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Vibration of a membrane having a circular outer boundary and an eccentric circular inner boundary
Journal of Sound and Vibration (1977) 50(4), 545-55 I
VIBRATION OF A MEMBRANE HAVING A CIRCULAR OUTER BOUNDARY AND AN ECCENTRIC CIRCULAR INNER BOUNDARY K. NAGAYA Faculty of Engineering, Yamagata University, Jyonan, Yonezawa, Japan (Received 1 July 1976, and in revisedform 21 October 1976)
In this paper, a method of solving vibration problems for a membrane having a circular outer boundary and an eccentric circular inner boundary is presented. The frequency equation for the membrane is given, and the dependence of the natural frequency on the eccentricity is obtained. It is shown that the result obtained from this analysis in the limit of the eccentricity being zero coincides with that of a membrane with a concentric circular inner boundary. 1. INTRODUCTION This paper deals with the vibration of a membrane having a circular outer boundary and an eccentric circular inner boundary. Problems of membrane vibration can be discussed by using the results obtained in certain acoustical waveguide problems, because the equation of motion of the membrane corresponds to that governing the dependence of the motion on the cross-section space variables in acoustical waveguides. Methods for waveguide problems involving complicated crosssections include the following: conformal mapping [l-4]; finite differences [S, 61; point matching [7, 81; Rayleigh-Ritz [9] ; Galerkin method [lo] ; finite element [l 11. As for the problems with which the present analysis is concerned, the finite element, finite difference, and point matching methods [5, 8, 1l] are probably the most appropriate general approaches; however, in general, a large size digital computer is needed. The conformal mapping method can also be used; Laura, Romanelli and Maurizi [4] have discussed the problem of waveguides of doubly connected cross-section of general shape by means of conformal mapping and Hine [3] has presented results for an eccentric-annulus cross-section, as obtained by the conformal mapping and Galerkin methods. In this paper, a method of solving vibration problem for a membrane having a circular outer boundary and an eccentric circular inner boundary is presented. In the analysis, the exact solution which satisfies the inner boundary condition is used, and the outer boundary condition is satisfied by means of the Fourier expansion method. Thus the method developed in this paper also gives only an approximate result; however, the technique can be more rigorous and straightforward than those given by the previous authors because of using the exact solution of the equation of motion which satisfies one boundary condition. The solution and technique are also applicable to problems of acoustical waveguides. The result given by setting the eccentricity equal to zero in the present analysis coincides with that for a concentric-circular-annular membrane which is obtained by exact methods [12,4]. 2. ANALYSIS A membrane with a circular outer boundary and an eccentric circular inner boundary, r’, of an eccentricity d is shown in Figure 1, in which the origin 0 of Cartesian co-ordinates x and 545
K. NAGAYA
Figure I. A circular membrane having an eccentric circular inner boundary.
y is chosen in the center of the membrane, and the z axis is normal to the surface of the membrane. The polar co-ordinates r. 0 and r’, 8’are taken as in the figure, and the radii of the outer edge r and the inner edge r’ are denoted by R and CI,respectively. If one denotes the displacement of the membrane in the z direction byw:, the equation of motion for the membrane, in the co-ordinates r’, 0’ is [ 131
T{a%4qarf2 + (Ijr')aw/ar' + (i/P)a%4~/aB'2} - pazW/at2 = 0,
(1)
where Tis the tension of the membrane, p the mass density and t the time. The displacement w for the steady-state vibration can be shown to be of the form w(r’, Q’, t) = W(r’, 0’) sin it.
Substituting
equation
(2) into equation W=
i j
~1
(2)
(l), one obtains
z s,{Aj,Jn(ar’)
+ Bi,Yn(ctr’)}@jn
(3)
n.sO
where Aj,, Bjn are the constants of integration, w is the circular frequency, J,(ar’) is the Bessel function of the first kind of order n and Y,(ar’) that of the second kind, and a2 = po2/T.
E,=
Iforn=O,and=2fornz
@jjn= cos n6’ for ,j = 1 and:
1,
sin n0’ forj = 2.
(4)
Since the geometrical symmetry exists in the membrane about the x axis, the motion of the membrane can be separated into two types of symmetric and antisymmetric vibrations about the axis of symmetry. 2.1. SYMMETRIC VIBRATIONS If one considers the symmetric vibration about the x axis (the axis of symmetry), displacement W is obtained by puttingj equal to 1 in equation (3):
w=$
n-n
E,{A~~J~(c(~‘) + B,,Y,(ar’)}cosnO’.
the
(9
The displacement should vanish around the boundary circle when the inner edge is fixed. Then the boundary condition at the inner edge is denoted as ( W),? =L1= 0.
(6)
VIBRATION
OF A MEMBRANE
547
Substituting equation (5) into equation (6), and eliminating B1,, one has W=~~Oe.A,.{J.(~r’) + y,Y,(crr’)} cosnel,
(7)
where yn = -J.(cxa)/Y, (au). One has following relations between the co-ordinates r, I3and r’, 8’: r sin e
rcose+d=r’cos@,
= r’sin 0’.
(8)
When the outer edge r = R also is fixed, the boundary condition at the outer edge is
( WLR = 0.
(9)
The displacement at an arbitrary point in the membrane can be expressed in terms of the co-ordinates r’, 8’; however, the boundary condition (equation (9)) around the outer edge cannot be satisfied directly by using equations (7) and (8). Therefore, in this analysis, the displacement around the outer edge is expanded into a Fourier series to satisfy this boundary condition, in the form
W%R=
2i
%Jl,Q.,,,cosm4 n-Om=0
(10)
where
Qn, = U/4
i [(J,(orr’) + y.y.(ar')}cod'],,,cos &de --x
(11)
and E,,,= 1 for m = 0 and = 2 for m >=1. By putting (IV),,, to zero, the boundary condition of the outer edge can be satisfied, and if the terms n, m are taken up to N + 1, one obtains the following simultaneous equations in matrix form :
,
(12)
where
IAl=
(13)
Since equation (13) is a determinant of the coefficients of equations (12), the frequency equation is IAl =O.
(14)
The ratios All/AI,, A12/Alo, . . ., AIN/AIO are obtained from equation (12). Thus each mode of vibration can be determined by substituting these ratios into equation (5). The values of the coefficients given by equation (11) can be derived by using the following relations : r’= (r2 + 2rdcos 8 + d2)l12,
8’= sin-‘{(r/r’) sin ej.
m
54x 2.2.
K. NAGAYA ANTISYMMETRIC
VIBRATIONS
The displacement of the membrane for the case of antisymmetric axis is obtained by takingj= 2 in equation (3):
vibration
about
W = i &“{AZnJ,(ctr’) + B,,Y, (w’)} sin no’.
(16)
n ~=1
The boundary condition around the inner edge is the same as equation equation (16) into equation (6) and eliminating B2,,, one obtains
(6). Substituting
W= .t, EnA2,,{Jn(ar’) + y,Yn(ar’)}sinn&. The frequency tion; it is
equation
the x
(17)
can be found in the same way as for the case of the symmetric
vibra-
(18)
where QT,, Q&, . . . . Q& have the same forms as in equation (11) but in which the term cos 0’ is replaced by sin W, and also cos me by sin me. As a limiting special case of the problem, consider the case where the eccentricity is zero. In such a problem, the relations r’ = r,
are obtained,
and equation
equation
(19)
(11) yields
Qnm= (J&R) + ~,Y&W Substituting
8’ = e,
(20) into equation
forn = m, and Q,, (14), one has the following
= Oforn # m. frequency
(20)
equation:
Qoo.Qll. Qx . . . QNN = 0.
(21)
It is noted, from equation (21), that the modes of vibration for the case of concentric boundaries can be characterized in terms of the number of nodal diameters, n. Thus the frequency equation for the case of concentric boundaries yields Qnn= 0. Equation (22), of course, coincides with the frequency equation of a circular membrane a concentric circular inner boundary, as obtained by exact methods. 3. NUMERICAL
(22) having
EXAMPLES
Numerical calculations have been carried out for the natural frequencies of a circular membrane with an eccentric circular inner boundary. The order N in equations (13) and (18) is taken so that nine terms are included in the calculation, because the maximum discrepancy between the result for N = 8 and that for N = 12 can be shown to be confined to within 0.5%. The values of the integrals in equation (11) were obtained by using the Gaussian formula for numerical integration. Figures 2(a)-(c) show the relations between the non-dimensional natural frequency, aR, and the non-dimensional eccentricity, d/R, for various values of a/R. In the figures, S,, S,, . . ., show the results for symmetric vibration and A,, AZ, . . ., those for antisymmetric. It is seen that the results for the symmetric vibration, except those for an
549
VIBRATION OF A MEMBRANE
0
0.1
0.2
0’3
0.4
0.5
0.6
0.7 d/R
Figure 2. The natural frequency uersusthe eccentricity for (a) a/R = O-1,(b) a/R = 0.3, and (c) a/R = 0.5. S, Symmetric; A, antisymmetric.
axisymmetric vibration with respect to the q axis, coincide with those for the antisymmetric in the eoncentric case (d/R = 0). However, both results separate rapidly with increasing values TABLE
1
Non-dimensional natural frequencies versus eccentricities for a/R = O-5 (the values in the bracket denote the results obtained by the point matchhg method [8]) Symmetric vibration dlR 0 0.2 O-5
rk 01 6246 (6.26) 4.811 (4.76)
11
11
6.393 (6w 6.172 (614)
6.814 (6.84) 7.391 (7.42) 6.836 (6.88)
Antisymmetric koiR 6-393
(64) 5511 (5.46) 4.569 (4.54)
550
K.NAGAYA
of the eccentricity. The numerical result of this analysis for d/R = 0, of course. coincides with that for the concentric case which is obtained by the exact method [4, 131. The non-dimensional natural frequencies of the first through the fourth modes in this analysis correspond to the non-dimensional cut-off frequencies &,a, k, ,a, k,,a and k,,a as obtained for acoustical waveguides in references [4] and [8]. Numerical results for this problem of the waveguide have been shown in Figures 9 and 10 in reference [g], as obtained by the point matching method. Comparison between the results obtained and those of the point matching method shows good agreement, as shown in Table 1, although there exist some reading errors from the figure in reference [8].
4. CONCLUSIONS A method of solving vibration problems for a circular membrane having an eccentric circular boundary has been presented. The dependence of the natural frequencies on the eccentricities has been shown, and the results for the case of an eccentric inner boundary compared with those for a concentric one. The results show that the dynamical behavior of a membrane with an eccentric inner boundary is quite different from that of a membrane with a concentric inner boundary.
ACKNOWLEDGMENT The author wishes to heartily thank Professor for his kind guidance in the present work.
Yoshitaro
Hirano,
Yamagata
University,
REFERENCES 1964 Institution of Electrical and Electronic Engineers Transactions on Microwaue Theory and Techniques MTT12, 248-249. Approximate method of determining the
1. M. CHI and P. A. A. LAURA
cutoff frequencies of waveguides of arbitrary cross section. 2. H. H. MEINKE,K. P. LANGEand J. F. RURGER 1963 Proceedings of the Institution of Electrical and Electronic Engineers 51, 1436-1443. TE and TM waves in waveguides of very general crosssection. 3. M. J. HINE 1971 Journal of Sound and Vibration 15, 295-305. Eigenvalues for a uniform fluid waveguide with an eccentric-annulus cross-section. 4. P. A. A. LAURA,E. ROMANELLI and M. J. MAURIZI 1972 Journalof Soundand Vibration 20,27-38. On the analysis of waveguides of doubly-connected cross-section by the method of conformal mapping. 5. J. B. DAVIESand C. A. MUILWVK 1966 Proceedings of the Institution of Electrical andElectronic Engineers 113,277-284. Numerical solution of uniform hollow waveguides with boundaries of arbitrary shape. 6. C. W. STEELE1968 Journuf of Computational Physics 3, 148-153. Numerical computation of electric and magnetic field in a uniform waveguide of arbitrary cross section. 7. H. Y. YEE 1965 Proceedings of the Institution of Electrical and Electronic Engineers 54, 64. On determination of cut-off frequencies of waveguides with arbitrary cross section. 8. H. Y. YEE and N. F. AUDEH 1966 Institution of Electrical and Electronic Engineers Transactions on Microwave Theory and Techniques MTT-14,487-493. Cut-off frequencies of eccentric waveguides. 9. R. M. BULLEYand J. B. DAVIES1969 Institution ofElectricalandElectronic Engineers Transactions on Microwave Theory and Techniques MTT-17, -6. Computation of approximate polynominal solutions to TE modes in an arbitrarily shaped waveguide. 10. P. A. A. LAURA 1966 Proceedings of the Institution of Electrical and Electronic Engineers 54, 1495-1497. A simple method for the calculation of cut-off frequencies of waveguides with arbitrary cross section.
VIBRATION OF A MEMBRANE
551
Il. P. L. ARLETT, A. K. BAHRANI and C. C. ZIENKWJICZ 1968 Proceedings of the Znstirurionof rhe Electrical and Electronic Engineers 11, 1762-1766. Application of finite elements to the solution of Helmholtz’s equation. 12. M. ABRAMOWITZ and I. STEOUN 1965 Handbook of iUathematica1 Functions, 390432. New York: Dover Publications. 13. LORD RAYLEIGH 1877 Theory of Sound, 306-351. New York: Dover Publications, second edition, 1945 re-issue.
VIBRATION OF A MEMBRANE HAVING A CIRCULAR OUTER BOUNDARY AND AN ECCENTRIC CIRCULAR INNER BOUNDARY K. NAGAYA Faculty of Engineering, Yamagata University, Jyonan, Yonezawa, Japan (Received 1 July 1976, and in revisedform 21 October 1976)
In this paper, a method of solving vibration problems for a membrane having a circular outer boundary and an eccentric circular inner boundary is presented. The frequency equation for the membrane is given, and the dependence of the natural frequency on the eccentricity is obtained. It is shown that the result obtained from this analysis in the limit of the eccentricity being zero coincides with that of a membrane with a concentric circular inner boundary. 1. INTRODUCTION This paper deals with the vibration of a membrane having a circular outer boundary and an eccentric circular inner boundary. Problems of membrane vibration can be discussed by using the results obtained in certain acoustical waveguide problems, because the equation of motion of the membrane corresponds to that governing the dependence of the motion on the cross-section space variables in acoustical waveguides. Methods for waveguide problems involving complicated crosssections include the following: conformal mapping [l-4]; finite differences [S, 61; point matching [7, 81; Rayleigh-Ritz [9] ; Galerkin method [lo] ; finite element [l 11. As for the problems with which the present analysis is concerned, the finite element, finite difference, and point matching methods [5, 8, 1l] are probably the most appropriate general approaches; however, in general, a large size digital computer is needed. The conformal mapping method can also be used; Laura, Romanelli and Maurizi [4] have discussed the problem of waveguides of doubly connected cross-section of general shape by means of conformal mapping and Hine [3] has presented results for an eccentric-annulus cross-section, as obtained by the conformal mapping and Galerkin methods. In this paper, a method of solving vibration problem for a membrane having a circular outer boundary and an eccentric circular inner boundary is presented. In the analysis, the exact solution which satisfies the inner boundary condition is used, and the outer boundary condition is satisfied by means of the Fourier expansion method. Thus the method developed in this paper also gives only an approximate result; however, the technique can be more rigorous and straightforward than those given by the previous authors because of using the exact solution of the equation of motion which satisfies one boundary condition. The solution and technique are also applicable to problems of acoustical waveguides. The result given by setting the eccentricity equal to zero in the present analysis coincides with that for a concentric-circular-annular membrane which is obtained by exact methods [12,4]. 2. ANALYSIS A membrane with a circular outer boundary and an eccentric circular inner boundary, r’, of an eccentricity d is shown in Figure 1, in which the origin 0 of Cartesian co-ordinates x and 545
K. NAGAYA
Figure I. A circular membrane having an eccentric circular inner boundary.
y is chosen in the center of the membrane, and the z axis is normal to the surface of the membrane. The polar co-ordinates r. 0 and r’, 8’are taken as in the figure, and the radii of the outer edge r and the inner edge r’ are denoted by R and CI,respectively. If one denotes the displacement of the membrane in the z direction byw:, the equation of motion for the membrane, in the co-ordinates r’, 0’ is [ 131
T{a%4qarf2 + (Ijr')aw/ar' + (i/P)a%4~/aB'2} - pazW/at2 = 0,
(1)
where Tis the tension of the membrane, p the mass density and t the time. The displacement w for the steady-state vibration can be shown to be of the form w(r’, Q’, t) = W(r’, 0’) sin it.
Substituting
equation
(2) into equation W=
i j
~1
(2)
(l), one obtains
z s,{Aj,Jn(ar’)
+ Bi,Yn(ctr’)}@jn
(3)
n.sO
where Aj,, Bjn are the constants of integration, w is the circular frequency, J,(ar’) is the Bessel function of the first kind of order n and Y,(ar’) that of the second kind, and a2 = po2/T.
E,=
Iforn=O,and=2fornz
@jjn= cos n6’ for ,j = 1 and:
1,
sin n0’ forj = 2.
(4)
Since the geometrical symmetry exists in the membrane about the x axis, the motion of the membrane can be separated into two types of symmetric and antisymmetric vibrations about the axis of symmetry. 2.1. SYMMETRIC VIBRATIONS If one considers the symmetric vibration about the x axis (the axis of symmetry), displacement W is obtained by puttingj equal to 1 in equation (3):
w=$
n-n
E,{A~~J~(c(~‘) + B,,Y,(ar’)}cosnO’.
the
(9
The displacement should vanish around the boundary circle when the inner edge is fixed. Then the boundary condition at the inner edge is denoted as ( W),? =L1= 0.
(6)
VIBRATION
OF A MEMBRANE
547
Substituting equation (5) into equation (6), and eliminating B1,, one has W=~~Oe.A,.{J.(~r’) + y,Y,(crr’)} cosnel,
(7)
where yn = -J.(cxa)/Y, (au). One has following relations between the co-ordinates r, I3and r’, 8’: r sin e
rcose+d=r’cos@,
= r’sin 0’.
(8)
When the outer edge r = R also is fixed, the boundary condition at the outer edge is
( WLR = 0.
(9)
The displacement at an arbitrary point in the membrane can be expressed in terms of the co-ordinates r’, 8’; however, the boundary condition (equation (9)) around the outer edge cannot be satisfied directly by using equations (7) and (8). Therefore, in this analysis, the displacement around the outer edge is expanded into a Fourier series to satisfy this boundary condition, in the form
W%R=
2i
%Jl,Q.,,,cosm4 n-Om=0
(10)
where
Qn, = U/4
i [(J,(orr’) + y.y.(ar')}cod'],,,cos &de --x
(11)
and E,,,= 1 for m = 0 and = 2 for m >=1. By putting (IV),,, to zero, the boundary condition of the outer edge can be satisfied, and if the terms n, m are taken up to N + 1, one obtains the following simultaneous equations in matrix form :
,
(12)
where
IAl=
(13)
Since equation (13) is a determinant of the coefficients of equations (12), the frequency equation is IAl =O.
(14)
The ratios All/AI,, A12/Alo, . . ., AIN/AIO are obtained from equation (12). Thus each mode of vibration can be determined by substituting these ratios into equation (5). The values of the coefficients given by equation (11) can be derived by using the following relations : r’= (r2 + 2rdcos 8 + d2)l12,
8’= sin-‘{(r/r’) sin ej.
m
54x 2.2.
K. NAGAYA ANTISYMMETRIC
VIBRATIONS
The displacement of the membrane for the case of antisymmetric axis is obtained by takingj= 2 in equation (3):
vibration
about
W = i &“{AZnJ,(ctr’) + B,,Y, (w’)} sin no’.
(16)
n ~=1
The boundary condition around the inner edge is the same as equation equation (16) into equation (6) and eliminating B2,,, one obtains
(6). Substituting
W= .t, EnA2,,{Jn(ar’) + y,Yn(ar’)}sinn&. The frequency tion; it is
equation
the x
(17)
can be found in the same way as for the case of the symmetric
vibra-
(18)
where QT,, Q&, . . . . Q& have the same forms as in equation (11) but in which the term cos 0’ is replaced by sin W, and also cos me by sin me. As a limiting special case of the problem, consider the case where the eccentricity is zero. In such a problem, the relations r’ = r,
are obtained,
and equation
equation
(19)
(11) yields
Qnm= (J&R) + ~,Y&W Substituting
8’ = e,
(20) into equation
forn = m, and Q,, (14), one has the following
= Oforn # m. frequency
(20)
equation:
Qoo.Qll. Qx . . . QNN = 0.
(21)
It is noted, from equation (21), that the modes of vibration for the case of concentric boundaries can be characterized in terms of the number of nodal diameters, n. Thus the frequency equation for the case of concentric boundaries yields Qnn= 0. Equation (22), of course, coincides with the frequency equation of a circular membrane a concentric circular inner boundary, as obtained by exact methods. 3. NUMERICAL
(22) having
EXAMPLES
Numerical calculations have been carried out for the natural frequencies of a circular membrane with an eccentric circular inner boundary. The order N in equations (13) and (18) is taken so that nine terms are included in the calculation, because the maximum discrepancy between the result for N = 8 and that for N = 12 can be shown to be confined to within 0.5%. The values of the integrals in equation (11) were obtained by using the Gaussian formula for numerical integration. Figures 2(a)-(c) show the relations between the non-dimensional natural frequency, aR, and the non-dimensional eccentricity, d/R, for various values of a/R. In the figures, S,, S,, . . ., show the results for symmetric vibration and A,, AZ, . . ., those for antisymmetric. It is seen that the results for the symmetric vibration, except those for an
549
VIBRATION OF A MEMBRANE
0
0.1
0.2
0’3
0.4
0.5
0.6
0.7 d/R
Figure 2. The natural frequency uersusthe eccentricity for (a) a/R = O-1,(b) a/R = 0.3, and (c) a/R = 0.5. S, Symmetric; A, antisymmetric.
axisymmetric vibration with respect to the q axis, coincide with those for the antisymmetric in the eoncentric case (d/R = 0). However, both results separate rapidly with increasing values TABLE
1
Non-dimensional natural frequencies versus eccentricities for a/R = O-5 (the values in the bracket denote the results obtained by the point matchhg method [8]) Symmetric vibration dlR 0 0.2 O-5
rk 01 6246 (6.26) 4.811 (4.76)
11
11
6.393 (6w 6.172 (614)
6.814 (6.84) 7.391 (7.42) 6.836 (6.88)
Antisymmetric koiR 6-393
(64) 5511 (5.46) 4.569 (4.54)
550
K.NAGAYA
of the eccentricity. The numerical result of this analysis for d/R = 0, of course. coincides with that for the concentric case which is obtained by the exact method [4, 131. The non-dimensional natural frequencies of the first through the fourth modes in this analysis correspond to the non-dimensional cut-off frequencies &,a, k, ,a, k,,a and k,,a as obtained for acoustical waveguides in references [4] and [8]. Numerical results for this problem of the waveguide have been shown in Figures 9 and 10 in reference [g], as obtained by the point matching method. Comparison between the results obtained and those of the point matching method shows good agreement, as shown in Table 1, although there exist some reading errors from the figure in reference [8].
4. CONCLUSIONS A method of solving vibration problems for a circular membrane having an eccentric circular boundary has been presented. The dependence of the natural frequencies on the eccentricities has been shown, and the results for the case of an eccentric inner boundary compared with those for a concentric one. The results show that the dynamical behavior of a membrane with an eccentric inner boundary is quite different from that of a membrane with a concentric inner boundary.
ACKNOWLEDGMENT The author wishes to heartily thank Professor for his kind guidance in the present work.
Yoshitaro
Hirano,
Yamagata
University,
REFERENCES 1964 Institution of Electrical and Electronic Engineers Transactions on Microwaue Theory and Techniques MTT12, 248-249. Approximate method of determining the
1. M. CHI and P. A. A. LAURA
cutoff frequencies of waveguides of arbitrary cross section. 2. H. H. MEINKE,K. P. LANGEand J. F. RURGER 1963 Proceedings of the Institution of Electrical and Electronic Engineers 51, 1436-1443. TE and TM waves in waveguides of very general crosssection. 3. M. J. HINE 1971 Journal of Sound and Vibration 15, 295-305. Eigenvalues for a uniform fluid waveguide with an eccentric-annulus cross-section. 4. P. A. A. LAURA,E. ROMANELLI and M. J. MAURIZI 1972 Journalof Soundand Vibration 20,27-38. On the analysis of waveguides of doubly-connected cross-section by the method of conformal mapping. 5. J. B. DAVIESand C. A. MUILWVK 1966 Proceedings of the Institution of Electrical andElectronic Engineers 113,277-284. Numerical solution of uniform hollow waveguides with boundaries of arbitrary shape. 6. C. W. STEELE1968 Journuf of Computational Physics 3, 148-153. Numerical computation of electric and magnetic field in a uniform waveguide of arbitrary cross section. 7. H. Y. YEE 1965 Proceedings of the Institution of Electrical and Electronic Engineers 54, 64. On determination of cut-off frequencies of waveguides with arbitrary cross section. 8. H. Y. YEE and N. F. AUDEH 1966 Institution of Electrical and Electronic Engineers Transactions on Microwave Theory and Techniques MTT-14,487-493. Cut-off frequencies of eccentric waveguides. 9. R. M. BULLEYand J. B. DAVIES1969 Institution ofElectricalandElectronic Engineers Transactions on Microwave Theory and Techniques MTT-17, -6. Computation of approximate polynominal solutions to TE modes in an arbitrarily shaped waveguide. 10. P. A. A. LAURA 1966 Proceedings of the Institution of Electrical and Electronic Engineers 54, 1495-1497. A simple method for the calculation of cut-off frequencies of waveguides with arbitrary cross section.
VIBRATION OF A MEMBRANE
551
Il. P. L. ARLETT, A. K. BAHRANI and C. C. ZIENKWJICZ 1968 Proceedings of the Znstirurionof rhe Electrical and Electronic Engineers 11, 1762-1766. Application of finite elements to the solution of Helmholtz’s equation. 12. M. ABRAMOWITZ and I. STEOUN 1965 Handbook of iUathematica1 Functions, 390432. New York: Dover Publications. 13. LORD RAYLEIGH 1877 Theory of Sound, 306-351. New York: Dover Publications, second edition, 1945 re-issue.