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9 Spheroidal Cavity fitting the fnSo, 5 evaluated onto an equation of its derived form, the first four using the least squares method.. The method used to obtain these coefficients 245

9

Spheroidal Cavity

fitting the fnSo, 5 evaluated onto an equation of its derived form, the first four

using the least squares method The method used to obtain these coefficients

245

Spheroidal Wave Functions in Electromagnetic Theory

Le-Wei Li, Xiao-Kang Kang, Mook-Seng Leong Copyright  2002 John Wiley & Sons, Inc ISBNs: 0-471-03170-4 (Hardback); 0-471-22157-0 (Electronic)

246

The prolate spheroidal body under consideration is shown in Fig 9.1 In view

of the fact that Mathematics handles only vector differential operations in the prolate spheroidal coordinates in accordance with the notations used in the book by Moon and Spencer [9, pp 28-291, a temporary change of coordinates

is necessarv The new notation used is shown in Fig 2.1

a

Fig 9.1 Geometry of the spheroidal cavity

As noted by Moon and Spencer (91, the vector Helmholtz equation is more complicated than the scalar counterpart, and its solution using the variable- separation principle may sometimes cause new problems This is especially true in rotational systems like that of the spherical coordinates or spheroidal coordinates In spheroidal coordinates, the solving of vector boundary value problems is further complicated by the fact that the vector wave equation

is not exactly separable in spheroidal coordinates Although another more general analysis has been performed using the vector wave functions, formed

by operating on the scalar spheroidal wave functions with vector operators, the validity of the results obtained is doubtful In view of these limitations,

THEORY AND FORMULATION 247

several assumptions are made in the formulation of the current boundary problem in order to provide a truer, more accurate picture

9.2.2 Derivation

With axial symmetry assumed, it is possible to separate the field components into Et, Eq, and I$ for the TM mode and Ht, &, and E4 for the TE mode First, the TM mode is considered With axial symmetry, I74 can be as- sumed simply as

By applying the Maxwell equations

dB

dD

and using the formulation of V x X in the spheroidal coordinates where

Q77 = d2K2 - v2) 4(1 - 72) ’ iI< = d2(C2 - v2)

(9.4a) (9.4b) (9.4c) the following equations can be obtained:

a2;p (C2 _ 1) + ,py

-

(9.5a)

(9.5b)

248 EM NGENFREQUENCIES IN A SPHEROIDAL CAVITY

In the case when the semimajor axis of the spheroidal surface is close to

in the summation with anzn will diminish due to the decreasing value of d2 Thus Eqs (9.5a) and (9.5b) will be reduced to

- [ Qrmn

- c2y2 + $-+ 1 G(c,C) = 0

(9.6a)

(9.6b)

Comparison of Eqs (9,.5a) and (9.5b) with Eqs (2.8a) and (2.8b) indicates that the solutions to the differential equations are in fact given by

J”(c, <) = BnRil,) (C, C>,

G(c 7) - C ‘) - n s(l)(c 7) In 7

(9.7a) (9.7b) (i.e., the radial and angular functions with m = 1) In the equations above,

tions Hence, the magnetic field component for the TM modes can in fact be expressed as

H4 = BnCnRln (C, 5)Sln (C7 V) Y (9 8)

and the electric field is therefore expressed as

jwq/iq

J&&G 5)

X BnCndw

[

~Sln(G rl) d

37 + +&7 7) 1 7 (9.9a)

To obtain the resonance condition, & must be zero at the surface 5 = 50

of the perfectly conducting spheroidal cavity From Eq (9.9b), this requires that

’ [Rln(CJf)j/~] 1

at t=t0 = 0 (9.10) Thus, by finding the roots of the equation above, the eigenfrequency of the

TM mode can be found

249

By principle of duality, the fields components for the TE mode can be

Hence, the resonance condition for the TE modes can be obtained by setting E+ = 0 at 5 = 50 From Eq (9.8), the boundary condition requires that

Using the package created in previous chapters, the zeros of the radial func- tion, as required by the resonance condition in Eq (9.11) can be found in a straightforward way This is because coding the radial function into a package

(1)

and sine Hence, the command FindRoot in Mathematics can be employed

Newton-Raphson method in the software program

In our program, the iterations will stop when the next iterative value com-

initial guess is required The spherical Bessel function zeros of various orders are assigned as the first guess This will provide faster convergence since in the case considered, the spheroidal coordinates can actually be approximated roughly by the spherical coordinates And from Stratton [7], the resonance condition is given by j, (/w) = 0 in the spherical coordinates Under the circumstance considered, 4 (in spheroidal coordinates) + kr (in spherical coordinated), thus the required values of g must be in the region around zeros of the spherical Bessel functions

It is observed from practical calculations that in the region when @ is large, FindRoot using Newton’s method is capable of evaluating the zeros accurately

at a very high speed However, at the same time, it is also observed that the rate will decrease drastically in the region where @ is small This can be explained by the proximity of the initial guess A series of zeros, spanning the range from t = 100 to < = 1000, were collected at irregular intervals

From the work done by Kokkorakis and Roumeliotis [65], it can be shown, after some manipulations, that the series of values of g that satisfies Rln(c, 5) =

0, are, in fact, governed by an equation of the form

are unknown coefficients to be determined

250 EM NGElVFREQUENClES IN A SPHEROlDAL CAWTY

From Eq (9.12), the following equation relating the eigenfrequency of the spheroidal cavity can be obtained:

Thus, by determining the coefficients, gl ,g2,g3, , a closed-form equation for the eigenfrequency of a spheroidal body is obtained For a given spheroidal dimension expressed in terms of d and &, the eigenfrequency of a spheroidal body can be computed quickly and accurately using Eq (9.13)

Hitherto, the coefficients have been solved only by Kokkorakis [65] How- ever, only the first two expansion coefficients (91 and 92) of the series in (9.12) are given in his work Moreover, except for the first coefficient g1 which can

be obtained directly, the second coefficients can only be obtained by using a relatively complicated equation Furthermore, the equation is obtained after

a very lengthy derivation that spanned over than 50 equations

For the purpose of numerical comparison, a more direct and simpler ap- proach to solving the coefficients is employed in the present work First, the

of c mentioned earlier are collected and placed in a list Then, by means of the least squares method, these values of < and { are fitted onto a function

can be determined readily In Mathematics, this is accomplished simply by two short statement commands

9.3.2 Results and Comparison

The values for the coefficients go, 91, g2, and g3 for the TE modes are calcu- lated and tabulated in Tables 9.1 and 9.2 Kokkorakis solved for the same set

of coefficients in a lengthy and complicated manner A complete but smaller table has been published in his work [65]

By comparing the present tables and Kokkorakis’s tabulated results, it

is observed, first, that the first two coefficients produced with this method agree with Kokkorakis’s evaluations to a minimum of five significant digits This shows the capability of the method to produce equally accurate results

by means of a simpler way Second, it is almost impossible to produce the coefficients g3, g4, and g5 using Kokkorkis’s method The amount of analytic computation required using the method makes it impractical On the other hand, the method presented here can be used to produce these coefficients effortlessly and almost instantly, without sacrificing any accuracy Finally, in Kokkorakis’s paper [65], it is claimed that the coefficients are valid in the case when c >> 1 However, there is no definite definition of how small < must

be for the coefficients to be valid In this chapter, the valid range of < has been determined, numerically, to be l/S < 0.01 for rz = 1,2 and l/c < 0.005

NUMERICAL RESULTS FOR TE MODES 251

Table 9 I Coefficients go, 91, 92, and g3 for TE,,o Modes (s = 1, 2, and 3)

Table 9.2 Coefficients go, 91, 92, and g3 for TE,,o Modes (s = 4, 5, and 6)

252 EM EIGENFREQUENCIES IN A SPHEROIDAL CAVITY

for n = 3,4 For other higher-order n, the valid range of < will have to be reduced further

9.4 NUMERICAL RESULTS FOR TM MODES

9.4.1 Numerical Calculation

Closed-form solutions of the eigenfrequencies for TM modes are obtained in

a similar fashion The variation of cc with 5 bears an identical form to the

Eq (9.13); that is, the eigenfrequency for the TM modes can be expressed

in a form identical to those shown in (9.13)

changed to satisfy the equation

except that now, go has to be

= ‘3

where js(x) represents the spherical Bessel functions

By comparison with the TE modes, two differences need to be considered

in the programming aspect First, the resonance condition has to be altered Previously, for the TE modes, the condition stated in Eq (9.11) is satisfied In the TM modes, the boundary condition requires that Eq (9.10) be satisfied

At the surface < = 50, the boundary condition becomes

(9.15)

With the new boundary condition, the zeros of the left-hand term of (9.10) have to be found instead of that of the radial function In the program, the zeros of the radial derivative expression in Eq (9.10) are evaluated using the same Newton’s method However, the function is now different, and so is the initial guess For the TE modes, the various orders of zeros of the functions

in Eq (9.14) are used instead

9.4.2 Results and Comparison

Employing the same technique to determine the expansion coefficients gi , 92,

zero is collected and fitted into an equation of the form in Eq (9.13) In this way, the various expansion coefficients are determined Tabulations of various values obtained using this method for the TM modes are made and shown in Tables 9.3 and 9.4

The same observation and the same conclusion as for the TE modes can

be drawn upon comparing of the two tables for the TM modes with those for the TE modes Hence, they are not repeated here

253

Table 9.3 Coefficients go, 91, g2, and g3 for TM,,0 Modes (S = 1, 2, and 3)

n m S =l S =2 s=3

93/m 1 0 0.000047 0.000050 0.000064

Table 9.4 Coefficients go, 91, g2, and g3 for TM,,0 Modes (s = 4, 5, and 6)

n m S =4 s=5 S =6

254 EM EIGENFREQUENCIES IN A SPHEROIDAL CAVITY

In this chapter, one of the many possible applications of the spheroidal wave function package is presented in detail, (i.e., solving of an interior boundary value problem) The convenience of coding in Mathematics package is man- ifested by the ability of this program to find the zeros of complex functions such as radial functions simply with one statement

This problem, by itself, is a highly interesting topic Due to the preoccupa- tion with the more important issue of completing the Mathematics package, the axial symmetry is assumed so as to reduce the complexity of the prob- lems The more general and practical problem in which the assumption of axial symmetry is removed is a topic worth looking into for future investiga- tions

As indicated in

can be achieved in

previous chapters, the study of oblate spheroidal cavities

a similar way or by sy pmbolic transfer between the oblate and prolate coordinates However, it should be noted that the assumed axial symmetry is kept in the z-direction and the assumed field components are not changed in the symbolic programming