natural resonant frequencies of an oblate acoustical resonator

Received 3 September 1971 5.4

Natural Resonant Frequencies of an Oblate Acoustical Resonator *

C. T. M. CHANG

Argonne National Laboratory, 4rgonne, Illinois 60439

Natural resonant frequencies of oblate acoustical resonators with either perfectly soft or perfectly rigid boundaries are calculated by locating the zeros of R,,,,•(x)(--ih, it) and R,,,,• {•)' (--ih, it). These frequencies are normalized to those of the corresponding modes of a spherical cavity with the same volume and presented for various ratios of major to minor axes.

INTRODUCTION

Natural resonant frequencies of cavities with different shapes are of some importance. They are con- nected with the designs of certain liquid and gas chambers. Among the acoustic cavity problems which

can be solved exactly, the resonances of rectangular boxes, circular cylinders, and spherical cavities have been known for quite some time; the resonances of prolate spheroidal cavities were studied in a recent articleX; thus the corresponding problem in oblate

z

•7=cos •r/12 'r/= cos w6

•7= COS •r/4

= COS

=COS s'rr/I 2

=0 x,y

•=-COS 5•/12

-COS 'rr/3

,'r/=-COS

=-COS w'/6

'r] =-COS 'rr/• 2 I q ---1

Fro. 1. The oblate spheroidal coordinate system.

The Journal of the Acoustical Society of America

Copyright ̧ 1972 by the Acoustical Society of America. Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 131.91.169.193 On: Sat, 22 Nov 2014

05:42:36

C. T. M. CHANG

TABLE I. Zeros of

n=0 n=l n=2

a/b • m--0

1.0 oo 3.14159 1.5 0.894427 2.48472 2.0 0.577350 2.19947 3.0 0.353553 1.95386 4.0 0.258199 1.84647 5.0 0.204124 1.78639 6.0 0.169031 1.74793 7.0 0.144338 1.72116 8.0 0.125988 1.70144 9.0 0.111803 1.68630

10.0 0.100504 1.67432

m -- 0 m-- 1 m -- 0 m -- 1 m -- 2 ,

4.49341 4.49341 5.76346 5.76346 5.76346 3,94333 3.33604 4.32339 4.72890 4.15465 3.70880 2.80650 3.51617 4.27107 3.39954 3.50058 2.33237 2.76715 3.85846 2.70735 3.40486 2.12044 2.42833 3.66769 2.39304 3.34964 2.00115 2.23834 3.55748 2.21520 3.31365 1.92458 2.11714 3.48558 2.10081 3.28831 1.87122 2.03314 3.43494 2.02101 3.26949 1.83188 1.97151 3.39733 1.96214 3.25495 1.80167 1.92436 3.36828 1.91691 3.24339 1.77773 1.88713 3.34517 1.88106

spheriodal geometry becomes the only problem in the above group which was not presented in the literature. This is partially due to the complexity involved in cal- culating these functions and their derivatives. In the present article we present the calculations of the resonant frequencies of several acoustic modes for oblate spheroidal cavities with either perfectly soft or perfectly rigid boundaries. Since they are the limiting cases of zero and infinite normalized acoustic im-

pedance z,, they are useful for estimating the corresponding resonances for the case with general Further, since the oblate spheroid becomes a sphere as t/---,oo and a thin pillbox as t/---,0, the present studies may serve as a generalized treatment for

3.0

2.0

1.0

0.0 I . 1.0 2.0

I I I I

I I 1

4.0 6.0 8.0 I0.0

a/b

Fzo. 2. Normalized resonant frequency of n=0 mode for cavity with perfectly soft boundary.

2 Volume 51 Number i (Part i) 1972

cavities with their shapes varied from that of a sphere to that of a thin pillbox.

I. FORMULATION OF THE PROBLEM

Inside the cavity the acoustic pressure p satisfies the scalar Helmholtz equation

(V'+k')p =0, (1)

where k=w/Co, w is the angular frequency, and co is the speed of sound of the media inside the cavity. A harmonic time dependence of e -i•t is assumed.

2,0-

1,0

0.0 I 1.0 2.0

I I I I

I I I

4,0 6.0 8.0 I0.0

a/b

Fzo. 3. Normalized resonant frequencies for n= 1 modes for cavity with perfectly soft boundary.

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05:42:36

RESONANT FREQUENCIES OF OBLATE RESONATOR

TABLE II. Zeros of R=,(x)' (--ih, i•) in (h•)'=,•.

n--O n--1 n--2

a/b • m--0 m--0 m-- 1 m--0 m-- 1 m--2

1.0 ,o 4.49341 2.08158 2.08158 3.34209 3.34209 3.34209 1.5 0.894427 3.90017 2.01824 1.39982 2.58244 2.78325 2.25152 2.0 0.577350 3.61884 1.95627 1.05349 1.99399 2.50882 1.69632 3.0 0.353553 3.46119 1.86004 0.704159 1.35080 2.21944 1.13496 4.0 0.258199 3.38323 1.79787 0.528617 1.01821 2.06406 0.851357 5.0 0.204124 3.33603 1.75671 0.423081 0.816412 1.96766 0.682318 6.0 0.169031 3.30430 1.72791 0.352652 0.681167 1.90243 0.563972 7.0 0.144338 3.28149 1.70674 0.302318 0.584281 1.85548 0.487643 8.0 0.125988 3.26430 1.69056 0.264553 0.511485 1.82013 0.426746 9.0 0.111803 3.25088 1.67780 0.235174 0.454799 1.79256 0.379366

10.0 0.100504 3.24010 1.66749 0.211666 0.409413 1.77047 0.341453

The oblate spheroidal coordinates (shown in Fig. 1) are related to the rectangular coordinates by the transformation s

and

x =q[(1-- n•')(•'-{ - 1)]• cos•, y =q[(1-- n•-)(•--{ - 1)]• sin4•,

z =q•,

(2a)

(2b)

(2c)

where q is the half of the interfocal distance and

--1_<7_<1,

0<$<oo, (3)

0_<•<2•r,

3.0

2.0

1.0

0.0 I I.O 2D

I i i I

•.0

,I I I 4.0 6.0 8.0 I0.0

Fro. 4. Normalized resonant frequencies for n--2 modes for cavity with perfectly soft boundary. They are grouped according to whether (n-m) is even or odd.

and the solution of Eq. 1 inside an oblate spheroidal cavity is

p=Am,,•Sm,,(t)(-ih, •)Rm,,(x)(--ih, i•)e i•*' (4)

and the normal velocity u, inside the cavity is defined as

u.=(1/•oo)(Op/On)

=(A•,ff•po)S•,(x)(--ih, •)R•,(x)'(--ih, i•)e 'mq', (5)

where h=kq; S•,(x)(--ih,•), R•,(x)(--ih, i•) are, respectively, the oblate angular and radial func-

3.0--

2.0

1.0 -

o.o I I.O 2.0

I I I

4..0 6.0 8.0 I0.0

a/b

FIG. 5. Normalized resonant frequency for n--0 mode for cavity with perfectly rigid boundary. ,

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C. T. M. CHANG

3.0

2.0

1.0

i I I I

0.0 i I I I 1,0 2:.0 4.0 6,0 8.0 I0.0

o/b

Fro. 6. Normalized resonant frequencies for n= 1 modes for cavity with perfectly rigid boundary.

tions of the first kind; Rmn(•)'(--ih, i•) is the derivative of Rmn(Z>(--ih, i•) with respect to •, n is the normal to the boundary, and t>0 is the mass density of the media.

According to the definition of classical acoustics, a the normalized impedance of an obIMe spheroidal cavity with boundary at •-•0 is

a, =•p0[R=,(z)(--ik, i•o)/Rm,(X)'(--ih, i•0)]. (6)

For perfectly soft boundary an=O, one has

i0)=o, (7)

and for perfectly rigid boundary an -ooo, one has

i0) =o. (8)

In the limit of • -o oo and q• --o r, a spheroidal cavity degenerates into a spherical cavity. Equations 4 and 5 become

p = A ,•n•jn (kr) Pn'n(cosO)e i=*, (9)

A mnl un=•j'.(kr)pn'n(cosO)e i=*, (10)

ipoco

where jn(kr) and j'n(kr) are, respectively, the spherical Bessel function and its derivation with respect to the argument. Prim(COS0) is the associated Legendre function of the first kind.

II. NUMERICAL RESULTS AND DISCUSSIONS

As was true in the prolate case, z the volume of the oblate spheroidal cavity is to be held constant and equal to that of a spherical cavity with radius r0. By equating volumes of the spherical and spheroidal cavities, one obtains

ro/q = I-t/0 (•0•'+ 1) It. (I I)

The resonant frequencies of the oblate resonantor will also be normalized to the corresponding resonant frequencies in a spherical resonator; hence

•rnn -- [ (h•O)mnl/ (krO)nl][ 1 •- •0--2] }, (12)

n' + (13)

where (h•O)mnl and (h•0)'•nl are zeros of Eqs. 7 and 8, and (kro),, and (kro)'nZ are roots of jn(kro) and j',(kro), respectively.

The index l is neglected from f•,•,• and fi'mn• since only the l= ! case is considered. Zeros are obtained by using Descartes' rule of sign (i.e., check the change of signs between two adjacent values of h at a given •). Rmn(•)(--ih, i•) are calculated using expansion into

3.0

2,0

1.0

0.0 i I.O 2.0

I I I I

1 I i

4.0 6.0 8.0 IO.O

o/b

Fro. 7. Normalized resonant frequencies for n=2 modes for cavity with perfectly rigid boundary. They are grouped according to whether (n-m) is even or odd.

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RESONANT FREQUENCIES OF OBLATE RESONATOR

sum of spherical Bessel functions of different orders and normalized after Flammer." The oblate char-

acteristic root Xm• is calculated by Bauwkamp's method 4 with the asymptotic value of large or small h as the first trial value. Calculation was done on IBM 360/75 at Argonne with double-precision arithmetic. The result agrees with the existing tables ".5 and is believed to be accurate to more than seven significant digits. Zeros of Eqs. 7 and 8 are presented in Tables I and II for various ratios of a/b (major to minor axes of the spheroid). From Table II one finds that 9'm becomes the lowest resonant frequency for oblate cavity with perfectly rigid wall, whereas 9'0u is the lowest one for its prolate counterpart. The spherical cavity is presented as a special case with a/b= 1. It possesses an n- fold degeneracy in resonant frequency. As in the case of prolate cavity, this n-fold degeneracy splits as the spherical cavity is squashed. However, unlike the prolate case, flm• and 9'• do not increase with m but

are grouped according to whether (n--m) is even or odd. This behavior can be understood with the aid of the

asymptotic expansions of spheroidal wave functions given in Chap. 8 in Flammer's work." The normalized frequencies of a number of modes are presented in Figs. 2-7.

* Work performed under the auspices of the U.S. Atomic Energy Commission.

• C. T. M. Chang, J. Acoust. Soc. Amer. 49, 611 (1971). • Carson Flammer, Spheroidal Wave Function (Stanford U. P.,

Stanford, California, 1957). a Philip M. Morse and K. Uno Ingrid, "Linear Acoustic

Theory," Encyclopedia of Physics, S. Flfigge, Ed. (Springer, Berlin, 1961), Vol. XI/1, p. 16.

4 C. J. Bauwkamp, J. Math. Phys. 26, 79 (1947); G. Blanch, J. Math. Phys. 2S, 1 (1946).

• S. Hanish, Carolyn Shely, and R. V. Baier, "Table of Spheroidal Wave Functions, Part 1--Oblate Radial Functions of the First and Second Type," NRL Rep. 6472 (Naval Res. Lab., Washington, D.C., 23 Dec. 1966).

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natural resonant frequencies of an oblate acoustical resonator

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