A Mode-Matching Tailored-Galerkin Approach for Higher Order Interface Conditions and Geometric Variations

Abstract

The current study focuses on the modeling and analysis of acoustic scattering from an elastic membrane disc located in a cylindrical waveguide that may involve structural discontinuities. The physical problem is governed by Helmholtz’s equation and involves higher order boundary conditions at the interfaces. The Mode-Matching (MM) method in conjunction with Galerkin formulation is developed to solve the governing boundary value problems. The solution procedure is first applied on two prototype problems to formulate the theoretical frame work, which is then used to analyze the structural response of the elastic membranes attached at the mouth of the cylindrical expansion chamber. The aforementioned solution method yields the linear algebraic systems containing infinite equations. These systems are truncated first and then are numerically solved. From the numerical experiments, it is found that geometrical and material properties of the structure significantly affect the transmission loss as well as the scattering energies.

1. Introduction

Noise control is a subject of major concern of this modern society. Numerous sources such as automobiles, airplanes, power stations and heating, ventilation, and air conditioning (HVAC) systems contribute a lot to raise levels of noise pollution. To control noise pollution, which is basically produced by structural vibrations and propagates via the material medium, numerous materials and structural designs are applied. The noise control devices that are commonly used to attenuate noises in the high-frequency range are based on the inclusion of sound absorbent materials, for instance see [1,2,3,4]. On the other hand, to mitigate low-frequency range noises active devices, wherein the noise is reduced through internal reflection are used. In reactive devices, the noise is minimized by the introduction of elastic segments and structural variations. Often such structures comprise rectangular or cylindrical geometrical configuration and have different material properties. Practically, the simplest structure that minimizes noise is one that contains a circular cylindrical shape together with structural variations and different material properties; for instance, see [5,6,7,8].

The current study will focus on the reflection and transmission of acoustic waves from the elastic membranes attached at the mouth of the chamber at interfaces together with the step-discontinuities. The physical system of the considered setting governs a differential system, which includes Helmholtz’s equation, elastic membrane conditions, rigid conditions, and edge conditions. To solve the aforementioned differential systems, an analytic and/or semi-analytic techniques are found to be useful. One of the main studies for solving such differential systems in the absence of elastic membrane conditions for a cylindrical channel was offered by Miles [9]. Later, Levine and Schwinger [10] considered a semi-infinite, acoustically rigid cylindrical duct and applied the Wiener-Hopf technique to solve the governing differential equations. Likewise, Ingard [11] used the Wiener-Hopf technique to discuss the scattering of piston radiation in a cylindrical duct containing one open side. The envisaged problems usually involve the governing equation to be either a Helmholtz or Laplace equation, and the boundary conditions to be Dritchet, Nuemann or Robin types. Until now, numerous analytical, semi-analytical, and numerical techniques have been advanced to address a variety of different physical problems; for instance, see [12,13,14,15].

Recently, Hassan [16] utilized the idea of channel partitioning and investigated the acoustic attenuation from trifurcated and pentafurcated waveguides with the Mode-Matching technique (MMT). Their proposed scheme projects the solution of governing problems on orthogonal basis functions, and the resulting eigen-systems are of Sturm-Liouville (SL) type. On the other hand, Lawrie et al. [17] considered flexible components in the guiding structure to attenuate noise, wherein they found that the systems associated with the flexible components underlie in the non-SL category and satisfy generalized orthogonal properties. Later, the MMT with generalized properties was found to be useful to investigate the attenuation of the fluid-structure coupled mode in different structures; for instance, see [18,19,20,21,22,23,24,25,26,27,28,29].

The research conducted so far in the particular direction was focused on the projection of the solution on transverse coupled modes. Nevertheless, practically, in many cases, the contribution of longitudinal modes is important as well. The present study will focus on the development of MMT in connection with the Galerkin formulation to address the dynamical response of elastic structures. The Galerkin approach is conceptually simpler and is based on the assumption of a basis function to define the dynamical response along the boundaries. This approach has been extensively studied and applied to solve a wide range of problems; for instance, [30,31,32]. The study is significant because of both the mathematical and physical prospective. Physically, the analysis of elastic membranes at interfaces of the chamber is useful in modeling the silencing devices and in reducing the tonal fan noise of the HVAC systems. Whereas, mathematically, the study aims to advance the MM method in conjunction with the Galerkin approach, which enables the solution of a wide range of problems involving elastic components attached at the mouth of the expansion chamber.

The article is arranged as follows: Section 2 includes the physical problem comprising a membrane disc in planar and non-planar structures. The Galerkin procedure is contained in Section 3, whereas, the MM solution is explained in Section 4. In Section 5, an expansion chamber having elastic membranes at interfaces is discussed. The numerical results and discussion is given in Section 6. Finally, the summary and concluding remarks are presented in Section 7.

2. Problem Formulation

Consider an elastic membrane in planar and non-planar cylindrical waveguides is located at interface $\overline{z}=\overline{0}$. The physical configurations of these waveguides are as shown in Figure 1a,b, respectively. Note the over-bar with a variable here, which henceforth expresses the dimensional setting of the coordinates. A compressible fluid with mass density $\rho$ and sound speed c is filled inside of these waveguides.

The incident radiation from the left hand region is reflected and transmitted after interaction with the membrane and a structural discontinuity. In a dimensional cylindrical coordinate setting $(\overline{r},\overline{z})$, the dynamical response of the membrane can be expressed as

$$\frac{{\partial }^2\overline{W}}{\partial {\overline{r}}^2}-\frac{1}{c_m^2}\frac{{\partial }^2\overline{W}}{\partial \overline{t^2}}=\frac{1}{T}({\overline{P}}^{+}-{\overline{P}}^{-}),$$

(1)

where, $\overline{W}$ denotes the membrane displacement, and ${\overline{P}}^{+}$ and ${\overline{P}}^{-}$ are used to specify the acoustic pressures in sections located at $\overline{z}>\overline{0}$ and $\overline{z}<\overline{0},$ respectively. Here, quantity $c_m=\sqrt{T/\left({\rho }_m\right)}$, membrane tension T, and membrane mass density ${\rho }_m$ determine the speed of the waves on the membrane. The acoustic response in the fluid medium can be formulated in terms of dimensional field potential $\overline{\mathrm{\Phi }}(\overline{r},\overline{z},\overline{t})$, which satisfies the wave equation

$${\overline{▽}}^{\mathbf{2}}\overline{\mathrm{\Phi }}(\overline{r},\overline{z},\overline{t})-\frac{1}{c^2}\frac{{\partial }^2\overline{\mathrm{\Phi }}}{\partial \overline{t^2}}=0.$$

(2)

The membrane displacement $\overline{W}$ and acoustic pressure $\overline{P}$ are related to the fluid potential $\overline{\mathrm{\Phi }}$, such as $\partial \overline{W}/\partial \overline{t}=\partial \overline{\mathrm{\Phi }}/\partial \overline{z}$ and $\overline{P}=-\rho \partial \overline{\mathrm{\Phi }}/\partial \overline{t}.$ It is assumed that the waveguides are bounded by acoustically rigid boundaries, which give

$$\overline{▽}\overline{\mathrm{\Phi }}·\overline{\mathit{n}}=\overline{0},$$

(3)

where $\overline{\mathit{n}}$ is the unit vector directed into the surface. The harmonic time dependence $exp\left(-i\omega \overline{t}\right)$ is assumed and is suppressed throughout the article ahead, while $\omega =ck$ is the radian frequency and k is the wave number. Moreover, the formulated boundary value problems are made non-dimensional with the length scale $k^{-1}$ and the time scale ${\omega }^{-1}$. With reference to duct sections at $z<0$ and $z>0$, we may write the time independent, dimensionless fluid potential $\varphi$ as

$$\varphi \left(r,z\right)=\left\{\begin{array}{c}{\varphi }_1\left(r,z\right),z<0,0\le r\le a,\hfill \\ {\varphi }_2\left(r,z\right),z>0,0\le r\le b,\hfill \end{array}\right.$$

(4)

where a and b are constants and are being used here to represent the length of the radii of sections lying along $z<0$ and $z>0,$ respectively. Clearly, for planar geometrical configuration, which is shown by Figure 1a, we take $a=b$, whilst, for non-planar geometrical configuration, which is shown by Figure 1b, we consider $a<b$.

3. Membrane Response: A Galerkin Approach

The non-dimensional membrane displacement w at $z=0$ satisfies the membrane equation

$$\left(\frac{{\partial }^2}{\partial r^2}+{\mu }^2\right)w=\alpha ({\varphi }_2-{\varphi }_1),$$

(5)

where the quantities $\mu =c/c_m$ and $\alpha =c^2\rho /\left(kT\right)$ respectively represent the non-dimensional membrane wavenumber and fluid loading parameter. It is assumed that the membrane at $r=a$, is fixed or fastened tightly, i.e.,

$$w\left(a\right)=0.$$

(6)

The idea of Garekin formulation is adopted here to express the response of the elastic membrane. The formal technique is based upon the assumption of priori solutions that satisfy the homogeneous part of the membrane Equation (5) along with the edge condition (6). Then the entire dynamical response of the elastic membrane is expressed by the Fourier series as

$$w\left(r\right)=\sum _{n=1}^{\infty }{G}_{n}sin\left(\frac{n\pi r}{a}\right),$$

(7)

where $sin\left(\frac{n\pi r}{a}\right),n=1,2,3\cdots$ are orthogonal and satisfy the orthogonality relation:

$${\int }_0^{a}sin\left(\frac{m\pi r}{a}\right)sin\left(\frac{n\pi r}{a}\right)dr={\delta }_{mn}\frac{a}{2}.$$

(8)

Here, the quantities $G_n,n=1,2,3\cdots$ are unknown Fourier coefficients and would be found from (5) in the proceeding section, once the acoustic response in the duct region becomes known.

4. Acoustic Response: A Mode-Matching Approach

In duct sections, the fluid potential ${\varphi }_j$ satisfy the Helmholtz equations

$$\left\{{\displaystyle \frac{{\partial }^2}{\partial r^2}}+\frac{1}{r}\frac{\partial }{\partial r}+{\displaystyle \frac{{\partial }^2}{\partial z^2}}+1\right\}{\varphi }_j=0,j=1,2,$$

(9)

along with the boundary conditions

$$\begin{array}{c}\hfill \frac{\partial {\varphi }_1}{\partial r}(a,z)=0=\frac{\partial {\varphi }_2}{\partial r}(b,z).\end{array}$$

(10)

By using the separation of variable method, from (9) and (10), one may write the acoustic mode propagating in respective duct regions in the term of eigenfunction expansions as:

$${\varphi }_1(r,z)=\sum _{n=0}^{\infty }\left\{A_n{e}^{i{\eta }_{n}z}+B_n{e}^{-i{\eta }_{n}z}\right\}{R}_{1n}\left(r\right).$$

(11)

$$\begin{array}{c}\hfill {\varphi }_2(r,z)=\sum _{n=0}^{\infty }\left\{C_n{e}^{i{s}_{n}z}+D_n{e}^{-i{s}_{n}z}\right\}{R}_{2n}\left(r\right).\end{array}$$

(12)

where ${\eta }_n=\sqrt{1-{\tau }_n^2}$ and $s_n=\sqrt{1-{\gamma }_n^2}$ represent the nth mode wave numbers in which eigenvalues ${\tau }_n$ and ${\gamma }_n$ are roots of relations

$$\begin{array}{c}\hfill J_0^{\prime }\left({\tau }_{n}a\right)=0=J_0^{\prime }\left({\gamma }_{n}b\right).\end{array}$$

The associated eigenfunctions satisfy the respective orthogonality relation

$$\begin{array}{c}\hfill {\int }_0^a{R}_{1m}\left(r\right)R_{1n}rdr={\delta }_{mn}{F}_n,\end{array}$$

$$\begin{array}{c}\hfill {\int }_0^b{R}_{2m}\left(r\right)R_{2n}rdr={\delta }_{mn}{M}_n,\end{array}$$

(13)

where

$$\begin{array}{c}\hfill F_n={\int }_0^a{R}_{1n}^{2}rdr,M_n={\int }_0^b{R}_{2n}^{2}rdr.\end{array}$$

Note that the coefficients $\left\{A_n,B_n,C_n,D_n\right\}$ involved in (11) and (12) are unknowns, and are calculated from the interface conditions. Let us assume the incident excitation from the negative z-direction propagating towards the positive z-direction with unit amplitude such that $A_n={\delta }_{n0}$; and as there is no reflection in the region at $z>0$, therefore, $D_n=0$. To determine the remaining coefficients, we fix the geometrical configuration and impose matching conditions. Two configurations are depicted in Figure 1: (1) Planar configuration when $a=b$ and (2) Non-planar configuration when $b>a$. The values of coefficients $B_n$ and $C_n$ in both of the cases are calculated in the subsequent subsections.

4.1. Planar Configuration When $a=b$

In the planar structure, there is no step-discontinuity involved, and the matching conditions at interface can be defined as follows:

$$\frac{\partial {\varphi }_1}{\partial z}=w\left(r\right)=\frac{\partial {\varphi }_2}{\partial z}.$$

(14)

By making use of (11), (12) and (7) into (14), after some mathematical rearrangements, one may get

$$B_m+C_m={\delta }_{m0},$$

(15)

where

$$C_m=-\frac{i}{{\eta }_m{F}_m}\sum _{n=1}^{\infty }{G}_n{P}_{mn},$$

(16)

in which

$$P_{mn}={\int }_0^a{R}_{1m}\left(r\right)sin\left(\frac{n\pi r}{a}\right)rdr.$$

Whereas, to find $G_n$, we substitute (11), (12), and (7) into (5) and simplify it with the aid of the orthogonality relation (8). Finally, we find that

$$G_m=\frac{-2\alpha }{a{△}_m}{Q}_{m0}-\frac{2\alpha }{a{△}_m}\sum _{n=0}^{\infty }\left\{B_n-C_n\right\}{Q}_{mn},$$

(17)

where, ${△}_n={\mu }^2-{\left(\frac{n\pi }{a}\right)}^2$ and

$$Q_{mn}={\int }_0^{a}sin\left(\frac{m\pi r}{a}\right)R_{1n}dr.$$

(18)

Thus, we get a linear algebraic system defined by (15) and (16) along with (17), which can be solved numerically after truncation.

4.2. Non-Planar Configuration When $b>a$

In this setting, $R_{1n}\left(r\right)\ne R_{2n}\left(r\right)$; therefore, $F_n\ne M_n$ and the matching conditions can be written as

$$\begin{array}{c}\hfill \frac{\partial {\varphi }_1}{\partial z}=w\left(r\right)\end{array}$$

(19)

and

$$\frac{\partial {\varphi }_2}{\partial z}=\left\{\begin{array}{c}w\left(r\right)0<r<a\hfill \\ 0a<r<b.\hfill \end{array}\right.$$

(20)

Using (11), (12) and (7), (19) and (20), after simplification yield, we obtain

$$B_m=\frac{{\delta }_{m0}}{{\eta }_m}+\frac{i}{{\eta }_m{F}_m}\sum _{n=0}^{\infty }{G}_n{P}_{mn},$$

(21)

$$C_m=\frac{-i}{s_m{M}_m}\sum _{n=1}^{\infty }{G}_n{Q}_{mn},$$

(22)

where

$$Q_{mn}={\int }_0^a{R}_{2m}\left(r\right)sin\left(\frac{n\pi r}{a}\right)rdr.$$

Now, to obtain $G_n$, we invoke (11), (12), and (7) into (5), after some mathematical manipulation, one can easily find

$$G_m=\frac{-2\alpha }{a{△}_m}{X}_{1m0}-\frac{2\alpha }{a{△}_m}\sum _{n=0}^{\infty }\{B_n{X}_{1mn}+C_n{X}_{2mn}\},$$

(23)

where

$$X_{1mn}={\int }_0^a{R}_{1n}\left(r\right)sin\left(\frac{m\pi r}{a}\right)dr,X_{2mn}={\int }_0^a{R}_{2n}\left(r\right)sin\left(\frac{m\pi r}{a}\right)dr.$$

In this way, we get a linear algebraic system defined by (21) and (22) along with (23), which can be solved numerically after truncation.

5. Application to an Expansion Chamber Enclosed by Membranes

Consider energy transmitted through an expansion chamber having membrane discs on the ends of the chamber at $z=-L$ and $z=L$. The physical system includes two semi-finite shells which are connected to the expansion chamber of dimensionless length $2L$ at $z=-L$ and $z=L$. The semi-infinite shell located in the region $0<r<a$, $z<-L$ serves as inlet and the region at $z>L$, $0<r<a$ is designated as outlet; whereas, the expansion chamber contains the space between $0<r<b,z>L$. At $z=\pm L$, two membrane discs are located. The physical configuration of the duct is as shown in Figure 2.

The inside of the regions is filled with a compressible fluid and bounding circular surfaces are assumed to be acoustically rigid. The governing differential equations are the same as was given in previous sections. Consider that the inlet region is excited with the fundamental duct mode that will be reflected and transmitted from membrane interfaces on interaction. The field potentials in the inlet, expansion chamber, and outlet take formulations:

$${\varphi }_1(r,z)=e^{i(z+L)}+\sum _{n=0}^{\infty }{A}_n{R}_{1n}\left(r\right)e^{-i{\eta }_n\left(z+L\right)},$$

(24)

$${\varphi }_2(r,z)=\sum _{n=0}^{\infty }{B}_n{R}_{2n}\left(r\right)e^{i{s}_n\left(z\right)}+\sum _{n=0}^{\infty }{C}_n{R}_{2n}\left(r\right)e^{-i{s}_n\left(z\right)},$$

(25)

$${\varphi }_3(r,z)=\sum _{n=0}^{\infty }{D}_n{R}_{3n}\left(r\right)e^{-i{\eta }_n\left(z-L\right)},$$

(26)

where, $\{A_n,B_n,C_n,D_n\}$ are unknown amplitude and are calculated by using the mode-matching procedure. However, to make the problem more simple, we may break it into symmetric and anti-symmetric sub-problems by using the symmetry assumed in the configuration about $z=0$. The following sections explain the associated settings.

5.1. Symmetric Case

The symmetric setting can be achieved by assuming the rigid disc at $z=0,$ and it reveals $B_n=C_n$, which thereby yields the field potential formulations as

$${\varphi }_1^s(r,z)=e^{i(z+L)}+\sum _{n=0}^{\infty }{A}_n^s{R}_{1n}\left(r\right)e^{-i{\eta }_n\left(z+L\right)},$$

(27)

$${\varphi }_2^s=\sum _{n=0}^{\infty }2B_n^{s}cos\left(s_{n}z\right)R_{2n}\left(r\right),$$

(28)

where, the superscript “s” shows the variables in symmetric setting, and quantities $A_n^s$ and $B_n^s$ are unknown amplitudes. To express the membrane response, the priori solution in symmetric configuration is

$$w^s\left(r\right)=\sum _{n=1}^{\infty }{G}_n^{s}sin\left(\frac{n\pi r}{a}\right),$$

(29)

where $G_n^s$ are Fourier coefficients. To determine these coefficients, we substitute (27)–(29) into the membrane equation given by

$$w_{rr}^s+{\mu }^2{w}^s=\alpha ({\varphi }_2^s-{\varphi }_1^s),\mathrm{at}z=-L,$$

(30)

which after mathematical simplifications yields

$$G_m^s=\frac{2\alpha }{a{△}_m}\left\{-X_{1m0}-\sum _{n=0}^{\infty }{A}_n^s{X}_{1mn}+\sum _{n=0}^{\infty }2B_n^{s}cos\left(s_{n}L\right)X_{2mn}\right\}.$$

(31)

Accordingly, the velocity conditions at $z=-L$ in the symmetric case conclude to

$$A_m^s={\delta }_{m0}-\frac{1}{i{\eta }_m{F}_m}\sum _{n=0}^{\infty }{G}_m^s{Q}_{mn},$$

(32)

$$B_m^s=-\frac{1}{2sin\left(s_{m}L\right)s_m{M}_m}\sum _{n=1}^{\infty }{G}_n^s{P}_{mn}.$$

(33)

5.2. Anti-Symmetric Case

The anti-symmetric problem can be formed by considering the soft disc at $z=0,$ which gives, $B_n=-C_n$ and field potentials take formulations:

$${\varphi }_1^a(r,z)=e^{i(z+L)}+\sum _{n=0}^{\infty }{A}_n^a{R}_{1n}\left(r\right)e^{-i{\eta }_n\left(z+L\right)}.$$

(34)

$${\varphi }_2^a=\sum _{n=0}^{\infty }{B}_n^{a}2isin\left(s_{n}z\right)R_{2n}\left(r\right),$$

(35)

where superscript “a” denotes the anti-symmetric. Note that the quantities $A_n^a$ and $B_n^a$ are unknown amplitudes. On applying the conditions at the interface as explained for the symmetric case, the unknown amplitudes are found through solving the following linear algebraic system:

$$A_m^a=\frac{{\delta }_{m0}}{{\eta }_m}-\frac{1}{i{\eta }_m{F}_m}\sum _{n=0}^{\infty }{G}_m^a{Q}_{mn},$$

(36)

$$B_m^a=\frac{1}{2icos\left(s_{m}L\right)s_m{M}_m}\sum _{n=1}^{\infty }{G}_n^a{P}_{mn,}$$

(37)

where

$$G_m^a=\frac{2\alpha }{a{△}_m}\left\{-X_{1m0}-\sum _{n=0}^{\infty }{A}_n^a{X}_{1mn}-\sum _{n=0}^{\infty }2i{B}_n^{a}sin\left(s_{n}L\right)X_{2mn}\right\}.$$

(38)

In this way, the symmetric and anti-symmetric system can be solved separately after truncation and yield the amplitudes $\{A_n^s,A_n^a,B_n^s,B_n^a\}$. Finally, the amplitudes of complete problem $\{A_n,B_n,C_n,D_n\}$ can be obtained from the symmetric and anti-symmetric amplitudes as

$$A_n={\displaystyle \frac{A_n^s+A_n^a}{2}},D_n={\displaystyle \frac{A_n^s-A_n^a}{2}},$$

$$B_n={\displaystyle \frac{B_n^s+B_n^a}{2}},C_n={\displaystyle \frac{B_n^s-B_n^a}{2}}.$$

6. Numerical Results and Discussion

The linear algebraic systems achieved against different geometrical configurations are truncated here by considering $m=n=0,1,2,\cdots N$, where N is a truncation parameter, and then are solved numerically after fixing the values of involving parameters. To perform numerical computation, the membrane is assumed to be made of stainless steel, with a density of $0.2{\mathrm{kgm}}^{-2}$, and having a tension of $T=350$ N; whereas, the density of the fluid is $\rho =1.2{\mathrm{kgm}}^{-3}$ in which sound speed $c=344{\mathrm{ms}}^{-1}$ is considered. These numerical values of parameters have been taken from [13] and will remain the same for all of the numerical results. However, the rest of the parameters involved are different in different settings, and would be defined along with the respective figure. To obtain the graphs depicted in Figure 3, Figure 4, Figure 5 and Figure 6, the systems of equations are truncated by considering $N=50$ terms at frequency $f=500$ Hz. For the planar configuration shown in Figure 1a, the linear algebraic systems are solved by fixing the dimensional radii $\overline{a}=\overline{b}=0.05$ m; whereas, for non-planar configuration ($b>a$), we consider the the dimensional radius of the region located at $z<0$ and $z>0$ in Figure 1b as $\overline{a}=0.05$ and $\overline{b}=0.1$ m, respectively.

To see how the truncated solution is adequate, the matching conditions are reconstructed numerically in Figure 3, Figure 4, Figure 5 and Figure 6. In Figure 3 and Figure 4, the real and imaginary parts of normal velocities and membrane displacements against non-dimensional radii at $z=0$ are shown, respectively. From these curves, it is seen that the membrane displacements exactly match with normal velocities in regime $0<r<a$ in both the cases when $a=b$ and $b>a$, and the normal velocities tend to zero when $a<r<b.$ The graphs reflect the situation that is mathematically considered in Equations (14) and (20).

Moreover, to confirm the mode-matching response of the elastic membrane to the pressure difference as assumed in (5), we rewrite it as follows: $\mathrm{\Theta }\left(r\right)=\mathrm{\Pi }\left(r\right)$, where

$$\mathrm{\Theta }\left(r\right)=\alpha ({\varphi }_2-{\varphi }_1)$$

(39)

and

$$\mathrm{\Pi }\left(r\right)=w_{rr}+{\mu }^{2}w.$$

(40)

Note that r in subscript denotes differentiation with respect to r. On substituting the truncated form of expansions from (8), (11), and (12) into (39) and (40), and then plotting the resulting $\mathrm{\Theta }\left(r\right)$ and $\mathrm{\Pi }\left(r\right)$ against r, where $0<r<a$, we achieve Figure 5 and Figure 6. From these figures, it is found that the real and imaginary parts of $\mathrm{\Theta }\left(r\right)$ and $\mathrm{\Pi }\left(r\right)$ coincide in the assumed regime, which assures that a sufficient number of modes in the truncated solution have been retained.

The physical investigation of the problems can be made by considering the power propagation in the duct regions. The expressions for the powers reflecting and transmitting in duct regions can be determined from the definition as given in [13]; that is,

$$\mathrm{Power}={\displaystyle \frac{1}{2}}\mathrm{Re}\left[{\int }_{\mathrm{\Sigma }}i\varphi {\left({\displaystyle \frac{\partial \varphi }{\partial n}}\right)}^{*}d\mathrm{\Sigma }\right],$$

(41)

where $\mathrm{\Sigma }$ denotes the finite domain and n is in normal direction to the bounding surface. On substituting the incident field, reflected field, and transmitted field into (41), one can obtain the expressions of scattering energies. By applying the energy conservation statement that the energy entering and leaving the system must be equal yields the following power-conserving identity:

$${\mathcal{E}}_1+{\mathcal{E}}_2=1={\mathcal{E}}_3,$$

(42)

where the incident power is scaled at unity and

$${\mathcal{E}}_1=\sum _{m=0}^{{\mathcal{K}}_1-1}\mid B_m{\mid }^2{F}_m{\eta }_m,$$

(43)

and

$${\mathcal{E}}_2=\sum _{m=0}^{{\mathcal{K}}_2-1}\mid C_m{\mid }^2{M}_m{s}_m,$$

(44)

represent the reflected and transmitted powers, respectively. Here, the quantities ${\mathcal{K}}_1$ and ${\mathcal{K}}_2$ express the number of cut-ons in regions having radii a and b, respectively. In Figure 7 and Figure 8, the scattering powers ${\mathcal{E}}_1$ and ${\mathcal{E}}_2$ and their sum ${\mathcal{E}}_3$ are plotted against frequency varying from 1 Hz to 1000 Hz. The radii of duct regions in Figure 7 and Figure 8 are changed. To obtain the graphs of Figure 7, we choose the dimensional radii of duct regions, such as, for planar configuration, $\overline{a}=\overline{b}=0.05$ m, whereas, for non-planar configuration, $\overline{a}=0.05$ m and $\overline{b}=0.15$ m. Accordingly, to obtain the graphs of Figure 8, we choose the dimensional radii of duct regions, such as, for planar configuration, $\overline{a}=\overline{b}=0.15$ m, whereas, for non-planar configuration, $\overline{a}=0.15$ m and $\overline{b}=0.25$ m. From these graphs, it is observed that the scattering powers fluctuate periodically against frequency and depend upon the size of the membrane at the interface as well as the radius of structural discontinuity. Note that parts (a) of Figure 7 and Figure 8 depict the results against frequency of the planar setting ($a=b$) whilst the parts (b) of Figure 7 and Figure 8 show the results for non-planar configurations ($b>a$). From Figure 7a and Figure 8a, it is seen that by increasing the dimensional radii from $\overline{a}=\overline{b}=0.05$ m to $\overline{a}=\overline{b}=0.15$ m, the variation in scattering powers is changed and a smooth increasing behavior after $f=550$ Hz is seen; see, for instance, Figure 7a and Figure 8a. The fluctuations in reflected and transmitted curves occur because of the resonant frequencies. As the non-dimensional radii a and b are related to their dimensional form $\overline{a}$ and $\overline{b}$ as: $a=k\overline{a}=(\omega /c)\overline{a}=(2\pi f/c)\overline{a}$ and $b=k\overline{b}=(\omega /c)\overline{b}=(2\pi f/c)\overline{b}$. When $\overline{a}=\overline{b}=0.05$ m, the resonant frequencies lie within the considered regime ($1<f<1000$ Hz). By increasing the height to $\overline{a}=\overline{b}=0.15$ m, the resonances of elastic membranes are shifted in the range of less than 550 Hz. After this limit, the membranes become stiff and reflection increases smoothly by increasing frequency.

The scattering powers behave conversely, such that the sum of both the reflected and transmitted powers remain at unity, and the conservation law holds. On the other hand, when the configuration contains structural variation at the interface, the maximum of the radiated power goes on reflection, and transmission behaves conversely to it, with the satisfaction of the power-conserving identity (Figure 7b). However, when the radii of duct regions are increased to obtain Figure 8b, the reflection decreases with fluctuations in the start of the observation, and then increases; and vice versa for transmission.

Now, to analyze the physical impact of elastic membranes attached at the interfaces of the chamber, the power reflected in inlet region $z<-L$ and the transmitted power in outlet region ($z>L$) are respectively given as

$${\mathcal{E}}_1=\sum _{m=0}^{\mathcal{K}-1}\mid A_m{\mid }^2{F}_m{\eta }_m,$$

(45)

$${\mathcal{E}}_2=\sum _{m=0}^{\mathcal{K}-1}\mid D_m{\mid }^2{F}_m{\eta }_m,$$

(46)

where ${\mathcal{E}}_1+{\mathcal{E}}_2=1={\mathcal{E}}_3$ and $\mathcal{K}$ is the number of cut-on modes in the inlet/outlet. Further, to measure the performance of the chamber device, the transmission loss is given by [13]

$$\mathrm{Transmission}\mathrm{Loss}=-10{log}_{10}\left({\mathcal{E}}_2\right).$$

(47)

The reflected power, transmitted power, and transmission loss found in (45), (46), and (47), respectively, for the expansion chamber problem, are numerically shown against various parameters in Figure 9, Figure 10, Figure 11 and Figure 12. In Figure 9, the scattering power components and transmission loss versus frequency in regime $1\mathrm{Hz}\le f\le 1000$ Hz, while, $\overline{a}=0.05$ m, $\overline{b}=0.1$ m and $\overline{L}=0.05$ m, are displaced. In Figure 9a, the reflected power is reduced up to about 20 percent at $f=200$ Hz, and then reaches to its maximum position at $f=300$ Hz. After this point, the reflected power varies sharply and then varies with dips occurring at membrane cut-ons to maximum level; whereas, the transmitted power behaves conversely such that sum of reflected and transmitted energies at each point is unity, see Figure 9a. The transmission-loss profile against frequency is displaced in Figure 9b. The peak value of transmission loss appears at $f=450$ Hz; whereas, the associated stopband in this regime is narrow. However, the width of stopbands appearing on the next frequencies is comparatively higher than the width of the previous stopband. Nevertheless, an increasing behavior in the transmission-loss curve along with spikes on the variation of frequency is evident.

To see how the variation in size of the inlet/outlet affect the the scattering power components and transmission loss, Figure 10 is portrayed. To obtain the plots of Figure 10, the chamber radius and half the length of the chamber are receptively contained at $\overline{b}=0.2$ m and $\overline{L}=0.05$ m while the dimensional radius of the inlet/outlet is varied from 0.01 m to $\overline{b}=0.2$ m at $f=550$ Hz. From the reflected and transmitted energies shown by Figure 10a, it is seen that all of the radiated energy goes on reflection until the elastic membranes attached at the interfaces of the chamber start propagating and transmission behaves conversely, such that the energy conversation law prevails. However, by increasing the radius after this limit, the reflected power is reduced rapidly while that transmitted energy goes higher with the same speed. Clearly, transmission loss in this case decreases with abrupt variations; see, for instance, Figure 10b.

The impact of variation of the radius of the chamber, which, in fact, vary the size of structural discontinuity, is displayed in Figure 11. The scattering powers and transmission loss against non-dimensional radius b by varying its dimensional radius from a to 0.5 m, where $\overline{a}=0.05$ m, $f=550$ m and $\overline{L}=0.05$ m remain fixed, as shown in Figure 11. In Figure 11a, the reflected power is its lowest value when the structure does not involve any variation, i.e., when $a=b$. But as the radius of the chamber varies, the reflection is shifted towards its peak value and vice versa for transmission. However, when the second mode of the membranes start propagating, a dip in reflection and, alternatively, spikes in transmission are evident. A dome-like behavior in the transmission-loss curve between the first two cut-ons of membranes is revealed, see Figure 11b.

Figure 12 depicts the scattering power components and transmission loss versus non-dimensional half chamber length L, which is achieved through the variation of dimensional half length from $\overline{L}=0.001$ m to $\overline{L}=0.5$ m along with fixed values of $\overline{a}=0.05$ m, $\overline{b}=0.1$ m and $f=550$ m. A periodic variation in scattering power components as well as transmission loss against chamber length is seen. It is observed that, with the variation of the chamber length, a periodic variation in scattering powers and transmission is found. Moreover, the sum of reflected and transmitted power remains at unity in the whole regime.

7. Summary and Conclusions

In the current study, the analysis of acoustic scattering from an elastic membrane disc placed at an interface inside of cylindrical duct that may comprise structural variation is presented. The inside of the cylindrical channel is filled with compressible fluid and contains an elastic membrane disc, which divides the guiding channel into two regions. The structure is radiated by the fundamental mode, which, on the interaction with fluid, generates the fluid-structure coupled modes in the waveguide. The MM method is advanced by the Galerkin procedure to incorporate the membrane displacement in the interface conditions. Nevertheless, in the absence of such an elastic membrane disc, the governing boundary value problem can be solved in a straight forward manner by using a classical mode-matching method. The proposed scheme is useful to incorporate the scattering of a structure-borne mode on the application of ring conditions at the joint of a cylindrical duct and the membrane disc edge. Moreover, the scheme is capable of encountering the response of higher-order modes. The accuracy of the solution is confirmed through the reconstruction of matching interface conditions and by the satisfaction of the energy conservation law. From the numerical experiments, it is concluded that the variation in the size of radii of duct regions as well as the chamber length and the structural discontinuities have a significant impact on attenuation of the device. Moreover, it is observed that the fluid-structure coupled modes can be controlled through the change of material properties of the elastic membrane disc and by the variation of the geometric setting. The sharp variations in sound transmission loss and scattering power components can be controlled by the stiffness of the elastic membrane disc, configuration setting, and ring conditions, especially in a low-frequency regime.