 0 201

(i) -r /2 <arg x< -6(X) X In this case -r < arg 2- j - 2 - 6(X), and arg 2X (2ikn) = argX e (-r/2, -6(X0 C (-2r, 2r). Thus the arguments leading to equations (C. 13), (C. 14) and (C. 15) can be repeated as I | — oo on this sector. But -r/2 < arg: < -6(X) implies ReX > O0 and -w/4 < argX A -6(X)/2, which in turn implies Be? > 0, Im (X < 0, Thenusing(C. 9) and (C. 15) together with the exponential representation of the cosine in (C. 9), w find - e2in 2iY n f,v 2; k ' since rn o n This together with the previous estimate of f/yT(nr, X) shows that on -w/2 < argX,<-6(X) f - -2Rea 2Re '~7 e e e ~ which is exponentially decreasing as IXI - oD since Im b' < 0, Re / > 0, and 92 >"l' (11) argX = -r/2 In this case X - l|L and = -J. Thus, we can use equation (20), In this case X= -ikL and 2ik 2k' page 100 of Buchholz (1953) to assert that as IXI — cao on argX M -r/2 2(r. X) -V exp- 2log e'i kIe' ~r/4 (C.32) 2 2k 2ke and by differentiation rixl lxl~ -2J1In7 coo/4 Y2(o X) l exp - log (C. 33) 2 ~ log 2k But equation (C. 17) holds for the r-function P (- 2 +. Hence, substltuting (C. 17) together with (C.3), (C.4), (C.32) and (C.33) into (C. 1), we find fe 2 1-coo,/4 e2 I co r/4 ince. Therore, awayfrom a pole of r + since n <no Therefore, away from, pole of 2 (' +i) 202

f 2 jFl 2cos r /4 2 x coso /4 2 rIX' coosr/4 ') 'f e e which vanishes exponentially if and only if + < I7. This equation also shows that as |xi -o a on argX = -i/2, the integrand of equation (1.16) is exponentially increasing if / + 2 > fT'. and X is away from a pole of (\2ik 2/ ' (tii) -3r/4 <argX < -r/2 On this saator, we cannot directly obtain a representation for y2(r, X) as | I -+oo. However, we recall that for 0 < argX < r /2 we derived equation (C. 10) for y2(C2, -X) as I x -I o. Thus if we consider the substitution v = Xe and recall that in 0 < arg X < r/2, -X = e X, then as I v -oo on -r < argv < -i/2, (C. 10) implies y2(N v) r(v V e, +2ik e2e But since v, C2 are dummy variables, we can write for -r < argX < -I/2, and in particular for -3ir/4 < argX < -r/2, y2(M X) + X/2k e 2 e Y TI. (C. 34) Y2 P + 2-le /5 (C. 34) and by differentiation y (no, r X. e (C.35) Substituting these equations together with (C.3) and (C.4) into (C. 1), we obtain r~"oX — X ) 2.'. 2 +' -~p -1 )+ A ) e- e +e e. But -3wr/4 < argX<-w/2 implies -3vr/8 < argj' < -r/4, which in turn implies Re T > 0, Im f < 0, RefT <-Im f. Then using (C. 9) together with the exponential representation of the cosine in (C. 9), we find 203

-2i 15 21i X since n K no. Therefore, 0 f 2Re k/i 2Re /2 2Im kj In addition, 2Re 2Be -2Im 2Im e e e Im < e e e 2 This implies that as |x -| 4- on -3r /4 < argX < -r/2, the integrand of equation (1.18) is exponentially decreasing if T~ + ^ < '. But as arg X approaches -7r/4, -Im\/f approaches Re /T. Hence, this integral is exponentially decreasing only if C,1+ C2 < ". Finally, -Im fT approaching Re V' also implies that if J1 + ~2 - = > 0, the integrand of equation (1,18) is exponentially increasing as |A I-co on -3 /4 < arg < -r/2. We continue to study equation (1.18) by considering the implication of having shown that if and only if 1 + Y2 < (c, the integrand is exponentially decreasing as |I A -k- o in the lower half-plane, and in particular for argAX — /2 and away from a poleof r( +) Thus suppose + < Since the poles of (e-ik + 2 lie at the points X = -ik(2n+1), n = 0, 1,2,... then the residue theorem implies y [y iy k(2n+1 y [ ik(2nl+ i VN(D v7. 0) = -(2ik)-/2 (.1)n Y n=0 yl [n,-lk2n+1) {j(n+ [- 2n+ i E,-ik(2n+ l) -y1 [n -Ik(2n+ li y [,-ik(2nl+ 1j (C. 36) where we have used the fact that the residues of +.) at the poles X =-k(2n+1) are 204

2k (-)n (-i) n. Moreover, the previously developed asymptotic forms can be used (-i) Ml to investigate the convergence of the series in (C. 36). By arguing as above, it is seen without difficulty that this series is absolutely convergent if and only if 1+?2</i. A representation of the type given in (C. 36) may be found for a less -icR restrictive inequality. Using the integral representation of -e /R (Appendix A), we observed that (1.16) could be written as M - _ ia vi 0) - e' + (2ik)'3+2 c d rx + r > +1 v to nsR.o) - 2- \ d+1r 2 2+ -00- ia (O <a < k) yl(~l, -X)y2('2o -X) Y... y (nyo(M, y( OX) (C.37) The arguments above show that the integrand of this latter integral is exponentially decreasing as |XI -o in the lower half plane if and only if {1 + 2< 2 'n - F. Thus, suppose +; < 2 /I - Then the residue theorem gives -ikB 01 y [- ik(2n+l] y2 [ ik(2n+ 1 e + / (2k)1/2 j( n Y12 n=0 y [.-k(2n+1] y n" -lk(2n+ y2 ro -ok-(2.+1, (C.38) and J1 + j < 2 fj - {i implies the series in (C.38) converges absolutely. It is instructive to consider the behavior of the integrand of (1. 1) ((C.37)) if f- + = ' f ( y + 2-. 2 Vi - tW). Then the exponential amplitude factors are equal to 1, and the behavior of the integrand as I|X l o is 205

governed by the powers of X which appear. It is not difficult to show that as fxJ -Oc in the lower half plane, the integrand behaves as 1/|Ix 3/. Consequently, the absolute value of the integrand of (1. 18) ((C.37)) is approaching zero as lxI -GoD. Therefore, the residue theorem may not necessarily be used to obtain (C.36) ((C.38)). At the poles, IXi k(2n+ 1). Hence, the residue series, whose terms behave as I / for large X, does not converge absolutely. 206

APPENDIX D UNIFORM ASYMPTOTIC REPRESENTATIONS In this appendix we will consider separately the two asymptotic representations of the Whittaker functions which are needed in Chapters II and II. The cases studied all correspond to problems investigated in those chapters. The results obtained are based on the work of Langer (1932, 1935). More detailed results are given in the memoir of Erd6lyi and Swanson (1957), who discuss the necessity for the two representations, and the paper by Taylor (1939). D. 1 Airy Function Representations In this section we study the Whittaker functions Mi (41s) and W (41s) o Et o for scR, R1 being the region illustrated in Fig. D-l, and I f -co subject to the restriction that arg ( s) =?r/2. These functions satisfy the differential equation d +2 41 u =0 (D. 1) ds 4s If we define j = -2i/, (D. 1) becomes d u F-2(s-1 1 d2LiU = 0 (D.2) For p a complex parameter such'that ( >>1 and s belonging to the closed domain R1, equation (D. 2) is of the type studied by Langer (1932) and his results may be applied. In our applications (Chapters I and II) we have adopted the convention args~ Erf/4, 7r/4). For a closed domain containing s = 1 in the region defined by argse f1/4. r,/4 Langer's results are applicable provided Is I~ 1/[p |/3 throughout the domain in question (Taylor, 1939). Such a region of validity is the region R shown in Fig. D-1. Let 0 2(8) = -8 where T is the root of 2(s) such that lim (f 1 1. s-l _( 1)1/2=1 207

s-plane FIG. D-1: REGION OF VALIDITY OF ARY FUNCTION REPRESENTATION. We also define tr8 ~8) = \ (t)dt 1?=p 8) - g 3 p g8) (D.3) and s 7) (s)J / ](s) with '(1) = limr (s); (D. 4) 8 —1 these are the functions used in Langer's theory. For Is |> 1 we can write 8s) = f(s-l) -og(~ + s), (D.5) while for | s < 1 we observe that =8) dt 1. (D.6) 1 We can derive the expansions (s) = (s- 1)1/2 2( - 1) + Os- 1)2)] -'(-1 as s —1, (D.7) 208

and (s) = (s-1)3/2 ((8-1)5/2+ O - 1) 7/2)as s —1. (D.8) We note from (D. 8) that as s - 1, I-0(1) if and oly if Is-l= - (f). / 1 1/2 We also need the behavior of the above functions as s -oo. Expanding - ) about s = oo gives 1 (s) = l- + (1/s2) as s->, (D.9) and using (D. 5) and (D. 3), respectively, yields ~(s) = s- -log2s'+0(1/s) as s —o0, (D.10) = p-2- log 2s +0(1/s) as s —a. (D.lOa) We wish to find asymptotic representations of the functions Ml ((4s) and Wl o(41s) for |lI —ao (argls = t/2) valid for all seR1. In deriving them we shall use the known asymptotic representations of these functions for Ils ~>>II |. Thus for ls| >> 1 I we use the result (Buchholz, 1953, Chapter 7) -1 U1s - Us -lri(- 1 M10(41s) o o + - e r. -1 i +1> Ms-J e+ (4 *) e ) (-r/2 < argis < 3n/2) Similarly for Itsl ~ I I and argis = /2, W, o(418) s (41s8) e - 1, 0 209

Finally argls = fr/2 implies arglse = -r/2 e (-3r/2, 37r/2); hence for?,i ) e- 7 i -1 21s W-l o(41s e ) (41s. -, o These latter two representations enable us to directly find the desired asymptotic representations for W1 o(41s) and W l o(41s e ). Once we obtain them we can use the relation (Buchholz, 1953, Chapter 2, equation 20a) -IfiW /AS ^-l -l7i f1i/2 e iW o(4se -i) e - ie W (41s) Ml0 + (D. 11) to obtain an asymptotic representation for Ml 0(41s). We derive the desired asymptotic representations for W, (41s) and W i. (41s e ) by comparing their behavior for Its I >>I I with the asymptotic behavior of the functions v(( (jr/2)1/2 e+5ri/12( )?1/3H() ( (j = 1,2) "1/3( which are solutions of the related differential equation d Y - 2 s1) - A]. = ds L / 5) The asymptotic behavior of these functions for large? is given by () = () -1/6 + - j = 1, -T < arg 5 < 2r ve) = e- +o(1/?) j =2, -2fr <arg<ffr But since argls = 7r/2, we see from equation (D. 1Oa) that as s- aoo arg - 0. Therefore either of the above expressions is valid as s -. oo; (D. lOa) implies that 210

v (s) has the correct exponential behavior in s for W (41s) and that v ()s) has the correct exponential behavior for W I (41s e ). Hence by Langer (1932), we see that for any? such that argIs = ir/2 and s e R1, — ri - - (1) (41s e ) = C v (8)+0(1/W,., 0 1 P) if I|kN. (D. 12a) -1/6 E(l) - " - 7r i — v- - (1).28)? M w (41s e ) = C v (s) + Oft if I'I>N, 1" 0 1 p c W, 0(41s) = C2 (a) +o(1/p)1 if I?| IH N (D. 12b) (D. 13a) (D. 13b) W (4s) C -(2)(s) + C( ) E ( ) *o 2Lp _ if |I| >N. where N is a large positive number, the E(j)(?) (j = 1, 2) are of the form E) -a i? T e-l? E(J=(X) = A(J)e e with A(J)= 0(1) or A(j) = 0, B( = 0(1) or B() = 0. depending on the value of arg?, and the Cj (j = 1, 2) are determined by the relations C = lir 8 — JOD W (4 se -Ti) -(1) v (8) (se-r i)-I 2s (_ots e ) e lim _/ei 5 -P-OD j(s) C1/ei W, 0(48ts) (4, 9)e-26s 2=.im -(2) = lim (-1/6 ie 2 s V(2) s) 8 - a -( 1/6s) ie t Using the definitions (D. 3) and D. 4) together with the expansions (D. 9) and (D.10), we find 211

1/6 = (-2)6e-log lIe eli C2=('2i el log l/e Thus (D. 12) and (D. 13) become, for argls = fr/2 and s R1 W (41-i ( 2i)l1[6e-ll gllo I (s)+Tj (ir N) * (1)(D.14a) /6 _11o/ee/r i) E( l v-(1)(E ).(4ts ) = (-2J)1/6 e-log/e Elv )(s) + Qr1>N) (D. 14b) Wl,0(41s) = (-2il)1/6elogl/e (2)(s) + 0(1/ ([jr1N). (D. 15a) W (4)(e)+ =(2)5 i E (+)] lOmN)I> (D. 15b) We now use (D. 11) to estimate Mi, o(41s) for argls = r/2 and se R1. Since jargli- r/2<1 r/4 r(+l)= illogl/e[1+o0(1/1] (1 >>1) (D. 16) (Erdelyi et al, 1953). Using (D. 11), (D. 14), (D. 15), (D. 16) and the relation (Erdelyi et al, 1953) r( -I)r(j+1 c) r r' (D. 17) we find that for arges = ir/2 and seR1 M2((a/j l [-() +e(ii/()(s+)+ O u.(1/pi if fl< N, (D.18a) 1/6 -r'i (1)2). (- 11/6,(1)() +(2)(l' M2 o(4 ) = ( 21 l/) + E, If kI>N. (D.18b) 212

For purposes of calculation, it is often more convenient to represent the Hankel functions H(1 (?) in terms of the Airy function Ai(z) defined by the 1/3 integral representation Ai(z) = cos( s3+z)ds. 1 1 3 coos + z)ds. We can use the relations (Abramowitz and Stegun, 1964) () -3 Hi/3() = e -. / [Ai(- )- iBi(-_a) (. (3)2/3). 1/3 2 () - iBi(-)] = 2 eT i/3 Ai(-a e2 i/3) H(13() = e7i /; [(7 [Ai(-.)+iBi(-;I [Ai(-)+ iBi(-o] 2eri/3 Ai(-Aie-2r i/3) (Bi(z) is an Airy function linearly independent of Ai(z) which does not enter in the final result and so is not defined here) to find J( e 1/2 4/3 1/6 (1) = ei/2 (2) 4/3(31/6 i/3 H23() i (2) (3) Ai(e-;2i/3) 1/3 11/3 Therefore equations (D. 14) and (D. 15) become, for argts - w/2 and s R1, i e' -/12 tlogt/ e irl AM 2tr i/ W (4.se -) C e' e l A-a e-3 (/) ({jr N). (D.19a) 213

W ri - r C le-/12e-llogl eeli )Al(-oe /)+.p.... (l > N), (D. 19b) W C 1/loe ) A l-ae21/ (le 1 (<N). (D.20a) W4 o(41s) = Cle'1/12 elogl/e [)A(-ae-2i /3) "E/6E P (lI>N). (D.20b) with C I 2(3)al/6 /2(u)l/6 ad a (3)2/3. Finally, since (Abramowitz 1 2 and Stegun, 1964) Ai(-o) = e Ai/3A(-a e2 )/3)+e l/3Al(-a / e ) equations (D. 18) becomn M (41s) = D, e ii/'4 eIA-iF (s)Ai(-) If I N (D.21a) I 0 L 1 Ml 0(4s) = D,. /4 te;(s)Ai(-7)1 +.r176^ + E2)] if 1j|>N (D.21b) where D, = C 21/, isasb, i as aboveargs r/2, and seR1~ D.2 Bessel Function Representations In this section we study M1 o(41s) and W, o(4es) for s Ro, the closed domain illustrated in Fig. D-2. We now define = 21 in equation (D. 1) obtaining 2du2 = 0( 8. )u(D.22) ds + 214

FIG. D-2: REGION OF VALIDITY OF BESSEL FUNCTION REPRESENTATION. Suppose we consider p to be a complex parameter such that Up | ~ 1. and s to belong to a simply connected, closed domain R of the complex plane which includes the point s = 0, but excludes the point s = 1. Then equation (D. 22) is of the type studied in Langer (1935) provided R has the required properties. For the s-plane described by arg s e r /4, 77/4), Langer's results may be applied in a closed domain containing s = 0 of the region described as follows (Taylor, 1939): Let s be a real, positive point in the s-plane such that (1 -s ) >> 2 o 1 P Then the region R is the union of the sub-region = -~{s.......X and 0<:args~< r/4, or 11 I - n( -arg s) s sin0 4 st<...p.. —ar — g and -rargs /4, or Is - 0 4 and -w/4 args O sin( +args) or r/4 < args < 3r/4 215

and the sub-region 2 =s { 3r/4 s args 7r/4. (To emphasize the arguments of s in the regions 1 and Q2 we shall simply refer to these subregions as argsef E /4,3r/4) and argse [3r/4,7?r/4), respectively.) It should be noted that R and the region R1 for the Airy function representation are not unique regions of validity for the respective representations. We have chosen them simply to ensure a valid representation in any portion of the s-plane arising from the applications. Let p2(s) = (1 - s), where p(s) is to be the root of 2(s) determined by the relation lim l1/2 }s) = l. 0 — We also define s) = \ p(t)dt, p(a) (D.23) ) = [(8)()0 1/2 with (04) =lim (s). (D.24) 0 —10 We show below that for large s we can write 11/2 + if arg~ acr/4.,, /4). (s).t= i t — dt, (D. 25) t - if arge [3ir/4,7r/4) (The relation (t..{.f/2 4(s) = i t-1 dt 0 also holds for any s such that args is in a neighborhood of r. ) Integrating by parts, and then following with an elementary integration, we obtain for any s 216

idt = s (1- 1 + log Q - ~, (D.26) the square root being defined above. In order to derive the necessary asymptotic representations the behavior of these functions as s- 0 and s- o a must be investigated. Clearly, () 1 1 1/2- + ( 3/2) as s-*0. (D.27) -1/2 - 2 + Therefore equations (D. 23) and (D. 24) yield (s) = 212 - s32+ 0(85/2) as s- 0 (D.28) = p(s) = 2ps1/2 [1- O(8+ s2 as s- 0,(D.28a) (s) = p [+8s+ 0(s2)] as s- 0, (D. 29) and -= 2s+ + 0(s2 as s- 0. (D.30), = s) + s 3 In examining the behavior at infinity we note that for argse [-r/4, 3fr/4) fi ini/2 r,,7 -7r -s=e s; hence as s- oo, 0(s) — e /.For argse [3 /474, -s=e - n i/2 1 1/2 hence as s - a), O(s) —e. Therefore expanding (1 — ) about s =O yields 0(s) =i1- 1 + 0(1/s2 as Is — oco argsE[- 7/4,37r/4) (D.31a) (s) = - 1 + O(l/s2)] as I s -.oo, argse[37/4, 7r/4). (D.31b) The definitions (D. 23) and(D. 26) then imply 217

=p(s) = ip[s- 1-log(2isi)+0(1/s] as |s-moo argse -r/4, 3r/4, (D. 32a) = p(s) = -p [s - -log(-2if)+ O(l/sIas Isj-o. args [3r/4.7ir/4) (D. 32b) We can now directly find an asymptotic representation of Ml 0(4Is) which is valid in R. Because of the regularity of M o(41s) at the origu s = 0, the o Io theory in Langer (1935) asserts that for any r such that argls = w/2 and seR M (41) = E(s)+ () 0(1)(s if | ir1N (D. 33a) Os+ I(s);12 0p ) + e 0(1 M (41s) = c (s) 2 )/ O(l) +e i(l i 1>N (D. 33b) ~ where v~(s) = i(s) Jo(r), N is a large positive number and c is a function of I to be determined by a comparison of the behavior of v (s) and Ml o(l4s) as s- 0. v (s) is a solution of the differential equation d2 Y 21 + 1 - '"(s) Y= 2 2 ds 4s which is called the related equation of (D. 25). We find M, (41s) c lim -, s-0 v~(s) and thus (41)1/2 1 21 /22 Therefore, equations (D. 33a) and (D. 33b) become 218

M.o-() = P [ 1 2 o() 9 (lI< N). P (D. 34a) M (s) / (a] (.)]+..... I>N) Ml o(41e)= PL / J(r+ o/2.)+eo(1 (D. 34b) for argls = /2 and seR. In Chapter II we are primarily concerned with the case where Ips << 1 while IpI - ao with argp - arg - r/2. In this case s- 0, and we can use expansions (D. 28) through (D. 30) to find for | C N M (41X) = (2p8)1/2J ()0f+o(sJ + O(ps) 1/ 2) Upon retaining only the order term of the lowest order n s this becomes Ml (4b) = (2ps) /2J (r)[ l+(sl " 0 o (argis = r/2, seR ), 0 (D. 35a) except at a zero of J0o(), where then the additional term is the required estimate. For I(|>N and r not a zero of J (r) we find M 1(4s) = (2ps)l/2 J( ) [1 +0(s)+0(1/p) while at a zero of J (r) the estimate 0 M. (41) = 0( ) /4 (argls = r/2, sER ) is valid. Upon comparing order terms we find the next to last result reduces to M 0(4) = (2ps)l/2J() [l+o(l/p (argbs = r/2, sR ). (D.35b) 0 219

In order to derive an asymptotic representation for W (41s) for s e R lto 0 we consider the two sub-regions (Q and 2 separately. In these two sub-regions we can directly find an asymptotic representation of W o(41s) by a procedure analogous to the one used in Section D. 1. We again make use of the fact that argis = /2 e (-3ir/2, 3r /2) implies W (4b), (41s) 2 e 8 1,0 for Ils j >>~l 1. We compare this asymptotic representation with the following solutions of the related equation v~ (s) = (,/2) 1/2 + tl/4P(sB) H(J) (j = 1 2) whose asymptotic behavior for large | | is given by v ) = 1/2 ir [+i j9l, -r <arg < 2ir v(a) = fs)?1 [+ Od//PH j=2, -27r< arg < r. Consider then arg s -r /4, 3r /4). Since argis i= r/2, equation (D. 32a) implies that as —. ao, arg C -4 r; consequently only v (s) has the asymptotic representation given above as s — oa. But we see from equation (D. 32a) that v (s) has the correct exponential dependence in s for WI o(41s) and hence according to Langer (1935) we see that for any C such that argis = ir/2 and s e SI (see the beginning of Section D. 2 for the definition of gl) W, (41) = D[V()(s)+ 182) logO(1)] if I lN. (D.36a) 1,0 where N is a large positive number, E(r) is of the form 220

E(r) = Ae + Be with A = 0(1) or A = 0, B = 0(1) or B = 0, depending on the value of arg, and D is determined by the relation W, (41s) 4 Is -21s D= lim lim (41s) e s — o v (s) s- 1(s) /) e s Using the definitions (D. 23), (D. 24) and the expansions (D. 31a), (D. 32a), we obtain D (i)1/2 llog-l/e D 1/2 e (21)1/2 therefore equations (D. 36) become () 1/2 _ _log-_/e v(1)(_ _(s)_logO_(1, N W (41s) = -1 2 e g/ (S>+ ) K ( (D.37a) 1, 0 (21)1/2 P W (41s) = ()1/2 elg-e (1 )+ (s ( >N) (D.37b) J90 ~(21)1/2 P for arg s = 7/2 and s e. As in the study of Ml o(41s), we further consider, for use in Chapter II, equations (D. 37) if 21s << 1. We can again use expansions (D. 28) through (D. 30) to findfor I|r$N W, (4s) = i(r )1/ e log- /e (21s)1/2 H( 1) 3)o(s)+0(l [ except at a zero of H (). (there are no zeros of H (r) on the principal branch, 0 0 Erd6lyi et al, 1953) where the additional term in (D. 37a) is the required estimate. Upon comparing order terms we find the last result becomes W (41s) = i(/r)1/2 Ilog -/e (21s)/2H(1)() [l+ 0f / (arg1 = 2, s 1 (D. 38a) 221

Similarly for | | > N (with th above exception for zeros of H(1)(C)) 0 ^(4s) =i(,)1/2e log-1/e (2,1)l/2H(l)(C)[1+0(1/ 1 (argIs = /2, se ). (D. 38b) To complete the investigation of W o(4s) we now consider argsc [3/4, 7,/4). In this region we also need an asymptotic representation of W o(41s e ). As previously, we make use of the fact that arg se e"1 - /2 e (-3 /2, 3 /2) implies W (4e i) ~ (4jse- i)-I 21s 1, o for ls| >>~t I. From (D. 32b) we see that as s — o, arg C->0; thus both asymptotic representations for v(j)(s), j = 1, 2 are valid as s-aoo. Hence, (D.32b) implies that v(2)(s) (v(1)(s)) has the correct exponential dependence in s for W (41s) (W (4se -i)). Therefore, according to Langer (1935), for,or Wo, o any C such that argts = rn/2 and a 2 W (41 - D (2, )+ )1/2 + W, o(4 Is)=D2 L(2)(s)+ ]/E(2)('~ if IC lkN. if IC>N l if I | I N if |IJ>N, (D. 39a) (D. 39b) (D. 40a) (D. 40b) where N is a large positive number, the E (()(, j = 1, 2 are of the form E(j)(r) = A(j)ei+ B(j) e-ic 222

with A Q = O(1 or A W= 0,B( = 0(1 or - j 0 dependng on the value of arg C. and the Di, j = 1, 2 are determined by the relations D 2 rni 2 l->G W1,(41s e T ) v(1)a (4lise. r) e21 l~m 1/2 lc 1 — 2O1Ps) (41s)1 e2s A o v(2) ) lim 5 -+) OD Using thd definitions (D. 23), (D. 24) and the expansions (D. 31b),, (D. 32b) we obtain (-O /2 log-1/ Irin Di (2t)1/ (l) 1/2 1lg-/ 2 (21)1 Thus equations (D. 3 9) and (D. 40) become, for arg Is = in /2 and aSE ~22 (4sei) = 01/ - log -1/e Ir i[(1)+ 'I(S) K10 rOM (D. 41a) W1(4lse ) (-l) 1/2 (21)/2 e-1 log -'t/e 01IT i V(S)+ J~)1/2 E(1)(~] >) (D. 41b) W (41 - = 1/2L 1 log -1/eF(2)() 1)510 (OM <N t,o (241)12 e 7s)lg0 W1(41) - log -.1/e 2 sc122)( W,o 1/2)-L' e V SI p,p 0 ~(2-,1/2 (D. 42a) (D. 42b) 223

APPENDIX E ASYMPTOTIC REPRESENTATION OF -e-ikR /R In Appendix A, as well as in Buchholz (1953), an integral representation -ikR' for -e /R-..(R being the distance from (:, 0) to (, rl)) is derived. The result is -ikR- 0D-.. -e = 2i r( + 2) r k 2+ 2) v( -X)v2(2, -X)v2(r, ). A ia (|a| <k) (BI) In a similar fashion we can derive -ikRH Po-ia -e d (k + + )V -X)vll2 R 27i 'i 222 D- ia (la I < k) (E. 2) where RH is the distance from (0, H) to (5, vl). If we refer to the asymptotic representations of Appendix C, we see that the contour of integration in (1. 1) ikRH can be closed around the top (bottom), and thus -e -/R_ can be replaced by a convergent residue; series expansion, provided the inequality J" - g2 + fi7 < 0 (fi7+ f- fj< 0) is satisfied. The same is true for (B.2) provided the inequality -<f+ 4i;j+ TV; <O (A+ fi; - ii < 0) holds. We note that there are regions of the interior of the paraboloid of revolution in which none of the inequalities listed above (top or bottom) are satisfied. Thus there are no convergent residue series representations in these regions. Hence if k >>~1, kg > 1, k~7 >>1 (k71 >>1, ky2 >>1, k ~>>) the derivationof an asymptotic representation of ikR- / ikRH -e (eI H directly from the integral representation (E. 1) ((E. 2)) is considered in order to provide some insight into the derivation of an asymptotic representation of the field. 224

E. 1 Asymptotic Representation of -e ikR/R for the Source at (E, 0) Since we only want to use this derivation as a guide, we assume a = 0 for brevity. Let us define I( ' 2' ) (2 + ) + r vl(l -A)v -)v, ) (E.3) so that (E. 1) becomes ikR oo e dR- d IX, 1,2. r) (E 4) We write (E.4) as -ikR_ 5 r e " dR- 2P L d7 2,7) * (E,4a) -. j= Cj where for some M > 0 (i = 1,..., 7) with M3>> 1 and M5 >>1 Cl QX-wo<x<-Mlk2rn<o 0 C2 = i'(Ml+M2)k2<<M3k} C = { -(M3*M)k< X< (M4+M5)k C4= 4 iM5k < < (M6+M7)k, C = { J M7k < X < OD} and we examine each integral separately. The behavior of the Whittaker functions over C1 and C5 is governed by the asymptotic representations of Appendix C (equations (C. 6), (C, 12), (C. 13)J 225

respectively for C1 and (C. 6), (C.10), (C. 13) respectively for C5). Using these representations, we see that the integrand over C1 and C5 is exponentially small. The behavior of the Whittaker functions over C2 and C4 is governed by the asymptotic representations of Appendix D. We consider first C2. Since argX = x, 2 -1 argsg = r (sp = k2 -). Then from (D.34b)and(D.42b), respectively, we find sj j j X (2 g(E. 5) v,(5,-A) -1(2ikgl)1 K-() 1 ( ). (E.5) L I 2 ik 2ik (s2)1/2 v2(F2, -X) ~ (2ik2) /2 (-i)1/2 exp { ~]log 2ie (, a.6) 2 with 2 1 2(s )= (l-s ) J JJ sS 1/2 (S) = j 1/2dt sj j In addition, argX = 7r implies args = 0. Thus, from (D. 15) we obtain v2(,X) ~ (2ikn)1/2 (-X/k)/6 exp2iklog 2i (r/2)1/2 ~-57 i/12 - ) 1/3 H(2 ) i 7) 226

with 2 s -1 2(s = -- -- s r] e (s )=i S1 1 2ds if s <>1 (2) - If s >1, then arg? =0, and a fixed distance away from s =1, H (/ ) is proportional to e xn', where f(r],X) is real. Hence, substituting (s.,5), (E. 6) and (E.q into the integrand, we see that it is exponentially small for s > 1 and bounded away from s =1. On the other hand, s < 1 implies arg? = 3ir/2, This exponential increase is proportional to exp k dt(l-t ) } Consequently, the integrand has order ~s t -- rXk-/2k If n >1,the ar 0 an afixd dstnceawy fom 1 1/ 7 prpotonltoeWn wh fX srel Hnc ubttuig t,5'O/E.6 227

But in the range of C2, s = 0(1). Since the trigonometric substitution t = sin 2 shows that ( ) dt = 7r/2, the integrand is also exponentially small for J0 s < 1 and bounded away from s = 1. Finally, H(1/3 ) is bounded around f1 H1l/3( r s = 1, and so the integrand is exponentially small there. Therefore, the integral over C2, as well as the integrals over C and C5, does not contribute materially to the asymptotic representation of -e -/R. To estimate the contribution of C3, we first recall (Buchholz, 1953, Chapter 7) the asymptotic representations valid in the range I X/k < 0(1). They are./2.ik ikl + )-X/2ik 71/2k eri/2 v1/2 (L 1 (2k ik1i e e -ike vl(l -X) " (2ik)-1/2 + e2 r V X r )X1 \2+2 ik. 2,'~ (E.8) -1/2 -X2k -ik2 v2(2 -X) - (2ikg2) (2ik )-/2ik e (E. 9) v2(., X) (2ikr)-1/2 (2ik)l/2ik -ik (E. 10) Substituting (E. 8), (E. 9) and (E. 10) into (E. 3), we find IX, 1, 2' rl) I+(X, 1, 02~ r) + I-(,1.n) (E 1) where + ikgl -ik(~2 +) r ) I- X,12,n17) = C(k.1.)e 1e 2 r lk + ' (2ike )X/2 (2ikC )-X/2ik(2ik,)X/2ik (E, 12) 228

I. 7 =Ck. l le -ike -ik(~2+,a) - 2); e r 2k+e * (2ik1) -A/2ik (2ik2 )-/2ik (2ik)X/2ik e7r /2k e 7ri/2 (E.13) with (2 ik )-1/2 (2 ik ) - - 1 --- - -- 2 C1/2 (2ikrd- 1/2 C(k, 1, r1,) = 27ri We consider first the integral Inserting (E. 12) we obtain dX I+ x - I lp 2-177) - c 3 ik 1 -ik( 2+n) 1 e C (k., 1.9 2 't T7) E C d(2ik 2). (2ik 1)/2ik(2ik2) -X/2ik (2iki7) X/2ik We estimate this integral by using the relation c3 I_(k, ~ 2' 3 5 -= j=l 5 dXM. j2,45?)C j=l,2,4, 5 C' S xi+(k, E 14 t), C' J (E. 14) where C1 is a straight line path from IXI = oo, 37r/4 < argX < 7r, to a point where X/k~ >> O(kTr), arg X =T, C = C2, = C4= C4 2 2' 3 3'1 4 4'0 229

CI is a straight line path from a point where X/k >> O(kn), argX = 0, to 5 \X1 = oo 0 < argX < 7/4. We evaluate the sum Ei 5 dX I((X.E 2 ) j=l 2j 1 as a single contour integral 5 j=l \ duAI-(Al~.?n) (C' = _c ). In order to estimate the integral over C1, we observe that 7/2 < argX < r implies -X = e X; thus arg(-X/2ik) = -a e (-t, -7r/2). Hence (C. 17) applies to r- -i- —, and the dependence of the integrand on X can be written as I(X, 1 2' r) ' exp { r- log - 2 exp { log -52 e which is equal to.+,5, X, f X, X I.X 2k, 177 X7r /4k Xa/2k I i -- i lo2i Ike }exp 2 log e eXa 2k (E.15) By choosing the end point of C' so that I X/k is large enough, namely so that Ak ' 4e > 1 (such a choice is implied in the original definition of C1, C2..., C k 4e 12 2 we then observe that the integrand is exponentially small over C'. Therefore, 1 the integral over C' does not contribute materially to the second sum of (E. 14). To estimate the integral over C', we observe that 0 < argX K ir/2 implies arg 2 = - 3 e (-7i/2,0]. Thus we use(C.9)tofind r2 +2ik 2ik. 230

and the dependence of the integrand on A can be written as +A,,./.. X ~ x4k UX.s 1 92' -) X lr-/ 2k rX -Ti exp log T e e exp.iik llg 2 {2ikl/4 2i expX 1,1 2ike l- ik 2 Ajk Q exp lo exp log eX (E By choosing the end point of CI so that I|/k is large enough, namely so that Ix/kl 1 /kh* 4 > 1, (such a choice is implied in the original definition of C1, C1,..., C, we then observe that the integrand is exponentially small over C. Consequently, the integral over C' does not contribute materially to the second sum of (E. 14). 5 + The last two paragraphs also show that Ie(X,. 92 r) i exponentially - ' "2 1k12 small for all values of l, 12 Tr on a large arc from CI to CI in the upper half 2 +1 5 plane. Thus we can evaluate \, dX I(X, 2 n) as a sum of residues of L(X, -1 72.) at the poles of r 2ik+). The residues of r(- 2ik +) 2k (2I)1n at the poles X = ik(2n+1), n = 0, 1,2,..., are 2 (1)n. Hence, the residue 1i n theorem yields r8 ikl -ik(S2+r1) w A dI'(.? l,77) (2k 2sk (1 )0 *n (2ikg )n(2ikg ) -(2ikn)n, 2 dX-X, e e2ikg 1 2 which reduces to n /-ikeln ikl -ik(%2+n) ao.I3c -2 n=0 Ikg -ik( 2+77) -2ikIr7/6 -e I e e(E (E. I?) 231

Inserting (E. 17) into (E. 14) we obtain ike1 -ik(e2+1) -2iklrn//2 1 X r2 1- 2 ) ~ J B. 18) since the integrals over C' and C' do not contribute. 1 5 In order to estimate the coatributionirom C' and C', we first note that 2 4 in these ranges of X the dependence of the integrand on X can be written as X r\ 2ikg X 2ike which, upon using the definition of s, becomes I+(X1. 2' ) exp 2log 4s e exp ~-i log 2 g (E19) If we4s e define ) then substituting in (E. 19) yields + i1 (X) I(Xe 12 )7) e. (E.20) For argX=7r, args4 = rT; thus log4s e = logi4s e[+ ir, implying 1 1 1 + x Tir1 52 1 7X/2k I(X, 1 2 ) 1 exp{ [ log2 -- log 4s ej e k which is exponentially small. Therefore the integral over C' does not contribute materially. However, for argX = 0, i (X) is real valued, and since X/k = O(kr), we study the integral over C, by using the method of stationary phase (Jeffreys and Jeffrey, 1. Differentiating X) we obtain Jeffreys, 1956). Differentiating b (X) we obtain 232

+ d0(X) 1 1 1 do = 1. 1 log - 4s (E.21) dX k 2 r k 2 Therefore, the stationary point equation dfo(X)/dX = 0 has a solution given by o 1 0= 4kr~ 2 This value is contained in C. because of the choice of end points of C. Since 44 d2 +(X) o 1 d12 2kX ' dX + + 2kr?1 ~ ) = ) - 2 0 o 2g I the method of stationary phase give-m SCt -1 ik1 -ili(2+r) -2ikrE1/2 1 21) e 2 (.(k) 22) (E. 22) 4 Substituting (E.22) into (E. 18), we find dX Ix, 1 2rl) =,(k )1/2) o (E 22a) We now wish to consider 5 dXI:(X,,.12 0 3 I g2, n). C3 Inserting (E. 13), we find this integral becomes 233

0(,- 2 ik~e 1ie" 2+" ~rj i/2 dX (*k2 c3 (2ikg ) X./2ik (2ikF )-XI2ik(2ikfl)) /2ik e 7X/2k 1,2 We estimate it by using the relation (s imilar to (E. 14)) 5 ScdL(,112j) 1 50 j=1,2,4,5S0, 3~(9. 2 3) where 0'I is astraight line path from IjXJI=wOD, ir <arg X< -St1/4, to apoint 1 where I XfkI~>>O(k7), arg X= r 01I=0C C 0=0C C = C 2 2' 3 3' 4 4 ' C I is a straight line path from a point where IjX/k I~ O>(kil), argXA = 0,9 5 to IX1I =OD, -7r /4 <arg X<O0 and evaluating the sum as a single contour integral CT 5 1 '=ZC 0 234

In order to approximate the integral over C', we note that ir <argX.< r6/4 (-37r/4 <argX < -r/2) implies -X = e X (-X = e X); thus arg(-X/2ik) = -7 e (-ir/2, -r/4] ((-ir/4, ). Hence we use (C.9) to approximate r ik +i2)' and the dependence of the integrand on X can be written as - ~xip f t 2k2 X-'XI//4k eo ~ log - e" - e ep - 2 ik log 2ikeJ er/, X X X-X/2k log 2" e lg exp log -2ik lo|2ike -2ik lg 7 k3/4k ' (E.24) If the end point of C{ does not satisfy the condition I k 4eklg >1 then we chose the original definition of C1, C2,..., C5 so that it does. Accordingly, the integrand is exponentially small over C1 and C; does not contribute materially to the second sum of (E. 23). To estimate the integral over Cy, we note that -r/2 < argX < 0 implies argX/2ik = -6 E (-, -r/2). Hence (C.8) applies to +), and the dependence of the integrand on X can be written as IL(X, I.,>2) ~- exp{ k log ke exp log 1 2} er/4k X r k 2k_ 1~ X7r/4k -6X/2k,,- g exp- log 2ie e 2ike U e (E.25) If the end point of C' does not satisfy the condition I X/kI * - l > 1, then we 5 4ekg1 2 choose the original definition of C1, C,..., C5 so that it does. Thus the integrand is exponentially small over C' and C' does not contribute materially to the second sum of 23). sum of (E. 23). 235

The last two paragraphs also show that I-(X, g ri) is exponentially small 1 to' in e lower half pla for all values of g1, 2' rl on a large arc from C' to C5 in the lower half plane. 1 1 5 Hence we can evaluate dXIX, Q 1' 2' 1 ) C' as a sum of residues of I-(X,,1 2 r) at the poles of r(-k + ) The residues of 2 \\ + 2 ( - 1)n of +) at the poles X = -ik(2n+ 1n=0 1, 2,..., are (-i) n. There-iki n fore, the residue theorem yields X -ik?1 -ik(n2+ ) o d e e 2k (-1) 1 2 2k. - (-i)-% n Jc' n=O ' C1.(2ikl) n(2ik2)n (2ikn)-n(-l)n (-i), -ikg, -ik(92+ r) 2ik( 2 /rl) e, - e 1 (E.26) T) Inserting (E.26) into (E.23) we obtain d\ I-(X,ee~2 ) e e 2 dX I (, 5' r2), r C- 3 -. \ dXI(Xl.e 5.n), (B.27j j=2,4 C' since the integrals over C' and C' do not contribute. 1 5 We consider now the integral over C'. Since argX = 7r implies argX/2ik = r/2, 236

1 n), exp.k log i eX /4k exp x log j3 / eAf/2k (E.28) which is exponentially small. Therefore, the integral over C' does not contribute materially to the second sum of (E. 23). To estimate the integral over C' we observe that for argX = 0, -X = e i X. Hence, I(X, 1 v2' rn) ~-' exp { - log 4s e exp j- log (1/.29) If we define -(X)= _i_ l /log l/+ 1 4s2 e then substituting in (E. 28) yields 0 a) Is(XA 1' 2' 7 ) 1 e For argX = 0, b (X) is real valued, and since X/k = O(kr7), we again study the integral over C' by using the method of stationary phase. Clearly k 2 ogi+- log4s. (E,31) Therefore, the stationary point equation do (X)/dX = 0 has a solution given by X;/k = 4kl 2/r1. This value is contained in C; because of the above choice of end points. Since d2 1 — 2k1 2 -.__ -..L oX2 2kX '(Xo o the method of stationary phase gives 237

CdiI(X 1 g2 1 -iki1 -ik( + 1) 2ik(l 12/n) 1 1 w I 2 - (kyl) (. 32) Substituting (E.32) into (E. 27), we find S\ d'I0(3,1' 2' ) -=; o((k1/2) consequently, by (E. 11) and (E.22a), dX I(X., 1' ) = 2 n):(>O) If we now substitute the result of the previous paragraphs into equation (E.4a), we obtain -ikR e R S dX I-r ( Ol 01/ R (X, gis 92 t1)+; 2 k7) */Z (ES,33) 4 This equation shows that if k1 >> 1, k2 >> 1, k) >> 1, the derivation of the -ikRT asymptotic representation of -e -/R. (from its integral representation) depends on the behavior of the Whittaker functions in the region C4. If the point X/k = kgl is not already contained within C4, we choose the original definition of C1, C2..., C so that it is. In this range argX = 0; thus argsg = 0, and for 8 = 0(1) (we discuss later the case where this is not true for Sg2) equations (D. 21) and (D. 20) yibld, respectively, 238

(2 ikg )-1/2 v Q -X) -,-, - 1 c(-x 7r i/4 Xv /2k /2ik) e e (s ) Ai(-a ).9 F7-r 1 9, v 1/2 7ri/12 x x 2Q 2,s -X) ~ (2ikg 2)- C(-)L/2ik) e exp 21 log - 2ike (s9 )4 2 (R,34),a Ai(- -a e -27r i/3 )JO 9 2 (1;.,,35) with C(-X/2ik) = 23) 1/6Tr1/2 (X/2k) 1/6 -2 ( - 1 is - 1)1j s0. 9 3 (s9 ) = i 5 S1 G) d-S j 3~s x, ( J) or* = 3 - )2/3 2 i 13 3 1/ Also, arg X = 0 implies arg s = IN, and since s = Ti VI (see equation (3.32)), equation (D. 42b) gives O(1 implies =7 O(ki7) ),T 1 / 2 1/2 x x v 2 67, X) -—, (2 W7 W) exp ik log - 2 Ee e Ti (s i))1/2 - (E. 36) with 2 ( 0 7 = 1(1 - Ti) Ti 7 1 239

s 12 tl k a da = 1 1-(1 s) )-log i- -V- ) (. (E37) 0o As discussed in Sections 3.1 and 3.2, the behavior of the Airy functions in equations (E. 34) and (E. 35) depends strongly on the value of A with respect to the turning point value given by 4/k = kg., j = 1,2. If X < k g1and k2 - -1 2/31 (kn 2/3 (kri) ( -1 (l)12/3) {1 (kT})2/3 then equations (E.34) and (E. 35) become, since by definition X <k k2 and 2 k2 k2 1 -1 >> 2/3 ( x ((hk~ 2i/)3 (2ikg )-1/2 vl(, -X) -r e. - 1 -/ 92 (to) 2/ -X. eir1 + ie d. (s ))1/2 1 a (E.38) (E.39) v2( -) - (2ik2)-1/2 exp {-k log 2 ike 2 9 2 If X is in a neighborhood of k2 l governed by k2 _ -1 = o / 1 ) x 2 u3. then we use (E. 34) for v1(~1' -X), with 240

b) = (1-L ) db for s <1 1 Since both in this appendix and in Chapter III we ae studying a Green's function, we do not consider g2 in a neighborhood of g1. Thus for these latter values of X, (E. 39) applies to v2( 2,-X). For 2 > k 1 (s1 < 1) we see that arg 3 = 37/2. If, in addition, k2 1 1 (kr) 2/3' the Airy function Ai(-Ca ) is exponentially decreasing: Ai(-a )- exp (t dt /.,40) 1Js If X <k k2 (s2 >1) and k2 2 1 -i — (kn) 2/3 (kr) 2 (E.39) applies to v2( 2, -X). If X is in a neighborhood of k 92 governed by k2 2-1 =O (1) then we use (E. 35) for v2(Q2, -X), with ( ) i (2 ) 1 ds for s <1 241

Finally, for X > k 2 (sB < 1) we note that arg 2 = 3/r2. If, in addition, k2~2 1 the Airy function Al(-a e / ) is exponentially increasing: we denote this by 2 Ai(- e-2r i/3) exp{ (-Xs ds (E.41) 2 We now consider equation (E. 33). For values of X satisfying X > k2 and k2l 1- >> x ()2/3 (k we assert that I.(,?1 2 r1) is exponentially small. If the values of A also 2 2 -1= -l1 = o (kil)2,3) this follows immediately from equation (E. 34) together with (E. 40) and equations (E.36), (E.39) or (E.35). However, for X > k 2 and k2 2 11 1- >> 2/3 (krp) we must show that the product of the exponential factors (E.40) and (E.41) remains exponentially small. This is true since C1 < g2 implies s l< s8 which in turn implies S 1-t /t 2 < 1 1/j2 \ 1 -t dt < \ 1 -)/ dt \ \ Sgl 242

or that the product is always less than one. But we do not consider g2 in a neighborhood of g1; thus the product is always exponentially very small. Therefore, we can write equation (E. 33) as -ikR- (k2 + R-kR e + ( R- - X d X,, n)+ dXL,(. 1 2' )+ m ~( - i) R- 1 V- 2 C; (k2 " (- ) (E,42) where C" = e C4 X < (k2) and (k2 ) X (k2 1) denotes the neighbor2 hood of k 1 defined by k2 1 2 kk - -1=o0( 2,) since the integral over the remainder of C4 is negligible. To estimate the integral over C', we begin by using equations (E. 36), 4' (B. 38) and (E. 39) to find that in this range i-C(k P' r +2 ) T (O + ) " 2er/2k(-)z1/2 L iE1+ i+ i-il]-ie 2 Wr where C(k, M, e A) was defined earlier in this appendix. Inserting (C.9) for r 2i +5i P i ) together with the exponential form of the cosine, this becomes n- -i | -i? i I,^- C(k1/, 2..n)(-i)1/2 Le +ie 1] e 2c, 2T1 2) )s 1/2(E 43) IL~1,~,2,gs~lj2 243

with given by (E. 37). Since s] >1, j = 1,2, equation (D. 5) yields = - log + j 1, 2. (44) =k j Thus we can write (E. 43) as 1/2 A I~ [- - ]/2 [e + ie.45 1 2 with + x x (o +o(X) =_-k o( -k -( ko( 8 ' 0 ((I8 46) 1 2 (s) = Vs(s-) -.log (G-T+ \IF) (Bl, 48) ~ (s)= — s(1-s) -log(J 1- - f). (i 49i From these last equations we see that over CI both O(X) and CO(X) are real valued. Since, in addition X/k = O(kr), we estimate the integral over C4 by the method of stationary phase. + Let us first consider the term containing e i~() We note that 1 dqo(s) s- 1 (Fs+ l/s -) _ 1 o.^i _____= I F - s-i ds ~s(s-) 1- i 2 Fj- (Since in 0(s) both args = 0 and s > 1, this becomes 244

ds ) s- 1 ds s (. 50) Similarly d)o(s) ds 1 1 -s-F 1 -.s(1 - (1- s) q-s(: - s) Since in o(s) arg s = vt and therefore arg(l - s) = 0, this becomes d4 (s) - ds -s Then differentiating J:(X) and using (E. 50), (E. 51), we obtain (E.51) do+(X) dx 1 k -1 - (I)(s l s92 k k X k ds 1 T But since k2. vj args =0 (j = 1,2), 3 rsj 2 S - Ti, and arg s = T rl (which implies arg-s = 0), this becomes dX = k )-1 )k n dX'. - (s - 1) k 1 1 k f2 S2 +1 -s(1-s). k Ti T7 (E. 52) 245

Substituting (B. 48) and (E. 49) into (E. 52), we observe that the stationary point equation d+oo()/dX = 0 is 0 = -l+ + + log )+ log)+ log s) or (. s.. ). (1E. 53) + If we make the substitution X = zk, where z has the dimensions of rl, and WE< zi. 1, theil (E. 53)- cab be Written as/ +~W _5 L (E.54) Prior to finding an explicit solution of (E.54), we investigate under what conditions a solution exists. Since z < 1, the functions w(z), u(z) defined by W(Z) U(z)= are real valued functions of z. Differentiating w(z) we obtain wl(z) =-1 L 2 Z 2si- f+ im - I51-z + But 2 > 1' thus 2-z > f -i and 2%-z + J > Z +i%. Hence w'(z) < 0, and w(z) is a decreasing function of z with minimum value approaching w(= )= 21 - 246

Differentiating u(z) we obtain u'(z) = 2z z z- F (Z - or u'(z) 2- r=z nI Thus u'(z) > 0, and u(z) is an increasing function of z with maximum value approaching + 1+ Since u(0 ) - 0, equation (E. 54) has exactly one real solution z in the considered 0 range of z if and only if w( 1) < u(1) or if and only if 1 uM )7)oE - y (E. 55) A direct solution of (E. 54) entiils quite lengthy and cumbersome calculations. However, Ivanov (1962) encounters a similar equation in his paper considering diffraction by the convex paraboloid of revolution. Using his solution as a guide, we produce a solution of (E. 54) which is verifiable by direct substitution. This solution is + 4g1 27 Z = ----. (g 5} o (l+ g+q7)2_g 4 1 Before indicating the substitutions to show that it satisfies (E. 54), we note that (1+ +)2- = ( 2- 2r( 1+2)+172 > 0, which implies that z is indeed positive. This can also be written O 247

22 (Q1+ 2+n1)2- 4912 = 2-9 1) - - + 492 > 42r7 ' Thus + 4_192'T Zo < 4I' =1; 21 the choice of z does not contradict the assumptions on z. That (E.56) satisfies O (E. 54) follows by inserting it together with + n(e1+e2+n)2 + = (E.57) 0 1 2 1 e + 2 + Y ) 58',) 1 o (e1+2+r)2_ 4f1i2 e(2 - 1 n)2 + = 12 1 E..589) (2 1+2+n) 2- 4F1l2 122-2 into (E.54) and reducing the equation to an equality. Since (E.54) contains the square roots of (E. 57), (E. 58), (E.59), we demand that the terms being squared are positive. This is evident for (E. 57) and (E. 59). However, for (-E. 58) to hold we need 92 > 1+rl' This follows since 2 N< 1+n implies 2I + 2< { + n,+ <1, which in turn implies (J 2-+ + i)(- -", ) 1, or that no solution of (E. 54) exists. We now estimate the term containing e i~() using the solution (E.56). We observed that equation (E. 52) implied d log lT7+logQ s - + - - dX g Thus using the derivatives of log (Fs- + )s, log ( T- ) previously obtained, we find 248

dX2+ -1 91 91 2,,1 + 1 2k j 7l 2 -k + k 1 2k~TTWFJQ Th~i.-.an be written as d 2 +(A.) 1 2ikXA.(s )( )( (S iO( 77) O~ )i~s 7 + ~s9O(S)2 Since 1 = 0 s ) + 0 92- 91-77 (91 + g2+ - 4g 192 +z?7+Z0 Ti 22 2(17 +g19 12 - 0 I we see that + 0 1 0 (Il4, 60 In addition,, 0 0 0 ~l 2 ~2- Ti TiX=X 0 1+X+) =flg + 1 + p(X = kI1(- ) - 92 (9 - 0) - /n+z0)I or 249

which upon using (E. 57), (E. 58) and (E. 59) becomes (XA) = -k /1+2+T7)2 - 41, (E.61) Then using the method of stationary phase, we obtain dX\C(k,1 2,T)(: /e 1 -1/2 220( 27C(k, 1' g2' r )(-i) /2 O + /4 1 2 ) (d2+)]1/2) _ _ + O ' dlk 2 C(k, V1J' 2' 9i)(-)1/2 i ) -ik(l+ 2+)2 4 0 Before attempting to interpret this result, let us consider the term conO= log 1sl2s)+log-1sl )+2log/-1- l taining e 0 It follows immediately from the calculations already performed that the stationary point equation d (~X)/dX = 0 is or (s -1+ + s^)il-s f )= 1 (E.63) 91 91i v i/ 2 v 2/v T? 7 250

If we make the same substitution as previously, then (E. 63) can be written as z3/2 ( >z + z + F> - A (E. 64) Prior to finding an explicit solution of (B. 64), we again investigate under what conditions a solution exists. Since z < 1, the functions f(z), g(z) defined by f(z) = (1 7+ ( + r) g(z) = 7+ - are real valued functions of z. It is obvious that f(z) is a decreasing function of z with minimum value approaching fAel g (ty +A) Differentiating g(z) we obtain 1/2 g'(z) = 32 —,Li+z (3 + z ) - z or 1/2 WU/2 2 P3i+2z - 3 \/17 + g'(z) = /- 2.3.+2z-3J. 2ynTz(}z" -vn Then the sign of g'(z) follows that of h(z) = 3r)+2z- 3^rin+z). But h(O) = 0 and h'Iz) =2- 2 n >0; thus h(z)> 0 and g'(z)> 0. Therefore, g(z) is an increasing function of z with maximum value approaching 3/2 1g( yn+ = ~-_ 251

Since g(O ) -- 0, equation (E. 64) has exactly one real solution z- in the considered 0 range of z if and only if f(1) <g(1) or 1 > 2 + (E. 65) We see at once that this condition is complementary to (E. 55). Either one or the other may be satisfied, but not both. We continue to defer any interpretation and consider the solution of (E. 64). As above, a direct solution entails quite lengthy and cumbersome calculations. We will again produce a solutionwhich is verifiable by direct substitution. This solution is 1 2 z (E. 66) (1+ 92+r)2) - 4g1 2 + which is, of course, equal to z (the use of the different symbol corresponds to 0 the notation in Section 3.4) and thus does not contradict the assumptions on z. That (E. 66) satisfies (E. 64) follows by inserting it together with r(l+?2+ n)2 "+Z = ' " ^ 2 ' )67 z - z- =. 1-. 6) 0 (+ -+r 4f) 21 2 2(f2- f,+ r)2..- 2 (E269)? =O 2 ) 1 -- ~. (Q1+ 2+ r1) - 4g1 92 into (E. 64) and reducing the equation to an equality. Since (E. 64) contains the square roots of (E. 67), (E.68), (E. 69), we demand that the terms being squared are positive. This is evident for (E. 67) and (E. 69). However, for (E. 68) we 252

need g2 < ~1+ir. This follows since g2 ~ 91 + r implies 21 + 9 / 2 F + fi/1 which in turn implies ( 2-~1 + g2 )(+ l- ~ i) 1' or that no solution of (E. 64) exists. We now estimate the term containing e i(k) using the solution (E. 66). We immediately find d2 o(X) dX2 =2 f(s) -)(S)0s -] E-(s )i0(s )- (s1 )i(s ) + (8 s) 2] 1 2 7 Since 1 ==z ( 2 )) =.l+2 -2 ' l 1+2+) g 4+ (i0(s)) = J+ = 0 1 ( 2+n)- 4 12 we see that -1 - 2ikX O(s )O(s ) O(s )] 0o 1 2 '7X =X o (E.70) In addition, }(X) = k -/( ) - / o ) - (rl+ Zo which upon uasbg (E. 67), (E. 68) and (E. 69) becomes 253

~o(X-) = -k ( +2+ n)2 412 Thenuifl gthe method of stationary phase we obtain (e, i) 54 dX 2C(k, g1' 2 r)(-i) e7 2 e o( [s )0(9 )O'P 7'] 27r C(k, 91' 2' r)(- i)/2 e /2 ( s )- W )])1/2 Q 1 '2 =X2 i -(X ~) /4 e 0 A -i0r/4 Inserting (E. 70), (E. 71) and the definition of C(k, 1 '2' rn), this reduces to 54 dX>2r C(k, 1' 2' r)(-i) e 2 e ~ 1 2 -ik (+1 f2+ ) - 41 2 -e ^v, (1+ 2+)- 4g12 (E. 72) Since the conditions (E. 65) and (E. 55) are complementary, both (E. 72) and (E. 62) cannot be true simultaneously. If there is a solution of (E. 54) ((E. 64)), then e i~() (e i X)) does not have a stationary point in the interval C Therefore, (E. 62) ((E. 72)) is valid while (E. 72) And (E',62) are replaced by 4[ ds 2; C(k, 1 2, n)(-i)/. ei2 e 1, [?1.-s2(sr~)]i/2 lot U~s9 )O's )O(s n)],F ((k), 1/2 (E.3 ~1 2 k) which is obtained by integration by parts (here the upper sign in (.(X) corresponds to (E. 72) and the lower sign to (E. 62). 254

In deriving equation (E. 72), as well as (E. 62), we tacitly assumed that the stationary value X =k z = k z = X lies within the range C". We now o 0 0 0 4 investigate the necessary and sufficient condition that this be true. We also com-ikRz plete the derivation of the asymptotic representation of -e /R- by considering the second integral in equation (E. 42). Finally, we interpret the result. We first observe that it is indeed possible for the stationary value to lie outside CY. To show this, we recall that g2 < 1+rl implies (1 + ~)(in+ f < or that there exists a solution of (E. 64). Similarly g2 > g1+r implies (\2- 1 + 2 )() > >1 or that there exists a solution of (E. 54). Thus if we set f2 = f1 +t+x with x small, then x positive implies we can solve (E. 54) and x negative implies we can solve (E. 64). If we use (E.58) for x positive or (E. 68) for x negative together with the value of z = z = z ((E. 56) or (E.66)), we find -z 2 x 1 0+ K =. (E.74) o 2 Using the Taylor expansion for + x/(l+r1, we obtain,-o _ 91 -1= x + - x2-1 z z 4 ) + O(x2) 0Z z0 410(+17) L (1+n) j or Z = f[-4. + )+(x+]. (E. 75) Consequently if x is small enough, then z is outside C. The necessary and sufficient condition that z lie inside C" follows immediately from (E.74) too 4 gether with the definition of C" It is 4 * 1 1 2(E 76) (kn)2/3 255

We now complete the study of equation (E. 42) if the condition (E. 76) is satisfied. If it is, __k +72_ 4_, f -k/( 2t7)2- 42 1 2 C + + t7- 4 (E, It remains to estimate the second integral in (E. 42). As in C4, equation (E. 36) applies to v2(r, X). Since we do not consider g2 in a neighborhood of g1 equation (E. 39) is still valid for v2(T2, -). However, for vl(1,-X) we need to use (E. 34). Then (E. 43) becomes S2 C(k, 1. t2') -i C IJ, 2 e ". (]g 78) /2 C(-X/2ik)( )Ai(- e e ( 78 Using (E. 37) together with (E. 49), (E. 44) together with (E. 48), and the definition of l(s ) together with (D. 7) and (D. 8), we find 2 2 C(k,g g2 r) 6(O)(X) -( )( (X/k) 1/6 (- )e (E.79) S2 r} with (x) ko ) % ( (E.80) ))k:2 k s i' But in this range — X' — - =0 -1=0 )2/ 1 3 ^ 1 (kt7) rkrl)2/ thus | = 0(1), which indicates that Ai(-o ) is bounded independently of kr and does not oscillate rapidly. Therefore, the integral 256

d\ X I_(., 1' 2' n) J(k 1)can be estimated by the method of stationary phase. It follows immediately from the previous calculations that the stationary point equation d(O (X)/dX = 0 is 0 = log 1 + j )+ log ((/s p ), or (. -1+- s )= i. (E'. 81) If we make the substitution X = zk2, where z has the dimensions of rl, and O <az< 2' then (E. 81) becomes 2- + 2 i-rz )= z. (E.82) ~2- + 2)(~+z- ~-~=z. Hence, the integrand IL(X, g 1%. 2 ) (given by (E. 79)) has a stationary point within the interval of integration if and only if (E. 82) has a solution in the z interval defined by 1) ( 2/3<)z. 1 + k~)2/3 Before investigating an explicit solution of (E. 82), we first consider under what conditions a solution exists. Since z < g2' the functions W(z), U(z) defined by W(z) = + z-+, U(z) = are real valued functions of z. W(z) is obviously a decreasing function of z with minumum value approaching W( 2) = I 2. Differentiating U(z), we obtain 257

2 +z (in+z - or U1(Z) U'(z) = 2 in+zz r+z- F)Y with U (z) = 2rl+z - 21ri(rn+z) But U(z)= 1- 1- >O; thus Ul(z) is an increasing function of z. Then U1(0) = 0 implies that Ul(z) >0 for z >0, or that U'(z) >0 for z >0. Therefore, U(z) is an increasing function of z on 0 < z < g2 with maximum value approaching f2 U(9))=. In addition, U(O ) — >2 while W(O+) —a 2. Hence if r} > 2', there is no solution of (E. 82) on 0< z < 2. If r1 < 52, there is one solution of (E. 82) in 0 < z < e2 if and only if W(U2) < U(2) or if and only if But J+i2 <;+ J2 implies I+g2- < g2. which in turn implies 52 > 2 \ q-Jt 17 > g2 258

or that the inequality is valid for all values of f2 and n. Consequently, 71 < f2 implies there exists one solution of (E. 82) on 0 < z < 2. By direct substitution we observe that this solution is z = 2-rl. We now assert that if (E. 76) is true, the integrand I_(X, f l, 2 r) (given by (E. 79)) has no stationary point within the range of intr r-ation. For if the integrand did have such a stationary point, then it would necessarily follow that 0(12,3) k ) or ~2-(~1+ ) = ~10 2 2 1 1 ((kil)2,3) However, this implies 1 + )2 = 2 1 (f + r) 2 I1krl 4 2 1 4g2 ((kri) 4/3) which violates (E. 76). Therefore in this case, the integral (k2f )+ \ 2 9I - ' 2 (k2 )X can be estimated by integration by parts. Since (X/k)1/6= 0(kkl) /) in (E.79), this yields (k25 ) ~1\ d^-(JfJr m -( —)) (E.83) dXI_.(X. 91 ~2' n) 1/3 0 J(k29 I2 (kn)The case where the condition (E. 76) is not fulfilled and the integrand has a stationary point in the range of integration, i%.discussed further in Seotion 3.4. Thus if 259

(E. 76) is fulfilled, we can substitute (E. 83) and (E. 77) into (E.42) to obtain -ikR- -ik 1+ i2+-) _ 4!2 e e e e. - (E.84) \1+ 2+ 1) - 42 12 But according to Buchholz (1953, Chapter 4, equation 3) 2 2(^n)-::]2-. R2- _ = 0(1)_ ~2 + 4gj ' or 2 2 2 IR = 2+y+ +2gr+2rl — gg This can be written as R2 = 22 +rl 22+ 2 or R2 = ( + +rj)2 -4 2 2 Hence R1 = ( + g2+n) -4g1 2 which shows for the configuration of i, -, and rn 2 -ikR that satisfies (E. 76), the term of largest order reprodues e /R-. -ikR - We conclude the study of e ik/R- with some observations concerning the relative magnitudes of Y1, 2', rf. In the above derivation, we made no explicit assumptions regarding order of magnitude relationships between these variables. We tacitly assumed that 91/n = 0(1), g1/'2 = 0(1). We assert that this derivation can be repeated (with minor modifications) for any other order of magnitude relationships, provided these relationships are independent of the large parameters of the asymptotic analysis. This is to ensure that we are not considering a limiting case, and is best illustrated by an example. We consider g1/} = 0(1), g1/2 << 1; these conditions represent a possibility of considerable physical interest. Then 260

provided we can choose kor such that fl/g2 > 1/(kr) /3, the above derivation can be repeated, with the appropriate modifications, as will be shown below. If we could not make this choice of krl, then we would, in reality, be considering a plane wave which has a different integral representation than (E. 1). 1/3 In the case 01/. = 0(1), 1/ 2 << 1, 1/A2 >>1/kr) /, we begin by writing (E. 4) as -ikR- 7 -e d- - I,t dXIX 1, 2, ), (E, ib) J j= C. where for some Mi >0 (i = 1,2,...11) with M5 >1 and M >>1 C 5 7-a<A< -M7k 92 < C2 =- -(M1+ M2)k22 <X <-M3k2nJ C = {jI -(M+M4)k2r< < -M5k C 3 1 -(M5+M6)k < X < (M6+ M7)k C { | M k < X < (M6+ Mg)k2r}, C0 = i X Mk<X <(M'+M )k '2 C L'7 89 c7={XiMllkk2e2<<}jj The argument for the outer intervals C1 and C7 remains the same as the one above for the outer intervals. The argument for C must be modified only to the extent of redefining the intervals C' through C' so that the stationary points of (X) and Wo(X) lie within C'. The difference between the argument for C2 and 0 0 4 2 261

the one previously is in the changing of sl from s_ = 0(1) to 1 I Bs~ = |7k = A(~1/~2) << 1, and the changing of s from |s | = 0(1) to Is1.1= k! - (/12) << 1 However, (E. 5) is not affected by this change. In addition, s O= 0/9= A / ) >> ~ 1 > 1 2 2 1(kn)1/3 (kn)2/3 implies that (E. 7) together with the previous argument for s < 1 can still be used. Therefore, the derivation for C2 remains the same. This is also true for C3, 2 kg2 ls -I - i = (2/n) >>1 which does not affect the validity of (E. 6). We can use similar reasoning for the intervals C5 and C6. The only difference for C5 is the changing from sK2 = 0(1) to 121 = X = o(2/17) >> 1 Then (E. 39) is valid for v2(2, -X) throughout all of C and the derivation proceeds as before. Finally, for C6 we see that s changes from s | = 0(1) to kn lS i= k = o(r/%) << 1 and s changes from s| = 0(1) to "1"1I 262

k?1 1, IX/k = =(1/2) << 1 But (E. 36) is not affected by this change, while 1 1 1 1 2 (k)n1/3 (k)2/3 (k2)3 (k17) (krl) (kg ) implies that (E. 34) and the previous argument for sa < 1 can be used for vl(9, -X). 11 Consequently, the derivation of the stationary point equations remains the same. One of the reasons why this case is of interest is that in similar problems an approximate solution of the stationary point equations (or equation) can be readily found. If we examine equation (E. 54) we observe that provided a solution exists, an approximation can be found by making the appropriate expansions for small values of z. Moreover, there is no possibility of obtaining an approximate solution to (E. 64). Therefore, in order that this case produce an approximate solution of the stationary point equations it must imply that (E. 54) has a solution or, equivalently, that (E. 55) be valid. We can write (E. 55) as 1 < (J -1 + Y.21 - +). (E. 55a) But r/ 1 = O(1) implies 2/ +i +1 =0(1), and since 2/'1 >> 1, (E. 55a) can be approximated by 1 < 2 2/ 1 O(1). (E. 55b) This shows that if we consider the condition 1j << 1 added to the defining 263

relations, then (E. 55) will be fulfilled, (E. 54) will have a solution, and there will be an approximate solution of the stationary point equations. We find the approximation by writing (E. 54) as 71 [+ O(z] = i [1+ o(z which has the solution z = 4n / [1+ 0(1/2; 2 (E. 54a) (E.56a) We now evaluate the result of the stationary phase integration if we use the stationary point given by (E. 56a). Since I A+ z=z 0 ( 2 ) + z=z o Z=Z ^+ - z 1 1 - f+ 0(2 2) = 1- - + o(2) 2 1- + 0(3,/2) 2 = 1 + 0(2/f2 ) 1 n+z - r ~ 71 2+ - 0+ O(/E)= 2 1 + - + 0(21/2),) we see that (E. 60) becomes (d22x) 1+o ii ) i+ 8k3in 1 2 K =2 1 (E. 60a) In addition, ~O [1<= - / o 2( 2- Z) - +0] 264

or which reduces to 4+o(^+) = -k2 + k(- ) + k O(1/. Then using the method of stationary phase, we obtain 1- ~ 2 2' (E. 61a) dX C(k, 1' 0 2 -r)(-i)1/2e iO(A.) -ik2 iek(9l- n),-e e 92 (E. 62a) This immediately implies cil A -ikg2 ik(l- T) d.l -(,, 2', n) - e- - 1 2 (E. 77a) 4 But in this case (E.76) is true. Hence, (E.83) is also true. We write (E.83) as \j dxI (\,?? n.1) = - ~( 73. (2/~1)/) 1 J(k21) dI.(, /1' 2' n) = k/ Since 2/1 ~(k)1/3 implies (/9 )1/2 < (krl)1/6, this becomes 2/g +<(2~) (k 2 ) + k21 1 0( 1. (E. 83a) 265

Substituting (E. 83a) and (E. 77a) into (E. 42), we obtain -ikR, -ik (l+2+n) 2 - 412 -ikg2 ik(l- ti) e e e e 1. -.... (E.84a) 1+ ~2+l) -_ 41 2 Observing that we can write 1(+2+l)2- 41 E2 = 22 + 21 + o( 22, we see that the term of largest order reprodues the first two terms of the phase and the first term of the amplitude of e -/R-ikR E.2 Asymptotic Representation of -e /R for the Source at (0, H) We again assume a = 0; thus (E.2) becomes -ikRH ro eRH= l2i dx(ik+)r + - +2)v2( -X)v1('1)v2(, ) -oo (E.2a) If we make the substitution X = -Is in the integrand, then equation (E. 2a) can be written as -ikRH P R. 27r i d ik 2ik l(rll 22 -0o 0(E.85) Moreover, we can define IH(L, nr, n2 ) = i r(2i +2)r(2 + ) vln. -^)v2(2k - )v2(,. (E.86) Substituting ( E. 86) in ( E. 85) yields 266

-ikRH oo -e-RR = d I4itH(,, 112 ). (E.87) RIH -oo Upon comparing (E. 86) with (E. 3) and (E. 87) with (E. 4), we see that the derivation of the asymptotic representation of -e -iH/RH is entirely equivalent to that of -ikR -e ikR-R. We need only make the appropriate change of parameters to go from one to the other. The same is not quite true of the remarks concerning the relative magnitudes of n1i r;2' f. The difference arises in that for the source at (0, H), conditions that represent a possibility of considerable physical interest are r11/r)2 = 0(1), r12/g < 1. However, virtually the same argument can be employed to show that for these conditions, the derivation of the asymptotic representation proceeds as previously. We again find that the additional condition ~2/ << 1 will produce an approximate solution to the stationary point equations. It is interesting to note that in this case the analog of equation (E. 64) _3/2 -z + -z + (E. 88) with the condition for a solution given by the analog of (E. 65) n 1> (f/Yl + ) ( A) + (E. 89) produces the approximate solution by making the appropriate expansions for small values of z. The analog of equation (R. 64) it r =- * (E. 90) Fnw2- + c f (E5) with the condition for a solution given by the analog of (E. 55) 267

rl < -(li + s (+I - in (E.91) is now the one with no possibility of producing an approximate solution. 268

APPENDIX V COMPARISON OR CONE WITH PARABOLOID OF REVOLUTION We consider here a cone with vertex at the origin of our coordinate system (Section 1.3) extending along the positive z-axis. The half cone angle 0 is o defined to be less than z/2. Then the domain described by the interior of the cone is given by 0 6 0 < 9. Let p represent a possible solution to the Neumann potential problem for an interior point source on the cone axis. Then Gauss' theorem for the bounded volume defined by the intersection of the cone with a sphere (center at the origin) of radius r leads to the equation o\ 2 r sin ed0d = \ V2 r drd1d, (=.1) ar O O O o O since the outward normal over the sphere is the unit vector r. Taking the limit as r-*oo of both sides of equation (E; 1) we obtain the condition 0 rli r22 sinOd0 = 2. (E2) r -co 30 Therefore, choosing a solution of the type chosen for the Dirichlet potential (Section 2.3) does not violate this condition. The difference between the cone of half angle 0 and paraboloid of revolution rn = o0 can best be illustrated by considering the area intersected by both on a sphere (center at the origin) of radius r. For the cone this area is given by 2w dAc 2\ 0 dA ~r \ sin OdOdo or (F 3) dA = 2r -cosj rIt 269

while for the paraboloid 27rw I(r) r JO Jo where 0(r) is defined by cos 0(r) = - of+ rl' But -nro 20o 2rlo o- =1- — 0 +no 1 =- r 0 170 and hence dAp = 27rr2 -cos (r)+ r or (F. 4) dAP = 47r r r o Consequently dAc r lim r = constant, 2 r —o r while dAp lim - = 0 2 r -aco r 270

BIBLIOGRAPHY Abramowitz, M. and I. A. Stegun (1964) Handbook of Mathematical Functions National Bureau of Standards, Applied Mathematics Series No. 55. Adams, E. P. and R. L. Hippisley (1922) Smithsonian Mathematical Formulae and Tables of Elliptic Functions (Smithsonian Institution, Washington, D. C.). Buchholz, Herbert (1942/3) "Die Ausbreitung der Schallwellen in einen Horn von der Gestalt eines Rotationsparaboloids bei Anregung durch eine im Brennpunkt befindliche puntfodrmige Schallquelle", Ann. der Physik 42, Part 6, 423-460. Buchholz, Herbert (1953) Die Konfluente Hypergeometrische Funktion (Springer Verlag, Berlin). Erdelyi, A., W. Magnus, F. Oberhettinger, and F.G. Tricomi (1953) Higher Transcendental Functions (McGraw-Hill Book Co., Inc., New York). Erdelyi, A.,and.C A. Swanson\(1 957) "Asymptotic Forms of Whittaker's Confluent Hypergeometric Functions", Memoir of the American Mathematical Society No. 25. Fock, V.A. (1957) "The Theory of Diffraction of a Paraboloid of Revolution", appearing in Diffraction of Electromagnetic Waves on Certain Bodies of Revolution (Sovietskbie Radio). Ivanov, V. I. (1962) "The Asymptotic Expansion of Green's Function for the Diffraction of Short Waves by a Paraboloid of Revolution (Axisymmetric Case)", USSR Computational Mathematics and Mathematical Physics 1, 97-113. Jeffries, Sir Harold and B. S. (Lady Jeffries) Jeffries (1956) Methods of Mathematical Physics (Cambridge University Press, London) Kazarinoff, N. D. and R. K. Ritt (1959) "On the Theory of Scalar Diffraction and its Application to the Prolate Spheroid", Annals of Physics 6, 277-299. Kazarinoff, N. D. and R. K. Ritt (1960) "Scalar Diffraction Theory and Turning Point Problems", Arch. Rational Mech. Anal. 5, 177-186. Keller, J. B. and S. I. Rubinow (1960) "Asymptotic Solution of Eigenvalue Problems", Annals of Physics 9, 24-75. Kinber, B. Ye (1961a) "Diffraction of Electromagnetic Waves by Concave Surface of Circular Cylinder", Radio Eng. and Elec. Phys. No. 8, 1130-1139. Kinber, B. Ye (1961b) "Diffraction of Electromagnetic Waves on a Concave Spherical Surface", Radio Eng. and Elec. Phys. No. 10, 1474-1475. Klante, Klaus (1959) "Zur Beugung skalarer Wellem am Rotations-Paraboloid", Ann. der Phys. 3, 171-182. Kline, M. and I. W. Kay (1965) Electromagnetic Theory and Geometrical Optics (Interscience Publishers, New York, Div. of John Wiley and Sons, Inc.). 271

Koshlyakov, N.S., M. M. Smirnov and E.B. Gliner (1946) Differential Equations of Mathematical Physics (North Holland Publishing Company, Amsterdam) (Translation from Russian original). Lamb, H. (1906) "On Sommerfeld's Diffraction Problem; And on Reflection by a Parabolic Mirror", Proc. London Math. Society, Second Series, 4 190-203. Langer, R. E. (1932) "On the Asymptotic Solutions of Differential Equations, with an Application to the Bessel Functions of Large Complex Order", Trans. Amer. Math. Soc. 34, 447-480. Langer, R.E. (1935) "On the Asymptotic olutions ot Urdinary Differential Equations, with Reference to the Stokes' Phenomenon About a Singular Point", Trans. Amer. Math. Soc. 37, 397-416. Levy, B.R. and J. B. Keller (1959) "Diffraction by a Smooth Object", Comm. Pure and Appl. Math. 12, 159-209. Morse, P.M. and H. Feshbach (1953) Methods of Theoretical Physics (McGrawHill Book Co., Inc., New York). Phillips, R. S. (1952) "Linear Ordinary Differential Operators of the Second Order", New York University Report No. EM-42. Pinney, E. (1946) "Laguerre Functions in the Mathematical Founjdatons -of the Electromagnetic Theory of the Paraboloidal Reflector", J. Math. Phys. 25, 49-79. Pinney, E. (1947) "Electromagnetic Fields in a Paraboloidal Reflector", J. Math. Phys. 26, 42-55. Strutt, J. W. (Lord Rayleigh) (1896) The Theory of Sound-Volume II (Macmillan and Co., Strutt, J. W. (Lord Rayleigh) (1910) "The Problem of the Whispering Gallery", Philos. Mag. 20, 1001-1004.,Also published in Scientific Papers, Vol. 5, Cambridge University Press, (1912) 617-620. Sims, A.R. (1957) "Boundary Conditions for Linear Differential Operators of the Second Order", J. Math. Mech. 6, 247-285. Skalskaya, I. P. (1955) "The Electromagnetic Field of a Dipole Located in the Interior of a Parabolic Reflector", J. Tech. Phys. (USSR) 25. Translation in New York University Report No. EM-103. Taylor, W. C.' (1939) "A Complete Set of Asymptotic Formulas for the Whittaker Function and the Laguerre Polynomials", J. Math. Phys. 18, 34-48. Zachmanoglou, E.C. (1963) "The Decay of Solutions of the Initial-Boundary Value Problem for the Wave Equation in Unbounded Regions", Arch. Rational Mech. Anal. 14, 312-325. 272

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UNCLASSIFIED Security Classification r DOCUMENT CONTROL DATA R&D (Security clasatficatlon of title, body of abstract and indexing annotation must be entered when the overall report is classified) I 1. ORIGINATIN G ACTIVITY (Corporate author) aa. REPORT SECURITY C LASSIFICATION The Radiation Laboratory Unclassified Department of Electrical Engineering Zb. GROpP The University of Michigan.. 3. REPORT TITLE Diffraction by the Concave Surface of the Paraboloid of Revolution 4. DESCRIPTIVE NOTES (Type of report and inclusive date) Technical Report No. 3 5. AUTHOR(S) (Lost name, first name, inOniW..l) Stone, Stephen E. 6. REPORT OATF ' 7.. TOTAL NO. OF PAGES 7b. NO. OF REFS February 1967 272 31 as. CONTRACT OR GRANT NO. *. ORIGINATOR'S REPORT NUMBER(S) F 04-694-67 C-0055 8525-3-T b. PROJECT NO, c. Sb. OTHER RIEPORT NO($) (Any other nmbere th*t may be aaaigned this report) d. BSD-TR- 67-143 10. AV& IL ABILITY/tL!MITATION NOTICES. This document is subject to special export controls and each transmittal to foreign governments or foreign nationals may be made only with prior approval of Ballistic Systems Division (BSOMS), Norton AFB. Calif_ 92409 11. SUPPILEMENTARY NOTES 1. SPONSORING MIL-ITARY ACTIVITY Air Force Ballistic Systems Division, AFSC Norton Air Force Base ____California 92409 13. ABSTRACT Let ri denote the focal length of a paraboloid of revolution, and let D be the closure of lhe domain bounded by its concave surface. Then for a point source, with wave number k, located in D and on the axis of the paraboloid, the diffraction by the boundary of D is considered not only if kr ~>> 1 but also if kr ~<< 1. If kr ~>> 1, an asymptotic representation of the totaol field on the boundary of D is derived for the Neumann boundary condition provided the source is far (with respect to wavelength) from the focus and the field point is far from the tip of the paraboloid. This representation is interpreted in terms of geometric optics. If kr << 1, an asymptotic representation of the total field anywhere in D is derived foroth Dirichlet and Neumann boundary conditions and for the source (field) point in the near field and the field (source) point in the far field as well as for both source and field points in the near field, The near field result is compared with the solution of the corresponding potential problem. A necessary and sufficient condition for the existence of a solution to the corresponding Neumann potential problem is also derived. D D 1IN II UN L S IFIEDII l... _ I....... I I.I _ _ jIIII D D, JA 64 1473 UNCLASSIFIED Security Classification

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