On the analysis of waveguides of doubly-connected cross-section by the method of conformal mapping

Journal of Sound and Vibration (1972) 20 (I), 27-38

ON THE ANALYSIS DOUBLY-CONNECTED METHOD

OF WAVEGUIDES CROSS-SECTION

OF CONFORMAL

OF BY THE

MAPPING

P. A. LAURA, E. ROMANELLIAND M. J. MAURIZI Departamento de Ingenieria, Universidad National de1 Sur, Alem 1253,Bahia Blanca, Argentina (Received 1 September 1971) This paper deals with the determination of cutoff frequencies in waveguides of arbitrary, doubly-connected cross-section. The given domain is transformed into an annular region and the transformed partial differential equation is solved by means of a variational technique. Calculations are presented for acoustically soft waveguides of regular polygonal shape with circular inner boundaries. The problem is mathematically equivalent to the determination of TM modes in electromagnetic waveguides. 1. INTRODUCTION

Hollow waveguides of rectangular and circular cross-section are commonly used for acoustic propagation. The mathematical analysis of the resulting boundary and eigenvalue problems is accomplished in those cases by means of known transcendental functions. It should be pointed out that the resulting mathematical simplicity stems from the separability of the governing partial differential equation in rectangular Cartesian and circular coordinates. Waveguides of rectangular and circular cross-section, however, do not always offer the optimum solution for meeting modern design requirements [l]. Numerical techniques used in the solution of waveguide problems involving complicated cross-sections which have appeared in the technical literature are as follows [2] : (i) conformal mapping [3-71; (ii) finite differences [8--g]; (iii) point matching [lo]; (iv) Rayleigh-Ritz [l, 111; (v) Galerkin method [12] ; (vi) method of moments [ 131; (vii) perturbation techniques [l I] ; (viii) impedance methods [ 141; (ix) finite elements [ 151.The basic techniques used in the methods listed above differ considerably, each method having a certain number of advantages and disadvantages. Admittedly, the finite-element method is probably the most general approach at present for a very large variety of scientific and technological problems. In general a large size digital computer is needed. The conformal mapping technique yields very satisfactory convergence. The main disadvantage is that the mapping function must be known in advance and finding that function is not an easy task in general. Since the accuracy of the eigenvalues obtained by this method is very good, they become very useful for checking the accuracy and speed of convergence of other approximate methods. The present paper deals with the determination of cutoff frequencies of waveguides of doubly-connected cross-section by the method of conformal mapping. Limited information is available in the open literature on waveguides of doublyconnected cross-section having arbitrarily shaped boundaries. An excellent paper dealing with a particular family of waveguides has been published recently by Hine [7]. The method applied in this study is quite general, straightforward and accurate. Since the method requires knowledge of the techniques of conformal mapping of doubly-connected cross-sections, a brief discussion on this subject is included also (section 4). 27

28

P. A. LAURA, E. ROMANELLI AND M. J. MAURIZI

2. MATHEMATICAL

STATEMENT OF THE PROBLEM

The problem of wave propagation in a fluid reduces to the solution of the wave equation,

where V2 is the three-dimensional Laplacian operator, 4 is the displacement potential ,) , u, = &#/a~] and c,$= kb/p, where k,, and p are the bulk 1%= War,, %3= (llr‘)(We modulus and mass density of the fluid, respectively. For normal modes, equation (1a) becomes V2Ml,

494 + (4%)2 .Ml,

fl,, 4 = 0.

(lb)

In the case of a uniform tube the axial variable can be separated:

3

+[(o/Q)~ - k2] .Z(z)

= 0,

V2++k2#=0.

(ld)

The boundary conditions for # are 4JW(&u) = 01 =

(acoustically soft boundary),

0

g [L&y) = 0) = 0

(acoustically hard boundary),

where L&y) = 0 is the functional relation which defines the boundary of the configuration under study and k is the eigenvalue under investigation. Equation (Id) and the corresponding boundary conditions define the differential system which will be analyzed in the present study. If the boundary of the cross-section is natural to a coordinate system for which equation (1) separates, a closed form solution is possible and solutions of this type have appeared in the literature. For a complicated cross-section, the natural coordinate system would be exceedingly complicated to find and in general would not be amenable to treatment by the standard method of separation of variables. It is advantageous therefore to transform the given shape onto a simpler one. For convenience, equation (1d) is written now in complex form : 4

a’* mw+k2#=0,

where w = x + iy and @= x - iy. Let w =f(S),

(3)

where .$= reie = x1 + iy,, be the analytic function that maps the given simply- or doublyconnected region onto a simpler one (see Figure 1). From equations (2) and (3) one obtains

and the transformed boundary conditions now become

In this study, only the case of I$ = 0 at the boundary is considered. The case of acoustically hard boundaries requires a modification of the method as has been shown by Hine [7].

29

ANALYSIS OF WAVEGUIDES

I

w-plane

j

&-plane

,

l-plane

w-plane

I

(b)

Figure 1. Conformal mapping of simply- and doubly-connected regions. (a) Mapping correspondence a simply-connected region. (b) Mapping correspondence for a doubly-connected region.

for

3. METHOD OF SOLUTION

Solution of a differential eigenvalue problem requires finding solutions of the governing differential equation such that the applicable boundary condition will be identically satisfied. The boundary configuration is now quite simple. However the differential equation (4) is rather complicated. It becomes apparent that it cannot be solved by the method of separation of variables. Therefore, Gale&in’s method is used in this study for the determination of the approximate solution of the transformed differential equation. The function $(t, $) is expressed in terms of an infinite series : #JG 6 =J

hM,

a,

(6)

where each coordinate function identically satisfies the boundary conditions. Substituting equation (6) into equation (4) results in an expression, ~(6, f),, that does not vanish, in general, since the expansion defined by equation (6) is not an exact solution. Galerkin’s method requires that the “error function”, E((,(), be orthogonal with respect to each coordinate function Y,(e, 8) over the domain in consideration : 1 ME, fl.45

(7)

8. dD = 0.

This condition yields an in&rite dlterminant in the eigenvalues. Usually the lowest frequency can be calculated with enough accuracy by taking a few terms in equation (6). 4. CONFORMAL MAPPING OF DOUBLY-CONNECTED

REGIONS

The existence of the function which maps the unit circle in one plane onto any simplyconnected region is proved in the well-known Riemann theorem. The problem of the effective construction of the mapping function, is in general, a very difficult problem.

P. A. LAURA, E. ROMANELLI AND M. J. MAURIZI

30

It appears at this point that two of the more useful relations in the theory of conformal mapping are the integral equations [ 161

s

ecu) cosb?!l

Ii!

(j(s) =

77

da

+

q.3(s)

r

L

Equation (8) holds for interior regions and equation (9) is applicable to exterior regions (see Figure 1). Several authors have discussed the importance of these equations in the numerical determination of a mapping function [17]. Banin [18] was probably one of the first authors to follow this approach. He made an analysis of fluid flow around cylinders of arbitrary cross section; as a consequence he was only concerned with equation (9). Wilson [19] has developed very accurate techniques for solving equations (8) and (9). Since he was dealing with elastic problems where a slight deviation of the boundary could cause a large error in the calculation of stress and displacement fields, his main concern was the accuracy of the conformal mapping. With respect to the transformation of any doubly connected region onto an annulus, the following theorem is known [ 16, p. 3621: “Any doubly-connected region can be transformed, conformally and with reciprocal single-valuedness, into an annulus with the ratio of the radii of its bounding circumferences finite or infinite”. It was shown by Kantorovich and Muratov [20] that the problem of conformal transformation of an arbitrary, finite doubly-connected region onto a circular annulus can be reduced to the solution of two coupled integral equations : (j2(s2) = -@(s,)

+

s s

d e2(u2)c~‘ r)dg2 nt’ -i

(z&,);

C2

edsd= 43~

1

+G

sel(ul’c~s(nt’r’du,, s s

(loa)

Cl

e2ca2) C;S (4, r>

du2

_

1

_

5r

e,(udcoshr) r

du,, (=C,)

(lob)

Cl

CT

(see Figure 1). If one of the boundaries (say C,) is a circle of radius R, equation (lOa) becomes [19]

e2(s2) = -28(~~)+

2&1e2(u2)da2C2

1 G

4(udc0sh,r)du r

I*

(11)

Ct

Furthermore, if one of the boundaries (say C,) is a circle and the given configuration has one or more axes of symmetry, equation (lOa) simplifies considerably and becomes [21] e2(S2)=-2&,)

+??--t

1 el(ul)c;s(nt’r)du,e

(12)

Cl

Richardson [22] has developed very efficient techniques for the solution of (lOa) and (lob) and very accurate mappings have been obtained. It will be shown in this section that for a certain class of doubly-connected regions the transformation can be obtained in a very easy manner if the mapping function for one of the boundaries is known. Consider now a simply-connected region with several axes of symmetry [Figure 2(a)]. In general, the approximate mapping function which transforms this domain in the w-plane onto a unit circle in the f-plane is given by a expression of the form w =

2 a,+np(l+“P,

ll=O

t = rei8,

(13)

31

ANALYSIS OF WAVEGUIDES

where p is the number of axes of symmetry of the configuration. Assume now a doublyconnected region with the same external boundary and an inner concentric circular boundary [Figure 2(b)] of radius R. I i

1 w-plane

f-plane

4 (a) f

i

w-plane

! I

(b) Figure 2. (a) Simply-connected boundary.

cross-section.

(b) Doubly-connected

cross-section

with an inner circular

When r < 1 in equation (13) it is obvious that the first term is predominant since the exponents are given by (1 + np) and for n > 0 the terms will decay rapidly. In other words, circles in the f-plane will map into approximate circles in the w-plane. The deviations become smaller as I decreases in magnitude and also as a function of the number of axes of symmetry of the primitive domain. Take for instance the case of a square domain. In this case the mapping function is f

du (1 +

u4)1/2

=

a(l*08f - 0*108[5 + O*045c9- 0.026[‘3 + * - *).

(14)

For r = 0.10 the maximum deviations with respect to a perfect circle of radius equal to 0.108a are of the order of (10%). For r = 0.80 the maximum error is approximately equal to 3.5 %. Figure 3 illustrates the approximations involved. It is important to point out that deviations of the order of 5 % from the actual boundary introduced by the transformation function have practically no influence on the calculated lower eigenvalues [23]. If R is the radius of the inner boundary in the physical plane, the corresponding value of the polar coordinate in the [-plane is given, to a good degree of approximation by h = RJa,, where a, is the first coefficient of the mapping polynomial.

32

P. A. LAURA, E. ROMANELLI AND M. J. MAURIZI

Maximum

0.10

error-3.5%

0.50

lo

Figure 3. Approximate conformal mapping of a doubly-connectedcross-section(square with circular perforation). TABLE 1

Coejkients

Polygon

p

Square Pentagon Hexagon Heptagon Octagon

4 5 6 7 8

of transformation functions of simply-connected regions

aI

1.08 1.0526 1a0376 1.0279 1.0219

al+p

al+zp

-0.108 -0*0704 -0M96 -0.0361 -0.0282

0.045 0.0272 0.0183 0*0119 0+)092

a1+3p

-0.026 -0.0153 -0.0108 -0@060 -0W48

a1+4p

O-0174 0.0103 0~0100 0.0040 OW30

al+sp

-0.0127 -0GO79 -0*0053 -O@t)17 -0+X)22

al+6p

0@0997 0.0058 0%)17

Table 1 contains the coefficients of several regular polygonal shapes. If a better approximation is desired, one must use then the general approach by solving the integral equations.

5. APPLICATIONS Since the boundaries of the transformed domain are concentric circumferences and the boundary conditions are independent of 8 one can take for the lowest eigenvalue [24] W, f) = WY 6) r z1 b”M),

(15)

where Mr>lr=, = M)lr-h

= 0.

Since the determinantal equation yields N, eigenvalues, they will correspond to propagation modes which are independent of 6 (as a first approximation) in the &plane. Calculations have been performed using the following coordinate functions :

33

ANALYSIS OF WAVEGUIDES

where r&s(r) = (1 - r2) (r2 - A)*,

I$*&) = (1 - r4) (r4 - A”).

Galerkin’s method requires that the residual function be orthogonal with respect to each coordinate function. For the case of a two-term approximation the procedure is the following: 1 277

~,[~;.~,.~+~;.~,+B*If’(~l*~~.~~ld~.d~+

is A0

1 277 +j

j-

~,[~;.cb~.~+~;.~,+B21f’(5)12dz~~Id~.d~=0,

(i= 1,2).

(16)

h 0 Integrating with respect to 8 results in the following expressions : jA,[~;.~,.rt9;.~1+82.?(r).~*.~1IdT+jA2[~~.~I.T+d;.dl+~2.l)(r).~2.~~ldr=0, h h

(17)

where +n.p)2uf+np.r2”p=

q(r)=n$o(l

5 u,.r2np n=O

(18))

and un =(l +n.pya*

1+np*

Equation (17) can now be written as

It is convenient to now define gtj =

SU#++

(i,j = 1,2);

+ildr,

h hl=/ h

r)(r).g$.gb, .r.dr = j 5 un.r2np.$i.$j.r.dr. h O

Since the expression &. qSj.r is of the form

the coefficient h,, results: hi3 = j” 5 (um.r2np.$ cr,.r’)dr h O = j.

=n$o u,, / $ cck.rk+2nPdr

%-(;.uk.;;;n;;+;).

The determinantal equation then results:

3

‘g,,+k’.h,r

g,,+k’.h,*’

g21+

g22

k2’h2*

r?z +

k2ah22i

0,

Wa)

Wb)

2.8917 3.1032 3.3240 3.5599 3.8165 4.1002 4.4184 4.7805 5.1984 5.6884 6.2729 6.9839 7.8696 g-0055 10.5171 12.6304 15.7972 21.0718 31.6164 63.2425

01

3.1623 3.3287 3.5136 3.7203 3.9528 4.2164 4.5175 4.8650 5.2705 5.7496 6.3246 7.0273 79057 9.0351 10.5409 126491 15.8114 21.0819 3 1.6228 63.2456

6.5552 6.8848 7.2544 7.6701 8.1400 8.6745 9.2869 9.9950 10.8222 11.8011 12.9766 14.4144 16.2125 18.5251 21.6095 25.9285 324079 43.2079 64.8092 129.6078

$,.

k 02 34641 3.6051 3.7660 3.9500 4.1611 4.4045 4.6868 5.0165 54052 5.8687 6.4289 7.1178 7.9833 9.1006 10.5951 12.6927 15.8451 21.1062 31.6385 63.2431

01

pa 3.8108 3.8305 3.8900 39905 4.1345 4.3255 4.5686 4.8711 5.2429 5.6986 6.2588 6.9541 7.8311 8.9638 10.4764 12.5939 15.7669 21.0488 316013 63.2362 10.3296 lo-3582 104452 10.5938 10.8100 11.1026 1 l-4838 1 l-9706 12.5864 13.3633 14.3465 156016 17.2269 19.3770 22.3091 26.4862 32.8341 43.5128 65.0030 12.9708

4.472 1 4.4889 4.5398 4.6263 4.7507 4.9171 5.1308 5.3996 5.7341 6.1493 6.6667 7.3174 8.1490 9.2361 10.7038 12.7775 159087 21.1510 31.6665 63.2663

9lb

kol 7.2457 7.2458 7.2471 7.2530 7.2689 7.3023 7.3632 7.4639 7.6197 7.8500 8.1801 8.6446 9.2938 10.2046 11 a5038 13.4181 16.3998 21.5033 31.8907 63.3731

02

t2b

6.8576 7.7855

10443 25

15.694 84

31.4110

3.313 9387 3.815 956

5.183 072

7.828 440

15.698 088

Exact results7 k 02 k 01

t From Handbook of Mathematical Functions. (M. Abramowitz & I. Stegun, eds.) New York: Dover Publications Inc.

0.05 0.10 o-15 o-20 o-25 o-30 0.35 040 o-45 o-50 0.55 060 O-65 0.70 o-75 O-80 0.85 0.90 0.95

O-00

TABLE 2

of results in the case of the annular waveguide of outer radius = 1 and inner radius = 2

Aa + 42a

Comparison

040 0.05 0.10 0.15 0.20 0.25 0.30 0.35 040 0.45 0.50 0.55 060 0.65 o-70 0.75 0.80 0.85 090 o-95

h

26673 2.8602 3.0610 3.2749 3.5068 3.7622 4.0474 4.3703 4.7409 5.1725 5.6833 6.2993 7.0587 8.0214 9.2855 I 1.0264 13.5923 17.7895 26.0262 50.3118

5.9421 6.2361 6.5647 6.9332 7.3485 7.8192 8.3568 8.9760 9.6963 10.5445 1 l-5576 12.7885 14.3164 16.2641 18.8345 22.3896 276464 36.2635 53.1832 103.0142

Square kol .a h2.a 6.1669 6.4742 6.8182 7.2044 76403 8.1351 8.7011 9.3539 10-l 149 11.0128 12.0878 13.3977 15.0287 17.1155 19.8809 23.7232 29.4329 38.8405 57.3975 112.2038

ko2.a

Pentagon

.a

2.7433 2.9430 3.1513 3.3735 3.6148 3.8812 4.1794 4.5179 4.9075 5.3628 59036 6.5584 7.3693 84022 9.7653 11.6526 144482 19.0424 28G901 54.8095

kol 2.7852 2.9884 3.2006 3.4270 3.6731 3.9450 4.2497 4.5959 4.9949 54619 6.0177 6.6919 7.5289 8.5977 10.0125 1l-9776 14.8979 19.7133 29.2235 57.3553

kol.a 6.2857 66000 6.9521 7.3479 7.7947 8.3023 8.8832 9.5539 10.3364 1 l-2605 12.3683 13.7199 154056 17.5664 20.4361 24.4336 30.3910 40.2378 59.7230 I 174063

k0z.a

Hexagon

2.8124 3.0179 3.2324 34614 3.7105 3.9857 4.2942 46451 5.0498 5.5238 6.0885 6.7743 7.6270 8.7177 10.1646 12.1792 15.1823 20.1513 3ow43 59.2737

ko,.a 6.3603 6.6792 7.0364 7.4381 7.8918 8.4076 8.9981 9.6802 10.4764 11.4175 12.5465 13.9253 156468 17.8564 20.7960 24.8989 31.0285 41.1901 61.3709 121.3664

ko2.a

Heptagon

2.8293 3.0361 3.2520 3.4826 3.7333 4.0105 4.3213 4.6748 5.0826 5.5605 6.1300 6.8222 7.6833 8.7857 10.2495 12.2898 15.3352 20.3811 304001 60.1931

kol.a

64043 6.7256 7.0858 7.4908 7.9484 84686 9.0644 9.7528 10.5566 1 l-5069 126471 14mO3 15.7805 18.0154 20.9906 25.1467 31.3620 41.6779 62.1918 123.2489

k0z.a

Octagon

Eigenvalues for doubly-connected regions as a function of the ratio of radii in the S-plane

TABLE 3

36

P. A. LAURA, E. ROMANRLLI AND M. J. MAURI21

9 0

I 0.20

I

I 0.40

60

I

0.20

I

I

I

040

I

1 080

1 100

Figure 5. TM cutoff frequencies ofpentagonal waveguide with circular inner boundary as a function of the ratio (radius of inner boundary/ apothem of pentagon). Cutoff frequencies are quasi-independent of 0 in the f-plane.

Figure 4. TM cutoff frequencies of square waveguide with circular inner boundary as a function of the ratio (radius of inner boundary/ apothem of square). Cutoff frequencies are quasi-independent of 0 in the f-plane.

I

1 060

R/U

R/O

0

I

I

I

060

I

I

0 80

,

0

I.00

,

/

020

,

040

R/O

060

0.80

I.00

R/0

Figure 7. TM cutoff frequencies of heptagonal waveguide with circular inner boundary as a

Figure 6. TM cutoff frequencies of hexagonal waveguide with circular inner boundary as a function of the ratio (radius of inner boundary/ apothem of hexagon). Cutoff frequencies are quasi-independent of 9 in the &plane.

0

I

I

0 20

I

I

040

function of the ratio (radius of inner boundary/ apothem of heptagon). Cutoff frequencies are quasi-independent of 0 in the f-plane.

I

I

060

I

I

080

I

1.00

Figure 8. TM cutoff frequencies of octagonal waveguide with circular inner boundary as a function of the ratio (radius of inner boundary/apothem of octagon). Cutoff frequencies are quasi-independent of 6 in the &plane.

ANALYSIS OF WAVEGUIDES

37

where g,, = g,, and hi2 = h21; k2 is the eigenvalue of the problem [see equation (l)]. One of the advantages of this procedure is the fact that it is valid for any mapping function given in polynomial form. Table 2 illustrates the accuracy of the different coordinate functions which have been selected by comparing the calculated eigenvalues with the exact values in the case of the waveguide of annular cross section. Since Galerkin’s method provides upper bounds all the results are higher than the exact values. The calculated eigenvalues converge to the exact results as Ni increases. It is obvious that the expression 4a(r)=A,.(l

-r).(r-A)+A2.(1

-r2).(r-A)

yields results which are, in general, considerably more accurate than those obtained by use of &,(r)=Al.(l

-r2).(r2-X2)+A2.(l

-r4).(r4-h4).

As the parameter X increases, both expressions give similar results. Table 3 and Figures 4-8 show the calculated eigenvalues using expression #J~for regular polygons with concentric circular perforations as a function of the ratio: perforation radius to apothem of the polygon. As expected, the results approach the values of the annular waveguide as the number of sides of the polygon increases. 6. CONCLUSIONS A limited amount of information is available on waveguides of doubly-connected crosssection in the technical literature. The present paper uses conformal mapping and a variational technique to determine cutoff frequencies of a class of waveguides of doubly-connected cross-section. The method is straightforward and the stability of the numerical results is excellent. ACKNOWLEDGMENTS The present investigation has been sponsored by the Consejo National de Investigaciones Cientifmas y Tecnicas, Buenos Aires, Argentina. The co-operation of Professor Ascensio Lara is gratefully acknowledged. The authors are indebted to Professors Jorge A. Reyes and Ra61 E. Rossi who prepared the computer program and to Ing. F. Domini who provided assistance beyond the call of duty in the preliminary calculations and preparation of the final graphs.

REFERENCES 1. R. M. BULLEYand

on Microwave

J. B. DAVIES 1969 Institution of Electrical and Electronic Engineers Transactions Theory and Techniques MTT-17, 440-446. Computation of approximate poly-

nomial solutions to TE modes in an arbitrarily shaped waveguides. 2. P. A. LAIJRA 1969 Proceedings of the Institution of Electrical and Electronic Engineers 116, 1168. Correspondence on “Application of finite elements to the solution of Helmholtz’s equation”, by P. L. Arlett, A. K. Bahrani and 0. C. Zienkiewicz (1968 Proceedings of the Institution of Electrical and Electronic Engineers 115, 1762-l 766). and P. A. LAURA 1964 Institution of Electrical and Electronic Engineers Transactions on Microwave Theory and Techniques (Correspondence) MTT-12,248-249. Approximate method of

3. M. CHI

determining the cutoff frequencies of waveguides of arbitrary cross section. 4. H. H. MEINKE, K. P. LANGE and J. F. RUGER 1963 Proceedings of the Institution of Electrical and Electronic Engineers 51, 14361443. TE and TM waves in waveguides of very general cross section.

38

P. A. LAURA, E. ROMANELLIAND M. J. MAURIZI

5. F. J. TISCHERand H. Y. YEE 1964 University of Alabama Research Institute Report No. 12. Waveguides of arbitrary cross section by conformal mapping. 6. R. WOHLLEBEN,S. STIEF and K. KAMMANN1968 (June) Paper presented at a symposium on electromagnetic waves, Stresa, Italy. Characteristic impedance of complicated cross sections by the method of approximate conformal mapping of doubly connected regions. 7. M. J. HINE 1971 Journal of Sound and Vibration 15,295-305. Eigenvalues for a uniform fluid waveguidewith an eccentric-annulus cross section. 8. J. B. DAVIESand C. A. MUILWYK 1966 Proceedings of the Institution of Electrical and Electronic Engineers 113, 277-284. Numerical solution of uniform hollow waveguides with boundaries of arbitrary shape. 9. C. W. STEELE1968 Journal of Computational Physics 3, 148-153. Numerical computation of electric and magnetic fields in a uniform waveguide of arbitrary cross section. 10. H. Y. YEE 1965 Proceedings of the Znstitution of Electrical and Electronic Engineers 54, 64. On determination of cutoff frequencies of waveguides with arbitrary cross section. 11. R. F. HARRINGTON1961 Time-harmonic Electromagnetic Fields. New York: McGraw Hill. 12. P. A. LAURA 1966 Proceedings of the Institution of Electrical and Electronic Engineers 54, 14951497. A simple method for the calculation of cutoff frequencies of waveguides with arbitrary cross section. 13. T. W. BRISTOL 1967 (November) Syracuse University Electrical Engineering Department Technical Report TR-67-16, Syracuse, New York, U.S.A. Waveguides of arbitrary cross section by moment methods. 14. J. VINE 1966 In Field Analysis: Experimental and Compatutional Methods. (D. Vitkovitch, ed.) New York: Van Nostrand. Chapter 7 : Impedance networks. 15. P. L. ARLETT, A. K. BAHRANIand 0. C. ZIENKIEWICZ 1968 Proceedings of the Institution of Electrical and Electronic Engineers 115, 1762-1766. Application of finite elements to the solution of Helmholtz’s equation. 16. L. V. KANTOROVICH and V. I. KRYLOV1958 Approximate Methods of Higher Analysis. New York: Interscience. 17. G. BIRKHOFF,D. M. YOUNG and E. H. ZARANTONELLO1953 In Proceedings of Symposia in Applied Mathematics, Vol. IV. Numerical methods in conformal mapping. New York: McGraw Hill. 18. A. M. BANIN 1943 Prikladnaya Matematika Mekhanica 7, 131-140. Approximate conformal transformation applied to a plane parallel flow past an arbitrary shape. (in Russian). 19. H. B. WILSON1964 (July) Report S-46, Rohm and Haas Co., Huntsville, Alabama, U.S.A. Further developments of conformal mapping techniques. 20. L. V. KANTOROV,CHand V. MURATOV1937 from Works of the Scientific Research Institute of Mathematics and Mechanics, Leningrad State University, Leningrade, U.S.S.R. Conformal mapping of simply and multiply connected regions. 21. P. A. LAURA 1965 (January) Technical Report 8 to NASA. Conformal mapping of a class of doubly connected regions. 22. M. K. RICHARDSON1965 Ph.D. Dissertation, University of Alabama A numerical method for the conformal mapping of finite doubly connected regions with application to the torsion problem for hollow bars. 23. P. A. LAURA 1966 Proceedings of the Institution of Electrical and Electronic Engineers 54, 10781080. Conformal mapping and the determination of cutoff frequencies of waveguides with arbitrary cross section. 24. P. A. LAURAand M. J. MAURIZI 1971 Journal of Sound and Vibration l&445447. Comments on “Eigenvalues for a uniform fluid waveguide with an eccentric-annulus cross-section”.