# Acoustic scattering from a wave-bearing cavity with flexible inlet and
outlet

Muhammad Afzal111Department of Mathematics, Capital University of Science and
Technology, Islamabad, 46000, Pakistan<EMAIL_ADDRESS><EMAIL_ADDRESS>Hazrat Bilal111Department of Mathematics, Capital
University of Science and Technology, Islamabad, 46000, Pakistan
<EMAIL_ADDRESS><EMAIL_ADDRESS>Naveed Ahmed222Gulf University
for Science & Technology, Block 5, Building 1, Mubarak Al-Abdullah Area, West
Mishref, Kuwait<EMAIL_ADDRESS>Abdul Wahab333Department of Mathematics,
School of Sciences and Humanities, Nazarbayev University, 53, Kabanbay Batyr
Avenue, 010000, Astana, Kazakhstan<EMAIL_ADDRESS>444Address all
correspondence to A. Wahab at<EMAIL_ADDRESS>
###### Abstract

In this article, we substantiate the appositeness of the _mode-matching
technique_ to study the scattering response of bridging elastic plates
connecting two flexible duct regions of different heights. We present two
different solution schemes to analyze the structure-borne and fluid-borne
radiations in the elastic plate-bounded waveguide. The first scheme
supplements the mode-matching technique with the so-called _tailored-Galerkin
approach_ which uses a solution ansatz with homogeneous and integral parts
corresponding to the vibrations of the bridging elastic plate and the cavity,
respectively. In the second scheme, we supplement the mode-matching technique
with the _modal approach_ wherein the displacement of the bridging elastic
plate is projected onto the eigenmodes of the cavity. To handle the non-
orthogonality of the eigenfunctions, we invoke generalized orthogonality
relations. An advantage of the proposed mode-matching schemes is that they
provide a convenient way of incorporating a variety of edge conditions on the
joints of the plates, including clamped, pin-joint, or restraint connections.
The numerical analysis of the waveguide scattering problems substantiates that
edge connections on the joints have a significant impact on the scattering
energies and transmission loss.

Keywords: mode-matching technique, tailored-Galerkin approach, modal approach,
wave bearing-cavity, waveguide scattering, fluid-borne radiations, elastic
plate-bounded waveguide.

##  1 Introduction

The problems involving wave scattering at structural discontinuities are
challenging as well as intriguing. Because of their numerous applications,
they have piqued the interest of engineers and scientists. Abrupt geometric
variations, changes in the material properties of the medium and bounding
walls, and the varying conditions on joints or edges in acoustic and
electromagnetic waveguides significantly influence the propagation and
scattering of waves [1, 2, 3, 4, 5, 6]. Such characteristics are pertinent,
for instance, for designing active and passive noise control devices such as
silencers, where the aim is to reduce pressure fluctuations or attenuate
unwanted noise. These structural discontinuities also play a critical role in
the designs of solid structures to reduce the vibrations generated by engines,
air flows, and gearboxes and the low-frequency vibrations of thin elastic
plates that cover the structural components in submarines [7, 8, 9, 10, 11,
12].

To understand how energy propagates in complex structures and scatters at
abrupt variations, flanges, and joints, it is necessary to model and analyze
fundamental problems. The current research focuses on a fluid-structure
coupled wave scattering problem with elastic plates bounding a fluid matrix.
The geometric structure has multiple joints and scatters. Note that the
reflection and transmission of waves at a planar interface of two plates
having different material properties are well understood; see, for instance,
[13]. However, the situation becomes complicated if the plates are immersed in
a fluid, given the leakage of structural energy at the edges of the fluidic
medium. An important acoustic scattering problem involving the diffraction of
the acoustic waves from a semi-infinite plate embedded in a fluid was
considered by Cannell [14]. He used the Wiener-Hopf technique to solve the
resulting boundary value problem. Similar studies were conducted, for
instance, in [15, 16, 17, 18].

A framework for deriving the analytic solutions of boundary value problems
consisting of the Helmholtz equation and high-order boundary conditions (for
elastic membranes and elastic plates) subject to various edge conditions is
offered by Lawrie and Abraham [19]. Their technique, coined the mode-matching
technique (MMT), relies on a discrete description of the spectrum rather than
a continuous formulation, in contrast to the integral transform-based
approaches. It provides a convenient way to work with a solution ansatz in
terms of the eigenfunctions of the associated eigenvalue problems. It is
important to mention that the eigenfunctions corresponding to the elastic
membranes and elastic plates (with high-order boundary conditions) are non-
orthogonal and linearly dependent. Therefore, the so-called generalized
orthogonality relations are indispensably employed to achieve a convergent
solution, see, for instance, [20]. For a variety of interesting problems
catered for using the MMT, we refer interested readers to [21, 22, 23, 24, 25,
26, 27, 28]. Recently, Nawaz and Lawrie [29] used the MMT for a scattering
problem involving a rigid-soft flange at the discontinuous interface of two
semi-infinite elastic plates bounding a fluid matrix. The approach was
followed in [30, 31, 32] to analyze the impact of acoustic linings along the
flanges of a flexible expansion chamber.

The current study builds upon the ideas presented by Lawrie and Afzal [33] to
analyze the acoustic scattering from a bridging membrane-height discontinuity
at the interface of an infinite membrane-bounded waveguide. The MMT was
supplemented with Galerkin and tailored-Galerkin approaches to derive the
solution to the scattering problem. The Galerkin method is based on the
assumption of prior solutions that change depending on the conditions at the
bridging membrane’s edges. In the tailored-Galerkin approach, the fluid-
structure coupled modes of a duct region are used to model the vibration
response along the bridging membrane. A unique representation of membrane
displacement in the advanced setting can reveal the scattering response with a
variety of edge conditions. The aforementioned problem is a prototype set
involving two semi-infinite sections bounded by elastic membranes and bridging
membrane heights at the interface. Thus, the practical configuration comprises
a finite-length expansion chamber that includes two bridging-height
discontinuities, elastic plates, and sets of edge conditions.

The model problem is physically interesting given the consideration of elastic
plates together with a variety of edge conditions and their effects on
scattering. The elastic plates can be connected in a variety of ways. For
instance, if the plates are connected with a clamped connection, the
displacements and their gradients are zero at the joint. On the other hand, if
the plates are pin-jointed, the displacement and bending moment are zero at
the joint. Thus, each physical setup yields a different set of mathematical
conditions. In this article, the Galerkin and tailored-Galerkin approaches are
used to study their responses with more advanced formulations.

The article is organized as follows. The waveguide structure, governing
equations, and idea of the MMT are presented in Section 2. The MMT solution is
supplemented with the tailored-Galerkin approach in Section 3 and with the
modal approach in Section 4. The numerical results are presented and discussed
in Section 5. The findings of the article are summarized in Section 6.

##  2 Mathematical formulation

In this section, we present the geometric structure of the waveguide and
mathematically formulate the related scattering problem. We also introduce
some notation and give some preliminary material beforehand to facilitate the
ensuing study.

###  2.1 Waveguide structure

In this article, we consider an infinite two-dimensional rectangular waveguide
with an expansion chamber stretched along $\bar{x}-$axis in the
$(\bar{x},\bar{y})$ plane. The lower wall of the waveguide
($\overline{\mathbb{W}}_{0}:=\mathbb{R}\times\\{\bar{0}\\}$) is acoustically
rigid, but the upper walls
($\overline{\mathbb{W}}_{1}:=(-\infty,-\bar{L})\times\\{\bar{a}\\}$,
$\overline{\mathbb{W}}_{2}:=(\bar{L},\infty)\times\\{\bar{a}\\}$, and
$\overline{\mathbb{W}}_{3}:=(-\bar{L},\bar{L})\times\\{\bar{b}\\}$, for some
fixed $\overline{L}\in\mathbb{R}^{+}$) consist of elastic plates. The
waveguide is divided into three duct regions (the inlet, the expansion
chamber, and the outlet) by two vertical elastic strips along the line
segments $\\{\bar{L}\\}\times[\bar{a},\bar{b}]$ and
$\\{-\bar{L}\\}\times[\bar{a},\bar{b}]$. The waveguide is loaded with a
compressible fluid having density $\rho$ and speed of sound $c$ whereas the
outside of the waveguide is assumed to be in-vacuo. The structural
configuration of the waveguide is depicted in Fig. 1. In the above and what
follows, a superposed bar indicates a dimensional quantity, and we will drop
the bar for non-dimensional counterparts.

Figure 1: Structural configuration of the waveguide. The wavy boundaries
represent elastic plates.

###  2.2 Governing equations

The waveguide structure is excited by a duct mode from the region
$(-\infty,-\overline{L})\times(0,\bar{a})$ that propagates from negative to
positive $\bar{x}$-direction. The acoustic pressure, acoustic velocity, and
the fluid potential inside the waveguide ($\bar{p}$, $\bar{\mathbf{v}}$, and
$\bar{\Phi}$, respectively) satisfy the partial differential equations

$\displaystyle\bar{p}=-\rho\dfrac{\partial\bar{\Phi}}{\partial\bar{t}}\quad\text{and}\quad\bar{\mathbf{v}}=\bar{\nabla}\bar{\Phi}.$
(1)

Assuming a time-harmonic dependence with angular frequency $\omega$ and
wavenumber $k:=\omega/c$, we define length scale $k^{-1}$ and time scale
$\omega^{-1}$ so that $x:=k\bar{x}$, $y:=k\bar{y}$, and $t:=\omega\bar{t}$
become non-dimensional space and time coordinates. Thanks to the relations in
Eq. (1), the time-harmonic non-dimensional fluid potential,
$\phi(x,y,\omega)$, satisfies the Helmholtz equation,

$\nabla^{2}\phi+\phi=0.$ (2)

In the rest of this article, we do not mention the dependence on $\omega$ for
ease of notation. We complement Eq. (2) with the non-dimensional boundary
conditions,

$\displaystyle\begin{cases}\dfrac{\partial\phi}{\partial
y}=0,&\mathbb{W}_{0},\\\ \dfrac{\partial^{5}\phi}{\partial x^{4}\partial
y}-\mu^{4}\dfrac{\partial\phi}{\partial
y}-\alpha\phi=0,&\mathbb{W}_{1}\cup\mathbb{W}_{2}\cup\mathbb{W}_{3},\end{cases}$
(3)

for the acoustically rigid lower wall and elastic plate upper walls,
respectively. The quantities

$\displaystyle\mu:=\left(\dfrac{12(1-\nu^{2})c^{2}\rho_{p}}{k^{2}h^{2}E}\right)^{1/4}\quad\text{and}\quad\alpha=:\dfrac{12(1-\nu^{2})c^{2}\rho}{k^{3}h^{3}E},$
(4)

are the in-vacuo plate wavenumber and fluid loading parameter, respectively.
Here, $E$, $\rho_{p}$, $\nu$, and $h$, respectively, are Young’s modulus,
plate density, Poisson’s ratio, and the thickness of the plates.

###  2.3 Elements of spectral theory

It is easy to verify from Eqs. (2) and (3), that the eigenvalues to the
problems in the regions $\\{(x,y)\in\mathbb{R}^{2}|\,\,L<|x|,\,0\leq y\leq
a\\}$ (inlet and outlet) and $\\{(x,y)\in\mathbb{R}^{2}|\,\,|x|<L,\,0\leq
y\leq b\\}$ (expansion chamber) are the roots of the dispersion relation,

$\displaystyle\left((\varrho^{2}+1)^{2}-\mu^{4}\right)\varrho\sinh(\varrho\zeta)-\alpha\cosh(\varrho\zeta)=0,$
(5)

where $\varrho=\tau$ (resp. $\varrho=\gamma$) when elastic plates lie along
$\zeta=a$ (resp. $\zeta=b$). These roots are numerically obtained and the
computation of all the roots is essential for a successful implementation of
the MMT. We refer the reader to [20] for more details on the spectrum of the
subproblems.

The corresponding eigenfunctions,
$Y_{\jmath}(\varrho_{n},y):=\cosh(\varrho_{n}y)$, in the inlet/outlet (with
$\jmath=1$, $\varrho_{n}=\tau_{n}$ and $\zeta=a$) and the expansion chamber
(with $\jmath=2$, $\varrho_{n}=\gamma_{n}$, and $\zeta=b$) are linearly
dependent and non-orthogonal [20]. Indeed,

$\displaystyle\displaystyle\sum_{n=0}^{\infty}\Delta_{\jmath,n}Y_{\jmath}(\varrho_{n},y)=0,\qquad\displaystyle\sum_{n=0}^{\infty}\varrho^{2}_{n}\Delta_{\jmath,n}Y_{\jmath}(\varrho_{n},y)=0,\quad
0\leq y\leq\zeta,$ (6)
$\displaystyle\displaystyle\sum_{n=0}^{\infty}\Delta^{2}_{\jmath,n}\Gamma_{\jmath,n}=0,\qquad\qquad\displaystyle\sum_{n=0}^{\infty}\varrho^{2}_{n}\Delta_{\jmath,n}^{2}\Gamma_{\jmath,n}=1.$
(7)

Here,

$\displaystyle\Gamma_{\jmath,n}:=\dfrac{\alpha\zeta}{2}+\dfrac{\alpha
Y_{\jmath}(\varrho_{n},\zeta)Y^{\prime}_{\jmath}(\varrho_{n},\zeta)}{2\varrho_{n}^{2}}+2\left[(\varrho_{n}^{2}+1)Y^{\prime}_{\jmath}(\varrho_{n},\zeta)\right]^{2}\quad\text{and}\quad\Delta_{\jmath,n}:=\dfrac{Y^{\prime}_{\jmath}(\varrho_{n},\zeta)}{\Gamma_{\jmath,n}},$
(8)

where a superposed prime indicates differentiation. Accordingly, the use of
generalized orthogonality relations is indispensable to procure a convergent
solution. To that end, the generalized orthogonality relation

$\displaystyle\alpha\int_{0}^{\zeta}Y_{\jmath}(\varrho_{n},y)Y_{\jmath}(\varrho_{m},y)dy=\Gamma_{\jmath,n}\delta_{m,n}-(\varrho^{2}_{m}+\varrho^{2}_{n}+2)Y^{\prime}_{\jmath}(\varrho_{m},\zeta)Y^{\prime}_{\jmath}(\varrho_{n},\zeta),$
(9)

is useful, wherein $\delta_{m,n}$ is the Kronecker’s delta. Moreover, we have
the identity

$\alpha\sum_{n=0}^{\infty}\dfrac{Y_{\jmath}(\varrho_{n},\nu)Y_{\jmath}(\varrho_{n},y)}{\Gamma_{\jmath,n}}=\delta_{0}(y+\nu)+\delta_{0}(y-\nu)+\delta_{0}(y+\nu-2\zeta),\qquad
0\leq y,\quad\nu\leq b,$ (10)

where $\delta_{0}$ is the Dirac mass at the origin; see, e.g., [20].

It is worth mentioning that the eigenfunction expansion representation of a
sufficiently smooth function converges point-wise [34]. We also emphasize that
the MMT is capable of catering to strong singularities appearing at the
corners of the vertical elastic plates in velocity flux, as comprehensively
discussed in [29].

###  2.4 Mode matching technique

We assume an ensemble time-harmonic fluid potential ansatz of the form

$\displaystyle\phi(x,y):=\left\\{\begin{array}[]{lll}\phi_{1}(x,y),&x\leq-L,&0\leq
y\leq a,\\\ \phi_{2}(x,y),&|x|\leq L,&0\leq y\leq b,\\\ \phi_{3}(x,y),&x\geq
L,&0\leq y\leq a,\end{array}\right.$ (14)

in terms of the inlet, expansion chamber, and outlet fluid potentials,
$\phi_{1}$, $\phi_{2}$, and $\phi_{3}$, respectively. Note that these fluid
potentials can be expanded in terms of the eigenfunctions as

$\displaystyle\phi_{1}(x,y)$
$\displaystyle=F_{\ell}Y_{1}({\tau_{\ell}},y)e^{i\xi_{\ell}(x+L)}+\sum_{n=0}^{\infty}A_{n}Y_{1}({\tau_{n}},y)e^{-i\xi_{n}(x+L)},$
(15) $\displaystyle\phi_{2}(x,y)$
$\displaystyle=\sum_{n=0}^{\infty}\left(B_{n}e^{is_{n}x}+C_{n}e^{-is_{n}x}\right)Y_{2}({\gamma_{n}},y),$
(16) $\displaystyle\phi_{3}(x,y)$
$\displaystyle=\sum_{n=0}^{\infty}D_{n}Y_{1}({\tau_{n}},y)e^{i\xi_{n}(x-L)},$
(17)

where $A_{n}$, $B_{n}$, $C_{n}$, and $D_{n}$ are the complex amplitudes of the
reflected and transmitted modes and are to be determined. The first term on
the right-hand side of Eq. (15) is the incident field with force $F_{\ell}$.
The index $\ell$ suggests whether the incident field consists of a fundamental
mode or a higher-order mode. The quantities $\xi_{n}$ and $s_{n}$ are the
wavenumbers and are defined in terms of the eigenvalues $\tau_{n}$ and
$\gamma_{n}$ as

$\xi_{n}:=\left(\tau^{2}_{n}+1\right)^{1/2}\quad\text{and}\quad
s_{n}:=\left(\gamma^{2}_{n}+1\right)^{1/2},\qquad n\in\mathbb{Z},\,\,n\geq 0.$

Further, the non-dimensional vertical elastic plate displacements, say
$w^{\pm}(y)$, satisfy

$\displaystyle\frac{d^{4}w^{\pm}}{dy^{4}}-\mu^{4}w^{\pm}=\pm\alpha\phi_{2},\quad\\{\pm
L\\}\times[a,b],$ (18)

in terms of the solution ansatz (14).

The goal in the MMT is to find the amplitudes $A_{n}$, $B_{n}$, $C_{n}$, and
$D_{n}$ using the continuity of the pressure flux and the normal velocities at
the interfaces $x=\pm L$. To that end, the continuity of the pressure flux at
the interfaces furnishes

$\displaystyle\int_{0}^{a}\phi_{1}(-L,y)Y_{1}(\tau_{m},y)dy$
$\displaystyle=\int_{0}^{a}\phi_{2}(-L,y)Y_{1}(\tau_{m},y)dy,$ (19)
$\displaystyle\int_{0}^{a}\phi_{3}(L,y)Y_{1}(\tau_{m},y)dy$
$\displaystyle=\int_{0}^{a}\phi_{2}(L,y)Y_{1}(\tau_{m},y)dy,$ (20)

whereas the continuity of the normal velocity flux at the interfaces yields

$\displaystyle\int_{0}^{b}\phi_{2x}(-L,y)Y_{2}(\gamma_{m},y)dy=$
$\displaystyle\int_{0}^{a}\phi_{1x}(-L,y)Y_{2}(\gamma_{m},y)dy+\int_{a}^{b}w^{-}(y)Y_{2}(\gamma_{m},y)dy,$
(21) $\displaystyle\int_{0}^{b}\phi_{2x}(L,y)Y_{2}(\gamma_{m},y)dy=$
$\displaystyle\int_{0}^{a}\phi_{3x}(L,y)Y_{2}(\gamma_{m},y)dy+\int_{a}^{b}w^{+}(y)Y_{2}(\gamma_{m},y)dy.$
(22)

Equations (19)-(20) relate the acoustic pressures of the inlet and outlet
regions to the acoustic pressures of the central region at $x=\pm L$. Together
with the generalized orthogonality relation (9), Eqs. (19)-(20) render the
relations

$\displaystyle\Psi^{\pm}_{m}$
$\displaystyle=-F_{\ell}\delta_{m,\ell}+\left[U_{1}^{\pm}+(\tau_{m}^{2}+2)U_{2}^{\pm}\right]\Delta_{1,m}-\dfrac{2i\alpha}{\Gamma_{1,m}}\sum_{n=0}^{\infty}\Pi^{\mp}_{n}R_{m,n}\chi^{\pm}_{n},$
(23)

between the symmetric and anti-symmetric propagating mode amplitudes,

$\displaystyle\Psi_{n}^{\pm}:=A_{n}\pm
D_{n}\,\,\text{and}\,\,\chi_{n}^{\pm}:=B_{n}\pm C_{n},$ (24)

of the inlet/outlet regions and the central region, respectively. Here,

$\displaystyle\Pi_{n}^{+}$
$\displaystyle:=\sin(s_{n}L)\quad\text{and}\quad\Pi_{n}^{-}:=i\cos(s_{n}L),$
(25) $\displaystyle R_{m,n}$
$\displaystyle:=\int_{0}^{a}\cosh(\tau_{m}y)\cosh(\gamma_{n}y)dy,\quad
U_{1}^{\pm}:=e_{1}\pm e_{3},\quad\text{and}\quad U_{2}^{\pm}:=e_{2}\pm e_{4},$
(26)

with

$\displaystyle\left\\{\begin{array}[]{ll}e_{1}:=\phi_{1yyy}(-L,a),&e_{2}:=\phi_{1y}(-L,a),\\\
e_{3}:=\phi_{3yyy}(L,a),&e_{4}:=\phi_{3y}(L,a).\end{array}\right.$ (29)

We precise that $U_{j}^{\pm}$, for $j=1,2$, are auxiliary unknowns to be
determined later on. We refer the reader to Appendix A.1 for the derivation of
expressions (23). The amplitudes of the modes propagating in the positive and
negative directions of the waveguide can be procured subsequently as

$\displaystyle A_{m}=\dfrac{1}{2}(\Psi_{m}^{+}+\Psi_{m}^{-}),\quad
D_{m}=\dfrac{1}{2}(\Psi^{+}_{m}-\Psi^{-}_{m}),\quad
B_{m}=\dfrac{1}{2}(\chi_{m}^{+}+\chi_{m}^{-}),\quad
C_{m}=\dfrac{1}{2}(\chi_{m}^{+}-\chi_{m}^{-}).$

Before using the continuity of the normal velocities to derive another set of
equations for $\chi^{\pm}_{n}$ and $\Psi^{\pm}_{n}$, it is important to cater
for the effects of the vertical elastic plates to uniquely solve Eq. (18) for
$w^{\pm}$. Towards this end, we supplement the MMT with two different
approaches. We use a tailored-Galerkin approach in Section 3 and a model
approach in Section 4.

##  3 Tailored-Galerkin approach

The tailored-Galerkin approach is based on an assortment of trial functions
that determine the vibrational response of the incident energy along the
vertical boundaries and their edges. We look for the solutions to Eq. (18) in
the form

$\displaystyle w^{-}(y)=$ $\displaystyle a_{1}\cos(\mu y)+a_{2}\sin(\mu
y)+a_{3}\cosh(\mu y)+a_{4}\sinh(\mu y)$
$\displaystyle-\alpha\sum_{n=0}^{\infty}\frac{\left(B_{n}e^{-is_{n}L}+C_{n}e^{is_{n}L}\right)\cosh(\gamma_{n}y)}{\gamma_{n}^{4}-\mu^{4}},$
(30) $\displaystyle w^{+}(y)=$ $\displaystyle a_{5}\cos(\mu y)+a_{6}\sin(\mu
y)+a_{7}\cosh(\mu y)+a_{8}\sinh(\mu y)$
$\displaystyle+\alpha\sum_{n=0}^{\infty}\frac{\left(B_{n}e^{is_{n}L}+C_{n}e^{-is_{n}L}\right)\cosh(\gamma_{n}y)}{\gamma_{n}^{4}-\mu^{4}},$
(31)

where the coefficients $a_{j}$, for $j=1\ldots 8$, are the unknown constants
to be determined using the physical behavior of the vertical elastic plates at
the edges. To that end, by adding and subtracting Eqs. (30) and (31), we find
out that

$\displaystyle W^{\pm}(y)=$ $\displaystyle a^{\pm}_{1,5}\cos(\mu
y)+a^{\pm}_{2,6}\sin(\mu y)+a^{\pm}_{3,7}\cosh(\mu y)+a^{\pm}_{4,8}\sinh(\mu
y)$
$\displaystyle+2i\alpha\sum_{n=0}^{\infty}\dfrac{\chi_{n}^{\mp}\Pi_{n}^{\pm}\cosh(\gamma_{n}y)}{\gamma_{n}^{4}-\mu^{4}},$
(32)

in terms of the symmetric and anti-symmetric modes,

$\displaystyle W^{\pm}(y):=w^{-}(y)\pm w^{+}(y),\quad\text{where}\quad
a^{\pm}_{j,j+4}:=a_{j}\pm a_{j+4},\qquad j=1,2,3,4.$

Consequently, straightforward calculations indicate that $a_{j}$, for
$j=1\ldots 8$, are obtained as

$\displaystyle
a_{j}:=\dfrac{1}{2}\left(a^{+}_{j,j+4}+a^{-}_{j,j+4}\right)\quad\text{and}\quad
a_{j+4}:=\dfrac{1}{2}\left(a^{+}_{j,j+4}-a^{-}_{j,j+4}\right),\qquad
j=1,2,3,4.$

###  3.1 Edge conditions on the vertical plates

To ensure the uniqueness of a physical solution, we impose some edge
conditions depending on the type of connection of vertical plates at finite
edges $y=a$ and $y=b$. We entertain two different types of edge connections
below. Specifically, we consider clamped and pin-joint connections.

####  3.1.1 Clamped edges of the vertical plates

When the plates are clamped at the edges, both the displacements and their
gradients are zero at the connection, i.e.,

$\displaystyle W^{\pm}(a)=0=W^{\pm}(b)\quad\text{and}\quad
W_{y}^{\pm}(a)=0=W_{y}^{\pm}(b).$ (33)

The edge conditions (33) and expression (32) provide

$\displaystyle
a^{\pm}_{1,5}\cos(\mu\zeta)+a^{\pm}_{2,6}\sin(\mu\zeta)+a^{\pm}_{3,7}\cosh(\mu\zeta)+a^{\pm}_{4,8}\sinh(\mu\zeta)=-2i\alpha\sum_{n=0}^{\infty}\frac{\chi^{\mp}\Pi^{\pm}_{n}\cosh(\gamma_{n}\zeta)}{\gamma_{n}^{4}-\mu^{4}},$
(34) $\displaystyle
a^{\pm}_{1,5}\sin(\mu\zeta)-a^{\pm}_{2,6}\cos(\mu\zeta)-a^{\pm}_{3,7}\sinh(\mu\zeta)-a^{\pm}_{4,8}\cosh(\mu\zeta)=\dfrac{2i\alpha}{\mu}\sum_{n=0}^{\infty}\frac{\chi^{\mp}\Pi_{m}^{\pm}\gamma_{n}\sinh(\gamma_{n}\zeta)}{\gamma_{n}^{4}-\mu^{4}},$
(35)

for $\zeta=a$ or $b$. The unknowns $a_{1,5}^{\pm}$, $a_{2,6}^{\pm}$,
$a_{3,7}^{\pm}$, and $a_{4,8}^{\pm}$ can easily be recovered from Eqs. (34)
and (35).

####  3.1.2 Pin-jointed edges of the vertical plates

When the edges of the vertical elastic plates are pin-jointed, the
displacements and the bending moments of the plates are zero, i.e.,

$\displaystyle W^{\pm}(a)=0=W^{\pm}(b)\quad\text{and}\quad
W_{yy}^{\pm}(a)=0=W_{yy}^{\pm}(b).$ (36)

Imposing conditions (36) on expression (32) for $\zeta=a$ or $b$, we get

$\displaystyle
a^{\pm}_{1,5}\cos(\mu\zeta)+a^{\pm}_{2,6}\sin(\mu\zeta)+a^{\pm}_{3,7}\cosh(\mu\zeta)+a^{\pm}_{4,8}\sinh(\mu\zeta)=-2i\alpha\sum_{n=0}^{\infty}\frac{\chi^{\mp}\Pi^{\pm}_{n}\cosh(\gamma_{n}\zeta)}{\gamma_{n}^{4}-\mu^{4}},$
(37) $\displaystyle
a^{\pm}_{1,5}\cos(\mu\zeta)+a^{\pm}_{2,6}\sin(\mu\zeta)-a^{\pm}_{3,7}\cosh(\mu\zeta)-a^{\pm}_{4,8}\sinh(\mu\zeta)=\dfrac{2i\alpha}{\mu^{2}}\sum_{n=0}^{\infty}\frac{\chi^{\mp}\Pi_{n}^{\pm}\gamma_{n}^{2}\cosh(\gamma_{n}\zeta)}{\gamma_{n}^{4}-\mu^{4}}.$
(38)

Therefore, the unknowns $a^{\pm}_{1,5}$, $a^{\pm}_{2,6}$, $a_{3,7}^{\pm}$, and
$a_{4,8}^{\pm}$ can be found from Eqs. (37) and (38).

###  3.2 Continuity of velocity flux and horizontal edge conditions

The velocity flux conditions (21) and (22) link the acoustic velocity of the
central region to the velocities of the inlet and outlet regions at $x=\pm L$.
These conditions lead to the amplitudes of symmetric and anti-symmetric modes
propagating in the central region as

$\displaystyle\chi^{\pm}_{m}=$
$\displaystyle\dfrac{i\Delta_{2,m}\left(V^{\mp}_{1}+(\gamma^{2}_{m}+2)V_{2}^{\mp}\right)}{2s_{m}\Pi^{\pm}_{m}}+\dfrac{i\alpha}{2s_{m}\Gamma_{2,m}\Pi^{\pm}_{m}}\left(F_{\ell}\xi_{\ell}R_{\ell,m}-\sum_{n=0}^{\infty}\xi_{n}R_{n,m}\Psi^{\pm}_{n}\right)$
$\displaystyle+\dfrac{\alpha}{2s_{m}\Gamma_{2,m}\Pi^{\pm}_{m}}\sum_{j=1}^{4}a^{\mp}_{j,j+4}A_{j,m}+\dfrac{i\alpha^{2}}{s_{m}\Gamma_{2,m}\Pi^{\pm}_{m}}\sum_{n=0}^{\infty}\frac{\Pi^{\mp}_{n}T_{m,n}\chi^{\pm}_{n}}{\gamma_{n}^{4}-\mu^{4}},$
(39)

where

$\displaystyle\begin{array}[]{ll}A_{1,m}:=\displaystyle\int_{a}^{b}\cos(\mu
y)\cosh(\gamma_{m}y)dy,&A_{2,m}:=\displaystyle\int_{a}^{b}\sin(\mu
y)\cosh(\gamma_{m}y)dy,\\\ A_{3,m}:=\displaystyle\int_{a}^{b}\cosh(\mu
y)\cosh(\gamma_{m}y)dy,&A_{4,m}:=\displaystyle\int_{a}^{b}\sinh(\mu
y)\cosh(\gamma_{m}y)dy,\end{array}$

and

$\displaystyle T_{m,n}:=$
$\displaystyle\int_{a}^{b}\cosh(\gamma_{m}y)\cosh(\gamma_{n}y)dy,\quad
V_{1}^{\pm}:=e_{5}\pm e_{7},\quad V_{2}^{\pm}:=e_{6}\pm e_{8},$ (41)

The auxiliary unknowns $V_{1}^{\pm}$ and $V_{2}^{\pm}$ involve additional
constants,

$\displaystyle e_{5}:=-i\phi_{2xyyy}(-L,b),\quad
e_{6}:=-i\phi_{2xy}(-L,b),\quad e_{7}:=-i\phi_{2xyyy}(L,b),\quad
e_{8}:=-i\phi_{2xy}(L,b),$ (42)

due to the normalization of velocity flux conditions (21) and (22) with the
generalized orthogonality relation (9). We refer the readers to Appendix A.1
for a detailed derivation of Eq. (39).

We are finally in possession of the required number of equations involving
$\Psi^{\pm}_{m}$ and $\chi^{\pm}_{m}$, but we still need to cater for the
auxiliary unknowns, $U_{j}^{\pm}$ and $V_{j}^{\pm}$, for $j=1,2$. They can be
found by taking into account the physical behavior of horizontal elastic
plates at the edges. To that end, we consider the situations when the
horizontal elastic plates are both clamped and pin-jointed. The evaluation of
the constants, $U_{j}^{\pm}$ and $V_{j}^{\pm}$ for both types of connections
is discussed separately. It is worth pointing out that other physical
conditions or combinations can also be imposed and that the framework is not
limited only to the cases discussed below. However, for brevity and to avoid
redundancy, we do not report them here.

####  3.2.1 Horizontal elastic plates with clamped edges

Setting the horizontal plate displacements as well as their gradients at the
edges equal to zero, we obtain

$\displaystyle\phi_{1y\phantom{x}}(-L,a)=0,$
$\displaystyle\phi_{3y\phantom{x}}(L,a)=0,$ (43)
$\displaystyle\phi_{1xy}(-L,a)=0,$ $\displaystyle\phi_{3xy}(L,a)=0,$ (44)
$\displaystyle\phi_{2y\phantom{x}}(\pm L,b)=0,$ (45)
$\displaystyle\phi_{2xy}(\pm L,b)=0.$ (46)

Thanks to conditions (43), (44), (45), and (46), we have
$U_{2}^{\pm}=0=V_{2}^{\pm}$ and

$\displaystyle U^{\pm}_{1}=$
$\displaystyle\dfrac{2}{S_{1}}\left(F_{\ell}\xi_{\ell}\Delta_{1,\ell}\Gamma_{1,\ell}+i\alpha\sum_{m,n=0}^{\infty}\xi_{m}\Delta_{1,m}R_{m,n}\Pi^{\mp}_{n}\chi^{\pm}_{n}\right),$
(47) $\displaystyle V^{+}_{1}=$
$\displaystyle-\dfrac{i\alpha}{2S_{2}}\sum_{m=0}^{\infty}\dfrac{\Delta_{2,m}\Pi_{m}^{+}}{s_{m}\Pi_{m}^{-}}\left(F_{\ell}\xi_{\ell}R_{m,\ell}-\sum_{n=0}^{\infty}\xi_{n}R_{m,n}\Psi^{+}_{n}\right)$
$\displaystyle-\dfrac{\alpha}{2S_{2}}\sum_{m=0}^{\infty}\left(\dfrac{\Delta_{2,m}\Pi_{m}^{+}}{s_{m}\Pi_{m}^{-}}\sum_{j=1}^{4}a_{j,j+4}^{-}A_{j,m}\right)$
$\displaystyle-\dfrac{i\alpha^{2}}{S_{2}}\sum_{m,n=0}^{\infty}\dfrac{\Delta_{2,m}\Pi_{m}^{+}\Pi^{+}_{n}T_{m,n}\chi^{+}_{n}}{s_{m}\Pi_{m}^{-}(\gamma^{4}_{n}-\mu^{4})},$
(48) $\displaystyle V^{-}_{1}=$
$\displaystyle-\dfrac{i\alpha}{2S_{3}}\sum_{m=0}^{\infty}\dfrac{\Delta_{2,m}\Pi_{m}^{-}}{s_{m}\Pi_{m}^{+}}\left(F_{\ell}\xi_{\ell}R_{m,\ell}-\sum_{n=0}^{\infty}\xi_{n}R_{m,n}\Psi^{-}_{n}\right)$
$\displaystyle-\dfrac{\alpha}{2S_{3}}\sum_{m=0}^{\infty}\left(\dfrac{\Delta_{2,m}\Pi_{m}^{-}}{s_{m}\Pi_{m}^{+}}\sum_{j=1}^{4}a_{j,j+4}^{+}A_{j,m}\right)$
$\displaystyle-\dfrac{i\alpha^{2}}{S_{3}}\sum_{m,n=0}^{\infty}\dfrac{\Delta_{2,m}\Pi_{m}^{-}\Pi^{-}_{n}T_{m,n}\chi^{-}_{n}}{s_{m}\Pi_{n}^{+}\left(\gamma^{4}_{n}-\mu^{4}\right)},$
(49)

with

$\displaystyle
S_{1}:=\sum_{m=0}^{\infty}\Delta^{2}_{1,m}\Gamma_{1,m}\xi_{m},\quad
S_{2}:=\sum_{m=0}^{\infty}\dfrac{i\Delta_{2,m}^{2}\Gamma_{2,m}\Pi_{m}^{+}}{2s_{m}\Pi_{m}^{-}},\quad
S_{3}:=\sum_{m=0}^{\infty}\dfrac{i\Delta_{2,m}^{2}\Gamma_{2,m}\Pi_{m}^{-}}{2s_{m}\Pi_{m}^{+}}.$
(50)

We refer the reader to Appendix A.2 for the derivation of expressions (47),
(48), and (49).

####  3.2.2 Horizontal elastic plates with pin-joint edges

For pin-joint horizontal edge conditions, zero displacements and zero bending
moments of the horizontal plates lead to

$\displaystyle\phi_{1y\phantom{xx}}(-L,a)=0,$
$\displaystyle\phi_{3y\phantom{xx}}(L,a)=0,$ (51)
$\displaystyle\phi_{1yxx}(-L,a)=0,$ $\displaystyle\phi_{3yxx}(L,a)=0,$ (52)
$\displaystyle\phi_{2y\phantom{xx}}(-L,b)=0,$
$\displaystyle\phi_{2y\phantom{xx}}(L,b)=0,$ (53)
$\displaystyle\phi_{2yxx}(-L,b)=0,$ $\displaystyle\phi_{2yxx}(L,b)=0.$ (54)

Note that Eq. (51) immediately renders $U^{\pm}_{2}=0$. To obtain
$U^{\pm}_{1}$, we use Eqs. (23) and (52). We multiply the equation for
$\Psi^{\pm}_{m}$ with $\Delta_{1,m}\Gamma_{1,m}\xi^{2}_{m}$, take summation
over $m$, and then utilize edge conditions (51), after fairly easy
manipulations, it is found that

$\displaystyle U^{\pm}_{1}=$
$\displaystyle\sum_{m,n=0}^{\infty}\dfrac{2i\alpha}{S_{4}}\Delta_{1,m}\xi_{m}^{2}R_{m,n}\Pi^{\mp}_{n}\Psi^{\pm}_{n},$
(55)

where

$\displaystyle
S_{4}:=\sum_{m=0}^{\infty}\Delta^{2}_{1,m}\xi^{2}_{m}\Gamma_{1,m}.$ (56)

To determine $V^{\pm}_{1}$ and $V^{\pm}_{2}$, we multiply Eq. (39) with
$\Delta_{2,m}\Pi_{m}^{\pm}\Gamma_{2,m}$ and
$\Delta_{2,m}\Pi_{m}^{\pm}\Gamma_{2,m}s^{2}_{m}$, take summation over $m$, and
then use respective conditions (53) and (54), we find after fairly easy
manipulations that

$\displaystyle S_{2}V^{+}_{1}+S_{5}V^{+}_{2}=$
$\displaystyle-i\alpha\sum_{m=0}^{\infty}\dfrac{\Delta_{2,m}\Pi_{m}^{+}}{2s_{m}\Pi_{m}^{-}}\left(F_{\ell}\xi_{\ell}R_{m,\ell}-\sum_{n=0}^{\infty}\xi_{n}R_{m,n}\Psi^{+}_{n}\right)$
$\displaystyle-\alpha\sum_{m=0}^{\infty}\left(\dfrac{\Delta_{2,m}\Pi_{m}^{+}}{2s_{m}\Pi_{m}^{-}}\sum_{j=1}^{4}a_{j,j+4}^{-}A_{j,m}\right)$
$\displaystyle-i\alpha^{2}\sum_{m,n=0}^{\infty}\dfrac{\Delta_{2,m}\Pi_{m}^{+}\Pi^{+}_{n}T_{m,n}\chi^{+}_{n}}{s_{m}\Pi_{m}^{-}(\gamma^{4}_{n}-\mu^{4})},$
(57) $\displaystyle S_{3}V^{-}_{1}+S_{6}V^{-}_{2}=$
$\displaystyle-i\alpha\sum_{m=0}^{\infty}\dfrac{\Delta_{2,m}\Pi_{m}^{-}}{2s_{m}\Pi_{m}^{+}}\left(F_{\ell}\xi_{\ell}R_{m,\ell}-\sum_{n=0}^{\infty}\xi_{n}R_{m,n}\Psi^{-}_{n}\right)$
$\displaystyle-\alpha\sum_{m=0}^{\infty}\left(\dfrac{\Delta_{2,m}\Pi_{m}^{-}}{2s_{m}\Pi_{m}^{+}}\sum_{j=1}^{4}a_{j,j+4}^{+}A_{j,m}\right)$
$\displaystyle-i\alpha^{2}\sum_{m,n=0}^{\infty}\dfrac{\Delta_{2,m}\Pi_{m}^{-}\Pi^{-}_{n}T_{m,n}\chi^{-}_{n}}{s_{m}\Pi_{n}^{+}\left(\gamma^{4}_{n}-\mu^{4}\right)},$
(58) $\displaystyle S_{7}V^{+}_{1}+S_{8}V^{+}_{2}=$
$\displaystyle-i\alpha\sum_{m=0}^{\infty}\dfrac{\Delta_{2,m}s_{m}\Pi_{m}^{+}}{2\Pi_{m}^{-}}\left(F_{\ell}\xi_{\ell}R_{m,\ell}-\sum_{n=0}^{\infty}\xi_{n}R_{m,n}\Psi^{+}_{n}\right)$
$\displaystyle-\alpha\sum_{m=0}^{\infty}\left(\dfrac{\Delta_{2,m}s_{m}\Pi_{m}^{+}}{2\Pi_{m}^{-}}\sum_{j=1}^{4}a_{j,j+4}^{-}A_{j,m}\right)$
$\displaystyle-i\alpha^{2}\sum_{m,n=0}^{\infty}\dfrac{\Delta_{2,m}s_{m}\Pi_{m}^{+}\Pi^{+}_{n}T_{m,n}\chi^{+}_{n}}{\Pi_{m}^{-}(\gamma^{4}_{n}-\mu^{4})},$
(59) $\displaystyle S_{9}V^{-}_{1}+S_{10}V^{-}_{2}=$
$\displaystyle-i\alpha\sum_{m=0}^{\infty}\dfrac{\Delta_{2,m}s_{m}\Pi_{m}^{-}}{2\Pi_{m}^{+}}\left(F_{\ell}\xi_{\ell}R_{m,\ell}-\sum_{n=0}^{\infty}\xi_{n}R_{m,n}\Psi^{-}_{n}\right)$
$\displaystyle-\alpha\sum_{m=0}^{\infty}\left(\dfrac{\Delta_{2,m}s_{m}\Pi_{m}^{-}}{2\Pi_{m}^{+}}\sum_{j=1}^{4}a_{j,j+4}^{+}A_{j,m}\right)$
$\displaystyle-i\alpha^{2}\sum_{m,n=0}^{\infty}\dfrac{\Delta_{2,m}s_{m}\Pi_{m}^{-}\Pi^{-}_{n}T_{m,n}\chi^{-}_{n}}{\Pi_{n}^{+}\left(\gamma^{4}_{n}-\mu^{4}\right)},$
(60)

where

$\displaystyle\left\\{\begin{array}[]{ll}\displaystyle
S_{5}:=\sum_{m=0}^{\infty}\dfrac{i\Delta_{2,m}^{2}\Gamma_{2,m}\Pi_{m}^{+}(\gamma^{2}_{m}+2)}{2s_{m}\Pi_{m}^{-}},&\displaystyle
S_{6}:=\sum_{m=0}^{\infty}\dfrac{i\Delta_{2,m}^{2}\Gamma_{2,m}\Pi_{m}^{-}(\gamma^{2}_{m}+2)}{2s_{m}\Pi_{m}^{+}},\\\
\displaystyle
S_{7}:=\sum_{m=0}^{\infty}\dfrac{i\Delta^{2}_{2,m}\Gamma_{2,m}s_{m}\Pi_{m}^{+}}{2\Pi_{m}^{-}},&\displaystyle
S_{8}:=\sum_{m=0}^{\infty}\dfrac{i\Delta^{2}_{2,m}\Gamma_{2,m}s_{m}\Pi_{m}^{+}}{2\Pi_{m}^{-}}(\gamma^{2}_{m}+2),\\\
\displaystyle
S_{9}:=\sum_{m=0}^{\infty}\dfrac{i\Delta_{2,m}^{2}\Gamma_{2,m}s_{m}\Pi_{m}^{-}}{2\Pi_{m}^{+}},&\displaystyle
S_{10}:=\sum_{m=0}^{\infty}\dfrac{i\Delta_{2,m}^{2}\Gamma_{2,m}s_{m}\Pi_{m}^{-}}{2\Pi_{m}^{+}}(\gamma^{2}_{m}+2).\end{array}\right.$
(64)

The derivation of Eqs. (55), (57), (58), (59), and (60) follows the same
procedure as described in Appendix A.2 for the derivation of Eqs. (47), (48),
and (49) with appropriate changes.

In a nutshell, we have derived the required number of equations for
$\Psi_{n}^{\pm}$ and $\chi^{\pm}_{n}$ from the continuity of the pressure and
velocity fluxes, for coefficients $a_{j}$’s from the edge conditions on the
vertical elastic plates, and for the auxiliary unknowns $U_{j}^{\pm}$ and
$V_{j}^{\pm}$ from the edge conditions on the horizontal elastic plates. The
MMT - tailored-Galerkin solution of the scattering problems is obtained by
solving the aforementioned systems of linear equations simultaneously.

##  4 Modal approach

In this section, we supplement the MMT with the modal approach to find the
vertical elastic plate displacements $w^{\pm}$. The modal approach is useful
as it broadens the range of edge conditions that can be addressed and avoids
the need for additional root-finding. In this study, we will use the modal
approach for comparing the results obtained using the tailored-Galerkin
approach. In this approach, the vertical plate displacements are expressed as
linear combinations of a set of known basis functions and the corresponding
coefficients are sought. To that end, it is convenient to express the plate
displacements in terms of the eigenfunctions of the duct height $y=b$. The
properties of the eigenfunctions such as linear dependence and the identity
(10) are critical for the success of the modal approach. The details beyond
those presented in this section can be found in [33, 11].

In the model approach, we express the vertical elastic plate displacements as

$\displaystyle w^{\pm}(y)=$
$\displaystyle\sum_{n=0}^{\infty}G_{n}^{\pm}\cosh(\gamma_{n}y),\qquad
y\in(a,b),$ (65)

and determine the unknown modal coefficients, $G_{n}^{\pm}$, using prescribed
boundary and edge conditions. To that end, we substitute Eq. (65) into Eq.
(18) for $x=-L$, multiply the resultant equation by $\cosh(\gamma_{m}y)$, and
integrate over $(a,b)$ to get

$\displaystyle\sum_{n=0}^{\infty}G_{n}^{-}\left(\gamma_{n}^{4}-\mu^{4}\right)T_{n,m}=-\alpha\sum_{n=0}^{\infty}(B_{n}e^{-is_{n}L}+C_{n}e^{is_{n}L})T_{n,m},$
(66)

where $T_{n,m}$ is the integral defined in Eq. (41). Integration by parts
suggests that

$\displaystyle\gamma_{n}^{2}T_{n,m}=$
$\displaystyle\gamma_{m}^{2}T_{n,m}+\gamma_{n}\cosh(\gamma_{m}b)\sinh(\gamma_{n}b)-\gamma_{n}\cosh(\gamma_{m}a)\sinh(\gamma_{n}a)$
$\displaystyle+\gamma_{m}\cosh(\gamma_{n}a)\sinh(\gamma_{m}a)-\gamma_{m}\cosh(\gamma_{n}b)\sinh(\gamma_{m}b).$
(67)

Moreover, after fairly simple manipulations, Eq. (67) renders the identity

$\displaystyle\gamma_{n}^{4}T_{n,m}=$
$\displaystyle\gamma_{m}^{4}T_{n,m}+\gamma_{m}^{2}\gamma_{n}\cosh(\gamma_{m}b)\sinh(\gamma_{n}b)-\gamma_{m}^{2}\gamma_{n}\cosh(\gamma_{m}a)\sinh(\gamma_{n}a)$
$\displaystyle+\gamma_{m}^{3}\cosh(\gamma_{n}a)\sinh(\gamma_{m}a)-\gamma_{m}^{3}\cosh(\gamma_{n}b)\sinh(\gamma_{m}b)$
$\displaystyle+\gamma_{n}^{3}\cosh(\gamma_{m}b)\sinh(\gamma_{n}b)-\gamma_{n}^{3}\cosh(\gamma_{m}a)\sinh(\gamma_{n}a)$
$\displaystyle+\gamma_{m}\gamma_{n}^{2}\cosh(\gamma_{n}a)\sinh(\gamma_{m}a)-\gamma_{m}\gamma_{n}^{2}\cosh(\gamma_{n}b)\sinh(\gamma_{m}b).$
(68)

Substituting Eq. (4) into Eq. (66), it is found that

$\displaystyle\sum_{n=0}^{\infty}G_{n}^{-}T_{n,m}=$
$\displaystyle-\frac{L_{1}\cosh(\gamma_{m}a)}{\gamma_{m}^{4}-\mu^{4}}-\frac{L_{2}\cosh(\gamma_{m}b)}{\gamma_{m}^{4}-\mu^{4}}-\frac{L_{3}\gamma_{m}\sinh(\gamma_{m}a)}{\gamma_{m}^{4}-\mu^{4}}-\frac{L_{4}\gamma_{m}\sinh(\gamma_{m}b)}{\gamma_{m}^{4}-\mu^{4}}$
$\displaystyle-\frac{L_{5}\gamma_{m}^{2}\cosh(\gamma_{m}a)}{\gamma_{m}^{4}-\mu^{4}}-\frac{L_{6}\gamma_{m}^{2}\cosh(\gamma_{m}b)}{\gamma_{m}^{4}-\mu^{4}}-\frac{L_{7}\gamma_{m}^{3}\sinh(\gamma_{m}a)}{\gamma_{m}^{4}-\mu^{4}}$
$\displaystyle-\frac{L_{8}\gamma_{m}^{3}\sinh(\gamma_{m}b)}{\gamma_{m}^{4}-\mu^{4}}-\frac{\alpha}{\gamma_{m}^{4}-\mu^{4}}\sum_{n=0}^{\infty}(B_{n}e^{-is_{n}L}+C_{n}e^{is_{n}L})T_{n,m},$
(69)

where the auxiliary constants, $L_{j}$, for $j=1,\cdots,8$, are defined as

$\displaystyle\begin{array}[]{llll}L_{1}:=-w^{-}_{yyy}(a),&L_{2}:=w^{-}_{yyy}(b),&L_{3}:=w^{-}_{yy}(a),&L_{4}:=-w^{-}_{yy}(b),\\\
L_{5}:=-w^{-}_{y}(a),&L_{6}:=w^{-}_{y}(b),&L_{7}:=w^{-}(a),&L_{8}:=-w^{-}(b).\end{array}$

On multiplying Eq. 4 by $\alpha\cosh(\gamma_{m}y)/\Gamma_{2,m}$ and summing
over index $m$, we get

$\displaystyle\alpha\sum_{n,m=0}^{\infty}\frac{G_{n}^{-}\cosh(\gamma_{m}y)T_{n,m}}{\Gamma_{2,m}}=$
$\displaystyle-\sum_{j=1}^{8}L_{j}\varphi^{j}(y)-\alpha^{2}\sum_{n,m=0}^{\infty}\frac{(B_{n}e^{-is_{n}L}+C_{n}e^{is_{n}L})\cosh(\gamma_{m}y)T_{n,m}}{\Gamma_{2,m}(\gamma_{m}^{4}-\mu^{4})},$
(71)

where the functions $\varphi^{j}(y)$, for $j=1,\cdots,8$, are defined as

$\displaystyle\begin{array}[]{llll}\varphi^{1}(y):=z(y,a),&\varphi^{2}(y):=z(y,b),&\varphi^{3}(y):=z^{\prime}(y,a),&\varphi^{4}(y):=z^{\prime}(y,b),\\\
\varphi^{5}(y):=z^{\prime\prime}(y,a),&\varphi^{6}(y):=z^{\prime\prime}(y,b),&\varphi^{7}(y):=z^{\prime\prime\prime}(y,a),&\varphi^{8}(y):=z^{\prime\prime\prime}(y,b).\end{array}$

Here,

$z(y,\zeta)=\alpha\sum_{m=0}^{\infty}\frac{\cosh(\gamma_{m}y)\cosh(\gamma_{m}\zeta)}{\Gamma_{2,m}(\gamma_{m}^{4}-\mu^{4})},\qquad\zeta=a,b,$

and the superposed prime indicates the derivative with respect to $\zeta$.

Recall that our aim is to recover coefficients $G_{n}^{-}$ for constructing
$w^{-}(y)$ from Eq. (71). Unfortunately, no explicit orthogonality relation
exists for the functions $\cosh(\gamma_{n}y)$ over $[a,b]$. To that end, we
will make use of the identity (10) in lieu. On interchanging the orders of the
summation and integration on the left hand side of Eq. (71) and invoking
identity (10), we get

$\displaystyle w^{-}(y)=$
$\displaystyle-\sum_{j=1}^{8}L_{j}\varphi^{j}(y)-\alpha^{2}\sum_{n,m=0}^{\infty}\frac{(B_{n}e^{-is_{n}L}+C_{n}e^{is_{n}L})\cosh(\gamma_{m}y)T_{n,m}}{\Gamma_{2,m}(\gamma_{m}^{4}-\mu^{4})}.$
(73)

Similarly, adopting the same procedure for $w^{+}(y)$, we find out that

$\displaystyle w^{+}(y)=$
$\displaystyle-\sum_{j=1}^{8}M_{j}\varphi^{j}(y)+\alpha^{2}\sum_{n,m=0}^{\infty}\frac{(B_{n}e^{is_{n}L}+C_{n}e^{-is_{n}L})\cosh(\gamma_{m}y)T_{n,m}}{\Gamma_{2,m}(\gamma_{m}^{4}-\mu^{4})},$
(74)

where auxiliary constants, $M_{j}$, for $j=1,\cdots,8$, are defined as

$\displaystyle\begin{array}[]{llll}M_{1}:=-w^{+}_{yyy}(a),&M_{2}:=w^{+}_{yyy}(b),&M_{3}:=w^{+}_{yy}(a),&M_{4}:=-w^{+}_{yy}(b),\\\
M_{5}:=-w^{+}_{y}(a),&M_{6}:=w^{+}_{y}(b),&M_{7}:=w^{+}(a),&M_{8}:=-w^{+}(b).\end{array}$

Finally, we add and subtract Eqs. (73) and (74) to get

$\displaystyle W^{\pm}(y)=$
$\displaystyle-\sum_{j=1}^{8}N^{\pm}_{j}\varphi^{j}(y)+2i\alpha^{2}\sum_{n,m=0}^{\infty}\dfrac{\chi^{\mp}_{n}\Pi_{n}^{\pm}\cosh(\gamma_{m}y)T_{n,m}}{\Gamma_{2,m}(\gamma_{q}^{4}-\mu^{4})},$
(76)

where $N_{j}^{\pm}:=L_{j}\pm M_{j}$, for $j=1,\ldots,8$.

It should be noted that the displacements of the vertical elastic plates,
$W^{\pm}$, still involve auxiliary unknowns, $N^{\pm}_{j}$. These constants
can be determined by imposing physical conditions on the edge connections of
the vertical elastic plates, as in the tailored-Galerkin approach. After
determining $N_{j}^{\pm}$, it is simple to compute $L_{j}$ and $M_{j}$ as

$\displaystyle L_{j}=\frac{N_{j}^{+}+N_{j}^{-}}{2}\quad\text{and}\quad
M_{j}:=\frac{N_{j}^{+}-N_{j}^{-}}{2},\qquad j=1,\cdots,8.$

We discuss the recovery $N^{\pm}_{j}$ in Section 4.1. To derive the required
number of equations for the linear system of the MMT using the modal
expansions of the vertical displacements, we use the continuity of the
velocity flux and the physical conditions on the edge connections of the
horizontal elastic plates in Section 4.2.

###  4.1 Edge conditions on the vertical plates

In this section, we assume either clamped or pin-joint edge connections of
both the vertical elastic plates to determine $N^{\pm}_{j}$. Individual cases
are discussed separately. As mentioned for the tailored-Galerkin approach,
other physically relevant edge conditions can also be imposed, and the
framework is not limited only to both clamped and both pin-joint edge
connections.

####  4.1.1 Clamped edges of the vertical plates

For clamped edges of the vertical plates at $x=\pm L$, Eq. (33) for zero plate
displacements and zero gradients, together with Eq. (76), furnishes

$N_{5}^{\pm}=N_{6}^{\pm}=N_{7}^{\pm}=N_{8}^{\pm}=0,$

and, for $\zeta=a$ or $b$,

$\displaystyle\sum_{j=1}^{4}N_{j}^{\pm}\varphi^{j}(\zeta)=$ $\displaystyle
2i\alpha^{2}\sum_{n,m=0}^{\infty}\frac{\chi^{\mp}\Pi_{n}^{\mp}\cosh(\gamma_{m}\zeta)T_{n,m}}{\Gamma_{2,m}(\gamma_{m}^{4}-\mu^{4})},$
(77) $\displaystyle\sum_{j=1}^{4}N_{j}^{\pm}\varphi^{j}_{y}(\zeta)=$
$\displaystyle
2i\alpha^{2}\sum_{n,m=0}^{\infty}\frac{\chi^{\mp}\Pi_{m}^{\mp}\gamma_{m}\sinh(\gamma_{m}\zeta)T_{n,m}}{\Gamma_{2,m}(\gamma_{m}^{4}-\mu^{4})}.$
(78)

Solving Eqs. (77) and (78) simultaneously, one can easily find $N_{j}^{\pm}$,
for $j=1,\cdots,4$.

####  4.1.2 Pin-jointed edges of the vertical plates

For pin-joint edge connections of the vertical plates at $x=\pm L$, Eq. (36)
for zero plate displacements and zero bending moments, together with Eq. (76),
leads to

$N_{3}^{\pm}=N_{4}^{\pm}=N_{7}^{\pm}=N_{8}^{\pm}=0,$

and

$\displaystyle
N_{1}^{\pm}\varphi^{1}(\zeta)+N_{2}^{\pm}\varphi^{2}(\zeta)+N_{5}^{\pm}\varphi^{5}(\zeta)+N_{6}^{\pm}\varphi^{6}(\zeta)=$
$\displaystyle
2i\alpha^{2}\sum_{n,m=0}^{\infty}\frac{\chi_{n}^{\mp}\Pi_{n}^{\pm}\cosh(\gamma_{m}\zeta)T_{n,m}}{\Gamma_{2,m}(\gamma_{m}^{4}-\mu^{4})},$
(79) $\displaystyle
N^{\pm}_{1}\varphi^{1}_{yy}(\zeta)+N^{\pm}_{2}\varphi^{2}_{yy}(\zeta)+N^{\pm}_{5}\varphi^{5}_{yy}(p)+N^{\pm}_{6}\varphi^{6}_{yy}(\zeta)=$
$\displaystyle
2i\alpha^{2}\sum_{n,m=0}^{\infty}\frac{\chi_{n}^{\mp}\Pi_{n}^{\pm}\gamma_{m}^{2}\cosh(\gamma_{m}\zeta)T_{n,m}}{\Gamma_{2,m}(\gamma_{m}^{4}-\mu^{4})}.$
(80)

Solving Eqs. (79) and (80) simultaneously, one can find all the non-zero
constants $N_{j}^{\pm}$, for $j=1,2,5,6$.

###  4.2 Continuity of velocity flux and horizontal edge conditions

In order to complete the linear system of equations for MMT supplemented by
the modal approach, we use the continuity of the velocity flux. To that end,
Eqs. (21) and (22) lead to

$\displaystyle\chi^{\pm}_{m}=$
$\displaystyle\dfrac{i\Delta_{2,m}\left(V^{\mp}_{1}+(\gamma^{2}_{m}+2)V_{2}^{\mp}\right)}{2s_{m}\Pi^{\pm}_{m}}+\dfrac{i\alpha}{2s_{m}\Gamma_{2,m}\Pi^{\pm}_{m}}\left(F_{\ell}\xi_{\ell}R_{\ell,m}-\sum_{n=0}^{\infty}\xi_{n}R_{n,m}\Psi^{\pm}_{n}\right)$
$\displaystyle-\dfrac{\alpha}{2s_{m}\Gamma_{2,m}\Pi^{\pm}_{m}}\sum_{j=1}^{8}N^{\mp}_{j}\Phi^{j}_{m}+\dfrac{i\alpha^{2}}{s_{m}\Gamma_{2,m}\Pi^{\pm}_{m}}\sum_{n,q=0}^{\infty}\frac{\Pi^{\mp}_{n}T_{m,q}T_{n,q}\chi^{\pm}_{n}}{\gamma_{n}^{4}-\mu^{4}},$
(81)

where

$\displaystyle\Phi^{j}_{m}=\int_{a}^{b}\varphi^{j}(y)Y_{2}(\gamma_{m},y)dy,\qquad
j=1,\ldots,8.$ (82)

We refer the reader to Appendix A.3 for the derivation of Eq. (81).

It should be noted that we still have to recover the auxiliary unknowns
$U_{j}^{\pm}$ and $V_{j}^{\pm}$, for $j=1,2$. To that end, different physical
conditions on the edge connections of the horizontal elastic plates can be
used. Below, we discuss the cases when both the horizontal elastic plates are
either clamped or pin-jointed.

####  4.2.1 Horizontal elastic plates with clamped edges

For clamped edge conditions on horizontal plates, the quantities $U_{1}^{\pm}$
and $U_{2}^{\pm}$ are linked to the edges of elastic plates that have been
defined in terms of field potentials of inlet and outlet regions, and their
values remain the same as in Section 3.2.1. However, the auxiliary parameters
$V_{1}^{\pm}$ and $V_{2}^{\pm}$ depend on the field potential of the chamber
region and are different. To that end, we apply the conditions (45) and (46)
together with the field potentials (81). Accordingly,

$V_{2}^{\pm}=e_{2}\pm e_{4}=0,$

and

$\displaystyle V^{+}_{1}=$
$\displaystyle-\dfrac{i\alpha}{2S_{2}}\sum_{m=0}^{\infty}\dfrac{\Delta_{2,m}\Pi_{m}^{+}}{s_{m}\Pi_{m}^{-}}\left(F_{\ell}\xi_{\ell}R_{m,\ell}-\sum_{n=0}^{\infty}\xi_{n}R_{m,n}\Psi^{+}_{n}\right)$
$\displaystyle+\dfrac{\alpha}{2S_{2}}\sum_{m=0}^{\infty}\left(\dfrac{\Delta_{2,m}\Pi_{m}^{+}}{s_{m}\Pi_{m}^{-}}\sum_{j=1}^{8}N^{-}_{j}\Phi^{j}_{m}\right)-\dfrac{i\alpha^{2}}{S_{2}}\sum_{m,n,q=0}^{\infty}\dfrac{\Delta_{2,m}\Pi_{m}^{+}\Pi^{+}_{n}T_{m,q}T_{n,q}\chi^{+}_{n}}{s_{m}\Pi_{m}^{-}(\gamma^{4}_{n}-\mu^{4})},$
(83) $\displaystyle V^{-}_{1}=$
$\displaystyle-\dfrac{i\alpha}{2S_{3}}\sum_{m=0}^{\infty}\dfrac{\Delta_{2,m}\Pi_{m}^{-}}{s_{m}\Pi_{m}^{+}}\left(F_{\ell}\xi_{\ell}R_{m,\ell}-\sum_{n=0}^{\infty}\xi_{n}R_{m,n}\Psi^{-}_{n}\right)$
$\displaystyle+\dfrac{\alpha}{2S_{3}}\sum_{m=0}^{\infty}\left(\dfrac{\Delta_{2,m}\Pi_{m}^{-}}{s_{m}\Pi_{m}^{+}}\sum_{j=1}^{8}N^{+}_{j}\Phi^{j}_{m}\right)-\dfrac{i\alpha^{2}}{S_{3}}\sum_{m,n,q=0}^{\infty}\dfrac{\Delta_{2,m}\Pi_{m}^{-}\Pi^{-}_{n}T_{m,q}T_{n,q}\chi^{-}_{n}}{s_{m}\Pi_{n}^{+}\left(\gamma^{4}_{n}-\mu^{4}\right)},$
(84)

where $S_{2}$ and $S_{3}$ are defined in Eq. (50).

####  4.2.2 Horizontal elastic plates with pin-joint edges

For pin-joint connections of the horizontal elastic plates, $U^{\pm}_{2}=0$
and the values of $U_{1}^{\pm}$ are the same as in Eq. (55) derived for the
tailored-Galerkin approach in Section 3.2.2. To determine $V^{\pm}_{1}$ and
$V^{\pm}_{2}$, we apply conditions (53) and (54) together with the field
potentials (81). Therefore, we get

$\displaystyle S_{2}V^{+}_{1}+S_{5}V^{+}_{2}=$
$\displaystyle-i\alpha\sum_{m=0}^{\infty}\dfrac{\Delta_{2,m}\Pi_{m}^{+}}{2s_{m}\Pi_{m}^{-}}\left(F_{\ell}\xi_{\ell}R_{m,\ell}-\sum_{n=0}^{\infty}\xi_{n}R_{m,n}\Psi^{+}_{n}\right)$
$\displaystyle+\alpha\sum_{m=0}^{\infty}\left(\dfrac{\Delta_{2,m}\Pi_{m}^{+}}{2s_{m}\Pi_{m}^{-}}\sum_{j=1}^{8}N^{+}_{j}\Phi^{j}_{m}\right)-i\alpha^{2}\sum_{m,n,q=0}^{\infty}\dfrac{\Delta_{2,m}\Pi_{m}^{+}\Pi^{+}_{n}T_{m,q}T_{n,q}\chi^{+}_{n}}{s_{m}\Pi_{m}^{-}(\gamma^{4}_{n}-\mu^{4})},$
(85) $\displaystyle S_{3}V^{-}_{1}+S_{6}V^{-}_{2}=$
$\displaystyle-i\alpha\sum_{m=0}^{\infty}\dfrac{\Delta_{2,m}\Pi_{m}^{-}}{2s_{m}\Pi_{m}^{+}}\left(F_{\ell}\xi_{\ell}R_{m,\ell}-\sum_{n=0}^{\infty}\xi_{n}R_{m,n}\Psi^{-}_{n}\right)$
$\displaystyle+\alpha\sum_{m=0}^{\infty}\left(\dfrac{\Delta_{2,m}\Pi_{m}^{-}}{2s_{m}\Pi_{m}^{+}}\sum_{j=1}^{8}N^{-}_{j}\Phi^{j}_{m}\right)-i\alpha^{2}\sum_{m,n,q=0}^{\infty}\dfrac{\Delta_{2,m}\Pi_{m}^{-}\Pi^{-}_{n}T_{m,q}T_{n,q}\chi^{-}_{n}}{s_{m}\Pi_{n}^{+}\left(\gamma^{4}_{n}-\mu^{4}\right)},$
(86) $\displaystyle S_{7}V^{+}_{1}+S_{8}V^{+}_{2}=$
$\displaystyle-i\alpha\sum_{m=0}^{\infty}\dfrac{\Delta_{2,m}s_{m}\Pi_{m}^{+}}{2\Pi_{m}^{-}}\left(F_{\ell}\xi_{\ell}R_{m,\ell}-\sum_{n=0}^{\infty}\xi_{n}R_{m,n}\Psi^{+}_{n}\right)$
$\displaystyle+\alpha\sum_{m=0}^{\infty}\left(\dfrac{\Delta_{2,m}s_{m}\Pi_{m}^{+}}{2\Pi_{m}^{-}}\sum_{j=1}^{8}N^{+}_{j}\Phi^{j}_{m}\right)-i\alpha^{2}\sum_{m,n,q=0}^{\infty}\dfrac{\Delta_{2,m}s_{m}\Pi_{m}^{+}\Pi^{+}_{n}T_{m,q}T_{n,q}\chi^{+}_{n}}{\Pi_{m}^{-}(\gamma^{4}_{n}-\mu^{4})},$
(87) $\displaystyle S_{9}V^{-}_{1}+S_{10}V^{-}_{2}=$
$\displaystyle-i\alpha\sum_{m=0}^{\infty}\dfrac{\Delta_{2,m}s_{m}\Pi_{m}^{-}}{2\Pi_{m}^{+}}\left(F_{\ell}\xi_{\ell}R_{m,\ell}-\sum_{n=0}^{\infty}\xi_{n}R_{m,n}\Psi^{-}_{n}\right)$
$\displaystyle+\alpha\sum_{m=0}^{\infty}\left(\dfrac{\Delta_{2,m}s_{m}\Pi_{m}^{-}}{2\Pi_{m}^{+}}\sum_{j=1}^{8}N^{-}_{j}\Phi^{j}_{m}\right)-i\alpha^{2}\sum_{m,n,q=0}^{\infty}\dfrac{\Delta_{2,m}s_{m}\Pi_{m}^{-}\Pi^{-}_{n}T_{m,q}T_{n,q}\chi^{-}_{n}}{\Pi_{n}^{+}\left(\gamma^{4}_{n}-\mu^{4}\right)}.$
(88)

Here the constants $S_{j}$, for $j=4,\cdots,10$, are defined in Eqs. (56) and
(64).

##  5 Numerical results and discussions

In this section, we first validate the mode-matching-tailored Galerkin
approach (MTG) numerically. Then, we investigate the influence of the edge
conditions on the reflected power, transmitted power, and transmission loss in
different frequency bands and compare the results produced by the MTG with
those of the mode-matching-modal approach (MMA). To that end, we truncate the
infinite linear algebraic systems derived in Sections 3 and 4 and retain the
first $N$ terms for a sufficiently large $N\in\mathbb{Z}^{+}$.

The numerical experiments are carried out on elastic plates (assumed to be of
aluminum) with a thickness of $\bar{h}=0.0006$m and a density of
$\rho_{p}=2700$kg/m3. Following [29], the Young’s modulus, Poisson’s ratio,
speed of sound, and density of air are fixed at $E=7.2\times 10^{10}$N/m2,
$\upsilon=0.34$, $c=344$m/s, and $\rho_{a}=1.2$kg/m3, respectively. The
dimensional height parameters are taken to be $\bar{a}=0.06$m and
$\bar{b}=0.085$m. Two different incident fields, a structure-borne fundamental
mode ($\ell=0$) and a fluid-borne second mode ($\ell=1$), are considered for
the experiments.

###  5.1 Validation of the MTG

The MTG is validated by reconstructing the matching conditions using the
truncated form of the MTG solution. In Fig. 2, the real and imaginary parts of
the non-dimensional normal velocities and pressures are plotted at interface
$x=-L$. Simlar plots can be produced for the interface $x=-L$. The curves for
real as well as imaginary parts of the normal velocities and pressures
coincide at the interfaces in the range $0\leq y\leq b$. We conclude that the
matching conditions (19) and (21) are fully satisfied. Note that the curves
oscillate around their mean values near the corners $y=a$ and $y=b$, which is
the so-called Gibb’s phenomenon, created by the truncation of a poorly
convergent series [29]. However, as $y$ moves away from the corners $y=a$ and
$y=b$, the amplitude of the oscillation reduces significantly.

(a) Real parts of the velocities.

(b) Real parts of the pressures.

(c) Imaginary parts of the velocities.

(d) Imaginary parts of the pressures.

Figure 2: The real and imaginary parts of the normal velocities and pressures
versus height at the interface $\bar{x}=-\bar{L}$ in the presence of vertical
elastic plates for clamped edges using frequency $f=700$Hz,
$\overline{L}=0.045$m, and $N=80$.

###  5.2 Scattering analysis

In this section, we perform a power and transmission loss analysis to
understand the scattering of fundamental and fluid-borne mode incident fields
in the waveguide structure with a wave-bearing cavity and flexible
inlet/outlet.

The reflected energy flux or power in the inlet and the transmitted energy
flux or power in the outlet are defined as

$\displaystyle{\mathcal{E}_{r}:=\dfrac{1}{\alpha}\sum_{m=0}^{K-1}|A_{m}|^{2}\xi_{m}\Gamma_{1,m}\quad\text{and}\quad\mathcal{E}_{t}:=\dfrac{1}{\alpha}\sum_{m=0}^{K-1}|D_{m}|^{2}\xi_{m}\Gamma_{1,m},}$

respectively; see, for instance, [18]. The incident energy flux or power is
normalized in this equation, and $K$ denotes the number of cut-on modes in the
extended inlet/outlet region. The conservation of energy flux in a confined
system is expressed through the conservation of power identity,

$\mathcal{E}_{r}+\mathcal{E}_{t}=1,$

wherein the sum of the reflected and transmitted powers represents the total
incident power fed to the system. The transmission loss of the transmitted
power through the waveguide is defined as

$\displaystyle{\rm
TL}:=-10\log_{10}\left(\dfrac{\mathcal{E}_{t}}{\mathcal{E}_{i}}\right),$ (89)

where $\mathcal{E}_{i}$ is the incident power (being normalized to $1$
herein); see, for instance, [35]. Throughout this section, we fix
$\overline{L}=0.02$m.

####  5.2.1 Reflected and transmitted powers

Figures 3-4 furnish, respectively, the plots for the reflected and transmitted
powers versus frequency obtained by truncating and inverting the MTG and the
MMA linear systems obtained in Sections 3 and 4. A good agreement between the
scattering energy curves furnished by the MTG and the MMA in all frequency
bands can be observed.

(a) Same edge conditions with $N=40$, $\ell=0$.

(b) Same edge conditions with $N=40$, $\ell=1$.

(c) Mixed edge conditions with $N=25$, $\ell=0$.

(d) Mixed edge conditions with $N=25$, $\ell=1$.

Figure 3: Reflected power vs. frequency using the MTG (with solid lines and
markers $\blacksquare$, $\blacktriangle$, and $\CIRCLE$) and the MMA (with
dashed lines, without markers). Left: structure-borne fundamental mode
incidence. Right: fluid-borne second mode incidence.

(a) Same edge conditions with $N=40$, $\ell=0$

(b) Same edge conditions with $N=40$, $\ell=1$

(c) Mixed edge conditions with $N=25$, $\ell=0$

(d) Mixed edge conditions with $N=25$, $\ell=1$

Figure 4: Transmitted power vs. frequency using the MTG technique (with solid
lines and markers $\blacksquare$, $\blacktriangle$, and $\CIRCLE$) and the MMA
(with dashed lines, without markers). Left: structure-borne fundamental mode
of incidence. Right: fluid-borne second mode of incidence.

The results in Figs. 3(a)-3(b) correspond to the horizontal and vertical
elastic plates with both clamped or pin-joint connections. Clearly, the series
solutions for the structure-borne fundamental mode incidence ($ell=0$)
converge faster than the fluid-borne second mode incidence ($ell=1$). The
effects of clamped and pin-joint connections become apparent when the second
cut-on mode of the inlet/outlet starts propagating. Observe that the maximum
radiated energy goes on reflection at $f=1$Hz. However, the amount of
reflection decreases with frequency and reaches its minimum at the point of
initiation of the second mode of the chamber cavity. Remark that the
fundamental mode of extended inlet/outlet is always cut-on for the entire
frequency band due to the presence of the zero eigenvalue. That renders a
plane acoustic wave, a mode that always exists in a duct with a rigid
boundary. The other propagating modes appear at higher frequencies. For
example, the second cut-on mode of the extended inlet/outlet arises at
$f=2881$Hz, however, it lies outside the frequency band considered herein.
Similarly, the waveguide containing a cavity, whose lower wall is rigid but
the upper walls consist of elastic plates, always has a fundamental duct mode
throughout the bandwidth due to the presence of a real eigenvalue. The cut-on
frequencies of the second and third modes of the duct containing a flexible
cavity are $f=1161$Hz and $f=2301$Hz, respectively, which affect the
scattering energies. A list of the cut-on frequencies is provided in Table 1.
The results in Figs. 3(c)-3(d) correspond to the effects of mixed edge
connections of the horizontal and vertical plates (i.e., one is clamped and
the other is pin-jointed) on the power reflection at different frequencies.
Observe that the influence of the mixed edge conditions on the energy
reflection is prominent, especially, when the second mode of inlet/outlet
starts propagating at $f=191$Hz. Moreover, most of the energy is reflected for
a fundamental mode incidence (see Fig. 3(c)), in contrast to a fluid-borne
mode instance for which energy is not much reflected (see Fig. 3(d)).
Apparently, the series solution converges more rapidly for the structure-borne
mode incidence (see Fig. 3(c)) than the fluid-borne mode incidence (see Fig.
3(d)).

Cut-on frequency $f$ in Hz | Extended inlet/outlet | Chamber cavity
---|---|---
191 | 1 | 1
1161 | 1 | 2
2301 | 1 | 3
2881 | 2 | 3
3446 | 2 | 4
3996 | 2 | 5
4015 | 3 | 5
Table 1: Cut-on modes

Figure 4 delineates the transmitted powers versus frequency with the same edge
connections (Figs. 4(a)-4(b)) and different edge connections (Figs. 4(c)-4(d))
of the horizontal and vertical elastic plates. As with energy reflection, the
edge conditions significantly influence energy transmission, especially when
the first cut-on mode of inlet/outlet or the higher-order cavity modes start
propagating. The maximum energy goes on transmission for fluid-borne mode
incidence, and a negligible amount of scattering energy goes on transmission
for structure-borne mode incidence.

####  5.2.2 Transmission loss

Figure 5 elaborates on the transmission loss with respect to frequency in the
presence of the expansion chamber for different sets of edge conditions on the
horizontal and vertical elastic plates. Figures 5(a)-5(b) correspond with all
clamped or all pin-joint elastic plates. It is observed that maximum energy
propagates along the structure and a stop-band is produced in the regime
$2{\rm Hz}\leq f\leq 190{\rm Hz}$ as the next mode of inlet duct becomes cut-
on at $f=191$Hz. This is a fluid-borne mode. When both modes start
propagating, the transmission loss starts increasing with frequency. In other
words, we get more reflection and absorption with the participation of
additional modes. Interestingly, more leakage in the compressional waves is
observed for the pin-joint connections than for the clamped edges due to zero
bending moments at the edges. Therefore, transmission loss can be optimized by
choosing pin-joint connections for the plates.

(a) Same edge conditions with $N=40$, $\ell=0$

(b) Same edge conditions with $N=20$, $\ell=1$

(c) Mixed edge conditions with $N=20$, $\ell=0$

(d) Mixed edge conditions with $N=20$, $\ell=1$

Figure 5: Transmission loss vs. frequency using the MTG technique (with solid
lines and markers $\blacksquare$, $\blacktriangle$, and $\CIRCLE$) and the MMA
(with dashed lines, without markers). Left: structure-borne fundamental mode
of incidence. Right: fluid-borne second mode of incidence.

The transmission loss with respect to the frequency corresponding to different
sets of edge connections of the horizontal and vertical elastic plates is
provided in Figs. 5(c)-5(d). Once again, maximum energy propagates along the
structure and a stop-band is produced in the frequency band $2{\rm Hz}\leq
f\leq 190{\rm Hz}$. For different edge connections, more leakage in the
compressional waves and, consequently, increased transmission loss are
observed for the pin-joint-clamped structure (pin-joint at $y=a$, clamped at
$y=b$) than for the clamped-pin-joint structure (clamped at $y=a$, pin-joint
at $y=b$).

##  6 Conclusions

The mode-matching-tailored-Galerkin approach and the mode-matching-modal
approach have been discussed in this article to model fluid-structure couple
wave propagation in a flexible waveguide containing bridging elastic plates
with different edge conditions. The displacement of a bridging elastic plate
is assumed to satisfy the plate equation in the tailored-Galerkin approach, so
that its homogeneous part determines the plate modes and its integral part
links the flexible chamber vibration. The homogeneous part involves 8
auxiliary constants that are determined from the conditions on the edges of
elastic plates. This article considers clamped and/or pin-jointed edges, but
its scope is broad; we can consider restraint or any other type of connection
depending on the physical conditions on elastic plates at edges. In the modal
approach, the displacement of the bridging elastic plate is projected onto the
eigenspace of the cavity region. In flexible ducts, where the fluid-structure
coupled modes exist, the solution comprises non-orthogonal eigenfunctions, and
the application of generalized orthogonal properties is essential. The
coupling quantities are integrated to determine the auxiliary constants needed
for the unique solution. The presence of auxiliary constants is linked to the
connections on the finite edges of elastic plates and helps us incorporate
physical constraints. On the other hand, these constants help us derive a set
of linearly independent eigenfunctions from a set of linearly dependent
eigenfunctions and ensure the uniqueness of the solution as well. A minor
drawback of the modal approach is that it requires more computational time to
converge than the tailored-Galerkin approach. The tailred-Glaerkin approach
outperforms previous approaches in that it does not require the implementation
of additional algorithms to incorporate more practical edge conditions such as
spring-like or restrained edge conditions. The formulation of plate
displacements achieved via the tailored-Galerkin approach or the modal
approach is incorporated in the matching conditions of normal velocities at
the interfaces. The proposed schemes are validated by verifying the matching
conditions at the interfaces and the conservation of power. The numerical
results show that the variation of conditions at the joints has a significant
impact on scattering energies and transmission loss.

##  Appendix A Derivations

###  A.1 Derivations of Eqs. (23) and (39)

Substituting expressions (15), (16), and (17) of $\phi_{1}$, $\phi_{2}$, and
$\phi_{3}$ into the pressure flux conditions (19) and (20), and subsequently
using generalized orthogonality relation (9), we get

$\displaystyle A_{m}=$ $\displaystyle-
F_{\ell}\delta_{m,\ell}+{\Delta_{1,m}}\left(e_{1}+(\tau^{2}_{m}+2)e_{2}\right)+\dfrac{\alpha}{\Gamma_{1,m}}\sum_{n=0}^{\infty}R_{m,n}\left(B_{n}e^{-is_{n}L}+C_{n}e^{is_{n}L}\right),$
(90) $\displaystyle D_{m}=$
$\displaystyle\Delta_{1,m}\left(e_{3}+(\tau^{2}_{m}+2)e_{4}\right)+\dfrac{\alpha}{\Gamma_{1,m}}\sum_{n=0}^{\infty}R_{m,n}\left(B_{n}e^{is_{n}L}+C_{n}e^{-is_{n}L}\right),$
(91)

where $\Delta_{\jmath,m}$, $R_{m,n}$, and $e_{1},\cdots,e_{4}$ are defined in
Eqs. (8), (26), and (29), respectively. Adding and subtracting Eqs. (90) and
(91), we arrive at

$\displaystyle A_{m}+D_{m}=$ $\displaystyle-
F_{\ell}\delta_{m,\ell}+{\Delta_{1,m}}\left[(e_{1}+e_{3})+(\tau^{2}_{m}+2)(e_{2}+e_{4})\right]$
$\displaystyle-\dfrac{2i\alpha}{\Gamma_{1,m}}\sum_{n=0}^{\infty}R_{m,n}\Pi_{n}^{-}\left(B_{n}+C_{n}\right).$
(92) $\displaystyle A_{m}-D_{m}=$ $\displaystyle-
F_{\ell}\delta_{m,\ell}+{\Delta_{1,m}}\left[(e_{1}-e_{3})+(\tau^{2}_{m}+2)(e_{2}-e_{4})\right]$
$\displaystyle-\dfrac{2i\alpha}{\Gamma_{1,m}}\sum_{n=0}^{\infty}R_{m,n}\Pi_{n}^{+}\left(B_{n}-C_{n}\right).$
(93)

Finally, using the definitions (24) and (26) of $\Psi^{\pm}_{m}$,
$\chi^{\pm}_{n}$, $R_{m,n}$, and $U_{j}^{\pm}$, we get Eq. (23).

Similarly, substituting Eqs. (15), (16), and (17) into Eqs. (21) and (22), and
then using orthogonality relation (9), we obtain

$\displaystyle B_{m}e^{-is_{m}L}$ $\displaystyle-
C_{m}e^{is_{m}L}=\dfrac{\Delta_{2,m}}{s_{m}}\left[e_{5}+(\gamma_{m}^{2}+2)e_{6}\right]+\dfrac{\alpha}{s_{m}\Gamma_{2,m}}\left(F_{\ell}\xi_{\ell}R_{\ell,m}-\sum_{n=0}^{\infty}R_{n,m}\xi_{n}A_{n}\right)$
$\displaystyle-\dfrac{i\alpha}{s_{m}\Gamma_{2,m}}\sum_{j=1}^{4}a_{j}A_{j,m}+\dfrac{i\alpha^{2}}{s_{m}\Gamma_{2,m}}\sum_{n=0}^{\infty}\dfrac{T_{m,n}(B_{n}e^{-is_{n}L}+C_{n}e^{is_{n}L})}{\gamma_{n}^{4}-\mu^{4}},$
(94) $\displaystyle B_{m}e^{is_{m}L}$ $\displaystyle-
C_{m}e^{-is_{m}L}=\dfrac{\Delta_{2,m}}{s_{m}}\left[e_{7}+(\gamma_{m}^{2}+2)e_{8}\right]+\dfrac{\alpha}{s_{m}\Gamma_{2,m}}\sum_{n=0}^{\infty}R_{n,m}\xi_{n}D_{n}$
$\displaystyle-\dfrac{i\alpha}{s_{m}\Gamma_{2,m}}\sum_{j=1}^{4}a_{j+4}A_{j,m}-\dfrac{i\alpha^{2}}{s_{m}\Gamma_{2,m}}\sum_{n=0}^{\infty}\dfrac{T_{m,n}(B_{n}e^{is_{n}L}+C_{n}e^{-is_{n}L})}{\gamma_{n}^{4}-\mu^{4}}.$
(95)

Subtracting and adding Eqs. (94) and (95), we get

$\displaystyle B_{m}+C_{m}=$
$\displaystyle\dfrac{i\Delta_{2,m}}{2s_{m}\Pi_{m}^{+}}\left[(e_{5}-e_{7})+(\gamma^{2}_{m}+2)(e_{6}-e_{8})\right]$
$\displaystyle+\dfrac{i\alpha}{2s_{m}\Gamma_{2,m}\Pi_{m}^{+}}\left(F_{\ell}\xi_{\ell}R_{\ell,m}-\sum_{n=0}^{\infty}\xi_{n}R_{n,m}(A_{n}+D_{n})\right)$
$\displaystyle+\dfrac{\alpha}{2s_{m}\Gamma_{2,m}\Pi_{m}^{+}}\sum_{j=1}^{4}(a_{j}-a_{j+4})A_{j,m}$
$\displaystyle+\dfrac{i\alpha^{2}}{s_{m}\Gamma_{2,m}\Pi_{m}^{+}}\sum_{n=0}^{\infty}\dfrac{\Pi_{n}^{-}T_{m,n}(B_{n}+C_{n})}{\gamma_{n}^{4}-\mu^{4}},$
(96)

and

$\displaystyle B_{m}-C_{m}=$
$\displaystyle\dfrac{i\Delta_{2,m}}{2s_{m}\Pi_{m}^{-}}\left[(e_{5}+e_{7})+(\gamma^{2}_{m}+2)(e_{6}+e_{8})\right]$
$\displaystyle+\dfrac{i\alpha}{2s_{m}\Gamma_{2,m}\Pi_{m}^{-}}\left(F_{\ell}\xi_{\ell}R_{\ell,m}-\sum_{n=0}^{\infty}\xi_{n}R_{n,m}(A_{n}-D_{n})\right)$
$\displaystyle+\dfrac{\alpha}{2s_{m}\Gamma_{2,m}\Pi_{m}^{-}}\sum_{j=1}^{4}(a_{j}+a_{j+4})A_{j,m}$
$\displaystyle+\dfrac{i\alpha^{2}}{s_{m}\Gamma_{2,m}\Pi_{m}^{-}}\sum_{n=0}^{\infty}\dfrac{\Pi_{n}^{+}T_{m,n}(B_{n}-C_{n})}{\gamma_{n}^{4}-\mu^{4}}.$
(97)

Finally, using the definitions (24), (26), and (41), of $\chi^{\pm}_{n}$,
$R_{m,n}$, $T_{m,n}$, and $V_{j}^{\pm}$, we get the expression (39).

###  A.2 Derivations of Eqs. (48) and (49)

To determine $U_{1}^{\pm}$, we multiply Eq. (23) with
$\gamma_{m}\sinh(\gamma_{m}a)$ and use the definition of $\Delta_{1,m}$ to get

$\displaystyle\xi_{m}\Delta_{1,m}\Gamma_{1,m}\Psi^{\pm}_{m}=$
$\displaystyle-\xi_{m}\Delta_{1,m}F_{\ell}\delta_{m,\ell}\Gamma_{1,\ell}+\xi_{m}\Delta^{2}_{1,m}\Gamma_{1,m}U_{1}^{\pm}-2i\alpha\xi_{m}\Delta_{1,m}\sum_{n=0}^{\infty}\Pi^{\mp}_{n}R_{m,n}\chi^{\pm}_{n}.$
(98)

On the other hand, using expressions (15) and (17) for $\phi_{1}$ and
$\phi_{3}$ into edge conditions (44), we see that

$\displaystyle
F_{\ell}\xi_{\ell}{\Delta_{1,\ell}\Gamma_{1,\ell}}-\sum_{m=0}^{\infty}{\xi_{m}\Delta_{1,m}\Gamma_{1,m}}A_{m}=0\quad\text{and}\quad\sum_{m=0}^{\infty}{\xi_{m}\Delta_{1,m}\Gamma_{1,m}}D_{m}=0.$

Consequently,

$\displaystyle\sum_{m=0}^{\infty}{\xi_{m}\Delta_{1,m}\Gamma_{1,m}}\Psi^{\pm}_{m}=F_{\ell}{\xi_{\ell}\Delta_{1,\ell}\Gamma_{1,\ell}}.$
(99)

Finally, expression (47) is found by summing over $m$ in Eq. (98) and then,
using the relation (99).

To determine $V_{1}^{\pm}$ we proceed in a similar fashion. Towards this end,
we multiply Eq. (39) by $\Delta_{2,m}\Gamma_{2,m}\Pi_{m}^{\pm}$, sum over $m$,
impose edge conditions (45) to finally arrive at Eqs. (48) and (49).

###  A.3 Derivation of Eq. (81)

To obtain the equations of continuity of velocity flux for the modal approach
in the form of scattering coefficients, we substitute Eqs. (15), (16), (17),
(73), and (74) into Eqs. (21) and (22). After normalizing resultant with the
aid of orthogonality relation (9) and simplifying, we arrive at

$\displaystyle B_{m}e^{-is_{m}L}-C_{m}e^{is_{m}L}=$
$\displaystyle{\dfrac{\Delta_{2,m}}{s_{m}}}\left[e_{5}+(\gamma^{2}_{m}+2)e_{6}\right]+\dfrac{\alpha}{s_{m}\Gamma_{2,m}}\left(F_{\ell}\xi_{\ell}R_{\ell,m}-\sum_{n=0}^{\infty}R_{n,m}\xi_{n}A_{n}\right)$
$\displaystyle+\dfrac{i\alpha}{s_{m}\Gamma_{2,m}}\left(\sum_{j=1}^{8}L_{j}\Phi^{j}_{m}+\alpha^{2}\sum_{n,q=0}^{\infty}\dfrac{(B_{n}e^{-is_{n}L}+C_{n}e^{is_{n}L})T_{m,q}T_{n,q}}{\Gamma_{2,q}(\gamma_{q}^{4}-\mu^{4})}\right),$
(100) $\displaystyle B_{m}e^{is_{m}L}-C_{m}e^{-is_{m}L}=$
$\displaystyle\dfrac{\Delta_{2,m}}{s_{m}}\left[e_{7}+(\gamma^{2}_{m}+2)e_{8}\right]+\dfrac{\alpha}{s_{m}\Gamma_{2,m}}\sum_{n=0}^{\infty}R_{n,m}\xi_{n}D_{n}$
$\displaystyle+\dfrac{i\alpha}{s_{m}\Gamma_{2,m}}\left(\sum_{j=1}^{8}M_{j}\Phi^{j}_{m}-\alpha^{2}\sum_{n,q=0}^{\infty}\dfrac{(B_{n}e^{+is_{n}L}+C_{n}e^{-s_{n}L})T_{m,q}T_{n,q}}{\Gamma_{2,q}(\gamma_{q}^{4}-\mu^{4})}\right).$
(101)

Subtracting and adding of Eqs. (100) and (101) then lead to

$\displaystyle B_{m}+C_{m}=$
$\displaystyle\dfrac{i\Delta_{2,m}}{2s_{m}\Pi_{m}^{+}}\left[(e_{5}-e_{7})+(\gamma^{2}_{m}+2)(e_{6}-e_{8})\right]$
$\displaystyle+\dfrac{i\alpha}{2s_{m}\Gamma_{2,m}\Pi_{m}^{+}}\left(F_{\ell}\xi_{\ell}R_{\ell,m}-\sum_{n=0}^{\infty}R_{n,m}\xi_{\ell}(A_{n}+D_{n})\right)$
$\displaystyle-\dfrac{\alpha}{2s_{m}\Gamma_{2,m}\Pi_{m}^{+}}\sum_{j=1}^{8}(L_{j}-M_{j})\Phi^{j}_{m}$
$\displaystyle+\dfrac{2i\alpha^{3}}{2s_{m}\Gamma_{2,m}\Pi_{m}^{+}}\sum_{n,q=0}^{\infty}\dfrac{T_{m,q}T_{n,q}\Pi_{n}^{-}(B_{n}+C_{n})}{\Gamma_{2,q}(\gamma_{q}^{4}-\mu^{4})},$
(102)

and

$\displaystyle B_{m}-C_{m}=$
$\displaystyle\dfrac{i\Delta_{2,m}}{2s_{m}\Pi_{m}^{-}}\left[(e_{5}+e_{7})+(\gamma^{2}_{m}+2)(e_{6}+e_{8})\right]$
$\displaystyle+\dfrac{i\alpha}{2s_{m}\Gamma_{2,m}\Pi_{m}^{-}}\left(F_{\ell}\xi_{\ell}R_{m,\ell}-\sum_{n=0}^{\infty}R_{m,n}\xi_{\ell}(A_{n}-D_{n})\right)$
$\displaystyle-\dfrac{\alpha}{2s_{m}\Gamma_{2,m}\Pi_{m}^{-}}\sum_{j=1}^{8}(L_{j}+M_{j})\Phi^{j}_{m}$
$\displaystyle+\dfrac{2i\alpha^{3}}{2s_{m}\Gamma_{2,m}\Pi_{m}^{-}}\sum_{n,q=0}^{\infty}\dfrac{T_{m,q}T_{n,q}\Pi_{n}^{+}(B_{n}-C_{n})}{\Gamma_{2,q}(\gamma_{q}^{4}-\mu^{4})}.$
(103)

Finally, using the definitions (24), (26), and (41), of $\chi^{\pm}_{n}$,
$R_{m,n}$, $T_{m,n}$, $V_{1}^{\pm}$, and $V_{2}^{\pm}$, we arrive at the
expression (81).

## Competing interests

All the authors declare that they have no competing interests.

## Acknowledgment

A.W. would like to acknowledge the financial supported by the Nazarbayev
University, Kazakhstan through Faculty Development Competitive Research Grant
Program (FDCRGP) under Grant 1022021FD2914. N.A. would like to acknowledge
financial support from the Gulf University for Science and Technology for an
Internal Seed Grant (No. 278877).

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