32

数理解析研究所講究録
第2076巻 2018年 32-42

On localized trapped modes in a pipe‐cavity system
Mikael A. Langthjem†, Masami Nakano‡† Faculty of Engineering, Yamagata University,
Jonan 4‐chome, Yonezawa‐shi, 992‐8510 Japan‡Institute of Fluid Science, Tohoku University,
2‐1‐1 Katahira, Aoba‐ku, Sendai‐shi, 980‐8577 Japan
Abstract

The work is concerned with an analytical study of acoustic trapped modes in a cyhndricalexpansion chamber, placed in between two semi‐infinite pipes (waveguides). Trapped mode
solutions are expressed in terms of Fourier‐Bessel series, with the expansion coefficientsdetermined from a determinant condition. The roots of the determinant, expressed in termsof the wavenumber k , correspond to trapped modes. In the case of a shallow cavity, inthe sense that the cavity radius is only slightly larger than the pipe radius, asymptotic
approximations for the coeficients of the determinant can be apphed. The determinant thenreduces to a simple form with four‐rowed minors placed on a diagonal, enabling analyticalevaluation and a proof of existence of trapped modes. We consider here circumferential mode
numbers m\geq 1 . For a shallow cavity and for low values of the circumferential mode number
there is just one trapped mode in the allowable wave number domain k_{\min}<k<k_{\mathrm{m}\mathrm{w}} , wherek_{\min} is the cutoff frequency for acoustic waves in the cavity and k_{\max} is the corresponding
cutoff frequency in the pipes. This mode is symmetric about a radial axis in the center ofthe cavity.

1

Introduction

The existence of trapped modes in a variety of physical systems has received much attention
for more than sixty years and it is now known that trapped modes may be found across a wide
range of length scales, from waveguides in the ocean, over acoustic and elastic waveguides, toquantum waveguides [5]. A trapped mode, also called a localized mode, refers to an infinite
medium in a state of linear oscillations only in the vicinity of an object or a geometrical fea‐
ture which can act as a resonator. There is no radiation of energy to infinity; the energy isalso trapped, or localized. One reason for the attention these problems have received is math‐ematical: the trapped modes, characterized by a discrete wavenumber spectrum, co‐exist withthe continuous spectrum which is characteristic for guided waves in an infinite domain. The
solutions to these problems are thus non‐unique even though the problems are hnear.
The present paper gives an analytical solution to the problem of acoustic trapped modes ina cylindrical expansion chamber, placed in between two semi‐infinite pipes, i.e. waveguides, \mathrm{a}problem which previously has been investigated just numerically [2]. The existence of trappedmodes is proved in the case of a shallow cavity, shallow in the sense that the radius of the cavity
is only slightly larger than the radius of the connecting pipes.

33

2

Problem formulation
Consider a cylindrical cavity — an expansion chamber — connected with two semi‐infinite

cylindrical pipes (wave guides), as shown in figure 1.

zz=\mathrm{z}_{1}<0 z=\mathrm{z}_{2}>0

Figure 1: Sketch of the configuration.

Let the radius of the pipes be r_{0} and let the radius of the cavity be r_{1} . The length of thecavity L_{1}=z_{2}-z_{1} . Cyhndrical polar coordinates z, r, $\theta$ will be used. The propagation of soundthrough the system is governed by the wave equation\displayst le\frac{\partial^{2}$\phi$}{\partialt^{2}=c_{0}^{2}\nabl ^{2}$\phi$ ,(1)\nabla $\phi$\cdot n=0(2)

subject to the Neumann conditionon the sohd boundaries. Here $\phi$(t, z, r, $\theta$) is the velocity potential, with t being the time, c_{0} is
the speed of sound, and n is the inward pointing normal vector. The Laplacian in cylindricalpolar coordinates is given by\displayst le\nabla^{2}=\frac{1}r\frac{\partial}{\partialr}(\frac{\partial}{\partialr})+\frac{1}r^{2}\frac{\partial^{2}{\partial$\theta$^{2}+\frac{\partial^{2}{\partialz^{2} .

(3)The sound pressure p(t, z, r, $\theta$) is given by(4)

p=- $\rho$\partial $\phi$/\partial t,where $\rho$ is the fluid density. The acoustic particle velocity u=(u_{z}, u_{r, $\theta$}u) is given by\mathrm{u}=\nabla $\phi$ .

3(5)Construction of a trapped mode
We will here consider the construction of a trapped mode. By employing the method ofseparation of variables, the solution to (1) can be expressed in the form of a Fourier‐Bessel
series. In the cavity domain z_{1}<z<z_{2} the mode m solution can be written$\phi$_{1m}=\displaystyle\sum_{n=0}^{\infty}$\phi$_{\mathrm{i}mn}=\sum_{n=0}^{\infty}\{C_{mn}\mathrm{e}^{\mathrm{i}k_{1mn}z +D_{mn}\mathrm{e}^{-\mathrm{i}k_{1mn}z \}J_{m}(j_{mn}\frac{r}{r_{1} )

,

(6)

34

where m is the circumferential mode number, C_{mn} and D_{mn} are constants, J_{m}(z) is the Bessel
function of first kind and order m, j_{mn} are the zeros of J_{m}'(z) , where the dash denotes differen‐

tiation with respect to the argument z , andk_{1mn}^{2}=k^{2}- (\displaystyle \frac{j_{mn} {r_{1} )^{2} k=\frac{ $\omega$}{c_{\mathrm{O}.

(7)

A factor \mathrm{e}^{-\mathrm{i} $\omega$ t}\mathrm{e}^{\mathrm{i}m $\theta$} has been suppressed. The zeros j_{mn} are ordered such thatjm0\leq jm\mathrm{i}\leq jm2\leq.. .

(8)

It is noted also that m\leq j_{m0} [ 1 , p. 370]. The complete solution is given by $\phi$_{1}=\displaystyle \sum_{m}$\phi$_{1m} . In
the following a single, fixed value of m will be considered rather than the sum over m . This isequivalent to the approach in [8].
The function J_{m}(j_{mn}r/r_{1}) , 0 \leq r/r_{1} \leq 1 , is shown in figure 2 for n=0 , 1, 2, and 3, with
m=1 in part (a) and with m=3 in part (b). For any value of m the value of n corresponds tothe number of nodal points in the domain 0\leq r/r_{1}\leq 1 ; thus there are no nodal point for n=0,
one nodal point for n=1 , etc. As part (b) indicates, the larger the value of m the ‘flatter’ are
the graphs by small values of r/r_{1}.

J_{1}(j_{1n}r/r_{1})J_{3}(j_{3n}r/r_{1})r/r_{1} r/r_{1}Figure 2: The function J_{m}(j_{mn}r/r_{1}) plotted in the range 0\leq r/r_{1} \leq 1 with m=1 in (a) and
m=3 in (b). Plain lines: n=0 ; broken hnes: n=1 ; dotted lines: n=2 ; dash‐dot lines: n=3.

In order to construct a trapped mode we consider the following mode m solutions in the pipe

domains. In the left hand domain -\infty<z<z_{1} , the left‐going (outgoing) wave$\phi$_{0m}^{-}=\displaystyle \sum_{n=0}^{\infty}A_{mn}J_{m}(j_{mn}\frac{r}{r_{0} )\mathrm{e}^{-\mathrm{i}k_{0mn}z}, -\infty<z<z_{1}<0

,

(9)

where

k_{0mn}^{2}=k^{2}- (\displaystyle \frac{j_{mn} {r_{0} )^{2}(10)

and in the right‐hand domain z_{2}<z<\infty , the right‐going (and again outgoing) wave$\phi$_{0m}^{+}=\displaystyle \sum_{n=0}^{\infty}B_{mn}J_{m}(j_{mn}\frac{r}{r_{0} )\mathrm{e}^{\mathrm{i}k_{0mn}z}, 0<z_{2}<z<\infty

.

(11)

35

The functions k_{0mn} and k_{1mn} in (10) and (7), respectively, are specified as

k_{$\alph$7n}=\left{bgin{ary}l \sqrt{k^2}-(lr_{$\alph$}\mathfrk{B}^\underli {n})^2,k>\frac{j_mn}{r_$\alph$}\ mathr{i}\sqrt(_{$\alph$}^{\mathr{k})^2-{}um,k<\frac{j_mn}{r_$\alph$} \end{ary}\ight.

$\alpha$=0 , 1.

(12)

It is thus seen that a trapped mode may be present by mode (m, n) if the wavenumber k is in
the range\displaystyle \frac{j_{mn} {r_{1} <k<\frac{j_{mn} {r_{0} ,(13)where j_{mn}/r_{0} is the cutoff frequency for mode (m, n) for a pipe of radius r_{0} and j_{mn}/r_{1} is thecutoff frequency for mode (m, n) for a pipe of radius r_{1} . This is because, with this specificationof k , the solution has the form \mathrm{e}^{\pm \mathrm{i}|k_{1mn}z|} in the cavity domain and the form \mathrm{e}^{-|k_{0mn}z|} in the pipe
domains.

It is noted that the expansions (6), (9), (11) constitute complete sets of functions [7]. These
functions must satisfy matching and boundary conditions at the steps. At z=z_{1} these conditions
are given by$\phi$_{m0}^{-}(z_{1-}) = $\phi$_{1m}(z_{1+}) , 0<r<r_{0} ,(14)\displaystyle \frac{\partial$\phi$_{0m}^{-} {\partial z}(z_{1-}) = \frac{\partial$\phi$_{1m} {\partial z}(z_{1+}) , 0<r<r_{0} ,\displaystyle \frac{\partial$\phi$_{1m}}{\partial z}(z_{1+}) = 0, r_{0}<r<r_{1} .

(15)
(16)

Similar conditions apply at z=z_{2} . It is noted that the boundary conditions\displaystyle\frac{\partial$\phi$_{0m}{\partialr}|_{r= 0}=0for

z_{1}<z<z_{2},\displaystyle\frac{\partial$\phi$_{1m}{\partialr}|_{r= _{1}=0for

z<z_{1}, z>z_{2} ,(17)which apply as well, are exactly satisfied by the solutions (6), (9), and (11). The conditions (14)(at z=z_{1}, z_{2} ) are Dirichlet conditions, while (15) and (16) (also at z=z_{1}, z_{2} ) are Neumannconditions.

The mixed boundary value problem (6), (9), (11) and (14) -(16) (again with similar conditionsz_{2} ) can be discretized by converting the boundary conditions (14)‐(16) to aapplying at z=

Galerkin weak formulation, using the weight functions
residual equations take the forms

J_{m}(j_{mq}\displaystyle \frac{r}{r0})r and J_{m}(j_{mq}\displaystyle \frac{r}{r_{1} )r . The\displaystyle \int_{0}^{r_{0} \{$\phi$_{0m}(z_{1-})-$\phi$_{1m}(z_{1+})\}J_{m}(j_{mq}\frac{\prime r}{r_{0} ) rdr=0 ,\displaystyle \int_{0}^{r_{0} \{\frac{\partial$\phi$_{0m} {\partial z}(z_{1-})-\frac{\partial$\phi$_{1m} {\partial z}(z_{1+})\}J_{m}(j_{mq}\frac{r}{r_{1} ) rdr=0 ,

(18)

(19)

and

\displaystyle \int_{r0}^{r_{1} \frac{\partial$\phi$_{1m} {\partial z}(z_{1+})J_{m}(j_{mq}\frac{r}{r_{1} ) rdr=0 ,for q=0 , 1, 2,\cdots(20). Combining (19) with (20) we get\displaystyle \int_{0}^{r_{0} \frac{\partial$\phi$_{0m} {\partial z}\partial z(z_{1-})J_{m}(j_{m\mathrm{q} \frac{r}{r_{1} )rdr-\int_{0}^{r_{1} \frac{\partial$\phi$_{1m} {\partial z}(z_{1+})J_{m}(j_{mq}\frac{r}{r_{1} ) rdr=0 .

(21)

36

4

Nondimensionalization
Next we make the equations nondimensional by introducing the nondimensional parameters\displaystyle \tilde{r}=\frac{r}{r_{0} , \hat{r}=\frac{r}{r_{1} , \tilde{z}=\frac{z}{r_{0} , \tilde{k}=kr_{0}, $\alpha$=\frac{r_{0} {r_{1} .

(22)\tilde{k}_{0mn}=\mathrm{i}\sqrt{j_{mn}^{2}-\tilde{k} , \tilde{k}_{1mn}=\sqrt{\tilde{k}-$\alpha$^{2}j_{mn}^{2} .

(23)

Also, let

The three series solutions (6), (9), and (11) are now expressed in the forms$\phi$_{\mathrm{i}m=\displaystyle\sum_{n=0}^{\infty}\{C_{mn}\mathrm{e}^{\mathrm{i}\tilde{k}_{1mn}(\tilde{z}-\tilde{z}_{1})+D_{mn}\mathrm{e}^{-\mathrm{i}\tilde{k}_{1mn}(\tilde{z}-\overline{z}_{2})\} frac{1}{\tilde{k}_{1mn}\frac{J_{m}(j_{mn}$\alpha$\tilde{r}){J_{m}^{2}(j_{mn})\frac{2j_{mn}^{2}{j_mn}^{2}-m^{2}$\phi$_{0m}^{-}=\displaystyle\sum_{n=0}^{\infty}A_{mn}\mathrm{e}_{J_{m}^{2}(j_{mn})j_{mn}^{2}^{-\mathrm{i}\tilde{k}_{0mn}(1}\overline{z}-\overline{z})^{J_{m}(j_{mn}\tilde{r})2j_{mn}^{2}\overline{-m^{2}}’-\infty<\overline{z}<\tilde{z}_{1} <0,

,

(24)

(25)

and

$\phi$_{0m}^{+}=\displaystyle\sum_{n=0}^{\infty}B_{mn}\mathrm{e}^{\mathrm{i}\tilde{k}_{0mn}(\tilde{z}-\tilde{z}_{2}) \frac{J_{m}(j_{mn}\tilde{r}) {J_{m}^{2}(j_{mn}) \frac{2j_{mn}^{2} {j_{mn}^{2}-m^{2} ,0<\tilde{z}_{2}<\tilde{z}<\infty
5

.

(26)

Determinant condition

The matching and boundary conditions (18) and (21), evaluated at \overline{z}=\tilde{z}_{1} and \tilde{z}=\tilde{z}_{2} , canwith (24), (25), and (26) inserted be written\displaystyle\sum_{n=0}^{\infty}\{$\delta$_{qn}A_{mn}-\mathcal{F}_{1mqn}C_{mn}-\mathcal{F}_{1mqn}\mathrm{e}^{\mathrm{i}\overline{k}_{1mn}(\overline{z}_{2}-\tilde{z}_{1}) D_{mn}\ =0\displaystyle\sum_{n=0}^{\infty}\{$\delta$_{qn}B_{mn}-\mathcal{F}_{1mqn}\mathrm{e}^{\mathrm{i}\tilde{k}_{1mn}(\tilde{z}2-\overline{z}_{1}) C_{mn}-\mathcal{F}_{1mqn}D_{mn}\ =0,\displaystyle\sum_{n=0}^{\infty}\{ overline{f}_{2mqn}A_{mn}+$\delta$_{qn}C_{mn}-$\delta$_{qn}\mathrm{e}^{\mathrm{i}\overline{k}_{1mn}(\tilde{z}_{2}-\overline{z}_{1}) D_{mn}\ =0,\displaystyle \sum_{n=0}^{\infty}\{21,,for q=0 , 1,

\cdots(27), where $\delta$_{\mathrm{q}n} is Kronecker’s delta and\displaystyle\mathcal{F}_{1mqn}=\frac{2$\alpha$^{2}j_{mn}^{2} {j_{mn}^{2}-m^{2} $\alpha$^{2}j_{mn}^{2}-j_{mq}^{2}\tilde{k}_{1mn}^{-1}\frac{1}{J_{m}^{2}(j_{mn})(28)\displaystyle\mathcal{F}_{2mqn}=\frac{2$\alpha$^{2}j_{mn}^{2} {j_{mn}^{2}-m^{2} j_{mn^{-$\alpha$^{2}j_{mq}^{2} ^{2^{\tilde{k}_{0mn} \frac{1}{J_{m}^{2}(j_{mn})(29)\times\{ $\alpha$ j_{mn}J_{m+1}($\alpha$_{jmn})J_{m}(jmq)-J_{m}( $\alpha$ jmn)J_{m+1(jmq})\},\times\{j_{mn}J_{m+1(jmn})J_{m}( $\alpha$ jmq)- $\alpha$ J_{m}(jmn)J_{m+1}( $\alpha$ j_{mq})\}.

37

The fUmctions (28) and (29) have been obtained by employing the relations [9, Ch. V]\displaystyle \int_{0}^{1}J_{m}(j_{mn}\tilde{r})J_{m}(j_{m\mathrm{q} \tilde{r})\tilde{r}d\overline{r}=\frac{1}{2}\frac{$\delta$_{qn} {j_{mq}^{2} [j_{mn}^{2}-m^{2}]\{J_{m}(j_{mn})\}^{2}

,

(30)

and

\displaystyle \int_{0}^{1}J_{m}(j_{mn} $\alpha$\hat{r})J_{m}(j_{mq}\hat{r})\hat{r}d\hat{r}=\frac{1}{j_{mn}^{2}-j_{m\mathrm{q} ^{2}

(31)

\times\{-\}.

Written in matrix form, (27) can be expressed as
Ab =\mathrm{O} ,where \mathrm{b}=\{A_{m0}B_{m0}C_{m0}D_{m0}A_{m1}B_{m1}C_{m1}D_{m1} . . . \}^{T} .

denoted by a_{qn}, q, n=0 , 1,

\cdots(32)
The elements of the matrix A are. The condition for the existence of a trapped mode at a given

value of the wavenumber \tilde{k} is that the infinite determinant\triangle=\det \mathrm{A}=|a_{qn}|_{0}^{\infty}=0 .

(33)

In terms of the nondimensional variables a trapped mode may exist in mode (m, n) in thewavenumber range$\alpha$ j_{mn}<\tilde{k}<j_{mn} .

6

(34)

Symmetric and antisymmetric solutions

It is seen directly from (27) that two types of solutions are possible: symmetric solutions,$\phi$(z) = $\phi$(-z) , and antisymmetric solutions, $\phi$(z) =- $\phi$(-z) . By symmetric solutions, A_{mn}=B_{mn} , and C_{mn}=D_{mn} , for any m, n . By antisymmetric solutions, A_{mn}= -B_{mn} , and C_{mn}=-D_{mn} , for any m, n . In both cases, (27) reduces to the following system of just two equations:\displaystyle\sum_{n=0}^{\infty}\{$\delta$_{qn}A_{mn}-\mathcal{F}_{1mqn}C_{mn}(1\pm\mathrm{e}^{\mathrm{i}\overline{k}_{1mn}(\tilde{z} 21\displaystyle\sum_{n=0}^{\infty}\{ mathcal{F}_{2mqn}A_{mn}+$\delta$_{qn}C_{mn}(1\mp\mathrm{e}^{\mathrm{i}\overline{k}_{1mn}(\tilde{z}_{2}-\tilde{z}_{1}) \}=0,,

q=0 , 1,

\cdots(35). In each of these two equations the upper sign (before the exponential function)

gives the symmetric solution, while the lower sign gives the antisymmetric solution.

7

The case of a shallow cavity
Here we consider the case where $\alpha$\approx< 1 , that is to say, $\alpha$ is shghtly smaller than one. Use

will be made of the expansion [1, p. 363]

J_{m}( $\alpha$\displaystyle \tilde{r}j_{mn})=$\alpha$^{m}\sum_{p=0}^{\infty}\frac{1}{p!}(-1)^{p}($\alpha$^{2}-1)^{p}J_{m+p}(\tilde{r}j_{mn})

.

(36)

38

Let $\epsilon$=1- $\alpha$ . For 0< $\epsilon$\ll 1 we can write

J_{m}( $\alpha$\tilde{r}j_{mn})=$\alpha$^{m}J_{m}(\overline{r}j_{mn})+O( $\epsilon$) .

(37)

The functions \mathcal{F}_{1mqn} and \mathcal{F}_{2mqn} , defined by (28) and (29), then reduce to\mathcal{F}_{1mqn}\approx$\delta$_{qn}$\alpha$^{m+2}\tilde{k}_{1mn}^{-1}, \overline{J^{-} _{2mqn}\approx$\delta$_{\mathrm{q}n}$\alpha$^{m}\tilde{k}_{0mn} .

(38)

A comparison between the‘full’ functions \mathcal{F}_{1mqn} and \overline{J^{-} _{2mqn} and their asymptotic approximationsn
0 , and with \overline{k}1, q(38) is shown in figure 3. The graphs are obtained with m==\displaystyle \frac{1}{2}j_{10}( $\alpha$+1) . As would be expected, the agreements are very good when==$\alpha$is close to one, i.e.

when $\epsilon$=1- $\alpha$ is small, and less good otherwise. The trends are similar for any values of m, q,n, and \tilde{k}.

\mathcal{F}_{1mqn}, \mathcal{F}_{2mqn}
3

2

1

0

$\alpha$^{-1}=r_{1}/r_{0}Figure 3: Comparisons between the functions \overline{f^{\sim} _{1mqn} and \overline{f}_{2mqn} , defined by (28) and (29), andtheir asymptotic approximations, defined by (38).
Employing (38) the system (27) reduces to\displaystyle\sum_{n=0}^{\infty}$\delta$_{qn}\{A_{mn}-\frac{$\alpha$^{m+2}{\tilde{k}_{1mn}C_{mn}-\frac{$\alpha$^{m+2}{\tilde{k}_{1mn}\mathrm{e}^{\mathrm{i}\tilde{k}_{1mn}()\tilde{z}2-\overline{z}1D_{mn}\=0\displaystyle\sum_{n=0}^{\infty}$\delta$_{qn}\{B_{mn}-\frac{$\alpha$^{m+2}{\tilde{k}_{1mn}\mathrm{e}^{\mathrm{i}\tilde{k}_{1mn}()\overline{z}_{2-\tilde{Z}1C_{mn}-\frac{$\alpha$^{m+2}{\tilde{k}_{1mn}D_{mn}\=0,\displaystyle\sum_{n=0}^{\infty}$\delta$_{qn}\{$\alpha$^{m}\overline{k}_{0mn}A_{mn}+C_{mn}-\mathrm{e}^{\mathrm{i}\tilde{k}_{1mn}(\tilde{z}2-\tilde{z}_{1}) D_{mn}\ =0,\displaystyle\sum_{n=0}^{\infty}$\delta$_{qn}\{$\alpha$^{m}\overline{k}_{0mn}B_{mn}-\mathrm{e}^{\mathrm{i}\tilde{k}_{1mn}(\tilde{z}_{2}-\tilde{z}_{1}) C_{mn}+D_{mn}\ =0,,

q=0 , 1,(39)

. That is to say, the determinant decouples into four‐rowed minors [4, p. 20], i.e.,
blocks of size 4\times 4 , placed on a diagonal, and with zeros elsewhere. The (determinant of the)\cdots39

minor no. n is evaluated as$\Delta$_{n}=2\displaystyle\frac{\mathrm{e}^{\mathrm{i}\overline{k}_{1mn}(\tilde{z}-\tilde{z}_{1})2}{\tilde{k}_{1mn}[2\tilde{k}_{0mn}\overline{k}_{1mn}$\alpha$^{2(m+1)}\cos(\overline{k}_{1mn}(\tilde{z}_{2}-\overline{z}_{1})(40)-\mathrm{i}(\overline{k}_{0mn}^{2}$\alpha$^{4(m+1)}+\tilde{k}_{1mn}^{2})\sin(\tilde{k}_{1mn}(\tilde{z}_{2}-\tilde{z}_{1}) ].

The original determinant is evaluated as $\Delta$=\displaystyle \prod_{n=0}^{\infty}\triangle_{n}.
Consider first the case where \tilde{k}_{\approx}> $\alpha$ j_{mn} ( \tilde{k} is just slightly larger than $\alpha$ j_{mn} ), specifically let$\delta$^{2}=\overline{k}^{2}-$\alpha$^{2}j_{mn}^{2}, 0<$\delta$^{2}\ll 1 . Then \tilde{k}_{1mn}= $\delta$ and \tilde{k}_{0mn}\approx \mathrm{i}j_{mn}(1-$\alpha$^{2})^{\frac{1}{2} . Inserting this into(40) gives\displaystyle\frac{1}{2}\frac{\tilde{k}_{1mn}{\mathrm{e}^{\mathrm{i}\tilde{k}_{1mn}(\overline{z}_{2}-\overline{z}_{1}) \triangle_{n}\ap rox\mathrm{i}[\mathrm{S}(41)-(-j_{mn}^{2}$\alpha$^{4(m+1)}(1-$\alpha$^{2})+$\delta$^{2})\sin( $\delta$(\tilde{z}_{2}-\tilde{z}_{1}) ]\approx \mathrm{i} $\delta$[2j_{mn}(1-$\alpha$^{2})^{\frac{1}{2} $\alpha$^{2(m+1)}+j_{mn}^{2}$\alpha$^{4(m+1)}(1-$\alpha$^{2})(\overline{z}_{2}-\tilde{z}_{1})].

Consider next the case where \overline{k} \ap rox< j_{mn} ( \tilde{k} is just slightly smaller than j_{mn} ), specifically let$\delta$^{2}=j_{mn}^{2}-\overline{k}^{2}, 0<$\delta$^{2}\ll 1 . Then \tilde{k}_{0mn}=\mathrm{i} $\delta$ and \tilde{k}_{1mn}\approx j_{mn}(1-$\alpha$^{2})^{\frac{1}{2} . Inserting this into (40)gives\displaystyle\frac{1}{2}\frac{\tilde{k}_{1mn} {\mathrm{e}^{\mathrm{i}\tilde{k}_{1mn}(\tilde{z}_{2}-\tilde{z}1 )}$\Delta$_{n}\ap rox\mathrm{i}[2$\delta$j_{mn}(1-$\alpha$^{2})^{\frac{1}{2} $\alpha$^{2(m+1)}\cos(j_{mn}\sqrt{1-$\alpha$^{2} (\tilde{z}_{2}-\tilde{z}_{1})(42)-(-$\delta$^{2}$\alpha$^{4(m+1)}+j_{mn}^{2}(1-$\alpha$^{2}) \sin(j_{mn}\sqrt{1-$\alpha$^{2} (\tilde{z}_{2}-\tilde{z}_{1}) ]\approx \mathrm{i}[2 $\delta$ j_{mn}(1-$\alpha$^{2})^{\frac{1}{2} $\alpha$^{2(m+1)}-(-$\delta$^{2}$\alpha$^{4(m+1)}+j_{mn}^{2}(1-$\alpha$^{2}) (j_{mn}\sqrt{1-$\alpha$^{2} (\tilde{z}_{2}-\tilde{z}_{1})]\approx -\mathrm{i}j_{mn}^{3}(1-$\alpha$^{2})^{\frac{3}{2} (\overline{z}_{2}-\tilde{z}_{1}).

-\displaystyle\mathrm{i}^{\tilde{k} \frac{1}{2}\ovalbox{\t \smal REJECT}_{2}\triangle_{n}\mathrm{e}^{1} is real and >0 for \overline{k}_{\approx}> $\alpha$ j_{mn} , whilein the second case, (42) shows that -\displayst le\mathrm{i}^ \overline{k} \frac{1} 2 \ovalbox{\t smal REJ CT}_{1}\triangle_{n}\mathrm{e}^{\mathrm{i}k_{1mn}(\overline{z}_{2}-\overline{z}) is real and <0 for \tilde{k}_{\approx}<j_{mn} . It is noted,
In the first case, (41) clearly shows that

also, that neither \mathrm{e}^{\mathrm{i}\tilde{k}_{1mn}(\tilde{z}2-\overline{z}_{1})} nor \tilde{k}_{1mn} is zero in $\alpha$ j_{mn}<\tilde{k}<j_{mn} . Thus \triangle_{n} (and thus \triangle ) has,at least, one zero for $\alpha$ j_{mn}<\tilde{k}<j_{mn} . To locate one or more of these zeros, set \triangle_{n}=0 and let\tilde{k}=\tilde{s}j_{mn} . Using (40), the expression for \triangle_{n}=0 can be written as\displaystyle\mathrm{t}\mathrm{m}( \tilde{z}_{2}-\tilde{z}_{1})j_{mn}(\tilde{s}^{2}-$\alpha$^{2})^{\frac{1}{2})=$\alpha$^{2(m+1)}\frac{(1-\tilde{s}^{2})^{\frac{1}{2}(\tilde{s}^{2}-$\alpha$^{2})^{\frac{1}{2} {$\alpha$^{4(m+1)}(\tilde{s}^{2}-1)+(\tilde{s}^{2}-$\alpha$^{2})

.

(43)

Assuming that \tilde{s}\approx $\alpha$<1\approx we can use that \tan(\cdots)\approx(\cdots) . Using also that (1-\displaystyle \tilde{s}^{2})^{\frac{1}{2} \approx 1-\frac{1}{2}\tilde{s},(43) gives thator equivalently\displayst le\tilde{s}=\{ frac{$\alpha$^{2(m+1)}+(\tilde{z}_2}-\tilde{z}_1})j_{mn}($\alpha$^{2}+$\alpha$^{4(m+1)} {\frac{1}2$\alpha$^{2(m+1)}+(\tilde{z}_2}-\tilde{z}_1})j_{mn}(1+\grave{$\alpha$}^{4(m+1)} \^{\frac{1}2(44)\displaystyle\tilde{s}^{2}=1+\frac{\frac{1}2$\alpha$^{2(m+1)}-(\tilde{z}_{2}-\tilde{z}_{1})j_{mn}(1-$\alpha$^{2}){\frac{1}2$\alpha$^{2(m+1)}+(\tilde{z}_{2}-\tilde{z}_{1})j_{mn}(1+$\alpha$^{4(m+1)}

(45)

.

In order to satisfy the condition $\alpha$ j_{mn} < \overline{k} <j_{mn} it is necessary that $\alpha$ <\tilde{s}< 1 . It is clearlypossible to choose a cavity length \tilde{z}_{2}-\tilde{z}_{1} such that this condition is satisfied. In particular,

40\tilde{s}< 1 if \tilde{z}_{2}-\tilde{z}_{1} >$\alpha$^{2(m+1)}/\{(1-$\alpha$^{2})j_{mn}\} . (It is noted that the lowest possible value of j_{mn} isj_{10}=1.8412\cdots.)
We finaly discuss the limit $\alpha$\rightarrow 1 , where the cavity disappears altogether. In this hmit (13)degenerates into a single point, \tilde{k}=j_{mn} . Then, from (23) we obtain \tilde{k}_{0mn} =\tilde{k}_{1mn} =0 . Thus

there is no wave motion and no meaningful solution to the problem, as there shouldn’t be either.
In the following we check (45) in this limit. As $\alpha$ \rightarrow 1, \tilde{k} \rightarrow \overline{k}_{\max} , that is, \tilde{k} \rightarrow j_{mn} , thecutoff frequency of the pipes. This means that \tilde{s}\rightarrow 1 in (43). Then, in (44) and (45), the squareroot function (1-\tilde{s}^{2})^{\frac{1}{2} needs to be approximated not for \tilde{s} small but for 1-\tilde{s}^{2} small. Here use

can be made of Lanczos’s power expansion of y=\sqrt{x} in terms of Chebyshev polynomials [3, \mathrm{p}.
485]. The first approximation is simply y_{1} =x . Using this approximation in (43), (45) takes
the form\displaystyle\overline{s}^{2}\ap rox1-\frac{(\tilde{z}_{2}-\tilde{z}_{1})j_{mn}(1-$\alpha$^{2}) {$\alpha$^{2(m+1)}+(\tilde{z}_{2}-\tilde{z}_{1})j_{mn}(1+$\alpha$^{4(m+1)}

,

(46)

which correctly gives that \tilde{s}\rightarrow 1 , and thus that \tilde{k}\rightarrow\overline{k}_{\max}=j_{mn} , as $\alpha$\rightarrow 1.

8Symmetric and antisymmetric solutions in the case of a shal‐
low cavity
Employing the shallow cavity approximation (38) to the two equations (35), they reduce to\displaystyle\sum_{n=0}^{\infty}$\delta$_{qn}\{A_{\mathfrak{m}n-\frac{$\alpha$^{m+2}{\tilde{k}_{1mn}C_{mn}(1\pm\mathrm{e}^{\mathrm{i}\overline{k}_{1mn}(\overline{z}2-\overline{z}_{1}) \}=0
\displaystyle\sum_{n=0}^{\infty}$\delta$_{\mathrm{q}n \{$\alpha$^{m}k_{0mn}A_{mn}+C_{mn}(1\mp\mathrm{e}^{\mathrm{i}\overline{k}_{1mn}(\tilde{z} 2-\tilde{z}_{1}) \}=0.
,(47)

Here the upper signs correspond to the solution \mathrm{s}\mathrm{y}\mathrm{n}$\alpha$\mathrm{n}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{C}_{\backsla h} about the r axis, while the lower
signs correspond to the antisynnnetric solution. The determinant of the nth minor is$\Delta$_{n}=1\displaystyle\mp\mathrm{e}^{\mathrm{i}\tilde{k}_{1mn}(\overline{z}_{2}-\overline{z}_{1})+$\alpha$^{2(m+1)}\frac{\tilde{k}_{0mn}{\tilde{k}_{1mn}(1\pm\mathrm{e}^{\mathrm{i}\tilde{k}_{1mn}(\tilde{z}_{2}-\tilde{z}_{1})

.

(48)

We consider first the symmetric case, that is, the upper signs in (48). With these signs (4S)
can be written as$\Delta$_{n}= 2\mathrm{e}^{\mathrm{i}\frac{1}{2}\tilde{k}_{1mn}(\overline{z}_{2}-\tilde{z}_{1})}\times(49){ $\alpha$^{2(m+1)}\displaystyle\frac{\tilde{k}_{0mn}{\overline{k}_{1mn}\cos(\frac{1}{2}\tilde{k}_{1mn}(\overline{z}_{2}-\overline{z}_{1}) sin (\displaystyle\frac{1}{2}\tilde{k}_{1mn}(\tilde{z}_{2}-\tilde{z}_{1}) }.
-\mathrm{i}Consider first the case where \tilde{k} \ap rox> $\alpha$ j_{mn} ( \tilde{k} is just slightly larger than $\alpha$ j_{mn} ), as done in theprevious section. Then \tilde{k}_{1mn}=$\delta$ and \tilde{k}_{0mn}\approx \mathrm{i}j_{mn}(1-$\alpha$^{2})^{\frac{1}{2} . Inserting these approximations

into (49) it can be written\displaystyle \triangle_{n}\ap rox 2$\alpha$^{2(m+1)}j_{mn}\sqrt{1-$\alpha$^{2} \{-\frac{1}{2}(\tilde{z}_{2}-\tilde{z}_{1})+\mathrm{i}$\delta$^{-1}\} .

(50)

Consider next the case where \overline{k}_{\approx}<j_{mn} ( \tilde{k} is just slightly smaller than j_{mn} ). Then \overline{k}_{0mn}=\mathrm{i} $\delta$ and\tilde{k}_{1mn}\approx j_{mn}(1-$\alpha$^{2})^{\frac{1}{2} . Inserting these expressions into (49) gives\displaystyle \triangle_{n}\ap rox\frac{1}{2}j_{mn}^{2}(1-$\alpha$^{2})(\tilde{z}_{2}-\tilde{z}_{1})^{2}-\mathrm{i}j_{mn}\sqrt{1-$\alpha$^{2} (\tilde{z}_{2}-\tilde{z}_{1}) .

(51)

41

Thus both the real part and the imaginary part change signs. This shows that a symmetrictrapped mode exists in the domain $\alpha$ j_{mn}<\tilde{k}<j_{mn}.

Consider next the antisymmetric case, that is, the lower signs in (48). Then (48) can be
written\triangle_{n} = 2\mathrm{e}^{\mathrm{i}\frac{1}{2}\tilde{k}_{1mn}(\overline{z}-\tilde{z}_{1})}2\times(52){ \displaystyle\cos(\frac{1}{2}\tilde{k}_{1mn}(\tilde{z}_{2}-\overline{z}_{1}) -\displayst le\mathrm{i}$\alpha$^{2(m+1)}\frac{\tilde{k}_{0mn}{\tilde{k}_{1mn} sin (\displaystyle\frac{1}{2}\tilde{k}_{1mn}(\overline{z}_{2}-\overline{z}_{1}) }.

In the case where \tilde{k}_{\approx}> $\alpha$ j_{mn} (52) can be approximated as\triangle_{n}\approx 2+$\alpha$^{2(m+1)}j_{mn}\sqrt{1-$\alpha$^{2} (\tilde{z}_{2}-\overline{z}_{1})+\mathrm{i}O( $\delta$) .

(53)

In the case where \tilde{k}<\approx jmn we get\triangle_{n}\approx 2+\mathrm{i}j_{mn}\sqrt{1-$\alpha$^{2} (\tilde{z}_{2}-\tilde{z}_{1}) .

(54)

The positive real parts in both (53) and (54) indicate that there is no antisymmetric trappedmode in the case of a shallow cavity.

9

Conclusion
In this paper we have carried out an analytical study of acoustic trapped modes in a cyhn‐

drical expansion chamber, connected with two semi‐infinite pipes that are waveguides. The
problem has been solved by employing the method of separation of variables, giving Fourier‐Bessel expansion‐type of solutions in each of the three subregions left pipe, expansion chamber,and right pipe. The solutions for each of these three regions have been hnked via boundary‐ andmatching‐conditions, giving an infinite determinant for which the roots correspond to trappedmodes. For the case of a shallow cavity—i.e. the cavity radius is only slightly larger than thepipe radius—it has been found that the infinite determinant decouples into four‐rowed minors(sub‐determinants of dimension 4\times 4 ) placed along a diagonal, enabling analytical evaluation
into a simple expression. Asymptotic results predicting the existence of trapped modes havebeen given in this hmit for which there, with low values of the circumferential mode number m,
is just one trapped mode in the allowable wave number domain, k_{\min}<k<k_{\max} . It is shownthat this mode is symmetric about a radial axis in the center of the cavity.
While we have restricted the analytical studies of the present paper to the case of a shallow

cavity, there is mother interesting hmit case that may be studied analytically, namely that ofa narrow cavity ( \overline{z}_{2}-\tilde{z}_{1} small). It may be suspected that a deep, narrow cavity will, in effect,
approach a three‐dimensional Helmholtz resonator, that is, a three‐dimensional version of thefirst problem studied in [6]. This may be of interest to examine in future work.
Acknowledgements
We wish to thank Professor Nobumasa Sugimoto of Kansai University for suggesting to us
the problem studied. The financial support by Institute of Fluid Science, Tohoku University,

via a ‘Collaborative Research Project 2016‐17 (project codes J16073 and \mathrm{J}17\mathrm{I}030 ) is gratefully
acknowledged.

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References[1] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions. Dover Publications,
New York, 1965.

[2] S. Hein and W. Koch. Acoustic resonances and trapped modes in pipes and tunnels. J. Fluid
Mech., 605:401−428, 2008.

[3] C. Lanczos. Applied Analysis. Dover Publications, New York, 1988.

[4] L. Mirsky. An Introduction to Linear Algebra. Dover Publications, New York, 1990.

[5] J. Postnova and R. V. Craster. Trapped modes in elastic plates, ocean and quantum waveg‐
Uides. Wave Motion, 45:565−579, 2008.

[6] N. Sugimoto and H. Imahori. Localized mode of sound in a waveguide with Helmholtz
resonators. J. Fluid Mech., 546:89−111, 2005.

[7] G. P. Tolstov. Fourier Senes. Dover Publications, New York, 1976.
[8] $\Gamma$ . Ursel. Trapped modes in a circular cylindrical acoustic waveguide. Proc. R. Soc. Lond.
A, 435:575−589, 1991.

[9] G. N. Watson. Theory of Bessel Functions. Cambridge University Press, Cambridge, 1944.