Approximate Boundary Conditions, Part I Thomas B.A. Senior Radiation Laboratory Department of Electrical Engineering and Computer Science University of Michigan, Ann Arbor, MI 48109-2122 June 1990 Abstract Approximate boundary conditions can be very helpful in simplifying the analytical and numerical solution of scattering problems. One of the simplest is the standard impedance boundary condition, but in an effort to improve the accuracy, more general boundary conditions are now being considered. To establish the general form of these new conditions and to explore the role played by the geometry of the surface, boundary conditions are developed for an inhomogeneous dielectric body whose surface is a coordinate surface in an orthogonal curvilinear coordinate system. The treatment is based on an asymptotic expansion of the interior fields in powers of v where N is the complex refractive index of the dielectric. Boundary conditions through the second order are derived, and it is shown that beyond the zeroth order in I, the geometry of the surface affects the boundary conditions. RL-861 = RL-861

1 Introduction The impedance boundary condition attributed to Leontovich [1948] is widely used to simulate the material properties of a scatterer, and even for a nonplanar surface of a material whose properties may vary laterally as well as in depth. it is customary to use the boundary condition derived from a consideration of a homogeneous half space. In other words, the effect of curvature is neglected, and the local value of the impedance is assumed at every point of the surface. As we shall see, this can be justified to the leading order, but in recent years more general boundary conditions have been proposed [Senior and Volakis, 1989] including derivatives of the fields and purporting to improve the simulation. Here again, the derivation is carried out for a laterally-uniform half space, and there is the presumption that the resulting conditions can be applied without modification to a curved surface whose properties may vary. This is not true, and to provide a simulation in these circumstances, we here examine the effects of surface curvature and material variations using the method employed by Rytov [1940] applied to the formulation developed by Leontovich [1948]. The results presented are more general than those given by either of these authors, and reveal errors in some of the formulas quoted by Leontovich. 2 Formulation A lossy body is composed of a material whose complex permittivity e and permeability JL may vary as functions of position. The body is immersed in free space and is illuminated by an electromagnetic field. On the assumption that the external field varies slowly over the surface S, we seek a boundary conditions that can be applied at S to simulate the effect of the material. Inside the body Maxwell's equations are V X E11 = iwjiHi, V x HFn = -ziweEi where a time factor e"wt has been assumed and suppressed. Since V x (ve E) = V(Vi) x E + V x E 2

we have 1 i vr x E VEi) - x Ein= V = i^V/ H' and therefore V x (V/; Ein) + V E'i x V(ln VW) = ikoY/i H'i where ko is the propagation constant in free space and N= ' (1) 6V oo is the complex refractive index of the material. Similarly V x (/ Hin') + V- H'i x V(ln#) = -iko N v Ein Assume \NI is large everywhere inside the body, and on this basis set N=- (2) q where v is a functions of position and q is a small parameter. Writing vE'"i, - H'in, (3) the defining equations can be written as V x ~ + ~ x V(ln /;) iko-'H q (4) V x 1 + H x V(ln )= -iko,. q With geometrical optics as a guide, let E = Aeiko/q, 7 = Beiko /q (5) Then V x = V x A+ ZOV x A eiko/ q 3

with a similar equation for V x 'h, and (4) now become [Leontovich. 194S] q tzA + V x B -- x B + B x 7(ln /j)} iko (6) vB-V xA = -{V x A+A x V(lnV)}. 0 We seek a solution for A and B in the form of a power series in q, viz. A = Ao+qAi+q2A2+... (7) B = Bo+qBi+q2B2+... For this purpose we introduce the orthogonal curvilinear coordinates a, /3, with metrical coefficients h,, hp, h, such that the surface S of the body is 7 = constant with 7 in the direction of the outward normal. 3 Zeroth Order Solution Inserting (7) into (6) and retaining only the terms which are independent of q, vAo + V B x Bo = 0 (8) vBo - V A x Ao = O showing that Ao, Bo and VX7 are mutually perpendicular. Eliminating (say) Bo (v2- Iv2) AO =O 0 implying 1V1[2 =v2 analogous to the eikonal equation of geometrical optics. Hence Vi =vs 4

where s. is a unit vector in the direction of propagation in the body. At the surface the tangential components of the electric and magnetic fields inside and outside the body must be equal and, as already noted. the fields outside are slowly varying over S. Hence, the fields inside must also vary slowly, and this is only possible if, - 0 on S. It follows that TVe (and therefore;s) are normal to S, and consistent with propagation into the body, V =- -v. (9) From (8) and (9) Ao = 7 x B, Bo = — x Ao, (10) and to this approximation the local field inside the body looks like a plane wave propagating in the direction of the inward normal to S. In particular, just inside S, Ao =-Bo, = B, A Bo, =. (11) From the continuity of the tangential field components at S we have Aoi /E'n- iE, Bo VH'n-=4f H where E, H are the external fields, and (10) now implies the approximate boundary condition y x ( x E) = -Z^ x H (12) on the external field where Z - (13) is the intrinsic impedance of the material at the surface. This is the standard impedance boundary condition, customarily derived on the basis of a homogeneous half space. Thus, to the zeroth order, E > -ZH, Ea = ZHa (14) with E, = H, = 0 on S. 5

4 First Order Solution Equating the coefficients of q on both sides of (6) we obtain Al - x B = {V x B, + B, x V(ln /7)} B + x A = - {VxAo +Ao x V(ln V)} B x A1=ikV In terms of the chosen coordinates the components of (15) are Ai1 + B1 = I { hh - (hoBo) - 03- (ln /7) Zikv {hah h a ay h, j 17 v } 1 r i Bo a + h-0 -(ln V) h/ ( }( and ipon ho (6 ar and in view of (11) the components of (16) are (15) (16) Ba - A-3 B1o + A41 ikov h3h, <a-(hBo) - - io {hah ay(hoBo) - ~ko h^thl^"^0^ Bo 0 (in Ve)} Bo a 0(lnV) h, ay( BOPd,_ r} = ikoV hahp (hBoa) + d(haBo)] Boo a ha (In v) - Ba (Inv e()} Boha (a ha 0aQ fV The expressions for Al, + B1l are identical if ha, (hBop) + h a(haBop) - hahoBo- (ln v'E) = 0 (77 (77 a7 6

implying, {hh3B2} =0 from which we obtain aBo BO f1i a hah ) a- 29hah, 0a7 VJ i.e. B,= 1 (, -B0^- a In h (17) and therefore 0ho hBoo (h3 Bo3) Bo a h + h -1- o o ) =( in V i). (18) 2 d7 ( ha ) Similarly, the expressions for A13 - Bia are identical if ha -(h3Boa) + hop (hchBoc) - hh3Bo,, (ln /JI) = implying a9 { h }I =0 so that a = 2-B -(l hh (19) h -= aBoa (n h ( 2) o(hBoa) = hoBoo In V-. (20) <7 2 h ) 7

Hence Aic = -BO - 2ikovh (In Z) (21) 2kh 09 h( a Y h ) with Bi ikovh{oh + ( ) 31 ) (24) It is now a trivial matter to construct approximate boundary conditions for the tangential components of the external field on S. From (11) and (21) Aa = Ao + qAl+ O(q2) = -B3 -qBl - -B2i ( In hZ + (q ) =q (9 hOln zq2) -B 1 2ikovh 9 (ln ) } + 0( and thus, to the first order in q, Aa -B 1 + 2ikn 0 Z. (25) Similarly, from (11) and (22) A= B{ 1 + 2akN h Z)} (26) and to the first koNha conditions on S are and to the first order in 1/1|N the boundary conditions on S are 8

E, = -ZH 1 + 2i-koNh -y ha )} (27) E ZHo 1 + 2ik Nh ( ho Z)} where all quantities are evaluated at the surface. The factor 2 is missing from the formulas quoted by Leontovich [1948]. To the first order, only the normal variation of the impedance has any effect, consistent with the interpretation of the surface impedance as the local impedance looking in, and this provides justification for applying the standard boundary condition (12) at each point of a surface even when the properties of the material vary laterally. If ha $ h, the effective surface impedance implied by (27) is anisotropic: 7= raa + riF, (28) where r -=Z 1 + 2ik Nh ( h Z)} (29) 1 = {1 2i-Nh- ( hah )} and in terms of =7 a compact (vector) form for the boundary conditions (27) is x (5 x E)= -7. 5 x H. (30) This should be compared with the zeroth order condition (12). By taking the vector product of (30) with a we obtain x (7 x H) = (a + () -7& x E \li F / 9

and (30) therefore satisfies duality if 11 1 -= r implying = r-, where an asterisk denotes the dual quantity. Since N -^ N and Z -a 1/Z under the duality transformation, 1 Z{ 1 ( hi - = z{1+ koNha( z)+ ha = Z 1 - 1n -Nh. -Z7 showing that duality is satisfied to the first order in INL-1. We can make this explicit by writing - =raa + F:. (31) The anisotropy is a consequence of the curvature of S. In the special case when the coordinates a and j3 coincide with the directions of the principal curvatures at every point of 5, 1 i(lnh) 1 1 (32) h 0, (9 ~ ho Ro RO where R, and Rp are the principal radii of curvature, and if R, = RO (including a planar surface as a particular example) the impedance becomes a scalar. Thus, for a planar surface (a = x,I3 = y,y = z where x,y,z are Cartesian coorindates, implying h = h = h = 1) 7 1= 2{1 1+ 0 (lnZ)}I (33) where I is the identity tensor in the a, 3 coordinates. Likewise for a spherical surface (a = 0, / = X, r = r where r, 0, / are spherical polar coordinates, 10

implying h0 = r, h= - r sin 0, h?, = 1) 1+ 2ikoNr lnZ)I. 34) but for a circular cylindrical surface (a = 3 3 = z. 7 = p where p, 0.. - are cylindrical polar coordinates, implying h0 = p, h =- h. = 1) =Z{1+ 2ikN p(lnZ)I +2ikoN (35) In all of these results the derivative is evaluated at the surface. 5 Second Order Solution For the terms involving q2 the analysis is more tedious, and to keep it as simple as possible, it is helpful to group the terms. To this end we note that V x B + B x V(lnj)- /7V x B and when this is inserted into (6), equating the coefficients of q2 on both sides gives A2- x B x (36) Zkov v/ _l 37A1 B2 + x A2 = i v (37) The components of (36) are A2 d + B20 a f 9 (hdB<\ a (hoB \ A2ik-Bvhh a J - ~a iovh~ h 90}J = -kV h{ —(h a) 5(h~))} (38) A2= /ikVh7 { 9 (hfB) 0 (hcBi)} ikovhah9 a \ a ) / a ( 7 ) v 11

and similarlv, from (37 ), B2a,- -423 B23 + A42 c ik,,zh3h, 043 __jE-a ik~vh 0hll 101 (h-yAij (holo) (h3 e-g a (9h3AlJ) } - a (hcyAia)} (39) S V6 i ko v h,, h3 { a 9aa Consider first the expressions for A20 + B2.3. These can be written as A20 ~ B23 = 1 J' B13 ik v h. a1 and aY a + Al4w-, (In h) fii/ — a hyBi, ho ao3 ( /jY } (40) A2a + B23 = ik~vhl., { a1 in fi- a9 (h-A, ha aa 5,1, (41) and are identical if aBa aA10 a1~ + A10,-, C a h N ln of )7 V16 -hlyAj, VEI + a3 / (In j%) hVB> h0,E ac In view of (21), this serves to determine 'BBas =9^ 2 aY In ht, 2h E-a (hAj + a- h1r~i 2h a/3 \V/'/ - Bo03 I a (hh0 4i'k~vh-ya- 1"h,3 - 4ik~lvhy9lIn Z }(42) and substitution into (40) then produces a unique expression for A20' + B23. Alternatively, since a knowledge of.- is not needed for the second order 12)

boundary condition, a simpler (but equivalent) approach is to averag)e (40) and (41). gi1ving A~a B23- 1 r41a 2I? k~vh-,<4 - ( n h,) + Blo11+ (in>) hc. Oa (hl)Al) h I,3 a- hv'7 'aO 2i ko &, Boo a ln ha Z.. v h, a-y h,3 and on using (21), (23) and (24) we obtain A2 c + B213 2iov1 -Boa (ho B01,3 2 ik o L h t ho h0 __ 2)v ~ 9' -~vyina- ~ - + ikoheaaa - fiza -,( ikoh13 ao [Z h[Iv hah13 fa (h13Bo\ (9 l, - (9 h(hBoo)} _1 h-yfa h1Boc a ha Boo3lf LZv ha,,h4O aa fi}-8a3~ fv f (43) A similar procedure can be applied to A213 - B2a. From (38) A213- B2a, = ik-h { and from (39) OBicj 091 - Blota (In>)o ha — Oa j il ~ A213- B2 o =- ikh { These are identical if (9 h,3 ve- a- h-A- + Alo in7 - a- 6 ho (90 Ve (45) OBla 1 O a1nha ho = la1~In 01~ 2 &917 V/-i7 +2h13&fl (hYAj v7't a (h -~B<\ 2hotaO VP/i7 - Boo.a 0 I( h13 _Z (I In ho-~ 4i k,,v h a1~ h a J9- O- ~ i a 4i ko0 { Boce a In ha z.(146) v h —, a-f h at i 13

Also, by averaging (44) and (45). A42,- - B2~ = 2i1h. V,6 a h____ h, 3&a3 vfi 1 { Al3 a (n h3\ 1n7) a (In h ) Lv h, a9 hc, 11 y~Vi&, (h~BJ ha, aa ~V'7-J + ia 2I ko a, and on using (2,(23) and (24) we then obtain A23- -~ 2ik~vh B1 - (ln oZ) ~ 2Bk0 [21h {* (ino)2 a ria( h13 \ (471) + 7kh1a3 +k hkot aa [Z hll [7 h, h,3 fa (ho3Boc3) (hI3BE a~ (h Bo)}] [ ih-fa..Zv h, h13 & We are now in a position to construct the second order boundary conditions for the tangential components of the external field on S. From (1 1), (121) and (43) Ace = AOat + qA1ce + q2A2a + 0(q 3) - -(B013 + qB113 + q2B23) - q(B013 + qB113) - 2i'kvh,. &-y (n hc,2 hn-Z ~ q2Boa3 a In ha12 &y~ ho3 + 02 (In,OZ) - - (In Z) ~(In vhv)] - -a (haBoca)}]q2V/t 2k~vhlyhaa& [Z-hi a [7 ha h13 a (hjBo~) a~ [A2h{ a (hoBoct aa /VEJ 14

a hB 3)}]+O( 3) and thus, to the second order in q, A = {1 + i ak o nhZ) a 2iko(vh O - ( ho ) a-Y2 ( ho ) a- ( ho (2kvh,). 9- In h )} IZ) (lnvh,) }Bo O -{ J q)kFv he9a vzh~h 2k0 h L&~[vch { (hB ) 9 ( hhBo3 q2 V/ 0a + k2vh.h ao [ 1 h. Zv hah/4 * (48) In terms of the external fieldsi 1 C, h,3B,3 h,,, ho aa VI[-, 9 (hcBa) 9 J V I } 1 ho, ho { h ( HH) &&W, - a(h, a Hc,) a = * V x H = -ikoYoEE and therefore 1 a ho 0aa [v hh0 ( ho ) Lv h~ho [9a VIP-Z a (haBa\ -a VW 9: h 4 iko q * v (% h E) ~ Similarly, on using (11), 1 a h,3 90 1 hl fJ (hoB)e A h /- Ba vi = - / [Zvhh, { 9 a v J+ a3 V[ /J q - J) 1The process is valid only because Vp = 0 on S and the derivatives are tangential ones. Hence h Bcr = h, Hine-ik:~'/q = ho Hin = h Ha. Vi; 15

and since v = Nq, implying -(lnv) - (n.) the boundary condition becomes 1 _ E k + -1 io V( (h E) = - {rFH + 2iko h * (~ ) } (49) with p_ L, 1 + (, n h O9 n h \ F 2ikoNh, + (i h ) (2k, Nh)2 [I9-2 ( h ) 2 { (ln" hZ - (In h Z) a -(ln h.N) (50) We note that to the second order in q (49) can also be written as Ea + ik Nh E =-r + (2ik NhH) 7 ) (51) The analysis for the second boundary condition involving the components E0 and Ha is similar in all respects, and using (11), (22) and (47) we obtain E + 2iNh V (~hE,- =r 1 HF + 2i kNh, V hH }(52) 2ikoN h, \zekoly P where rl differs from r only in having ha and h, interchanged. Accordingly, 1 1rto the second order in q, where the asterisk denotes the dual quantity, and a compact (vector) form for the boundary conditions which makes the duality 16

self-evident is 5x x E E+ 2 k V(^ ) h -E {H 1 o 2 + 2ik Nh-, V h H) }3 with 7 =rad + /. (54) To the required order r - (1 )+ 2ak (5) where a= 2ikoNh. (0 h Z) (56) 6 Examination of the Solution The second order boundary condition (53) has several interesting features and it is worthwhile examining these as well as the forms taken in special cases. The most intriguing feature of all is that the form of the boundary condition is independent of any variation in the material properties of the surface, and in this respect (53) is similar to the zeroth and first order conditions. Any lateral variation of the properties is taken care of by the gradient operations, and a variation in the normal direction affects only the second order terms in the expression for the effective surface impedance 7I. Nevertheless, for simplicity we shall henceforth assume that the material is homogeneous. For the special case of the planar surface z = 0 we choose a = x, = y,= z implying ha = h= = h. = 1. Then F= = Z so that = ZI, and the boundary condition becomes ( E + 2N VEZ}) -Zz x H + 2ikN VHz (57 x xE2i'koN e ZkNj 17

The components of this are E + 2iko N e ax = Z + 2ikoV af (58) y+ 2ikoN e y1 = Z Hl + 2ioN a ox J' and since Z ko a ay kHo ( x ay (58) can be written as xa2 2Hx 2(k0N)2 Exay 2 { 2(koN)2 ( 2 axdy) 2(ko)2 xay 2 ) (59) E+ z i 1 (2H aH Y a2E, y2E a y Z x 2(koN)2 ay2 aOxy) 2(koN)2 x2 xay) On inserting the zeroth order approximations to the derivatives of the electric field on the right hand side, we then obtain E, -z Hy+ 1 \( 92 - a2 Hy - 2 2HX' y + 2(kN)2 [( 2 ay2) y Dzy (60) E -2 a2 azff 2H Ey -Z {H + 12( kN)2 - y2 H - ] } and these are analogous to the results cited by Leontovich [1948]. However, the coefficient of the terms O(NI-2) differs by a factor i/2 from his, and it appears that the error was made in extending the correct but specialized results of Rytov [1940]. Using Maxwell's equations and the divergence conditions, (60) can be expressed in terms of the normal field components: 18

Qz2 9J {&2 -i kA ko(2 - 1) E 0 (61) { a3 + 2 i (2 o+ 1)- - I(Ak)3 H, 0 and from these it is clear that (60) do not satisfy the duality condition. On the other hand, from (58) or (59) without any approximation at all, {02 a {22 + 2ikdN- - k (2N2 - 1)} E, = 0 (62) a2 a 2 { + 2ikoN k -2 N-=0 - i- + 2 /lNo - k~(2 -1) H -=0 which are quite different from (61). Equations (62) are second order generalized impedance boundary conditions in the form generally adopted [Senior and Volakis, 1989] for a planar surface and do satisfy duality. In view of the discrepancy between (61) and (62) it would seem that boundary conditions like this are very sensitive to the precise form of conditions such as (57) from which they are often derived. For the circular cylindrical surface p = constant we choose a = P, 3 = z, 7 = p, implying h, =o p, h = h, = 1. Then px(px {E e _V ) =- p x H + PO1 1H 3x IPX 2ikoN e ) H 2zkNV P (63) with - 1 i= rI` + -ZZ and I- Z {1 2ikoNp + 2(2koNp)2 19

so that r Z 1 21ko0P 2(2koXp)j The components of (63) are 1 e o1lEp __ 1 __ ~H Eo+ 2 e p &p -r Hz + kNo aI z ilzkN e p 0' ik N ~' Oz (64) E- 2iko^ e z r {H + 2ikoN - p 0 in agreement with the results in Appendix A, but without some approximation it is not possible to write these in terms of normal derivatives of the normal field components alone. However, they can be written in terms of tangential components. Since E iZo1 (1H _ &1H iY IaE, _ EO p ko p Q0 dz ) ' p ko p k kz ) ' we have = -z {H + 2(koN)2 1p2 02 p- pz) 2(koN)2Y p (O1 z 2z2 ) ' As regards the last term on the right hand side it is sufficient to insert the zeroth order approximations 1 E,= -FH, Ez, =-H with = Z and hence 1_ 1 1 2 1 E = - { [ + 2ikoNp + 2(2koNp)2 2(koN)2 (z2 p2 2 H] 1 0 2H } (65) ( koN)2p ao& 20

Sin-ilarly Z: { [ - 2iko -, 29(92kp)2 + 2(koNV)2 (2 a02) H a12 2H2 (koN)2p ao a (66) For the spherical surface r = constant we choose a = 0, 3 = Q. = r so that h, = r, h3 = r sin 0. h. = 1. The boundary condition (53) then is K'1 POE= rx x 2E+-k VEr}) =-7 r x H + 1 iko VH-} (67) with 1 - - re + r-*~ and (see (50)) 1 F-Z= r* By a method similar to that used to derive (65) and (66), it can be shown that the boundary conditions expressed in terms of the tangential field components are Eo -=si -Z.12 ] H [ 2(koNr)2 { k2 +sin r sin2 0 sin2 1 2 H) 1 H( 68) (koNr)2 sin 8 &909 j E = 1 C2 cos 8 1 H1 ] EO -z I- + Ho(kN.NVr)2 sin9 8 oa0 21

In general form at least. (65) and (66) and (68) and (69) are similar to the boundary conditions (60) for a planar surface, and it is tempting to ask whether, for a curved surface, we can derive the boundary conditions by treating x. y. z as local coordinates. To test this, consider the cylindrical surface p = constant. Replacing z by p, x by p9 and y by z in (60) we do indeed reproduce the differentiated terms in (65) and (66). but do not obtain the undifferentiated terms associated with F/Z and Y/P. This is not unexpected in view of the first order condition (35), and since the dominant (for large \NI) terms then differ from those in (65) and (66), it is clear that the procedure cannot be justified. Our final point concerns the sequence of boundary conditions through the second order for an inhomogeneous curved surface, and the implications for simulating other surfaces using boundary conditions of this type. In its most general form the zeroth order condition is y x (? x E) = -7rl x H (70) and involves only a single scalar parameter qr which is independent of any material inhomogeneity and the shape of the surface. The first order condition is - x (x xE) -- a. x H (71) with - ra a + rI, but if duality is satisfied, IF = 1/r*. There is still only one parameter (r) involved, but this may be a function of the variation of the material properties in the normal direction, as well as the curvature of the surface. Finally, the second order condition is E + A (hE) } 7 ix 1<{E+ 7^^^v x-y^))= j-7x{H 2ikoNhA V o hH) (72) involving the two parameters I and A, and all material variations in the normal direction are embedded in r. 22

7 Conclusions The method developed by Rytov [1940] has been used to derive approximate boundary conditions at a surface which may be non-planar and have ma11Lterial properties that vary laterally as well as in depth. The approximnations are based on the assumption that 1NI is large, where N is the complex refractive index of the material, and boundary conditions through the second order have been obtained. The lowest (zeroth) order one is the standard Leontovich impedance condition whose form is the same for any surface. planar or curved, and is independent of any variation of the material properties. This is not true of a higher order condition and, in particular, the surface curvature now affects the boundary condition. Because of this it should not be assumed that a generalized boundary condition developed for a planar surface is immediately applicable to a non-planar surface. References Leontovich, M.A., "Approximate boundary conditions for the electromagnetic field on the surface of a good conductor", in Investigations on Radiowave Propagation, Part II, pp. 5-12. Printing House of the USSR Academy of Sciences: Moscow, 1948. Rytov, S.M., "Computation of the skin effect by the perturbation method," J. Exp. Theor. Phys., vol. 10, p. 180, 1940. Translation by V. Kerdemelidis and K.M. Mitzner. Senior, T.B.A., and J.L. Volakis, "Derivation and application of a class of generalized boundary conditions," IEEE Trans. Antennas Propagat., vol. 37, pp. 1566-72, 1989. 23

Appendix A: Homogeneous Circular Cylinder When the exact solution of a scattering problem is known, it can be used to derive an approximate boundary condition applicable at the surface, and a case in point is a right circular homogeneous dielectric cylinder. In terms of the cylindrical polar coordinates p, 0, z the cylinder is defined as p = p, and is illuminated by an H-polarized plane wave incident in a plane perpendicular to the axis. The only components of the field are then H, Ep and E~, and their expessions are P > Po: 00 H = e(-i)n {J(kop) + RnH(1)(kop)} cos n n=O E -i- n(-i)n {J,(kop) + RnH )(kop)} sin n -iZo -) {J'(kp)+ n oP) oo E = -iZ en,(-i)n{aJ,(N kopR) cRp)os nn n=O - ikopZoel(-i)nafll(N kOp)sinnq$ Hz= -i z E-inainJ(Nkop) COS n n=O P7 oo p i (konaJkp) sin niYPJ(k) = (.4.2) 00 E = -iZ En(-zi)anJ(Nkop) cosno n=O where the prime denotes the derivative and RnOan are coefficients to be determined. From the continuity of Hz and Ex at p = po we find R - (oPo )+)+ Jn(kopo) (A. n with J.(Nlk,p, ) (A 24

On the assumption that ['A"kopo >> 1 with Im..V > 0 to prevent any penetration through the cylinder, P can be expanded in an asymptotic series for large t = VNk po. Since Jn(t) = - {H(l)(t) + H(2)(t)} it follows that J) e-i(t-2- ) 2 (n, m) 2n7 t m=O (2it)m e-i(t-n'-r) f 4n2- 1 (4n2-l )(4n2-9) v/2t + 8it 128t2 (4n2 - 1)(4n2 - 9)(4n2 - 25) 128 24it3 (4n2 - 1)(4n2 - 9)(4n2 - 25)(4n2 - 49) + O(t-5) (128)2 ~ 6t4J Hence e-i(t-n2-) f 1 3 4n - 1 Jn() - iJn(t) + t 2 t 8i t '2 t 8 it 5 (4n2 1)(4n2 -9) 7 (4n2 - 1)(4n2 - 9)(4n2 - 25) 1 + - +2-t' +0(t)> 2t 128t2 2t 128*24it3 e-i(t-n2-i) r 4n2 + 3 (4n2 - 1)(4n2 + 15) L/2. r 8it ~ 128t2 (4n2 - 1)(4n2 - 9)(4n2 + 35) 128 24it3 (4n2 - 1)(4n2 -9)(4n2- 25)(4n2 + 63) } + (128)2 6t4 + (t giving 25

1 4n2 - 1 4n2 - 1 P- Z 1+ -+ +2 9 - t2 + Sit3 (4n2 - 1)(4n2 - 25) 5 O( 128t4 - } An approximate boundary condition must reproduce P (and therefore Rn) to a specified order in 1/t. A standard impedance boundary condition has the form E = -FH, (A.4) and using the previous expressions for the field components in p ~ p0 we obtain (A.1) with P = F. Thus, to the zeroth order in 1/t, r=Z, ( 5) but (A.4) is also sufficient to reproduce the exact solution to the first order in 1/t if r is chosen as =Z{ 2iNkopo} (A.6) and (A.4) is then a first order impedance boundary condition. To reproduce terms of higher order in 1/t, it is necessary to generalize the boundary condition (A.4). In line with (72) we now consider the second order boundary condition E + 2iNkp,O (~E) = -rH (A.7) and using again the field expressions in p > po we find P =- AO ( n r- A (n\2 (A8) 2N E kopo 2 (A) Comparison with (A.3) now shows that to the second order in 1/t =Z {1+2iNkop +2(2Nkop)2} (A.9) 26

and A = 1 (A. 10) in agreement with (63). As a matter of fact, by choosing F = Z { + 2iNk~p0 + 1 ~ (-4.11) { 2iNokopo 9)(9.opo)2 i(')./'k opo)3} ( 4.11) and i -Vk0p0 (A.12) the boundary condition (A.7) is sufficient to match the third order terms as well, but this is peculiar to the geometry. In general, a third order boundary condition is necessary to achieve this accuracy, and as evident from the term in n4 in (A.3), a still higher order boundary condition is required to match the terms in 1/t4. 27