IEEE G-AP 8 -4 THE DIFFRACTION MATRIX FOR A DISCONTINUITY IN CURVATURE* T. B. A. Senior, Senior Member, IEEE''" Abstract- To enlarge the scope of the geometrical theory of diffraction, the diffraction matrix for a surface singularity where the curvature but not the slope is discontinuous is rigorously derived. The model that is employed consists of two parabolic cylinders of different latus recta joined together at the front, thereby creating a line discontinuity of the required form. For each of the two principal polarizations, asymptotic developments of the surface fields in the vicinity of the join are calculated, from which the diffraction coefficients are then obtained by integration. The results differ significantly from the physical optics estimates and are analogous to those for a wedge-like singularity. This analogy permits a trivial deduction of the complete diffraction matrix. I INTRODUCTION When a metallic object is illuminated by an electromagnetic wave, a; powerful method for I estimating its high frequency scattering behavior is the geometrical theory of diffraction, originated by Keller [, 2. The theory is basically an extension of ray techniques to include the concept of diffracted rays which arise from surface singularities of the body. The strength of each such ray contribution to the scattering is proportional to a diffraction coefficient which is determined, to the first order at least, by the local surface geometry,This work was supported by the Northrop Corporate Laboratories, Hawthorne California under Purchase Order 12122. The author is with The University of Michigan Radiation Laboratory, Department of Electrical and Computer Engineering,\Ann Arbor, Michigan 48105.

at the point of diffraction. In those cases where the diffraction coefficients are known, their expressions have been obtained from exact solutions of selected canonical problems displaying the geometry in question, and thus it is that the coefficients for an edge or wedge-like singularity (slope discontinuity) are deduced from the solution of the two-dimensional problem of plane wave scattering by a half-plane or wedge. The diffraction coefficients are the key to the GTD method and one particular but important case where they are not yet known is when the surface slope (first derivative) is continuous, but the curvature (involving the second derivative) is discontinuous. Their derivation for this geometrical feature is vital for an adequate treatment of scattering by bodies such as a cone-sphere or hemisphericallycapped cylinder, and in the absence of any exact canonical solution from which to deduce them, it has been necessary to rely (see, for example, [31 ) on the crude estimates offered by physical optics. Automatically, therefore, the polarization dependence has been suppressed [4. Although an exact canonical solution would be desirable, it is not, in fact, essential to the determination of a diffraction coefficient, and an adequate description of the surface field in a vicinity of the geometric feature can suffice. For a discontinuity in curvature, we can obtain such a surface field description using the model that was employed by Weston [5, 6] in studying the creeping waves launched by the discontinuity. Weston considered only the case of a plane wave incident with its magnetic vector parallel to the line discontinuity (H polarization). This is treated in Section III and the initial part of the analysis follows closely that given in [5]. The analogous case of a plane wave incident with its electric vector parallel to the discontinuity (E polarization) is discussed in Section IV. The corresponding diffraction coefficients for H and E polarized waves are derived in Sections V and VI respectively, and the general diffraction matrix is constructed in Section VII. The results differ from the physical optics estimates for almost all angles of incidence and diffraction, and some of the consequences of these new and rigorous formulae are explored. 2

II PRELIMINARY CONSIDERATIONS We consider a two-dimensional perfectly conducting surface consisting of two half parabolic cylinders of different latus recta joined at the front. In terms of the Cartesian coordinates (x, y, z ) with the z axis coincident with the join, the surface is defined as: 1 2 1 2 x -a2y, y>O; x-aly, y<O, (1) so that the positive x axis is in the direction of the outward normal to the surface at the join. For convenience we shall henceforth write (1) as 1 2 x = ay, (2) where a = a2 (y> ), a = a ( y < 0 ). It is easily verified that the surface slope is continuous at the join (it is infinite there), and that the curvature is discontinuous aty = unless a2 a1 A plane electromagnetic wave is incident with its propagation vector lying in the xy plane and making an angle a with the y axis, where' 0 < a < r (see Fig. 1). If the wave has its magnetic vector in the z direction (H polarization), we can write i A -ik(x sin a+y cos a) H =ze (3) i A A -ik(x sin + y cos a) E Z (x cos a - y sin a) e, -iwt where Z = 1/Y is the intrinsic impedance of free space and a time factor e has In the derivation of the surface field about the join it is, in fact, necessary to assume that a is bounded away from zero and r to ensure that the shadow boundary is sufficiently far removed from the z axis; however, see \Section VII. 3

been assumed and suppressed. Due to the presence of the perfectly conducting surface, a scattered field (E, H ) will be generated satisfying the boundary condition n^(E +Es) = 0 A at the surface, where n is a unit vector normal in the outwards direction. The initial task is to find the total (incident plus scattered) magnetic field at the surface, with particular reference to a region about the j oin. Since the problem is two-dimensional ( being independent of the coordinate z), it can be expressed as a scalar problem for the total magnetic field component, Hz -= u, which is required to satisfy the Neumann boundary condition (au/an) = 0 at the surface, with u-uo obeying the radiation condition, where u i H -ik(x sin a + y cos a ) u = z e (4) This is a hard-body problem and is treated in Section III. If, on the other hand, the incident plane wave has its electric vector in the z direction, then Ei A -ik(x sin a + y cos a ) E= z e -ik(x sin a + y cos a ) (5) H =-Y ( cos a - y sin a)e The task is again to find the total magnetic field at the surface, and since the problem is two-dimensional, it can be expressed as a scalar one for the total electric field component Ez=u. This is required to satisfy the Dirichlet boundary condition u=0 at the surface, with u-uo obeying the radiation condition, where u =E -ik(x sin a + y cos a ) The resulting soft-body problem is treated in Section IV.(6) The resulting soft-body problem is treated in Section IV. 4

III SURFACE FIELD, H POLARIZATION Maue's integral equation for the field on a two-dimensional acoustically hard surface at a point specified by the coordinate y is u(y)=2u (y)+2 + u(y H() (kR)dsl (7) where R (X —xl)2 + (y-y)2 (1) and Hm (kR) is the Hankel function of the first kind of order m. For the particular surface defined by (2), it is convenient to take y1 as the variable of integration. The line element dsl then becomes. = 2 2 dy1 ds1 (:1 + 22) dy where a = a(y >0 ), a = a1(y <0 ), and the integral equation takes the form 00 2 e a2y al( — ik t m 2 2) 1 R u(y)= 2u (y)- 7 i u(y)Hl (kR) ay{ (yy)- (yIay) } L (8) -00 with R = 2y-y) + 1 (ay - ayl ) -. '(9) (lay (9) Note that a has the value a2 or al depending on the sign of the variable of integration Y1, whereas a has the value a2 or al depending on the sign of the coordinate y of the field point. Although an exact solution of eq. (8) is impossible, an asymptotic expansion of u(y) for large k/a can be found by making use of the particular character of the surface. To this end, we note that if the incident field (3) were to impinge 1 2 on the complete and uniform parabolic surface x = - ay, an asymptotic expansion of the surface field could be obtained by the Luneberg-Kline method, and is u(y) = U (y, a) eikf(y, a) (10) 5

with \[7] U(y, a) = 2- (sin c + ay cos ) + O(k-2), k (11) (12) 1 2 f(y, a) = -y cos a + -ay sin a. 2 Equation (10) reduces to the result given in [8 for the special case of normal incidence, a = 7/2, and is valid only over the illuminated region of the surface. This includes the join by virtue of the restriction on a. Following Weston [5], the field on the conjoint surface of eq. (2) is now written as the sum of two parts: that which would exist on the section in question were the whole surface a continuation of it, plus a perturbation created by the join. Thus uy =Uy aeikf(ya) 1 iks(y, a) u(y)=U(y, (a)e + - I(y, a)e (13) k where IYI s(y, a)= 4 0 (14) (l+a272) dT The only unknown quantity in (13) is I(y, a), and since the discontinuity in U(y, a) is O(k-1) at y=0, it is clear that I(y, a) must be O(k~) for small y. On substituting the expression for u(y) into (8) and using the fact that the first part of (13) is, by definition, the field on a single parabolic cylinder formed by continuing that; portion on which the field point lies, the integral equation reduces to 00 iks(y, a)e ik | (ya)e f2 O-D iks(Y1 ) k2 I(y1, a)e K(y y a, a)dy1- - Q (15) (16) where K(ylya, ) ayl(yl-y)- Y 2 RY -ay.) JR 00 ikf(yl, ) ikf Q= U(y1, a)e K(y1, y, a, )-U(yl a)e -OD (y1, a) K(yl, y, a, a)} dy1. (17) 6

Although the integrand in (17) is a known function, so that in principle at least a precise evaluation of the integral is feasible, an asymptotic development suffices. Taking first the case y < 0, we note that the integrand vanishes for y <0, and hence c ikf(yl, a2) ikf(Y, al) 1 Q=f U(y1 a2)e K(1, Y, a a)-U(Y1, a)e K(Y1, y, a1, a1) dY1 0 But K(yyaa H1 1() +H- a2(l-_)+(a.a) l+0(k-2)} K(Y1,Y,a1,a2)-H 1 2 1 1 2 )1O(k) where l=kyli ' = ky, and since U (Yl a2)=2 + O(Ck-1) U(yl, al), ikf(y1, a2) -i~cosa 1 ikf(y, a) e = e l+O(k -1) = e we have a2-al 00 -i(l+O)cosa (1) ( 1) Q=- 2 e ) H2 I (2 - d)l+0(k-l)d. (18) kl J e 11 1 l+ I 1) -r A similar result holds for y > 0 and this can be combined with (18) to give Q 2 e i sL() l+(k-1)} (19) k where 00 ~i cosa 1 2 L( ()H(l)(() -)d 1 (20) and the upper (lower) signs apply according as y > 0 ( < 0). In addition, ) iks(yl, a) I(y, e K(Yl,Y, a, a )dyl=O(k-2) -oo00 since K=O(k-1) and I=O(k~), and the final expression for I(y, a) is therefore 7

-ik s (y, a) 4y cos a (y,a)= + 2-a)e L(r) + 0(k-l). (21) The perturbation field is all that is needed to specify the total surface field u(y). From eq. (13), 2 cosa r ia T- sina -icosa2 ) 2()a u(y)=e s U( a)e k (a2-a)L(r)+O(k-2) (22) = k 2k 2 1 2 and this proves adequate for determining the diffraction coefficient to the leading order in k. We note in passing that expansion of the right hand side of (22) for << 1 shows that for H polarization a discontinuity in curvature is characterized by a surface field 'singularity' of the form y2 log y | IV SURFACE FIELD, E POLARIZATION The problem of finding the surface field is now a soft body one but the basic character of the analysis is unchanged. The appropriate form of Maue's integral equation is a U(y)2= 2 aa 1 (23)1 where u (y) and R are given in eqs. (6) and (9) respectively, and if we define v(y)= (l+a y ) /2 u(y), (24) o(y)= (l+a)221/2 a () (25) v-~ Uo((Y) + y) the equation can be written as ik (1) 2 21 dY v(y)= 2vo(y)- f v (Y)H1 (kR) {a(Yl-Y)- (a y-ay2) (26) -00 We again postulate a representation of the surface field as the sum of that which would exist on a complete parabolic cylinder and a perturbation created by the join, viz. 8

i \ a)e 1 ~ iks(y, a) ] v(y)= -kj ik ( y, a a)e + I(y, a)e ( (27) where f(y, a) and: s(y, a) are given in eqs. (12) and (14) respectively. _The first term in (27) is the solution for a uniform parabolic cylinder of latus rectum i/a, and by application of the Luneberg-Kline method we have [7] V(y, a)=(sin a + ay cos a2) 2+ (sin a + ay cos a)- +O(k-2), (28) which reduces to the result obtained by Keller et al [8] in the particular case of normal incidence, a = 7T/2. The second term in (27) is due to the join, and it is evident that the unknown quantity, I(y, a) is O(k~) for small y. On substituting (27) into (26) and invoking the fact that the first term in (27) is the solution for a uniform cylinder, the integral equation becomes 00 ~ iks(y a) ik f iks( ) ik =y, a) - ~ II(y1, )e K(Yl, y, a, R)dy-1 2 ' -00 where H (1) 1 2 2 I(kR) K(yl y a, )= ay(yl-y)- 2(a y Y -ay 2 ) } R a y and Q is as shown in (17) but with U and K replaced by V and K. An asymptotic expansion of the expression for Q is Q = 2 sina e M() (k-(29) with t icos a (l) 1 )2 M()= e H1 ) l1 dl, (30) I. where 1 and 1 are as before, and the upper(lower) sign again holds according as y > 0 (< 0). In addition 00 iks(yl a). I(y, a)e K(y1, y, a, a )dy1= O(k-2 ) -00 9

and hence *, -ik {S (y, a)+y cos a} I(y,a)= +2~ (a2-al)sin a e M()+(k-1). (31) The total field v(y) on the surface is therefore 2 icosa ia 2k cosa. v(y) =-ike -ics{V( ~ a)e c (a2-a)cosa M()+O(k-2) } (32) and expansion of the right hand side for |I|<< 1 shows that for E polarization the join is characterized by a surface field 'singularity' of the form y log I y V DIFFRACTION COEFFICIENT, H POLARIZATION For a two dimensional geometry such as that shown in Fig. 1, the scattered magnetic field at a point r' is H ) = H r - H (k I-r J)ds, (33) S and in the far zone -l ikr'I-i 7/4 H S(r') e P z - 7kr H where k fAA -ik r P =H n * r H (r)e i - ds S is the far field amplitude. For the specific geometry of Fig. 1, y2 k -ikycosO +ika — sinO PH(C, 0) J (sin+aycos 0O)u(y)e dy in which the integration has been limited to a region about the join since our concern is only with the scattering originating there. In terms of the variable ~, however, the range of integration extends from -ki to kt, and because k is large, it is sufficient to replace the limits by oo. Hence 10

002 o 1 -iW cosO+ia - sinO 1 k) PH( )= (sine +a cose)u()e d] (34) _ 4 - k kcos-u -00 and this is the integral to be evaluated. An expansion for u ( ) accurate to two orders in k was given in eq. (22). k When this is inserted into (34), expansion of U (-,a) as well as both quadratic exponentials in powers of k-1 yields pH), sin0 e iP ia [ 3 _ ~2 0)~ p (a,)= O 1 — 2kosec a+2i cotO (sina+sinl -00 + L_ (a2-al)L( )+O(k-2) d~ (35) 4k 1 where p = cos a + os 0, and since oo 0 0 J m ii m+l m ei d ( = - ( m m! -00 it follows that OD 0 2a al l+cos(a-0) a2al 2 2i 3i c 3 e-ip~ 12k 3 8k p H(1r, 0)=-,2k + i i 8k sin --- CoseC a+ ( L(r)d +O)(k2). (36) The first term in (36) is contributed by the leading term in the expansion of u(k ) and is the physical optics result:.o. ( ) a21 l+cos(a-0) H 2k 3 P To produce an explicit expression for PH(a, 0)/ accurate to the first order H ' in k, it is necessary to evaluate precisely the integrals containing L(r). This is accomplished as follows. We first note that 11

( O0 0 O 1 1 -eip~ e 2 (~ )(- -~~d 11 ~ (38) and since eo +i~Cosa (1) 3 H I(~I)~1d~l cosec a' 2 i -+ i -+ -(sincecosa -a).0 7r 6 integration by parts applied to the ~ integral in (38) changes the right hand side into 2i C~ 3a +2i p p 1~~s (1) e H I(~I) d~ -00 - coscy 1 ip~ ea - (1) dI H (U )-)~ d~. 1 Il~ I E ach of the double (see[7] ). Thus ~ integrals can now be reduced by a further integrati o —n \by parts,& i~Icosa '1.1 H(1I) -d~ d~ I r= 1 -e~os icosai (l) d~ 1 * 1+I-ipb~e -i-os i. H I M p J with an analogous result for the other., and when these are inserted into (38) and thence (36), we obtain p (a. Q):ppP..(a0)- 2 1 1+cos(,a+-O) + ~-2) H' H i 2 p (39) 12

where we have used the formulae 00 sin(' cos )H ( )dU= cot ) 00 o 0 1 (1) =i(sind-sin ) sin( cos )-sin(cos y2)H() i(sin2 deduced from the discontinuous integrals of Weber and Schafheitlin [9]. The difference between P and its physical optics estimate is evident from eq. (39). H - VI DIFFRACTION COEFFICIENT, E POLARIZATION The derivation follows closely that in Section V, and only a brief description will suffice. In place of eq. (33) we now have E (r')= J ( E (r)H (k jr -rl)ds (40) S and in the far zone S ikr -i 7/4 E (r'), e P Z- Ywkr' E where i Ez -ikr' r - a e -ds S For the specific geometry of Fig. 1, 2 i D -i cos +ia - sin (a) k v ( e d] (41) PE 4k k -00 where v(y) is defined in eq. (24), and when the expression (32) for v(y) is substituted into (41), expansion in powers of kl1 gives 13

00oo 0 O 2 -1 1+cos (a+) +. a2 al f2i 3 \ ip E(. ) 2k 3 8k sina -pcosec c+J -J +e M(~)d +O)(k-2) (42) The first term is the physical optics approximation: P aPo2 a 1+cos(+0) (43) E 2k 3 and comparison with eq. (37) shows that p p.o. ( a, ) (44) as expected 4|. The integrals involving M(r) in (42) can be evaluated precisely using successive integrations by parts. The procedure is similar to that for H polarization, and when the results are inserted into eq. (42), we have a -a p (a 2)=po ( 2 1 1+cos(a+) +o(k-2) (45) E 2k The shortcoming of the physical optics approximation is again evident. VH THE DIFFRACTION MATRIX A derivation of the diffraction matrix associated with a discontinuity in curvature is essential for the incorporation of our results within the general framework of the geometrical theory of diffraction. For this purpose, it is convenient to collect together the results obtained so far. Using the model illustrated in Fig. 1 it has been shown that if i A -ik(x sin a+y cos a).E ze then A F7 i(kr?-7/4) E with P =F-G; (46) PE=F. 14

whereas if i A -ik(x sin a + y cos a) H = z e implying i. -ik(x sin a+y cos a) E =Z (xcos a- y sin ) e then i (kr '-T/ ) implying i(kr- ) E S - 2 (x cos 0-y sin 0) Jkr e PH with PH F + G. (47) In (46) and (47) F a l 1+cos (a+O) +O(k-2) (48) (cos a+cos 0) - al l+cos (- 0) +(k-2). (49) G k- 3 (49) 2k 3 (cos a +cos 0) As demanded by the reciprocity condition concerning the interchange of receiver and transmitter, the expressions for F and G are unaffected if a and 0 are interchanged. The terms shown on the right hand sides of (48) and (49) are the leading terms in high frequency asymptotic developments of F and G, and are valid provided 7r-( +a) is bounded away from zero (to separate the contribution of the specular point from that of the ciscontinuity). In the expansion of the surface field it was found necessary to assume that a is bounded away from 0 and Tr (to ensure that the discontinuity is fully illuminated), and on physical grounds it would also appear necessary to have 0 similarly bounded so that the discontinuity will be directly visible to the field point. However, the expressions for F and G are finite and continuous in the limits a, 0 - 0, 7; 0 + a ~= r, allowing us to replace the conditions on a and 0 individually by the less restrictive ones I a-7/2 I, 10-I/2 1 7/2 15

i Equations (46) and (47) can be written more compactly as P = F G. (50) E,H The physical optics approximation is p.o -. - PEH =+G and under many circumstances F is small compared with G: for example, in the particular case of backscattering (0 = c), a -a F = - sec a + O(k-2) 8k a -a G -— sec +O(k-2) 8k and F is less than G by a factor 10 or more for incidence within 180 of normal. Nevertheless, F is the sole source of polarization dependence in the scattering. Its inclusion is therefore important in any analysis which seeks to reproduce the polarization characteristics, and is vital in any study aimed at the cross polarized components, of the backscattered field. It is instructive to compare the scattering from a discontinuity in curvature with that from a wedge-like singularity involving a discontinuity in slope. For the metallic wedge of half angle 2 shown in Fig. 2, the diffraction coefficients PE, H are EM P E, (X Y), (51) where X = sin -cos - -cos - (a) 1 n n n n Y = -sin- cos + cos - ( —) n n c n n with n = 2 (1- Q /7r). On comparing (50) and (51), it is seen that the diffracted field produced by a discontinuity in second derivative is obtainable from that created by a first derivative discontinuity by replacing 16

2 a-o a -a sec 2 X by -2iF = - -- - -- X by 2ik cosa+cosO and 2 + (52) a -a sec - Y by -2iG = - - -- 2ik cosa+cosO This analogy between the two diffracted fields is rather interesting. In a physical sense a discontinuity in the second derivative is like a very 'subdued' version of a first derivative discontinuity, and the latter may occur in the form of a surface kink if insufficient care is taken in the fabrication of a model. From an examination of the surface in the immediate vicinity of the discontinuity, it could be hard to tell whether the discontinuity was in the second derivative (as, perhaps, intended) or was instead in the first derivative. Such a slight kink corresponds to a wedge of half angle close to 7/2, and if the expressions for X and Y are particularized to this case, the resulting diffracted field is identical to that produced by a discontinuity in curvature if = 2{1- ik- (cosa+cose) ) In addition to the expected dependence on ik, we observe that the effective wedge angle is also a function of a and 0. A further consequence of the analogy between the two types of surface discontinuity is that we can write down immediately the complete diffraction matrix for a discontinuity in curvature by making the substitution (52) in the known matrix for a wedge-like discontinuity. For this purpose we introduce a set of A A A base vectors T, N, B, where these are parallel, normal and 'binormal' to the A discontinuity respectively, with B pointed into the shadowed half space. The A A A A direction of T is chosen to make T, N, B a right-handed system, and in terms of the Cartesian coordinate system of Fig. 1, A A A A A A T = -z, N = x, B = -y Following Keller [i], we consider an incident plane wave having electric vector i A i ik i r E =e e - 17

. -0 where A A Ai = T cos /3 - N sin /3 sin a + B sin /3 cos a with 0 < 3 < r. At points far from the discontinuity in curvature and in a direction satisfying the previously'mentioned restrictions on a and 0, the diffracted electric field can be written as Ed d / e \ ik s r E =e -. e. sin 3/27-kr' where A ^ ^ A s = T cos + N sin 3 sin10 - B sin 3 cos 0, A A A and if e i is treated as a column vector in the base T, N, B, then Ad i e Ae in which A is a 3 x 3 matrix. An adequate expression for Anow follows by making the substitution (52) ip Eq. (A. 13) of [12] and is / 2i(F-G) 0 0 A= -2i(F-G)cot/ sinO -2i(F+G)cos0cosa -2i(F+G)cos0sina (53) A 2i(F-G)cot/ cosO -2i(F+G)sin0cosa -2i(F+G)sin0sina/ where F and G are given in eqs. (48) and (49) respectively. VIII CONCLUSION The derivation of a valid diffraction matrix for a discontinuity in curvature enlarges the scope of geometrical diffraction theory and permits the application of GTD to surface singularities other than the wedge-type to which it was restricted heretofore. Two particular targets for which more complete treatments are now possible are the hemispherically-capped cylinder and the cone-sphere, and in the latter case some experimental confirmation of our analysis has already been obtained. 18

REFERENCES l3 J.B. Keller, "Diffraction by an aperture," J.Appl.Phys., vol. 28, 1957, pp. 426-444. [2 J.B. Keller, "Geometrical theory of diffraction," J. Opt. Soc. Amer., vol. 52, 1962, pp. 116-130. L3] T. B.A. Senior, "The backscattering cross section of a cone-sphere," IEEE Trans. Antennas Propagat., vol. AP-13, 1965, pp. 271-277. E. F. Knott and T. B. A. Senior, "The polarization characteristics of scattered fields," Electronics Ltrs., vol. 7, 1971, pp. 183-184. 5]J V. H.Weston, "The effect of a discontinuity in curvature in high-frequency scattering," IRE Trans. Antennas Propagat., vol. AP-10, 1962, pp. 775-780. [6] V. H. Weston, "Extension of Fock theory for currents in the penumbra region," Radio Science, vol. 69D, 1965, pp. 1275-1270. [7J J. B. Keller, R. M. Lewis and B. D. Seckler, "Asymptotic solutions of some diffraction problems, " Comm. Pure Appl. Math., vol. 9, 1956, pp. 207-265. See also: Schensted, C.E., "Electromagnetic and acoustic scattering by a semi-infinite body of revolution," J. Appl, Phys., vol. 26, 1955, pp. 306-308. [8] T. B. A. Senior, "The diffraction matrix for a discontinuity in curvature," The University of Michigan Radiation Laboratory Report No. 494-1-T, June 1971. [] G. N.Watson, Bessel Functions, Cambridge University Press, 1945, pp. 398 et seq. 0 ] E. F. Knott and T. B. A. Senior, "Cross polarization diagnostics, " IEEE Trans. Antennas Propagat., vol. AP-20, 1972, pp [11] J. J. Bowman, T. B. A. Seniorl and P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes, North-Holland Publishing Company, Amsterdam, 1969; pp. 258, 263. 2] T. B. A. Senior and P. L. E. Uslenghi, "High-frequency backscattering from a finite cone, " Radio Science, vol. 6, 1971, pp. 393-406.

530 y XA Fig. 2. The geometry for a first derivative (wedge-like) discontinuity

FIGURE CAPTIONS 1. The diffraction coefficient model for a discontinuity in curvature. 2. The geometry for a first derivative (wedge-like) discontinuity.