Two-dimensional acoustic wave propagation in elastic ducts

Journal ofSound and Vibration (1981) 76(4), 517-528

TWO-DIMENSIONAL

ACOUSTIC

IN ELASTIC Y.

WAVE

PROPAGATION

DUCTS

L. SINArt

Department of Applied Mathematical Studies, Leeds University, Leeds LS2 9JT, England (Received 11 August 1980, and in revised form 17 December 1980)

Integral transforms are employed in order to obtain a formal solution to the twodimensional elastic-walled duct problem. The fluid inside the duct is stationary, inviscid and compressible, and is identical to the fluid outside the duct. A time-harmonic line source lies between the duct walls. With attention confined to the field inside the duct, an asymptotic analysis is implemented for high and low frequencies, yielding residues which are valid throughout the duct and branch-cut contributions which apply only in the far field.

1. INTRODUCTION

The literature on the general subject of wave propagation in ducts encompasses such diverse topics as blood flow, electromagnetic waveguides and sound propagation; this paper is concerned with one aspect of the ultimate area, namely the influence of elasticity of the duct walls on the waves produced in the compressible medium in which the duct is immersed. Much work has been published on the various aspects of sound propagation when an independent impedance condition is applied at the duct walls [ 1, 21, but such a condition is plainly inadequate when the walls are wave-bearing surfaces such as membranes, plates or elastic shells. A considerable number of articles have appeared in which the authors have examined the phenomena associated with elastic tubes [l-6], but these articles are restricted to waves possessing discrete axial wavenumber spectra, related to the form exp (ikx -iwt), where x is the axial co-ordinate, and to the scattering of an incident acoustic wave by the tube. In contrast, the present problem is that for a source inside the embodied in a Fourier transform. Whilst one may duct with a continuous spectrum anticipate some similarities between the behaviour of the planar two-dimensional and cylindrical ducts, one must acknowledge the cardinal differences, particularly with respect to the dynamic behaviour of the duct walls [7]. The rigid-walled case, in which the normal gradient of the velocity potential vanishes at the duct walls, is characterized by a sequence of eigenmodes, each of which fails to propagate at frequencies below its own “cut-off frequency”. The plane mode is nondispersive and possesses a vanishing cut-off frequency, whereas the higher modes are all dispersive and exhibit axial phase speeds which are greater than the sound speed. Presently, in the subsequent sections, an analysis is to be given of the two-dimensional problem in which a time-harmonic line source is situated between two parallel membranes, with a compressible, stationary, invlscid fluid occupying the space both inside and outside the duct (the situation arising when the fluid inside the duct differs from the external fluid is relevant under numerous circumstances but will not be pursued here). t Present address Research and Development Risley, Warrington WA3 6BZ, England.

Department

(RD4),

National

Nuclear Corporation

Limited,

517 0022-460X/81/120517+

12 $02.00/O

@ 1981 Academic

Press Inc. (London) Limited

518

Y. L. SINAI

Harding-Payne [8] has in fact examined a more complicated version of this problem, in which a low-Mach number mean flow exists in the duct, but that analysis is restricted to high (in an appropriate sense) frequencies and to sources lying on the duct centreline, and no attention is paid to any of the non-uniformities which are delineated in the present paper. The principles underlying fluid/structure coupling may be found, for example, in the work of Crighton [9-l l] and of Junger and Feit [ 121. Briefly, one need only mention here that as far as a single infinite plate is conerrned, two fundamentally important regimes may be identified: (i) at low frequencies (with respect to a “critical” frequency) the source excites subsonic flexural waves in the plate, associated with which is a trapped surface wave in which acoustic energy is transmitted only parallel to the plate (in the absence of dissipation, these waves are unattenuated); (ii) at high frequencies the source of disturbance can excite supersonic flexural plate waves which succeed in radiating a “Mach beam” towards the fluid; these supersonic flexural waves are weakly attenuated, however, and consequently the Mach beams do not reach infinity. As far as the duct problem is concerned, one may expect these phenomena to carry over, so that surface waves will appear next to each wall at low frequencies, and at high frequencies one can anticipate the appearance of near-vacuum flexural waves in the duct walls as well as small modifications to the familiar rigid-walled eigenmodes. It should, however, be pointed out that the duct parameters may be such that the coincidence frequency is much smaller than the cut-off frequency of the first higher duct mode, and one could then operate the duct at a “high” frequency in the presence of the plane mode alone. Be that as it may, at high frequencies, it transpires that whereas the flexural waves are attenuated at the expected rate, the duct modes are attenuated at a much lower rate, and an explanation is offered below. Furthermore, it is found that under certain circumstances, namely high frequencies and a “narrow” duct, the fields are no longer small perturbations on the rigid eigenmodes. The solution to the problem is easily obtained, in a formal sense, in the form of integral transforms, but it appears to be impossible to invert these transforms exactly and analytically in terms of tabulated functions, and consequently here only an asymptotic analysis at low and high frequencies is presented. Numerical inversion is, of course, another alternative, but asymptotics are sufficiently beneficial to merit their implementation, since they frequently provide a physical insight into the underlying process as well as supplying a useful check on the numerical calculations, particularly in those parametric regions where the numerical algorithms become unreliable. Furthermore, in this paper attention is confined to the duct field itself, although this field is actually intimately related to the external field in the sense that decay of the duct field in the axial direction is a consequence of acoustic radiation from the duct towards the external fluid. Finally, the results are generalized to account for duct walls which are made up of thin plates rather than membranes.

Figure

1. The co-ordinate

system.

ACOUSTIC

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IN ELASTIC

DUCTS

519

2. FORMULATION For the duct shown in Figure 1, and for time-harmonic behaviour of the form exp (-iwt), the velocity potentials in the various regions are satisfied by the following equations: regions I & III:

(v2+k;)c$

region II:

=o,

k. = w/ ao;

(2.1)

(V2+ k;)c$ = S(x)G(z - zo),

(2.2)

where S is the Dirac delta function. Applying a Fourier transform in the x-direction, where 4(k, z) = lrn 4(x, z)e -ikx dx, --a0 leads to (a2/az2 -A 2)&i = 6(z - zo),

(a2/az2 -A 2)~*,111 = 0,

A = (k2 - ki)1’2.

(2.3a, b, c)

The branch cuts from f k. are arranged in the usual manner, such that A - 1k 1as 1k I+ 00 on the real axis. The boundary conditions applied along z = *.h are the continuity of normal velocity as well as the dynamic membrane equations; the latter read, after translation into the k-plane, v,(k)=E(k)[&(k)-qS+(k)],E(k)=

vki/(k2-ki),

v=po/m,

ki =mw2/T,

where u,,, is the vertical membrane velocity, m is the membrane mass per unit area, T is its tension per unit length, and & refer to the values of the potential on the lower and upper sides of a given membrane. After invoking the radiation condition, the solutions to equations (2.3) become & = A ehz,

& =B ehz+ C eeA’- (1/2A) e--A’z-ZO’,

&II=

D e- *‘.

(2.4a, b, c)

Applying the four boundary conditions, one finds that A = (-2/2AU)(AA

e**O+a e-‘*O),

C = (A/2 U)(AA eAzo+ Lye-**O),

B = (A/2 V)(cu e *‘“+A A e-*‘“),

D = (-Z/2AU)

(2.5a, b)

e2Ah(CyeA’“+AA em”‘“), (2.5c, d)

where U=A2A2-a2,

LY= (AA -Z)

ezAh,

A=k2-k$,

Z = 2vk:.

(2.6)

These expressions simplify when to = 0, and the field is then even in z. It is convenient to non-dimensionalize the integrals, in terms of the wavenumber ko. The quantities defined in equations (2.6) then become U=A2A2-LY2=-e2K’$f,

k = kocL,

A = (p2- 1)1’2,

A =p2-N2,

77= Z/k;: = a/M4, Observe that n is inversely proportional interest in what follows, is given by a

--

1

m

47l I -me

E = 2vk,Jk;,

cy =(AA-q)e2K”, N = M-’ = k,lk,,, K = koh.

(2.7)

to w. The field in the duct itself, which is that of

cash [A(z f zo)] + AA cash [A(z -to)]} i@X dp ipx-Alz-rol

*

A’

(2.8)

520

Y.

L. SINAI

The last term in expression (2.8) corresponds, of course, to the Hankel function particular integral of equation (2.4b), but in performing an asymptotic analysis of expression (2.8) it is important to consider all the terms simultaneously. When the source is on the centreline the first integral in expression (2.8) simplifies to 1 O”A cash (AZ) exp (ipx) dp AA-CT 2lT I --oo

i&x-K,4

-

where W=2AAsinh(KA)-7

cash (AZ) dF,

(2.9)

eKA= Wo+qWl. 3. LOW FREQUENCIES

When o < 2paai/ T, n is greater than one, and in this section the asymptotic behaviour as 77+ 00 is to be evaluated. 3.1. SOURCE NEAR CENTRELINE First one can determine the locations of any poles which may exist. Putting V=AA-cx,

(3.1.1)

77=q3,

one searches for zeroes of V given by (3.1.2).

c20=/&?+a(CLcJ

One then finds that go is given by the zeroes of ~~(~~)1’2 -q3, and ‘indeed that the differences between the zeroes of AA - 77and the zeroes of V will be exponentially small as n + co. The zeroes of AA - TJcan be found exactly after the transformation to a cubic [9], and special care is required when considering the implications of various branch-cut arrangements. It transpires that if the two cuts are parallel and inclined at an angle relative to the positive real axis, which lies in the range 7r/3 to 2~/3, only two zeroes exist, lying on the real axis, Here vertical cuts are chosen, and &, is then given by

fro= *[q + {(N2+:)/w + OW3)1.

(3.1.3)

Clearly, to the condition n >>1 one must add the restriction q >>N. Remembering that the integration contour is indented above and below any singularities lying on the negative and positive real axes respectively, one can now deform the contour around the branch cuts, yielding a field determined by residue and branch cut contributions. In the present circumstances the far field branch-cut contribution is evaluated easily: &I-- -

B.C. &L>>l

The residue contributions

exp (i/x I+ irr/4) 2(2?r(x1)1’2

(3.1.4)

are, approximately,

1

cash (iz) exp (i$o]xI-2Ki)+

O(zo/z),

(3.1.5)

where i = (ki - 1)“2 -bo. Examination of the exponential terms in expression (3.1.5) reveals that the residues simply represent the surface waves associated with “image” membranes at z = *2h. These waves become exponentially small as q + 03, and their sum describes a field which possesses a minimum along the centreline and which increases monotonically towards the “real” duct walls. Naturally, these fields are allied to the subsonic flexural waves of the membranes [9].

ACOUSTIC

3.2.

SOURCE

WAVES

IN ELASTIC

DUCTS

521

OFF-CENTRE

It is helpful to write U as follows: l_J= -e4KA 9,

Q=(AA-n)2-A2A2e-4KA.

(3.2.1)

Initially, one might be tempted to disregard the last, exponentially small, term in c, but the resultant second-order pole leads to a field which grows like (x 1as (x (+ 00, which is unacceptable. The correct assessment indicates that there are four poles on the real axis, consisting of two pairs of poles, and that the two poles making up each pair are exponentially close to each other. Indeed, what is more important than the absolute position of each pole is the distance between the two poles constituting each pair, and the latter is easily estimated as $4 exp (-2Kq) in the limit r] + co. The positions of the four singularities are therefore given by fi = *cir,

-2Kq +$2[=*(&-f4e

1,

*(I_i"+&

ep2Kq)],

(3.2.2)

where & is the zero of AA - v, as discussed in section 3.1. The residue contributions therefore

(3.2.3a)

-fi~)IFG2)-~G1)1,

[-i/2($

are

where F(P) = (A/P) e

i#4KA[(hA

_ 77)e*KAcoshh(z+~~)+hAcoshA(z-zo)].

(3.2.3b)

It is interesting that expression (3.2.3a) resembles a first-order approximation to a derivative of F provided exp (ip Ix 1)varies sufficiently slowly, and one is consequently led to the conclusion that the field does indeed grow like 1x1,but only to ranges of the order (4-l exp 2Kq), beyond which the field ceases to grow in x. It should be noted that InI is multiplied by exponentially small terms which prevent the field from acquiring large values when 1x1approaches the critical region delineated above. The asymptotic branch-cut contribution is again adequately described by expression (3.1.4). 4. HIGH FREQUENCIES The term “high frequencies” q --, 0: i.e., o >>2p,az/T.

here means those frequencies

satisfying the condition

4.1. SOURCE NEAR CENTRELINE For the source near the centreline (see equation to (2.9)), it appears that under certain conditions, to be elucidated below, one may search for zeroes of W having the asymptotic form p -~(“)+77cL(1)+g(77)cL(2)+o(g).

(4.1.1)

These turn out to be P=Ci=fCLm, EL=&=fCLo, 1 EL=cL =*I&,

CL,,,=N+w!?+WI*), ~o=~o+~~~~+~3’2~~2~+o(~*), ~n=5n+77~~1)+772CLj12)+0(773),

(4.1.2)

where &, = [ 1 - (nr/K)‘]“’ , n c K/r, and A,, = -inr/K. The quantities CL,,,are analogous to the familiar membrane wavenumbers, and c(” are the duct-mode wavenumbers

522

Y. L. SINAI

modified by the non-rigid walls. Provided WA (p @))f 0, and W&L(~)), Wi (CL‘O))exist one has (1) Pi

=

-(WI/w;)

I*Jo)

(2)

CLI = -[(pjl)/w;)(&u;‘)w;

I 1

+ w;)],_,y.

(4.1.3)

For future reference, one can set down the generalization of this result. If W&A(~)), Wb (do)) . . , Wb”-” (p(O))all vanish, but Wb”’(p(O)) # 0, then there exists a cluster of u zeroes str’addling cc”), their positions being given by 6 --/&(0)+nVJ1)+O(n”), Furthermore,

~(“=[-a!W,/WIP’]1==.I~jI,

v = l/a.

(4.1.4a)

in that vicinity W acquires the asymptotic form

w-AWl”‘(p”“, fj (p -&), i-1

where fii is the set of u zeroes described in equation (4.1.4a). Then, from equations (4.1.2), ~2’ = {2NA,(l -e~2KhN)}~1, /.L::’= (4KQ))‘,

(4.1.5)

Q=l-N2,

(4.1.6)

&*I = 1/4(2KQ3)“2,

CL%,,= (2K5,&) -‘, P!?~o

AN = (N2 - 1)“2,

A,=&N2,

=[1/(2K~“A,)31{(l/tA,)[d,(1+5~lh~)+45~l-(K5,1A,)

(4.1.7a)

eei”7.

(4.1.7b)

The following observations may be made. When N > 1, CL!,!,) is real, and CL,,,describes the customary “trapped” surface wave; if N < 1, p !,? is complex, and its imaginary component quantifies the loss experienced by the supersonic wave due to acoustic radiation [9, lo]. The duct modes CL,,,however, exhibit a more complicated behaviour. The plane mode, n = 0, is unattenuated when Q > 0, but survives only to distances of the order n-‘/2 when Q < 0; the higher duct modes are invariably attenuated and survive to distances of order 77-*. The explanation for the relatively small attenuation rates of the duct modes lies in the fact that their contributions to the vertical perturbation velocity are small at the duct walls (they vanish identically when the walls are rigid), and the acoustic energy transmitted to the fluid outside the duct is therefore small compared to the energy transmitted by the walls themselves. It is clear from expressions (4.1.5)-(4.1.7) that a number of non-uniformities arise, and they will be considered below, but presently this exercise can be pursued on the assumption that the duct is operating well away from any such non-uniformity. The residue contributions are easily evaluated, done is an asymptotic assessment aforementioned assumptions, one branch point at p = 1, namely ko,

and will not be written here, and all that remains to be of the branch-cut (B.C.) contribution. In view of the need only consider the proximity of one pole to the and then expression (4.1.4) becomes

W-4KQCp The far field extimate

is therefore

i/xl -i7r/4

LD

77e

$:

-

(4.1.8)

-PO).

4~5 ?rKQ

r

-l/2

ep”‘T

dT

=-

61

l/2

( 2371

7+icj,

ei(1+‘1”x’+im’2r($,

iSljxj),

(4.1.9a)

where 6, = r&) and r is the incomplete

Gamma

function

= rj/4KQ,

(4.1.9b)

[ 131. When 6, /x I>> 1, this field decays like Ix )-1’2.

ACOUSTIC

WAVES

IN ELASTIC

One is now in a position to tackle some of the non-uniformities Specifically, these occur when

Q = O(T),

K = 0(17),

An = O(T),

523

DUCTS

in = OW2),

which may arise:

N = O(7j

I:*).

(4.1.10a-e) These conditions correspond, respectively, to plane-mode coincidence, higher-duct-mode coincidence, narrow duct, mode cut-off and “hypersonic” membrane. Of these, only the first and the third are to be considered in this paper, since the remainder can be dealt with in a similar, straightforward fashion. 4.1.1. Coincidence For coincidence one can write zV=1+7&,

(4.1.11)

and then W-2A3sinh(KA)-n(eK”

+2@A sinhKA).

(4.1.12)

It is assumed that the frequency is not so high that [i is close to p = 1; an alternative view of this restriction involves the specification of a duct width which is sufficiently small to render a value of [i which is well away from p = 1. Under such circumstances only two poles influence the B.C. contribution, but in any case additional poles in the vicinity of p = 1 can be accounted for in principle by involing expressions (4.1.4). After carrying out the various differentiations, one finds that the two pairs of poles near the branch points are given by -1 * ,I’/2P’i’,

1 f n l’*/.P,

P(‘) = (8K)-i/2.

The residues are easily calculated, and the B.C. contribution

(4.1.13)

is then

l/4

[e i’s2’x’-T’4)r($, iS21x])- e-i@Z’X’Prr’4)r(&- iS2]xj)], a2 =

771’2p% (7//8K)“2.

(4.1.14a) (4.1.14b)

When l<< Ix/<
411-

B C.

and when /xl >>8;’ this contribution

reverts to the cylindrical field,

45, - -e

B.C.

i/xl+:ff/4/2(2~lxl)1/2,

(4.1.16)

Naturally, the B.C. contribution cannot be considered in isolation, and the coupling between the poles and the branch points will be reflected in the behaviour of the sum of the residues and &_,. 4.1.2. Narrow duct On the basis of expressions (4.1.6) and (4.1.7) one expects a breakdown of the analysis when K = O(v ), and if one assumes a priori that IKA/is small near the zeroes of W, then

524

Y. L. SINAI

one may approximate

that function by

w=~[2R(&1)(/_Lz-N2)-1]-n2KA+O(r)3) =2K[/.L4-(1+N2)/L2+N2-1/212]-772&

(4.1.17a) (4.1.17b)

K=I$.

One can now write (1+N2)~=[(1-N2)2+2/R]1’2,

I’&= 1/2N2,

2 >o,

(4.1.18, 19)

and denote the zeroes of expression (4.1.17) by r; Z&ZCClyO) +77&+0(77).

(4.1.20)

The character of the field then turns out to depend on two fundamental conditions, as follows. (i) I?<&. Here E’>l, and the first approximations to @K are given by I.# =

f/u,,

d? = *pKz,

/,bK,

=

[:(I + N2)(Z + l)]“‘,

(4.1.21)

gKz

=

i&l + N*)(Z - l)]““.

(4.1.22)

Presently, ll;~~lies on the real axis, to the right of the branch point. (ii) k > Kc. Here E < 1, and all four zeroes lie on the real axis: CL:) =

*pK,,

p&=[:(1+N2)(1+z)]1’2,

(4.1.23)

&) =

*wKz,

,.‘K2=[$(1+N2)(1-E)]1’2.

(4.1.24)

As far as the small correction to these approximations

is concerned, the general result

is &,! = [A/4/~(2p*-

(4.1.25)

1 -N2)]r=Fp’.

This relation bears relevance to the question of how precisely the integration contour should be specified, and again it is convenient to consider the two fundamental configurations individually. (i) K I&. At a first glance, it might appear that j.&Kland @KZmay both lie on either side of the branch point p = 1, but if one observes (see expression (4.1.18)) that Z > I(1 - N*)/(l

+ N*)l,

(4.1.26)

one concludes immediately that @Kl may only acquire values greater than one, whereas p& is invariably smaller than one. Thus cKl lies on the real axis, as in case (i), and the contour is indented below this pole in the usual manner. As far as fi& is concerned, one notes that 2& - 1 -N* < 0, and consequently ~$1 has argument 1~12when pK, < 1, so that this pole represents a supersonic wave attenuated by acoustic radiation. Be sure that as it may, the residue field may be summarized by 411

=

res. K=Wr))

4Ksc~+N2j(CLK:[11

-gki[A

ew(i~bl-KA)

exp (ip]x]-KA)

coshAz],=&.

cosh~~l,=~,, (4.1.27)

ACOUSTIC

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525

DUCTS

For the B.C. contribution, it is clear that the far field is adequately described by expresssion (3.1.4), unless the poles which arise here happen to lie close to CL= 1, but this matter need not be pursued here, since it is accounted for in precisely the manner already outlined in section 4.1.1. 4.2. SOURCE OFF-CENTRE When the source is off-centre it is more convenient to examine the B.C. contribution before the residue contributions, and expanding @ near A = 0 shows that the first approximation to a meaningful representation of I@ in that vicinity is given by +--4KA2A3-4KvAA2-2r/AA

+q2

(4.2.1)

(indeed, neglect of the term n2 leads to a divergent integral). The cut contribution therefore approximated by that part of the integral eic”lxl

-- T2 41r

J

is

dp

h(4KA2h3-4KvAA2-2gAA

+T/~)’

which is wrapped around the vertical branch cut emanating from p = 1. Applying the transformations 1~= 1 +eilr’2 vu,

X = 771x1,

_xu J [

P(u)

(4.2.2)

one can derive the following far field integral: 3/2 eiixl

00

77

$:.-2561rK~Q~

~KQ(~u)“~ ei*‘4- (2u)-“2 e-i*‘4 du

o e

3

’

(4.2.3)

where P(u) = u3+au2+bu

+c,

a = i/2KQ,

b = -1/16K2Q2,

c = -in/128K2Q4. (4.2.4)

One can denote the three zeroes of P by z.+ and define li; = -uj. Remembering

(4.2.5)

that p-l

_ -

Pi

i j=l

U

-Uj’

p,’ = P'(Uj),

(4.2.6)

one sees that expression (4.2.3) may be expressed in the standard form related to the incomplete Gamma function, already noted in section 4.1. The zeroes of P are given by u1= -(JI1+

92)

u2 = -(ei2”‘3 JI1+ e-i2*‘3 +!j2) - a/3,

-a/3,

u3 = - (e-i2”‘3 +1+ ei2w’3+2) -a/3, f+hT = &{e+ [e2 +4(d/3)2]“2},

CL2 =

-Wih,

d =-$a2+b,

(4.2.7) e=&a3 -fab +c.

Hence 3/Z ei(x+irr/4 -77 411 B.C.

256rK2Q4

3

C pj[c7”’ e”Ix r($, CiX) -4iKQu’ff2 is1

e”j”f (-$, z&X)]. (4.2.8)

Ultimately, for large CiX, the first term in the square brackets will dominate the second, since the two terms then behave like X-1’2 and X-3’2 respectively.

526

Y. L. SINAI

For the poles, one sees first that the positions of the poles close to p = 1 are immediately given by expressions (4.2.2) and (4.2.7), and their residues are easily calculated by using expression (4.2.6). The membrane and waveguide modes are given by expression (4.1.2) (excluding pO), with two poles appearing close to N: (4.2.9) p%,

=

(2Kl,AJ1,

/L2L = -{(llfvb&P’*

ln = [l - (n7r/2K)*]“‘,

(4.2.10a)

(V~f’+~fV;l)+~L*‘(AV~)‘+~V~]}CL=5n,

(4.2.10b)

where W has been written as W=2fV,,+2qAV1+q2V2, V, = A2 sinh (2KA), V2 = exp (2KA),

VI = -A exp (2KA),

f=~2+2-N2)2~

(4.2.11)

With regard to attenuation of the membrane waves, it can be shown that when N < 1 (N real), the arguments of both values of WE’ supplied by expression (4.2.9) lie in the range 0 to rr, so that p,,, lies above the real axis, and as usual the attenuation rate is given by the imaginary component. The same comments apply to the higher duct modes, although it should be stressed that no coincidence condition is admitted in this subsection. The residues are easily calculated, and this completes the discussion of the off-centre problem, any of the non-uniformities listed in section 4.1 and introduced in this section (e.g., sinh KA = 0) being ignored.

5. EXTENSION

TO THIN PLATES

It is a straightforward matter to modify the theory to account for duct walls which are made up of thin elastic plates instead of membranes, and a brief summary of the results follows. The Fourier transform of the membrane velocity in section 2 is replaced by u,(k) = E(k)L#-(k) E(k) = vk;/(k4-

k;f),

-b+(k)l, k;l = inw*/9,

(5.1)

where $3 is the bending stiffness. Following these changes through the analysis, one finds that expression (2.8) stands, but A, N and n are now given by A =p4-N4,

N = M-’ = k,/ko,

7 = eIM6,

E =

2vkojk;.

(5.2)

E is a quantity which is independent of frequency and plate thickness, and is normally small when the walls are metallic and immersed in water.

5.1.

LOW

FREQUENCIES

For low frequencies one has 4 = $‘S

(5.1.1)

and the source-near-centreline results follow. When the source is off-centre, the distance between the adjacent poles on the real axis is estimated as 2 exp (--2Kq)/5q and

ACOUSTIC

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IN ELASTIC

527

DUCTS

expression (3.2.2) is replaced by :2=&+-e

1 5q

-2Kq .

(5.1.2)

This implies that the algebraic growth exists to ranges of the order q exp (2Kq). branch-cut estimates remain unaltered. 5.2.

HIGH

The

FREQUENCIES

For high frequencies one can define P=l+N’. Then expressions (4.1.5)-(4.1.7)

(5.2.1)

become (5.2.2)

p:’ = 4N3AN[1 -exp (-2K&)], (I’= (4KPQ)-‘, CL0

/.$ = (32KP3Q3)“‘,

A, = [: - N4.

(5.2.4)

In expressions (4.1.8) and (4.1.9), Q is replaced by PO. For coincidence, (4.1.13) becomes (I) = (16K)-‘l” CL and expression (4.1.14a) still applies, with (n/EM)”

(5.2.3)

expression (5.2.5a)

replaced by S:“, where

82 = (9/16I#“.

(5.2.5b)

For a narrow duct, express&n (4.1.17a) changes to W-2K(/.&~4-N4~2+N4-1/2k)-nKA and the first approximation

+O(n3)

(5.2.6a)

to the positions of the poles are given by n = 11/2.

[3-[2-N4[+N4-1/2k=0, The zeroes of this cubic may be determined

(5.2.6b)

in the usual way (cf. expression (4.2.7)).

6. CONCLUSIONS

An asymptotic analysis, at low and high frequencies (relative to 2poai/ T), has indicated that whilst the notions arising from the behaviour of a single fluid-loaded wave-bearing surface do carry over to the duct problem under some circumstances, situations do arise where the response of the system is new and, without the aid of hindsight, unexpected. The asymptotic calculations of the free waves are not confined to the far field, but the contributions made by the branch cuts have not been evaluated in general except in regions which are many acoustic wavelengths away from the source. The calculations have been executed both for sources on and off the duct centreline, although the latter are invariably more complicated algebraically. The results may be summarized as follows. (i) Low frequencies, source near centreline. The field comprises a geometrically decaying contribution in addition to a sum of two surface waves originating at the positions of the two “image” duct walls. (ii) Low frequencies, source off centreline. The residue contributions are now characterized by poles which are exponentially close to each other, resulting in a residue field which grows to O(1) values when Ix]= O[q-’ exp (2Kq)].

528

Y. L. SINAI

(iii) High frequencies, source near centreline. A preliminary appraisal produces the expected free wavenumbers. However, whereas the attenuation rate for supersonic flexural membrane waves scales on n, it appears that the attenuation rates for the plane duct mode scales on n3” (only when N > l), and for the higher duct modes on q2, and the explanation for this phenomenon hinges on the fact that the contributions of the duct modes to the normal velocity at a duct wall are much smaller than that made by the membrane itself, so that the duct modes sustain a smaller acoustic radiation loss than the membranes. These results indicate that the lowest mode is slightly supersonic when Q < 0 and they contradict earlier suggestions [3] that that mode is unattenuated. In any case, it transpires that the analysis breaks down in five parameter domains (see expressions (4.1.10)), and for the purpose of illustration two of these non-uniformities have been tackled, namely those of membrane coincidence and a narrow duct. In the latter case new modes arise which are not small perturbations on the rigid-walled duct eigenmodes. (iv) High frequencies, source off centreline. Here the residues are easily calculated as in case (iii), but the branch-cut contributions are more involved, and none of the non-uniformities has been dealt with. (v) Thin plates. The modifications to the results when the membranes are replaced by thin elastic plates have been outlined in section 5. ACKNOWLEDGMENT This work was supported by the Office of Naval Research, Code 222, Grant N0001477-G-0072. REFERENCES

1. P. M. MORSE and K. U. INGARD 1968 Theoretical Acoustics. New York: McGraw-Hill. 2. A. H. NAYFEH, J. E. KAISER and D. P. TELIONIS 1975 American Institute of Aeronautics and Astronautics Journal 13, 130-153. Acoustics of aircraft engine-duct systems. 3. M. C. JUNGER 1955 Journal of Applied Mechanics, Transactions of the American Society of Mechanical Engineers 77, 227-231. The effect of a surrounding fluid on pressure waves in a fluid-filled elastic tube. 4. P. W. SMITH 1957 Journal of the Acoustical Society of America 29, 721-729. Sound transmission through thin cylindrical shells. 5. E. L. SHENDEROV 1963 Soviet Physics-Acoustics 9, 178-184. Transmission of a sound wave through an elastic cylindrical shell. 6. V. N. MERKULOV, V. Yu. PRIKHOD’KOand V. V. TYUTEKIN 1978 SovietPhysics-Acoustics 24, 405-409. Excitation and propagation of normal modes in a thin cylindrical elastic shell filled with fluid. 7. M. C. JUNGER and P. W. SMITH 1955 Acoustica $47-48. Letter to the editor. 8. R. A. HARDING-PAYNE 1980 Presented at the 22nd British TheoreticalMechanics Colloquium, Cambridge. Radiation from a partly elastic infinite duct with flow. 9. D. G. CRIGHTON 1971 Journal of Fluid Mechanics 47, 625-638. Acoustic beaming and reflection from wave-bearing surfaces. 10. D. G. CRIGHTON 1979 Journal of Sound and Vibration 63, 225-235. The free and forced waves on a fluid-loaded elastic plate. 11. D. G. CRIGHTON 1980 Journal of Sound and Vibration 68, 15-33. Approximations to the admittances and free wavenumbers of fluid-loaded panels. 12. M. C. JUNGER and D. FEIT 1972 Sound, Structures and their Interaction. London. 13. A. ERDBLYI, W. MAGNUS, F.OBERHE'ITINGER and F. G. TRICOMI 1954 Tables oflntegrul Transforms Volume I. New York: McGraw-Hill. See p. 137.