Journal of Sound and Vibration (1978) 58(1), 15-26

ACOUSTIC PROPAGATION IN NEARLY ANNULAR DUCTS

S. SRIDHAR

West Virginia University, Morgantown, West Virginia 46506, U.S.A.

AND

A. H. NAY~rI

Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, U.S.A.

(Received 6 July 1976, and in revised form I November 1977)

The propagation of acoustic waves in a duct of nearly annular cross-section is inves- tigated. A method, based on a perturbation technique, is developed for calculating the wavenumbers of the propagating modes. Numerical results are presented for the case of (1) an annular duct having an elliptical outer boundary and a circular inner boundary, and (2) a cylindrical duct having an elliptical boundary. The results of the latter case compare favorably with those previously published, for eccentricities up to 0.6. Major advantages of the present method, as compared with previously used methods, are its generality and the relative ease of the computation involved, provided the boundaries do not deviate drastically from circles.

1. INTRODUCTION

The study of wave propagation in ducts is of current research interest because of its application to the problems of central air conditioning and heating installations, loudspeakers, wind tunnels, aircraft engine duct systems, and rocket nozzles. An essential piece of information in such a study is an accurate knowledge of the cut-off frequencies and the wavenumbers of the propagating modes. These wavenumbers can be readily determined for simple cross-sections of the duet, such as rectangular and circular. However, the problem of finding the wave- numbers in a duct of arbitrary cross-section is a more difficult one and has been the object of several investigations.

For the case of no mean flow, the problem is to solve the two-dimensional reduced wave equation (Helmholtz equation) for given geometry and given boundary conditions. The literature on the subject is considerable. An excellent survey paper is due to Mazumdar [1]. The techniques that have been used can be broadly classified as (1) separation of variables, (2) conformal mapping, and (3) numerics.

The separation of variables method has been used for problems involving elliptical boundaries. Examples are the works of Chu [2] and Kretzschmar [3] in the context of electro- magnetic waves, Daymond [4], and recently Lowson and Baskaran [5]. The method is, of course, restricted to special geometries and boundary conditions for which the variables can be separated. Even in such cases, the numerical work needed for finding the wavenumbers may be non-trivial (see, for example, reference [2]).

The basic idea of the conformal mapping technique is to map a complicated boundary into a more convenient symmetric one, usually a circle, and then solve the problem in the mapped region. Examples are the works of Segel [6], Roberts [7], Laura [8, 9] and Hine [10]. In this method there is the inherent difficulty of finding a suitable mapping function for mapping a

15 0022--460X/7810408--0015 $02.00/0 �9 1978 Academic Press Inc. (London) Limited

16 S. SRIDHAR AND A. H. NAYFEH

given arbitrary boundary. In addition, the governing differential equation becomes com- plicated due to variable coefficients and has to be solved by an approximate method, such as the Galerkin minimizing method or a perturbation method.

In the numerical method an integral form of the reduced wave equation is usually employed because of its suitability for numerical work. Examples are the studies by Banaugh and Goldsmith [! I] and Bell, Meyer and Zinn [12]. A disadvantage of a strictly numerical method is that the solution has to be repeated for every cross-section, resulting in excessive compu- tation time.

Torvik and Eastep [13] presented a method for calculating the natural frequencies of non-circular membranes. The method is based on expanding the boundary in a Fourier series about an "average" circle. However, no procedure was given for selecting the average radius. The frequencies are obtained as the roots of a modified (from the circular case) characteristic equation. Though this procedure may be convenient in special cases, its utility is limited due to the ad hoc nature of the approximations. It is felt that, for a consistent approximation, the frequency parameter must also be expanded in a perturbation series ('see, for example, the book by Nayfeh [14], section 3.1.6).

The purpose of this paper is to provide an alternative to the above techniques. The method developed here is a consistent and systematic procedure based on the method of strained parameters [14], a perturbation technique. The method enables one to find the wavenumbers in a duct of arbitrary cross-section (at least when the boundary does not deviate drastically from a circle). A general analysis is presented for the case of an annular duct having non- circular boundaries. In order to evaluate the effectiveness of the method, numerical examples involving an annular duct and a cylindrical duct are also presented. The results for the latter case are compared with those of reference [5].

2. STATEMENT OF THE PROBLEM

Consideration is given to the propagation of pressure waves in an inviseid, isothermal, static fluid contained in a nearly annular semi-infinite duct. The governing equations, in polar coordinates, are

V2p = (l/c 2 ) a2plat 2, (1)

av, ap avo i ap, av, ap p o . ~ = ar' po a t = r ao p o - ~ = - az' (2a-c)

wherep is the disturbance pressure, Po is the initial density, c is the undisturbed speed of sound, v,, vo and v~ are the components of the velocity, that is,

v = v,~, + ve~s + v , ~ , (3) and

V z =- aZl~r 2 + (l/r) a/ar + (l/r 2) a2/ao z + aZ/Oz 2.

The boundaries of the duct cross-section are assumed to be slight deviations from perfect circles, so that the region under consideration is given by a + el(O) >t r >~ b + eg(O), where a and b are the "average" outer and inner radii of the annulus, and e is a small dimensionless quantity which characterizes the deviation from perfect circles.

The appropriate boundary conditions are the so-called no-penetration conditions" that is,

v.n = 0, on F a n d G, (4a)

where

F = r - a - ~ . f = O , G = r - b - e g = O , (4b, c)

PROPAGATION IN NEARLY ANNULAR DUCTS 17

and n is the normal to the boundary. Equations (4) and (2) lead to

Op e afap ap e agap Or r 2 O0 O0 = 0 on r = a + ef, and Or - r -~ O-/~ O--O = 0 on r = b + e g .

(5a, b)

We seek a solution to the present problem in the form of a normal solution: that is,

p (r , 0, z, t) = P (r, 0) exp [i (kz - 090]. (6)

Substituting equation (6) into equations (1) and (5) yields

V2p + x2P = O, (7)

OP e Of OP OP e OgOP "Or r 2 0 0 0 0 = 0 on r = a + e f and Or r 2 0 0 0 0 = O on r = b + e g ,

(da, b) where t~ 2 = (o~2/c 2) - k 2.

3. METHOD OF SOLUTION

An approximate solution to equations (7) and (8) is obtained by using the method o f strained parameters. To accomplish this, we transfer equations (8) to r = a and r = b by expanding them in Taylor series. The result is

f Or 2 a2 ~.~ ~_~ j ~ f 2 + = 0 ~ r + e + e 2 "'" Or a "~fO-Offrr~O + ~ f 0"0 ~_]

on r = a, (9a)

OP [ OzP I OgaP] [1 2 03P 1 Og 02P 2 OgOP]

[ i g ~r a O0 Or O0 -~r+e[g Or 2 g2 ff~ ~-~j + e 2 ~-~g I- ~-~ g ~-~-ff] + --- = 0

on r = b. (9b)

Approximate solutions o f equations (7) and (9) are sought by expanding P and tc as follows"

P(r, O; e) = Po(r, O) + ~Pt(r, O) + e2p2(r, 0) + . . . . (10a)

~r = h'o + e~: t + e2~62 + . . . . (lOb)

Substituting equations (10) into equations (7) and (9) and equating coefficients o f like powers of e on both sides yields the following results:

order e ~ 01"o

V 2 p o + x o 2 P o = 0 , = 0 on r = a and b, Or

order e:

OP t _02 po =-f-Tp-r~ + Or

order e 2:

1 0 f O P o

V2pt + r 2 p t = -2Xo tq Po,

a 2 O0 O0

OP2 02 PI Or = - f ~r 2 +

OP2 02 Pt -t 1 Og OPt Or - g--~i'r~ b 2 O0 O0

on r = a, OP 1 OZPo I OgOP o Or = - g T -1 b 20000

o n

V2p2 + tc2p2 = -2~o ~t Pt - ( tr + 2tOo tr Po,

1 Of OP t [ 2 03 e o I 0 2 0Po a2 00 O0 ~ f Or 3 a3 aO ( f )-~-~ on

I 03Po I O OPo _ 2

2 g Or a b ~ ~ ( g 2 ) - ~ - on

(l la, b)

(12a)

r=b , (12b, c)

(13a)

r = a, (13b)

r = b. (13c)

18 S. S R I D H A R A N D A, H . N A Y F E H

4. FIRST-ORDER SOLUTION

The solution of equations (I 1) can be written as

Po = ~ ~ A..,c~.,.(r)exp(inO),

where

(14a)

q~nm = c.m[Jn(tCo.,.r) -- e.m Y.(tCo..,r)], e.., = J~(~:o.m a)]Y'.(h'o..,a), (14b, c)

primes denote differentiation with respect to r, c... is chosen such that

b

j rqbZ..,dr = 1, (14d) a

and rCo.,, are the roots of

t i t t . J.(Ko.., a) Y.(Ko.,. b) - J.0%.m b) Y.(•o.., a) = 0. (14e)

Since P is real, A_.~, = .4.., and 4-. , . = ~.m, where `4 is the complex conjugate of A. The solution given by equations (14) corresponds to the propagation of waves in a perfect

annulus. To determine the effect of the non-circularity of the boundary, consideration is given to a typical mode of equation (14): that is,

Po.,. = ~...(r) [A.,. exp (in0) + .4.,. exp (-in0)]. (15)

Substituting equation (15) into equations (12) yields

V~ PI + Ir = -2~:o.m~:l...ck..,[A.mexp(inO) + .4.,. exp (-in0)], (16a)

Or" = - fd?~,. - a2-- -ff~ d?.., A.,. exp (mO) - fd?~,~ + -~ ~ qS.., .4.., exp (-toO)

on r = a , (16b)

[ ] Or = - gck"~ "~ "~ d?.,. A.., exp (in0) - g~ , . + ~-i ~'~ 4.,. A'.., exp (-in0)

on r = b . (16c)

To solve equations (16), we expand P l , f a n d g in Fourier series in terms of 0 according to

P~= ~ ~b+(r)exp(isO), f = ~ f~exp(is0), g = ~ g, exp(isO). (17a-c)

Since P~, fand g are real, ~,_~ = i~+,f_, = ~ and g_~ = ~+. We chose the radii a and b such that fo = go = 0: that is, a and b are the averages of the outer and inner boundaries of the cross- section.

Substituting equations (17) into equations (16), multiplying the result by exp(-is0), inte- grating from 0 = 0 to 0 = 2n, using equations (14) and the recurrence relations of Bessel functions, one obtains

Iv . ~ + Ko~..,] r = -2~o... ~:,.., 4... x..,,

r = - ( l /a ' ) (a z K2.m + n2) ~)nmf2n J.m on

~ = -(11b2)(b 2 ~:g..~ + n 2) q~.., g~. 2 . . on

(18a)

r = a, (18b)

r - - b , (18e)

PROPAGATION IN NEARLY ANNULAR DUCTS 19

for s = n, where V 2 -- d2/dr 2 + (l/r) d/dr - n2lr 2, and

[V~ + Ko~."*] ~ = 0, (19a)

l tp~=-~dp..,[(a2h'~."* + ns)f~+.A.., + (a2h'2.m-ns)f~_.A."*] on r = a , (19b)

1 Ip; = ~-~ (~o."*[(b 2 tcg."*+ n s ) g s + . A.,. + (6 2 h'g.., - - n s ) g s _ . A."*] on r = b, (19c)

for s--# n. When s #- n, the solution of equations (19) can be written as

~, = ~(,',4."* + ~42'A."*, (20a)

where

~,(l) = Bl J~(h'o."* r) + B2 Y~(~o.,. r), ~,<2~ = BaJ~(~:o.mr) + B4Y.O%.,.r), (20b, c)

in which the constants B~ are determined from the boundary conditions (19b) and (19c) and are given in Appendix A.

When s = n, equations (I 8) have a solution if, and only if, a solvability condition is satisfied because the homogeneous parts of equations (18) have a non-trivial solution. To determine the solvability condition, we multiply equation (18a) by r~b."*, integrate the result by parts from r = b to r = a, use equation (14c), and obtain

a~."*(a) ~'n(a) -- bdpn"*(b) ~'.(b) = - 2 ~ o . . K~ . . . .4."*. (21 a)

Using the boundary conditions (18b) and (1Be), we rewrite equation (21a) as

KI."* = D."*A."*]A."*, (21b)

where

O."* = (l/2xL"*) [(l/a)(a 2 r~"* + n 2) dp2."*(a)f2. - (l/b)(b 2 ~o2."* + n 2) cb.X"*(b) g,.]. (21c)

To a~alyze equation (21b) we note that f2. and g: . and hence D."* are, in general, complex. Thus, we replace A.., and D.,. by their polar forms

A."* = a... exp (ict..,) and D."* = r/."* exp (itr.=) (22)

in equation (21b) and obtain

For/r to be real,

Hence

x~.,. = r/."* exp [i(tr."* - 2ct..,)]. (23)

x1.., = r/.,. when ct.., = -~u."*, (25a)

xl.,. = -aT."* when a.,. = -i(a."* + n). (25b)

Therefore, corresponding to each propagating mode (i.e., ran) of a perfectly annular duct, there are actually two modes in a nearly annular duct (i.e., the degeneracies are removed). They are given by

p( l ) (r ,O,z , t )= a."*dp."*(r)cos(nO +-~a."*)exp[i(k(l)z-oat)] + O(e), (26a)

where

k <') = [(toZ/c a) - (Xo,~ + erh"*)2] 't2, (26b)

a.,~ - 2#."* = 0 or n. (24)

20

and

where

S. SRIDHAR AND A. H. NAYFEH

p~2'(r,O,z,t)= a.,.q~.~(r)sin(nO+�89 (27a)

k r'~ = [ (oJ~/d) - (~o.,,, - ~rt.,.)=] "~ . (27b)

Equations (25) provide the first correction to rCo.~ unless

f2. = g2. = 0 (28a)

and thus set.,. = 0. (28b)

In this case, the degeneracy remains. To obtain a first correction to h'o.~ when both f2. and g2. are zero, we have to carry out the analysis to second order: that is, order e z. We note that the symmetric modes (i.e., n = 0) always fall into this category.

5. SECOND-ORDER SOLUTION

For the second-order analysis, the function Pl, as given by equation (17a) has to be fully determined. It is seen that though ~'s, for s :~ n, is completely given by equation (20), ~b. is still undetermined. However, when f2., g2. and tq.m are zero, equations (18) which govern ~. become homogeneous and identical in structure to the zeroth-order problem. Therefore, in this case, ~,. can be set equal to zero so that

e l = ~ ~.(r)exp(is0). (29) . g ~ - r

To solve equations (13), we expand P2 , f 2 and g2 in Fourier series in terms of 0 according to

P2= ~ 7,(r)exp(isO), f 2 = ~: J~exp(is0), g2= ~ ~.exp(is0). (30a--c)

Since P2,f 2, gZ are real, r-~ = 7~,f-. =J~ and g_, = L . Substituting equations (I 5), (17), (28) and (29) into equations (13), multiplying by exp (-in0)

and integrating from 0 = 0 to 0 = 2n, one obtains

2 2 _ _ [V. + Ko.~,] 7. - -2~Co..~ x2.m ~b.,. A.. , (31 a)

?___lzdp.,~(A.,~fo+A.,~A.)_2n2laSdp.,~d...j~_ ~u.. f~_. on r = a , (31b)

r ; ,=~4 , .%(a . . . bo+A.m~. ) -2n2 /b~ ,p . , .A . . .~ . - ~ ~'..g._. on r=b, (31c) $ - - r

where

~'.~(r) = r -- [s(s- n)/r 2] r (31d)

The homogeneous parts of equations (31) have a non-trivial solution and hence the solvability condition has to be satisfied. Multiplying equation (31 a) by rob.re(r), integrating the result by parts from r = b to r = a, and using equation (14c), one obtains

adp..,(a) 7" (a) - bqb..~(b) 7,~ (b) = -2to . . , r2.., A.,.. (32a)

Using the boundary conditions (31b) and (31c), we can rewrite equation (32a) as

~c~.., = (1/2m.,.) [El + (A.,./A...)E2], (32b)

PROPAGATION IN NEARLY ANNULAR DUCTS 21

where E, = �89 ~ ' , . (a)~ - bdp.,.(b) q~.~(b)~o]

+ i [a~b.m(a) " ' ~,c (b)g._s], (32c) 7:.s (a)f ._~ - bdp.,.(b) tl, $ = - e o

E2 = ~[aq~.,.(a) q~',.(a)j~. - bdp..,(b) c~;,.(b) g2.] + 2n2 [( 1/a 2) d?2.,.(a)~. - (1/b 2) d?2.,.(b) g2.]

+ ~ [ac~.m(a) 7Jc.~(a)f._s - bgp.,.(b) 7Jc.~'(b)g._s], (32d) s ~ - e o

and 7/c.~ ) is given by equation (31 d) with ~ replaced by Ip, ~k)whieh in turn is given by equations (20b) and (20c). Using equations (14), (31d) and the recurrence relations of Bessel functions, we can reduce equations (32c) and (32d) to

+ -~ (a ~ Kg.,. - ns) [C,(a 2 Kg.,. - ns) - a] 42..,(a)./',_.f,_.

s - - c o

1 - b-- 5 (b z K2o.., - ns) [C~(bZkg.., - ns) b] ck~.,.(b)g,_.g~_.

+ ~o.~ a 2 b ' ' ' ' ~ 2 ~.m(a) ~ .~(b) (a 2 Kg.~ - ns) (b 2 K~,~ -- ns) ( f~ ' .g~_ . + f ~ _ . ~ _ . ) , (32e)

1[ 1 2 2 2 2 ^ 1 ] E, = ~ ~i (a ~:O.m + n ~ ~..,(a)A. -- ~ (b 2 ~:o~.,. + n 2) 4.z,.(b)R,.

+ -~ (a 2 Kg~,~ + ns) [C,(a 2 xg~, - ns) - a] 41~(a)f~+.f~-. $ - - m

1 b3 (b: K~.,~ + ns) [C~(b ~ ~g.~, - ns) - bl ~ m ( b ) g~+. ~ - .

C3 q b2 ~b.,.(a) qS.,.(b) [a ~ ~:g.,. - ns) (b 2 ~:02.., + ns) ~ _ . g~+.

h 'On m ( l 2

(a ~ ~:~,~ + ns) (b ~ x 2 . , - ns) f .+. ~_.] }, (320 +

where C~ are real functions and are listed in Appendix A. It can be seen from equations (32e) and (32f) that E~ is real and/?2 is, in general, complex.

Replacing A~,. and/?2 by their polar forms

A.,. = 6.,. exp (ia.m), E2 = :/.,. exp (i#.,.) (33)

in equation (32b), one obtains

to2.., = (l/2~Co.m){E~ + 0.,. exp [i(0.,. - 2a.,.)]}. (34)

For ~:~.,~ to be real,

0.,. - 2c~.,, = 0 o r ~z. ( 3 5 )

22 s. SRIDHAR AND A. H. NAYFEH

Therefore, the two values of n2.,, are given by

~tt) = (l/2h'o~)(Ea + f i~) when ~.= = -!2~.,. (36a) 2/L~

and t2) _ . r2.,. - (1/2%.,.) (Ex - ~.,.) when ~.,. = 4~(8.,. + ~). (36b)

The two propagating modes (corresponding to each nm) are given by

p" ) ( r ,O , z , t )=d .~qS .~ ( r ) cos (nO + t S . ~ ) e x p [ i ( k ( l ) z - t o t ) ] + O(e), (37a)

where

and

where

k( l ) [(0-)2/c2) -( / (Onm + ~2 v(l)"~2 ]1/2 ~-- r ~ 2 n m j J (37b)

p(2)(r ,O,z , t )=d.md?.=(r)s in(nO+-12~. , . )exp[ i (k t2)z -ogt )]+O(e) , (38a)

= ~2 .c2) "~211/2 (38b) k ' " [(o~2/c ") - ( ~ : o . , . + o , , , . . , , J �9

For the special case of the symmetric modes (i.e., n = 0), we let A.,. = 0 in equations (32) and obtain

~2.,. = -~r {�89 + $~,.(b)~o] + .--~o ~ [(C, aK~.m- l)q~.Z,.(a)fs)~

-(C2b~:~.,. - I) $~n(b)g,g, + C3Ko..,cl,.,.(a)d~.,.(b)(.~g,+f~ao,)]}. (39)

We note from equation (39) that tc2.m is always real and thus single valued, in contrast with equation (34).

6. ANNULAR DUCT: AN EXAMPLE

The procedure developed in the previous sections is applied to the case of an annular duct having a circular inner boundary and an elliptical outer boundary. The outer boundary is considered as a deviation from a circle whose radius is determined during the course of the analysis. The equation of an ellipse having a semi-major axis ~ and a semi-minor axis r/can be written in polar form as

r = q(1 -- e 2 cos 2 0) - ' p , ( 4 0 )

where the eccentricity e is given by e2 = 1 - (r//r 2, the origin is at the center of the ellipse, and the angle 0 is measured counter-clockwise from the positive major axis. Equation (40) can be rewritten as

r = a + ef(O) (41)

where ef(O) = - a + !/(1 - e 2 cos 20) -~/2 and a is the average radius of the circular boundary to be determined. The value of a is chosen such that the constant term in the Fourier expansion off(0) is zero (i.e.,fo = 0). The Fourier coefficients of sf(0) are, in general, given by

2z

~f~= ~ f (I - e2cos20 ) - ' / 2cos sOdO, (42) 0

so that 2x

a = (1 - e2cos2 0)-ltZd0.

O

(,43)

P R O P A G A T I O N IN NEARLY A N N U L A R DUCTS 2 3

The Fourier coefficients ofeZf2(o) are, in general, given by 2x

2A 1 f e f~ = ~ [ - a + 7(1 - e2cos20)-l/2]2cossOdO. (44~

0

It is noted that, for the example under consideration, all the Fourier coefficients are real and can be determined by simple numerical quadratures.

For the case of the asymmetric modes, the first correction to Ko,m is obtained either from equation (21b), as

•,.,. = +(1/2a~%.,.) (a 2 K2.m + n 2) ~.2,.(a)lf2.1, (45)

when f2. is non-zero, or from equation (32b), as

x2.,. = (l/2Ko.m) (El + [E21), (46a)

when F2. is zero. Here

1 2 2 2 s ^ 1 2 E1 = ~ a 2 (a ro.m -- 3n ) ~.m(a)fo + ~-~ t~.,.(a)

x ~_ (a s r~.., -- us) [Cl(a 2 ~:o2.,. -- us) -- a]f~_.~_., (46b) $~-0o

E~ = ~ (a" ~L,~'+ n ~) ~ L ( a ~ A . + ~,~(a) za

x ~. (a s I<o2.,. + us) [C,(a 2 Kg.., - us) - a]A+,~Z-.; (46c)

in which q~.,. is given by equations (14) and C~ is given in Appendix A. For the case of the symmetric modes the first correction to ~<o.,. is obtained from equations

(39) as

Numerical results are presented in Tables I and 2 for the case of an annular duct having an inner radius b = I and an average outer radius a = 2, Wavenumbers for the asymmetric modes

TABLE 1

Wavenumbers f o r annular duct (outer elliptical and inner c ircular) for odd f imctions. Inner radius = 1, "average" outer radius = 2, e is the eccentricity, A is the anmdar area a n d n is the

circumferential mode number

e 0"0 0"1 0"2 0"3 0'4 0-5 0"6 0"7 n ~ 9"43 9-43 9"43 9"43 9~44 9"46 9"50 9"60

1 0-6773 0-6976 0"6866 0 " 6 9 8 8 0 " 7 1 7 0 0 " 7 4 2 8 0 " 7 7 8 7 0-8299 2 1-3406 1 " 3 4 0 6 1-3407 1 "3412 1 - 3 4 2 7 1 - 3 4 6 4 1 - 3 5 4 4 1-3718 3 1"9879 1 " 9 7 8 9 1 " 9 7 8 5 1 " 9 7 7 0 1 " 9 7 2 3 1 " 9 6 0 9 1 " 9 3 5 1 1.8778 4 2"5876 2 " 5 8 7 5 2 " 5 8 6 7 2 " 5 8 2 6 2 " 5 7 0 6 2 " 5 4 1 0 2 " 4 7 3 6 2"3193 5 3"1694 3 " 1 6 9 3 3 " 1 6 7 7 3.1604 3 " 1 3 8 1 3 " 0 8 1 5 2 " 9 4 3 1 2.5774 6 3"7311 3 " 7 3 1 0 3.7290 3 " 7 2 0 1 3 " 6 9 3 8 3 " 6 3 1 3 3 . 5 0 0 1 3"2512 7 4"2793 4 " 2 7 9 2 4 " 2 7 7 1 4 " 2 6 7 8 4.2401 4 " 1 7 3 6 4-0304 3"7397 8 4"8191 4 " 8 1 9 0 4 " 8 1 7 2 4 " 8 0 8 9 4 " 7 8 4 3 4.7253 4 " 5 9 8 0 4"3387 9 5"3535 5 " 3 5 3 4 5 " 3 5 1 8 5 " 3 4 4 3 5 - 3 2 2 1 5 " 2 6 9 1 5 " 1 5 5 5 4"9279

10 5"8844 5 - 8 8 4 3 5 " 8 8 2 8 5"876 1 5 " 8 5 6 1 5 " 8 0 8 7 5.7095 5"5214

24 S. SRIDttAR AND A. H. NAYFEtl

TABLE 2

tVavenumbers for annular duct (outer elliptical attd hmer circular)for even fimction.s, hmer radius = 1, "average"outer radius = 2, e is the eccentricity, A is the atmttlar area, and n is the

circumferential mode nunlber

n• 0.0 0"1 0"2 0"3 0"4 0"5 0"6 0-7 9"43 9"43 9"43 9"43 9 -~ 9-46 9.50 9-60

1 0-6773 0-6750 0.6680 0.6558 0.6376 0.6118 0.5759 0-5247 2 1-3406 1 . 3 4 0 6 1 . 3 4 0 5 1 - 3 4 0 0 1 . 3 3 8 5 1 - 3 3 4 8 1 - 3 2 6 8 1.3094

3 1.9879 1 - 9 7 8 9 1 . 9 7 8 5 1 . 9 7 7 0 1 . 9 7 2 5 1 - 9 6 1 6 1 - 9 3 7 9 1.8876 4 2.5876 2.5875 2.5867 2.5826 2.5707 2.5417 2 - 4 7 7 8 2-3408 5 3.1694 3.1693 3 - 1 6 7 7 3-1604 3 . 1 3 8 1 3.0822 2.9488 2.6210 6 3.7311 3.7310 3-7290 3 - 7 2 0 1 3.6938 3 . 6 3 1 3 3 - 4 9 9 7 3-2461 7 4-2793 4.2792 4 . 2 7 7 1 4-2678 4 . 2 4 0 1 4.1736 4.0303 3-7391 8 4.8191 4.8190 4.8172 4.8089 4.7843 4.7253 4-5980 4-3386 9 5.3535 5.3534 5 - 3 5 1 8 5.3443 5 . 3 2 2 1 5 . 2 6 9 1 5 - 1 5 5 5 4.9279

10 5.8844 5 - 8 8 4 3 5.8828 5 . 8 7 6 1 5 . 8 5 6 1 5.8087 5 - 7 0 9 5 5.5214

were computed for various eccentricities of the outer elliptical boundary. The first correction to rCo,r~ is given by either equations (45) or equations (46). In general, the choice depends on the relative magnitudes of the corrections obtained from these equations. For the first two circumferential modes the wavenumbers were Calculated by using_equations (45). For the higher modes equations (46) were used. It was found that the infinite sums in equations (46b) and (46c) converge rapidly: typically, in less than 15 terms.

7. CYLINDRICAL DUCT: AN EXAMPLE

In this section we specialize the analysis of the previous sections to the case of a cylindrical duct o f an elliptical cross-section. This is done, essentially, by letting the inner boundary go to zero and requiring the solution to be bounded at r = 0.

The first corrections to the wavenumbers are given by equations (45) or (46) and equation (47). However, for the case of the cylindrical duct certain simplifications are possible. The coefficient C1 reduces to

C l = --Js(Konm a)/J's(h'onm a), (48)

equations (14) reduce to

where the Ko.m are the roots of

and the C,m are chosen such that

~.m = c.., J.(h'o.m r), (49a)

J.(Ko.,. a) 0, (49b)

a

f r~2~ dr = I. (49c) 0

By using equations (49) it can be shown that (see, for example, the book by Hildebrand [15]) c,m is given by

C2m 2 2 2 = 2~o.m/(a Ir n21J.2(h'o.,, a), (50a)

PROPAGATION IN NEARLY ANNULAR DUCTS 25

and therefore

~bz,,,(a) = 2t<~,,,/(a ~ Kg, m - n2). (50b)

Wavenumbers for the asymmetric modes were computed for comparison with the results of reference [5]. For the first two circumferential modes the wavenumbers were calculated by using equation (45). For the higher modes equations (46) were used. It was found that the infinite sums in equations (46b) and (46c) converge rapidly: typically, less than 15 terms are needed.

The computed wavenumbers were compared with those of reference [5] and the percentage differences computed. The results are presented in Table 3. For eccentricities of less than 0.4, the difference was found to be less than 0.2 ~o; actually, it is much less than this for most of the values.

TABLE 3

Percentage difference hi the computed wavenumbers f o r elliptical duct as compared with those o f reference [5], e is the eccentricity, n is the circumferential mode number, and the duct area = rc

Odd functions Even functions C ~ f N

0"4 0"5 0"6 0"7 0"4 0"5 0'6 0"7

1 0'01 0'03 0"07 0"14 0"01 0"04 0"09 0"23 2 0"10 0"27 0"65 1-48 0"57 1"48 3"24 6-30 3 0"27 0"83 2"28 6"01 0"18 0"37 0-66 1"06 4 0"17 0'49 1-28 3"31 0"17 0"44 0"98 1"98 5 0"13 0"37 0'90 2-15 0"13 0"36 0-84 1"78 6 0"10 0"28 0'66 1"41 0"10 0"28 0"64 1"31 7 0"08 0"20 0"42 0"51 0"08 0"20 0"42 0"54 8 0-05 0-12 0"20 0'03 0"05 0"12 0"20 0'00 9 0"03 0"03 0"14 1"39 0"03 0"03 0"14 1"39

I0 0"04 0"34 2"12 11-26 0"04 0"34 �9 2"12 11"26

8. CONCLUDING REMARKS

For the annular and the cylindrical ducts considered in this work, the effect of increasing the eccentricity of the elliptical boundary can be summarized as follows:

(1) for a lower order mode (n = 1 or 2), the wavenumber of the odd function increases, from that of the circular case, and the wavenumber of the even function decreases, from that of the circular case;

(2) for a higher order mode (n > 3), the wavenumbers of both the odd and even functions decrease slightly, from those of the circular case.

For the case of the cylindrical elliptical duct, this is a confirmation of the results of reference [5]. Also, the computed wavenumbers compare favorably with those of reference [5] for eccentricities up to 0.6.

The method developed in this paper is an asymptotic technique and was not expected to yield accurate results for very high eccentricities. It should be emphasized that, though considerable analysis was required to arrive at equations (45) and (46), the only major effort needed to evaluate the wavenumbers is evaluating the Fourier coefficients for a given non-

26 S. SRIDHAR AND A. H. NAYFEH

circular boundary . Thus, it has been shown that the present method is a viable technique for calculating the wavenumbers in a duct. Fur ther , the present technique can be applied to lined as well as hard-walled ducts carrying general mean flows.

REFERENCES

1. J. MAZUMDAR 1975 The Shock and Vibration Digest 7, 75-88. A review of approximate methods for determining the vibrational modes of membranes.

2. L.J . CHU 1938 Journalof AppliedPhysics 9, 583-591. Electromagnetic waves in elliptical metal pipes.

3. J .G. KRETZSCHMAR 1970 Institution of Electrical and Electronic Engineers on Microwave Theory and Techniques MTT-18, 547-554. Wave propagation in holIow conducting elliptical wave- guides.

4. S. D. DAYMOND 1955 Quarterly Journal of Mechanics and Applied ~Iathematics VIII, 361-372. The principal frequencies of vibrating systems with elliptical boundaries.

5. M.V. LOWSON and S. BASKARAN 1975 Journal of Sound attd Vibration 38, 185-194. Propagation of sound in elliptical ducts.

6. L. A. SEGEL 1961 Archives of Rational )~[echanics and Analysis 8, 228-237. Application of conformal mapping to boundary perturbations problems for the membrane equation.

7. S. B. ROBERTS 1967 Journal of Applied Alechanics 34, 618-622. The eigenvalue problem for two-dimensional regions with irregular boundaries.

8. P. A. LAURA 1967 Journal of the Acoustical Society of America 42, 21-26. Calculations of eigenvalues for uniform fluid waveguides with complicated cross sections.

9. P. A. LAURA, E. ROMANELLI and M. J. MAtmXzl 1972 Journal of Sound attd Vibration 20, 27-38. On the analysis of waveguides of doubly-connected cross-section by the method of conformal mapping.

10. M. J. Hit~E 1971 Journal of Sound and Vibration 15, 295-305. Eigenvalues for a uniform fluid waveguide with an eccentric-annulus cross-section.

11. R. P. BANAUGH and W. GOLDSMITH 1963 Journal of the Acoustical Society of America 35, 1590-1601. Diffraction of steady acoustic waves by surfaces of arbitrary shape.

12. W. A. BELL, W. L. MEYER and B. T. ZINN 1975 Proceedings, Third Interagency Symposium on University Research in Transportation Noise, University of Utah, 476--495. An integral approach

�9 for determining the resonant frequencies and natural modes of arbitrarily shaped ducts. 13. P. J. TORWK and F. E. EAS'rEP 1972 Journal of Sound and Vibration 21, 285-294. A method for

improving the estimation of membrane frequencies. 14. A. H. NAYFEH 1973 Perturbation 3Iethods, New York: Wiley-Interscience. See section 3.1. 15. F. B. HtLDEBRAND 1976 Advanced Calcuhtsfor Applications. New Jersey: Prentice-Hall. See

p. 229.

where

APPENDIX A

Bx = 6[q~2(b) Y'~(tr a)g~_, - q'2(a) Y~(~Co,,, b~ fs- , ] ,

B2 = 6[~2(a) J'~(~r b)f~_, - ~2(b) J~(~Co,,, a) gs_,],

B3 = 6 [q,,(b) Y'~(h'o,,. a) g,+, - ~ , (a ) Yj(t%,,, b)f~+,], t t .

B, = 6[q' l(a) J,0r b)L+, - ,/,,(b) Js(Ko,,. a) g~+,],

Ca = t5 [J~(t%,,, b) Y,(tr a) - J; (xo,,, a) Y~(Ko,,, b)],

c~ = ,~[J~0.'o.., b) Y~(xo.., a) - J~(m.m a) Y.(m.., b)],

C3 = 2 f /n,

6 = 1/[J;(xo,,. a) Y~(h'o,,, b) - J;(~o,,, b) Y;(t%,,, a)],

q'5(r) = - ( l / r 2) (r 2 xJ,m + ns) ~,r,(r), ~2(r) = - ( l / r 2) (r 2 tc~,,, - ns) q~,,,(r).