IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 34, NO. 1 , FEBRUARY 1992 9

Mathieu Functions and Their Applications to Scattering by a Coated Strip

Richard Holland, Member, IEEE and Vaughn P. Cable

Abstruct- This article begins with a review of the elliptic- cylinder harmonics, known as Mathieu functions. These functions are then used to describe EM scattering by confocal elliptic cylinders where each cylinder’s dielectric constant is different. A peculiarity of this problem is that the Mathieu functions in different regions are not orthogonal at regional boundaries. Hence, each boundary couples all harmonics from both sides together, and infinite sets of coefficients must be simultaneously evaluated. Numerical results are given for the special case where the innermost region is a perfect conductor. We consider both TE and TM illumination. Only normal incidence is actually treated, although oblique generalization is conceptually easy.

I. INTRODUCTION LLIPTIC cylinder coordinates are perhaps the simplest to E work with excluding the “big three” (Cartesian, circular

cylinder, and spherical). They are one of the few coordinate systems for which the harmonic expansion of a plane wave is tabulated [I]. (Scattering of a vector plane wave by a conducting spheroid using spheroidal wave functions was an unsolved problem until 1977 [l], [2].)

In elliptic cylinder coordinates, separation of variables leads to Mathieu’s equation, which has solutions known as Mathieu functions [3]. These interesting entities have been somewhat plagued by a lack of standardized notation; at least five differ- ent symbologies are in use [4]. Perhaps because of notational confusion, their use is relatively infrequent in literature.

The propose of the present article is two fold. First, we shall present a somewhat tutorial introduction to the derivation and evaluation of Mathieu functions. We believe this is useful because nowhere else have we found such a tutorial. This difficulty initially caused us great inefficiency in dealing with the coated-strip scattering problem.

The second propose is to describe our application of Math- ieu functions to scattering by a dielectrically coated conducting strip. (The conducting strip and the conducting disk are the only scattering problems with sharp edges which can be solved canonically.) As such, the strip provides a most useful example for checking the accuracy of numerical techniques, such as time-domain finite differencing (FDTD). This problem is also of practical interest, as a conformally coated thin elliptic cylinder or strip is by far the closest representation of an

Manuscript received December 10, 1990; revised June 25, 1991. This work was supported by the U.S. Air Force under Contract F29601-87-C-0207 and by Lockheed Aeronautical Systems Company.

R. Holland is with Computer Sciences Corporation, Albuquerque, NM 87106.

V. P. Cable is with Lockheed Aeronautical Systems Company, Burbank, CA 91520.

IEEE Log Number 9104202.

aircraft wing which one could hope to solve without using some sort of approximating discretization.

Electromagnetic scattering by a purely dielectric elliptic cylinder leads to a curious mathematical phenomenon: the elliptic-cylinder harmonics inside the cylinder are different from those outside. Familiar orthogonality relationships at the interface do not apply. In this way, scattering by a dielectric elliptic cylinder is fundamentally different from scattering by a circular cylinder or sphere. Now there is a coupling between every internal harmonic and every external harmonic. This has been discovered by Yeh for the single-medium elliptic dielectric cylinder [SI, although applications to a multiply coated strip or multiple-medium elliptic dielectric cylinder were first worked out by Holland et al. [6], [7]. The same techniques have also been used by us to solve canonically the problem of scattering by an anisotropic circular cylinder [8]. More recently, some of these concepts have also been studied by Richmond [9], and Razheb et al., have treated the case of coating by an elliptic nonconfocal dielectric region [ 101.

11. DIFFERENTIAL EQUATIONS, DEFINITIONS, AND MATHIEU FUNCTIONS

The transformation from Cartesian (x. y, 2 ) to elliptic cylin- der coordinates ( U , 71; 2 ) is

z = 112 d cosh I L COS v y = 112 d sinh U sin v z = z

where d is the distance between the foci of the ellipse. Loosely speaking, v corresponds to the azimuth and d / 2 cosh I L to the radius. Additionally, some literature makes the definitions

E = coshu r) = cosv

so that

sinh U = d m . (6)

In this article, we shall use the notation of Uslenghi and Zitran [l] , that of Blanch [ l l ] where it does not conflict with the first notation, and that of Stratton [12] where is does not conflict with either of the above.

In all cylindrical coordinate systems, the TE solution for H, obeys the Helmholz equation

0 2 H , + K’H, = 0. (7)

0018-9375/92$03.00 0 1992 IEEE

1 0 1 t t . t TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 34, NO. 1, FEBRUARY 1992

In elliptic cylinder coordinates, this equation takes the form

1

(d/2)’ (cosh2 U - cos’ U)

@H, - + K ~ H , = 0. (8) dz2

Let us now assume (8) can be solved by separation of variables

H , = U(u)V(w)Z(z) . (9)

Substitution of (9) in (8) yields

normalized to be unity at w = 0, is the rth even Mathieu functions of period T

CO

Sen,(c. 7) = DeiL(c) cos 2kv. (19) k=O

Hodge [13] has a computer program available which evaluates these eigenvalues and the associated coefficient vectors. Typi- cally, (19) is truncated around the 40th term ( k = 40), the first 20 eigenvalues and eigenvectors are iteratively found, and the first 30 components ( r = 30) of the eigenvectors DeiL(c) are evaluated.

Alternatively, the r t h eigenfunction of this system having period 27r, also normalized to be unity at v = 0, is

1 x

( d / 2 ) 2 (cosh’ U - COS’ w) Se2r+l(c,v) = De;iz:(c) cos (2k + 1)v. (20) (10) k=O

Let C be the first separation constant

Z’/ - +C=O. Z

Now let us consider odd solutions of period T . The rth eigenvalue is now denoted b2‘ and the rth eigenfunction, normalized to have unity derivative at v = 0 is (11)

Then U and V must obey x

S O ~ , ( C . v) = Do;;(.) sin2kv. (21) U” - ( a - 1 / 8 . d 2 ( ~ 2 - C) cosh2u)U = 0 (12) k = l

”’ + + ‘ d2 ( K z - ‘Os 2v) = (13) Finally, the r t h odd eigenfunction of period 27r is

where a is the second separation constant. Dc:

In the rest of this article, we shall assume there is no z ~ o ~ ~ + ~ ( e , v) = DO;;’,: ( e ) sin (2k + 1)v. (22) dependence, so k=O

K’ - c -+ K 2 . (14) Any even Mathieu function is orthogonal to any odd Math- ieu function on the interval (0,27r). Likewise, different even or odd Mathieu functions are orthogonal to each other, provided c does not change. The normalization integrals are

TO conform with existing literature [13, we shall denote

(15)

where c is not the speed of light, but is the number of radians

of its foci. Then (12) and (13) become

c = 1/2 ( K d )

at the frequency of interest between the ellipse center and one

U” - ( a - 1/2 . e‘ cosh 2u) U = 0 (16)

Ni:)(c) =

= 7r(2(Deir(c))’ + (De;‘(c))’+...+] (23)

V” + (U + 1/2 . c2 cos 2U) v = 0. (17)

Equation (17) is Mathieu’s equation and (16) is the modified Mathieu equation.

The c of (15)-(17) equals c,X in Stratton and fi in Blanch. In some more modern works, such as [4] and [13] it equals 2Jq.

Mathieu Functions and CoefFcients

It turns out that Mathieu’s equation only permits periodic so- lutions for discrete eigenvalues of a, where these eigenvalues depend on c. If we assume

00

V = Desk(c) COS 2kW (18) k=O

we obtain even solutions of period T. The rth eigenvalue of this equation is denoted a2r, and the r t h eigenfunction,

3c

k = l nr

k=O

(25)

The four types of Mathieu functions defined by (17)-(26) describe the azimuthal dependence of the elliptic-cylinder harmonics. Equation (16), on the other hand, describes the radial dependence. There are a total of 16 kinds of radial Mathieu functions.

Let Z i ” ( z ) denote the kth Bessel function of the j th kind. For example, the Hankel function Hi1) (x) becomes

ZP’(z) = Jk(.) + iYk(.). (27)

HOLLAND AND CABLE: MATHIEU FUNCTIONS AND THEIR APPLICATIONS 11

Then the four radial Mathieu functions corresponding to then the desired expansion is [12] Se*,(c. q ) may be shown [4], [12] to be x

H~~~ = i, a: R~:)(c. E ) Se,(c, q ) 7r x (- 1)"'" De;;-, ( e ) (m=o Re!ji)(c.<) =

30 k=O € 3 D e ; m ) (37) + bg RoE)(c, 5) So,(c, q )

. [ Jk - s (E1 1 zi;: ( E 2 + J k + S (E1 1.4 2 (E2 )] m=l

(28) where

where

and s is any arbitrary integer, but is best selected if

De::(c) = max(De;L(c)). (32)

Similarly, the four radial Mathieu functions corresponding to Se2r+1 (c. q ) are

The radial Mathieu functions corresponding to 5 ' 0 2 ~ ~ ~ ( e , 7 ) are

-1 ( d H F . 3H:" EinC = - i, - - a , -) (40) twE,h dv 3 U

where h is the elliptic cylinder metric

1/2 . h = h, = h,, = ( d / 2 ) (cosh2 U - cos2 U) (41)

Expansion of (40) yields

30 1

L m=O

I) 00

+ bFRoE) ' ( c ,E )So , ( c .q ) . m=l

(34) Scattering by a Conducting Elliptic Cylinder

We shall first assume this plane wave is scattered by 30 2r+l a perfectly conducting elliptic cylinder. Thus, the scattered

solution must be in an infinite series of outgoing Mathieu- 7r Ro::i,(c. E ) = - ( - I ) ~ + ~

Do2k+1(c)

2 k=1 Do;::: (c) Hankel functions.

111. ELECTROMAGNETIC SCATTERING BY ELLIPTIC CYLINDERS

Expansion of a Plane Wave

Let us now consider the expansion of a plane wave in elliptic-cylinder harmonics. If we have a plane wave propa- gating at an angle & from the +z axis

3c 1

12 IEEE TRANSACTIONS ON ELECTROMAGNETlC COMPATlBlLITY, VOL. 34, NO. 1, FEBRUARY 1992

(44) I) o=

+ b z a t Rag)' (c, E ) Soh (c. q ) . m=l

The requirement that on 5 = E1

E F C + E:,Cat - - 0 (45)

then yields formulas for a r t and b T t

(47)

We are now in possession of enough information to evaluate the TE RCS of the elliptic cylinder. At large arguments, ReE)(c. e ) and RoE)(c, E ) both approach [12]

(48) - e i ( c c o s h u - - ) T .

- d Z G G Uslenghi and Zitran [ 11 then define the scattering function

,KG e i ( K T b -x/4) H s c a t p(4J = .J-i- 2 (+;) (49)

where w; approaches 4; at large r;. In view of (l), (2), and (15), we can show that

K r ; + ccosh U . (50)

Substitution of (43), (46)-(48), and (50) into (49) then yields the TE bistatic scattering function

. SO,(C, COS $ o ) S ~ m (c, COS U;) . (51) ) We can then evaluate the TE bistatic RCS

Uslenghi and Zitran [ 11 have published curves for I P (U;) I as functions of dol, et1 and tanh u1. The quantity

cE1 = (1/2) . Kd cosh ~1 (53)

is the number of free space radians at the frequency of interest along one half of the major axis of the ellipse.

Additionally

tanhul = d(f - 1/[1 = (ndsinhul)/(Kdcoshul) (54)

is zero if the ellipse is reduced to a conducting strip, is unity in the special case where the ellipse fattens to a circular cylinder, and in general is the ratio of the minor to the major axis length of the ellipse.

It is also useful to express the bistatic RCS in terms of the scattering coefficients a r t and b z t of (43). This is the case because it is not possible to find closed-form expressions like (46) and (47) for these coefficients in the dielectric scatterer case. Substitution of (46), (47), and (51) into (52) yields the result

It is interesting to note that in the case of scattering by a conducting strip, u1 is zero, El is unity, Re:)’(c. 1) is zero, and all the a T t vanish (see (46)).

We have also studied TM scattering by an elliptic cylinder. In this case, a derivation dual to that of the TE situation leads to the bistatic TM RCS

‘x \ 12

m=l / I

s c a t / s c a t

where a; and b,, are given by

In the case of scattering by a conducting strip, u1 is zero, (1

is unity, Roi ) (c , 1) is zero, and all the bzat vanish (see (58)).

Scattering by a Dielectric Elliptic Cylinder

For scattering by a dielectric cylinder, we encounter the nonorthogonality complication. In the TE case, the incident and scattering fields can still be represented by (37)-(44).

HOLLAND AND CABLE: MATHIEU FUNCTIONS AND THEIR APPLICATIONS 13

oc However, we must now also consider the fields which pene- trate the cylinder

The penetrating electric field EtranS is similarly expressed, with u p , bzns, and ~1 replacing u g , b g , and EO of (42).

Here 1c1 is the wavenumber in the dielectric

K 1 = (60)

and c1 is the number of radians between the ellipse center and a focus for a wave traveling in the dielectric

m=O "

m=O 00

= - ueE;l ~ e : ( ~ ) ~e : ) ' (~ , tl). (67) m=O

These equations must be satisfied for all n, as must a similar family of equations for the odd elliptic harmonics. Once such a solution has been performed, the RCS is still given by (43)-(52), except that (46) and (47) no longer apply.

The analogous equations for TM scattering are 30

a z a t De,"(c) Rez)(c. €1)

m=O 70

m=O oc

= - af" De;(c) ReE)(c, (1) (68) The boundary conditions at E = (1 are now the continuity m=O

of H, and E, 00

HFC + H s c a t - - H;rans (62) De;(.) Reg)'(c,<l) E"" + E;at = .Fans. (63) m=o

03

These conditions lead to the relationships

" = uzns ReE)(cl, 61) Se,(cl, 7 ) (64)

m=O

1 "

€1 m=O

= - a ~ ' R e ~ ) ' ( c l . 6 1 ) S e m ( c l . 7 ) (65)

plus an additional pair of relationships connecting the odd elliptic harmonics.

The complication now arising is that Se,(c,v) and Sem(cl.q) are not equal, since c and c1 are different. In other words, the dielectric cylinder causes each incident elliptic harmonic to couple to every scattered elliptic harmonic.

It is necessary to substitute (19) and (20) for Sem(c, 7 ) and Se, (cl. 7 ) into (64) and (65). Doing this, rearranging terms, and factoring out the trigonometric functions yield

3c

agat DeF(c) ReE)(c. t1) m=O

70

m=O oc

= - af" De:(.) Reg)(c, cl) (66) m=O

- aZansp;l D e r ( c l ) reg)'(^^,[^) m=O

00

m=O

The Two-Medium Problem

The two-medium TE elliptic cylinder problem is a fairly straightforward extension of (68) and (69). Unlike Bessel functions of the second kind, radial Mathieu functions of the second kind do not go to infinity at U = 0. However, they do not have both nonzero values and nonzero derivatives there. This means elliptic-cylinder harmonics with radial Mathieu functions of the second kind have either a sharp ridge or a step discontinuity on the line connecting the coordinate system foci. Consequently, their appearance is forbidden in the case of scattering from a uniform single-medium elliptic cylinder. Such is not the case, however, for the outer medium in the two-medium case.

Subject to this understanding, the TE elliptic-cylinder har- monic coefficients for the two-medium cylinder obey

00

m=O 03

30

m=O

14 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 34. NO. 1, FEBRUARY 1 Y Y 2

x

(a:S.t;'Re:)'(c.&) + aYtt ; ' Re2)'(c.&)) De,"(c) m=O

a o u t e r ( ~ ) -1 R (1)' = 53 c2 e, (c2. (2) + a ~ ~ ~ ~ ( ~ ) t ~ ~ m=O

.RC:)'(CZ.<~)) Der(c2). (73)

The odd elliptic-cylinder harmonic coefficients obey a similar equation family. This procedure can be extended to an arbitrary number of confocal regions in a straightforward manner.

If the inner region is a perfect conductor, we need only be concerned with the vanishing of the total tangential electric field at its surface. Thus, in that case, we can ignore (70) which pertains to continuity of H,. Moreover, we know that a:ner will vanish if the inner material is a perfect conductor. This has the effect of decoupling the elliptic harmonics from each other at the inner interface; (71) reduces to

aouter( l ) Re(')' ( c2. €1) + aEter(2) ReEj ' (~2 , (1) = 0 m (74)

Equations (72) and (73) do not simplify. The two-medium TM elliptic cylinder problem may be

solved in the same way as the TE problem. The TM elliptic- cylinder harmonic coefficients for the two-medium cylinder obey (70)-(73), with p replacing E everywhere. If the inner region is a perfect conductor, we can ignore the analog of (71) which pertains to continuity of H,. Moreover, we know that am will vanish if the inner material is a perfect conductor. This has the effect of decoupling the elliptic harmonics from each other at the inner interface; the analog of (70) then reduces to

/l""ei

The other two TM equations do not simplify.

Numerical Results

We first numerically considered a conducting strip d = 2a = 4 m wide, and infinitely thin (see Fig. 1). Figs. 2 and 3 show the bistatic RCS for the TE case when the strip is bare and when it is coated by a dielectric ellipse of cr = 2, with major axis = b = 2.02 m, minor axis = 0.283 m for c = 10 rads. We then reran the TE problem for c = 20 rads (see Figs. 4 and 5). (These dimensions correspond to the conducting strip being (1OX0/7r) or (20X0/7r) wide.) Finally, we reran both values of

Fig. 1. A metallic strip confocally coated by a dielectric.

RCS (BISTATIC) OF A BARE STRIP

TE Case 6004

+m 20 0

c = 10.0 c = 100 c( , = 10 0

600 400 a = 200 m

20 0

Normalization

Factor = 1 @e04 60 o

Fig. 2. Bistatic TE RCS of a bare conducting strip. Strip is 4 m = 10 X O / K wide.

RCS (BISTATIC) OF A COATED STRIP

GO 0 20 0 20 0

b = 2 0 2 3 Z O O

tanh u1 = O$OO I

Normalizotion

Factor = 1 0e04 GO o

Fig. 3. Bistatic TE RCS of a coated conducting strip. Strip is 4 m = 10 X o / a wide, and dielectric coating ( zJ . = 2 ) is 0.283 m on each side at midpoint.

c for the TM case (see Figs. 6-9). All these figures are based on illumination at 40 = 45", and clearly show both the optical shadow around 45" and the specular reflection at -45".

Typically, these calculations were performed by solving (70)-(73) with truncation occurring around 15 terms and with about 15 equations kept from each of the four sets. Equations (74) and (75) for a perfectly conducting inner region are not explicitly a part of the code. This means a system of 60 simultaneous equations has to be solved at each frequency and angle of incidence for both the even and the odd portions

HOLLAND AND CABLE: MATHIEU FUNCTIONS AND THEIR APPLICATIONS

__p 27 C

15

RCS (BISTATIC) or A BARE STRIP

TE Case @" = 45O

'Jormolization Factor = 1 0e04 60 0

Fig. 4. Bistatic TE RCS of a bare conducting strip. Strip is 4 m = 20 XO/X wide.

RCS (BISTATIC) or A COATED STRIP

TE Case

_m_r 6 0 ,

tanh u 2 = 0 140

Normalization Factor = 1 0e04 60 0

Fig. 5. Bistatic TE RCS of a coated conducting strip. Strip is 4 m = 20 A s / " wide, and dielectric coating (E' = 2 ) is 0.283 m on each side of midpoint.

PCS IEISTATIC) CIF A BARE STRIP

TM Case 2 7 0 4 I

. , , . , , . . , , , 2 7 0 180 9 0 5,- 9 18 0

1 Normalization Factor : 1 OeOl 27 o 2

Fig. 6. Bistatic TM RCS of a bare conducting strip. Strip is 4 m = 10 X O / T wide.

RCS (BISTATIC) OF A CONED STRIP

TM Case

@n = 45O c = l o 0 [ = 100 f = 101

c F = 10.1 E = 2 p = l

a = 200 m b = 2.02 m

c,[, = 100

3 2 - 2 7 0 180

tanhul=OOOO tanh u2 = 0 140 freq = 10/(47r(~,~J'~) '* Normalization

Factor = 1 OeOl 27 o

7 27

Fig. 7. Bistatic TM RCS of a coated conducting strip. Strip is 4 m = 10 &/a wide, and dielectric coating ( c r = 2 ) is 0.283 m on each side at midpoint.

RCS (BISTATIC) OF A BARE SlRlP

TM Case

Normalization Factor = 1 Oe02 45 o

Fig. 8. Bistatic TM RCS of a bare conducting strip. Strip is 4 m = 20 X o / a wide.

RCS (BISTATIC) OF A COATED STRIP

TM Case 6- = 45O C) = 20 0

CJ, = 20 0 c3f2 = 20 2

f = 100 [ = 101

E = 2 u = l

a = 200 m

1 Normalization Factor = 1.0e02 4 5 0

Fig. 9. Bistatic TM RCS of a coated conducting strip. Strip is 4 m = 20 A?/" wide, and dielectric coating (cr = 2 ) is 0.283 m on each side of midpoint.

16 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 34, NO. 1, FEBRUARY 1992

of the response. The even and odd portions of the response do not couple, and these two parts of the scattered field need only be added at the end of the computation to obtain the RCS.

It is very difficult to be absolutely sure convergence has been achieved in calculations of this sort. We tested convergence by adding two or three more terms to all the truncations and seeing if the third significant figure of the RCS changed. If it did not, we assumed we were close to true answer. Also, we made calculations in the limiting case of all surfaces becoming circular (tanh p. = tanh p2 = 0.995), and determined that the Mathieu-function based code gave results which agreed with circular cylinder codes. None of these tests gave us cause to question the validity of our numerical implementation.

Ideally, we would also have liked to treat the case where the dielectric layers were lossy. This, however, would have led to Mathieu functions of complex arguments which apparently, are not well understood [lo].

ACKNOWLEDGMENT

The authors wish to thank two anonymous reviewers who made numerous useful suggestions and pointed out that scat- tering of a vector plane wave by a prolate spheroid was no longer an unsolved problem.

REFERENCES

[l] P.L.E. Uslenghi and N.R. Zitran, “The elliptic cylinder,” in Elec- tromagnetic and Acoustic Scattering by Simple Shapes, J. J . Bowman, T. B. Senior, and P. L. E. Uslenghi, Eds. Amsterdam, The Netherlands: North Holland, 1969, pp. 129-180; and “The prolate spheroid,” in Elec- tromagnetic and Acoustic Scattering by Simple Shapes, J. J. Bowman, T. B. Senior, and P. L. E. Uslenghi, Eds. Amsterdam, The Netherlands: North Holland, 1969, pp. 416-471.

121 B. P. Sinha and R. H. MacPhie, “Electromagnetic scattering by prolate spheroids for plane waves with arbitrary polarization and angle of incidence,” Radio Sci., vol. 12, pp. 171-184, Mar. 1977.

[3] E. Mathieu, “Memoire sur le mouvement vibratorie d’une membrane de forme elliptique,” J. d. Math Pures et Appliquees, s. 2, vol. 13, pp. 137-203, 1868.

[4] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Func- tions. National Bureau of Standards, Washington, D.C., June 1964, pp. 721-750.

[5] C. Yeh, “Backscattering cross section of a dielectric elliptic cylinder,” J. Optical Soc. America, vol. 5 5 , pp. 309-314, Mar. 1965.

[6] R. Holland and K. S. Cho, “Radar cross-section evaluation of arbitrary cylinders,” Applied Physics, Inc., Albuquerque, NM, Rep. API-TR-129, Sept. 12, 1986.

[7] -, “Canonical solution of EM scattering by layered elliptic cylin- ders,” Applied Physics, Inc., Albuquerque, NM, Rep. API-TR-143, Jan. 15, 1988.

[SI __, “Scattering matrix calculation for obliquely illuminated cylin- ders,” Applied Physics, Inc., Albuquerque, NM, Rep. API-TR-136, Oct. 15, 1987.

[9] J. J. Richmond, “Scattering by a conducting elliptic cylinder with dielectric coating,” Radio Sci., vol. 23, pp. 1061-1066, Nov. 1988.

[ lo] H.A. Ragheb, L. Shafai, and M. Hamid, “Plane wave scattering by a conducting elliptic cylinder coated by a nonconfocal dielectric,” IEEE Trans. Antennas Propagat., vol. 39, pp. 218-223, Feb. 1991.

[11] G. Blanch, TablesRelating to Mathieu Functions. New York: Columbia Univ. Press, 1951.

[12] J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941, pp. 375-387.

[13] D. B. Hodge, “The calculation of the eigenvalues and eigenfunctions of Mathieu’s equation,” Ohio State University, Electro Science Lab., Rep. 2902-4, June 1971.

Richard Holland (S’62-M’66-M’75), photograph and biography not avail- able at the time of publication.

Vaughn P. Cable (S’69-M’75) was born in Santa Monica, CA, and received the B.S. and M.S. degrees in engineering from California State University, Northridge, in 1970 and 1972, respectively, and the Ph.D. degree in electrical engineering from the Ohio State University, Columbus, OH, in 1975.

He was an Assistant Professor in electrical engi- neering at California State University, Northridge, from 1975 to 1978, and a Summer Faculty Fellow at Hanscomb Air Force Base, Hanscomb, MA, in 1977. Between 1978 and 1988, he held positions

in aerospace with Hughes Aircraft Missile Systems Group and California Microwave Government Electronics Division, and a position in the field of medical electronics with Pacesetter Systems, Inc., Sylmar, CA. He has worked on airborne radar systems, airborne SIGINT systems, and has earned two patents for his work in programmable insulin delivery systems. Since 1988, he has been with Lockheed Aeronautical Systems Co. and more recently, with Lockheed Advanced Development Co., where he has been responsible for notable achievements in computational electromagnetics using massively parallel processing. His major technical interests include frequency and time domain electromagmtic scattering theories, numerical modeling, and the analysis and design of antenna systems.

Dr. Cable is presently a member of the IEEE Antennas and Propagation and Magnetics Societies, and Sigma Xi Research Society, and the Association of Old Crows. He has been a reviewer for the TRANSACTIONS ON ANTENNAS AND PROPAGATION.