The scattering analysis of trifurcated waveguide involving structural discontinuities

Abstract

The present article highlights the acoustic waves scattering in a trifurcated waveguide containing compressible fluid together with step discontinuities and change of bounding material properties. The study is important because of its applications in active noise control measures, especially used to control the low-frequency noise and related vibrations. The governing boundary value problem is solved by using the mode-matching technique. The solution is developed for the analysis of symmetric, uniform, and non-uniform cross-sections. The solution procedure begins by determining the expanded form of field potentials in various duct regions of the waveguide. Then, the pressures and the normal velocity modes across the waveguide regions are matched at interface. The energy flux against frequency and various duct configurations is plotted. The solution is validated altogether through the apposite analytical and numerical results.

Introduction

Over the years, a range of interesting and challenging problems that involve the wave scattering analysis in bifurcated and trifurcated waveguide channels have been discussed by many researchers. The interest to minimize the noise pollution impending from the heating, ventilation, and air conditioning (HVAC) system of buildings or automotive exhaust systems of vehicles or aircrafts has stipulated the continued interest. The dissipative silencers containing complex geometrical shapes and bulk reacting materials have been modeled to attenuate the broadband noise. Several analytic and numerical techniques have been developed so far to incorporate these theoretical models. The aim of these studies was to investigate the sound attenuation region against various configurations of channels and material properties.^1–5

The reduction of unwanted noise through the parallel plate setting of walls together with rigid, soft, or impedance type bounding characteristics has been discussed by many authors. For instance, the studies^6–11 include the acoustic scattering in planar trifurcated waveguides containing rigid, soft, or porous lining along with stationary and moving compressible fluid medium. In all such cases, the duct segments contain constant geometrical and physical properties for which the single field representation remains suitable. The integral formulation based on Fourier analysis yields the Wiener–Hopf (WH) solution of the problems. But for more complex geometrical and physical properties, where the wave number spectrum in the waveguide is not continuous, the use of WH technique is not appropriate. However, recently the mode-matching (MM) technique has been used to deal with more complex geometries along with different medium and material properties. Such methods were initially developed to tackle canonical problems represented by Laplace or Helmholtz equations and the ducts/channel boundaries depicted by Dirichlet, Neumann, or Robin conditions. The investigation of reflection and transmission of waves across the interface at discontinuities in pipes and ducts is hence performed generally by matching modes, for example, see literature.^12–15

The fundamental interest of the present study is to apply the hybrid matching scheme to analyze the acoustic scattering in non-uniform and/or uniform trifurcated waveguide that contains compressible fluid in its duct sections and different bounding wall conditions. For the duct region of constant physical and geometrical properties, the modal field representation is formulated. The key challenge is to encounter the geometric discontinuities at the plane connection (junction) of duct segments. The study of acoustic radiation from structural discontinuities goes back to Rayleigh ¹⁶ even though traditionally it has been merely feasible to look at rather a simple non-uniform geometry with modest dimensions. However, Boström, ¹⁷ Miles, ¹⁸ and, more recently, Nawaz and Lawrie ¹⁹ and Nawaz et al. ²⁰ discussed the sound scattering from structural discontinuities together with obstacles at finite junction. The continuity conditions of pressures and normal velocities at aperture are used to match the scattering modes. The geometric discontinuities are encountered in normal velocity fields and many modes are allowed to propagate.

Here, the MM scheme has been explored to solve a trifurcated waveguide problem involving geometric discontinuities. The inside of the waveguide regions contains compressible fluid and different bounding wall conditions. Whereas Hassan ²¹ and Hassan et al.^22,23 applied the MM technique to discuss the acoustic scattering in planar trifurcated and pentafurcated waveguide containing compressible fluid in the absence of step discontinuities. The physical problem is stated in the ”Mathematical formulation” section. The MM solution is sorted in the “MM solution” section. The energy conservation has been analyzed in the “Energy balance” section, while the numerical results and discussion is given in the “Numerical results and discussion” section. The concluding remarks are given in the “Conclusion” section.

Mathematical formulation

To formulate the boundary value problem, we consider an infinitely stretched trifurcated waveguide occupying the regions ${\mathcal{R}}_1:=\{\overline{x}<\overline{0},-\overline{a}\le \overline{y}\le \overline{a}\}$ , ${\mathcal{R}}_2:=\{\overline{x}<\overline{0},-\overline{b}\le \overline{y}\le -\overline{a}\}$ , ${\mathcal{R}}_3:=\{\overline{x}<\overline{0},\overline{a}\le \overline{y}\le \overline{b}\}$ , and ${\mathcal{R}}_4:=\{\overline{x}>\overline{0},-\overline{h}\le \overline{y}\le \overline{h}\}$ , where over-bar denotes the dimensional setting of coordinates. The inside of the regions ${\mathcal{R}}_j,j=1,2,3,4$ is filled with compressible fluid of density $\rho$ and sound speed c. The bounding wall conditions of the region may vary: (a) each of the region ${\mathcal{R}}_j,j=1,2,3,4$ is bounded by acoustically rigid boundaries, and (b) the regions ${\mathcal{R}}_2$ and ${\mathcal{R}}_3$ comprise soft boundaries at $\overline{y}=\pm \overline{b}$ . The outside of these waveguide is set into vacou. At $\overline{x}=0$ , there lies two rigid vertical step discontinuities aligned along with $-\overline{h}\le \overline{y}\le -\overline{b}$ and $\overline{b}\le \overline{y}\le \overline{h}$ . The physical configuration of the problem is shown in Figure 1.

Figure 1.

The physical configuration of the waveguide.

Consider an incident wave of harmonic time dependence propagating in ${\mathcal{R}}_1$ from negative x-direction toward $\overline{x}=0$ . At $\overline{x}=0$ , it will scatter into the infinite number of reflected and transmitted modes. The dimensional field potentials that characterize the scattering fields in different duct regions may be written as

$$\overline{\psi }\left(\overline{x},\overline{y}\right)=\{\begin{array}{cc}{\overline{\psi }}_1\left(\overline{x},\overline{y}\right),& \left(\overline{x},\overline{y}\right)\in {ℛ}_1\\ {\overline{\psi }}_2\left(\overline{x},\overline{y}\right),& \left(\overline{x},\overline{y}\right)\in {ℛ}_2\\ {\overline{\psi }}_3\left(\overline{x},\overline{y}\right),& \left(\overline{x},\overline{y}\right)\in {ℛ}_3\\ {\overline{\psi }}_4\left(\overline{x},\overline{y}\right),& \left(\overline{x},\overline{y}\right)\in {ℛ}_4\end{array}$$

(1)

Now, we assume the harmonic time dependence of $e^{-i\omega \overline{t}}$ in which $\omega =\mathrm{ck}$ is the angular frequency and $k=2\pi f/c$ is the wave number with frequency f. On non-dimensionalizing the boundary value problem with respect to the length scale $k^{-1}$ and the time scale ${\omega }^{-1}$ under the transformations $x=k\overline{x}$ , $y=k\overline{y}$ , and $t=\omega \overline{t}$ . Thus, the dimensionless fluid velocity potential $\psi (x,y)$ satisfies the Helmholtz’s equation

$$\left({\nabla }^2+1\right)\psi \left(x,y\right)=0$$

(2)

In ${\mathcal{R}}_j;j=1,2,3,4$ , the non-dimensional forms of rigid boundary conditions are

$$\frac{\partial {\psi }_1}{\partial y}=0,y=\pm a,x<0$$

(3)

$$\frac{\partial {\psi }_2}{\partial y}=0,y=-b,-a,x<0$$

(4)

$$\frac{\partial {\psi }_3}{\partial y}=0,y=a,b,x<0$$

(5)

$$\frac{\partial {\psi }_4}{\partial y}=0,y=\pm h,x>0$$

(6)

$$\frac{\partial {\psi }_4}{\partial x}=0,x=0,\mathrm{for}-h\le y\le -b\mathrm{and}b\le y\le h$$

(7)

Here, equation (7) represents the rigid conditions for two vertical step discontinuities. However, the acoustically soft wall conditions for ${\mathcal{R}}_2$ and ${\mathcal{R}}_3$ are

$${\psi }_2=0,y=-b,x<0$$

(8)

$${\psi }_3=0,y=b,x<0$$

(9)

Moreover, the continuity conditions of the fluid pressure and normal velocity at matching interface are

$${\psi }_2={\psi }_4,x=0,-b\le y\le -a$$

(10)

$${\psi }_1={\psi }_4,x=0,-a\le y\le a$$

(11)

$${\psi }_3={\psi }_4,x=0,a\le y\le b$$

(12)

and

$${\psi }_{4x}=\{\begin{array}{ccc}0\hfill & x=0,\hfill & -h\le y<-b\hfill \\ \multicolumn{1}{c}{{\psi }_{2x}}& x=0,\hfill & -b\le y\le -a\hfill \\ \multicolumn{1}{c}{{\psi }_{1x}}& x=0,\hfill & -a\le y\le a\hfill \\ \multicolumn{1}{c}{{\psi }_{3x}}& x=0,\hfill & a\le y\le b\hfill \\ \multicolumn{1}{c}{0}& x=0,\hfill & b\le y<h\hfill \end{array}$$

(13)

The subscript x here denotes differentiation with respect to x. In the following section, we solve the boundary value problem by using the MM technique.

MM solution

In order to solve the boundary value problem by using the MM technique, we first determine the eigenfunction expansions and related orthogonality conditions in each duct region. In different duct regions, these are found as

$$\mathbf{Region}{\mathcal{R}}_1:=\{x<0,-a\le y\le a\}$$

In this region, equations (2) and (3) yield the eigenexpansion form of field potential as

$${\psi }_1(x,y)=e^{\mathrm{ix}}+\sum _{n=0}^{\infty }{A}_n\cos \left[{\tau }_n(y+a)\right]e^{-i{\eta }_{n}x}$$

(14)

The quantity ${\eta }_n=\sqrt{1-{\tau }_n^2}$ be the $n\mathrm{th}$ reflected mode wave number in which ${\tau }_n$ , $n=0,1,2,\dots$ , are the eigenvalues, and these satisfy the dispersion relation

$$\sin \left[2{\tau }_{n}a\right]=0,\mathrm{for}n=0,1,2,\dots$$

(15)

The corresponding eigenfunctions $\cos [{\tau }_n(y+a)]$ , $n=0,1,2,\dots$ , satisfy the orthogonality relation

$$\underset{-a}{\overset{a}{\int }}\cos \left[{\tau }_m(y+a)\right]\cos \left[{\tau }_n(y+a)\right]\mathrm{dy}={ϵ}_{m}a{\delta }_{\mathrm{mn}}$$

(16)

where ${\delta }_{\mathrm{mn}}$ is Kronecker delta and ${ϵ}_m=2$ for $m=0$ and $1$ otherwise. Note that the first term in equation (14) denotes the incident wave, while the second term represents the reflected field in which $A_n$ , $n=0,1,2,\dots$ , are the reflected mode coefficients and are unknowns. These unknowns will be found later from matching conditions

$$\mathbf{Region}{\mathcal{R}}_2:=\{x<0,-b\le y\le -a\}$$

From equations (2), (4), and (8), the eigenexpansion form is obtained as

$${\psi }_2(x,y)=\sum _{n=0}^{\infty }{B}_n\cos \left[{\xi }_n(y+a)\right]e^{-i{\nu }_{n}x}$$

(17)

where ${\nu }_n=\sqrt{1-{\xi }_n^2}$ be the $n\mathrm{th}$ transmitted mode wave number in which ${\xi }_n$ , $n=0,1,2,\dots$ , are the eigenvalues. For rigid boundary condition at $y=-b$ , these satisfy the dispersion relation

$$\sin \left[{\xi }_n(b-a)\right]=0,\mathrm{for}n=0,1,2,\dots$$

(18)

However, for soft boundary condition at $y=-b$ , these satisfy

$$\cos \left[{\xi }_n(b-a)\right]=0,\mathrm{for}n=0,1,2,\dots$$

(19)

The corresponding eigenfunctions $\cos [{\xi }_n(y+a)]$ , $n=0,1,2,\dots$ , satisfy the orthogonality relation

$$\underset{-b}{\overset{-a}{\int }}\cos \left[{\xi }_m(y+a)\right]\cos \left[{\xi }_n(y+a)\right]\mathrm{dy}=\frac{\Theta \left({\xi }_n\right)(b-a){\delta }_{\mathrm{mn}}}{2}$$

(20)

where

$$\Theta \left({\xi }_n\right)=\{\begin{array}{c}{ϵ}_n,\mathrm{for}{\xi }_n=\frac{n\pi }{b-a}:n=0,1,2,\dots \hfill \\ \multicolumn{1}{c}{1,\mathrm{for}{\xi }_n=\frac{(2n+1)\pi }{2(b-a)}:n=0,1,2,\dots }\end{array}$$

(21)

$$\mathbf{Region}{\mathcal{R}}_3:=\{x<0,a\le y\le b\}$$

From equations (2), (5), and (9), the eigenexpansion form is found to be as

$${\psi }_3(x,y)=\sum _{n=0}^{\infty }{C}_n\cos \left[{\lambda }_n(y-a)\right]e^{-i{\varkappa }_{n}x}$$

(22)

where ${\varkappa }_n=\sqrt{1-{\lambda }_n^2}$ be the wave number of $n\mathrm{th}$ transmitted mode and ${\lambda }_n$ , $n=0,1,2,\dots$ , are the eigenvalues. For rigid surface at $y=b,$ these eigenvalues satisfy the dispersion relation

$$\sin \left[{\lambda }_n(b-a)\right]=0,\mathrm{for}n=0,1,2,\dots$$

(23)

whereas for the soft surface at $y=b$ , these satisfy

$$\cos \left[{\lambda }_n(b-a)\right]=0,\mathrm{for}n=0,1,2,\dots$$

(24)

The corresponding eigenfunctions $\cos [{\lambda }_n(y-a)]$ , $n=0,1,2,\dots$ , are orthogonal and satisfy the usual orthogonality relation

$$\underset{a}{\overset{b}{\int }}\cos \left[{\lambda }_m(y-a)\right]\cos \left[{\lambda }_n(y-a)\right]\mathrm{dy}=\frac{\Theta \left({\lambda }_n\right)(b-a){\delta }_{\mathrm{mn}}}{2}$$

(25)

$$\mathbf{Region}{\mathcal{R}}_4:=\{x>0,-h\le y\le h\}$$

From equations (2) and (6), the eigenexpansion form of transmitted field is revealed as

$${\psi }_4(x,y)=\sum _{n=0}^{\infty }{D}_n\cos \left[{\gamma }_n(y+h)\right]e^{i{s}_{n}x}$$

(26)

where $s_n=\sqrt{1-{\gamma }_n^2}$ be the $n\mathrm{th}$ transmitted mode wave number and ${\gamma }_n$ , $n=0,1,2,\dots$ , are the eigenvalues, and these satisfy the dispersion relation

$$\sin \left[2{\gamma }_{n}h\right]=0,\mathrm{for}n=0,1,2,\dots$$

(27)

The corresponding eigenfunctions $\cos [{\gamma }_n(y+h)]$ , $n=0,1,2,\dots$ , satisfy the orthogonality relation

$$\underset{-h}{\overset{h}{\int }}\cos \left[{\gamma }_m(y+h)\right]\cos \left[{\gamma }_n(y+h)\right]\mathrm{dy}=h{ϵ}_m{\delta }_{\mathrm{mn}}$$

(28)

Now, the unknown coefficients $\{A_n,B_n,C_n,D_n\}$ , $n=0,1,2,\dots$ , are found by using the matching conditions (equations (11)–(13)). On using equations (14) and (26) into equation (11), we get

$$1+\sum _{n=0}^{\infty }{A}_n\cos \left[{\tau }_n(y+a)\right]=\sum _{n=0}^{\infty }{D}_n\cos \left[{\gamma }_n(y+h)\right]$$

(29)

On multiplying equation (29) with $\cos [{\tau }_m(y+a)]$ , integrating from $-a$ to a and then using the orthogonality relation (equation (16)), we get

$$A_m=-{\delta }_{m0}+\frac{1}{a{ϵ}_m}\sum _{n=o}^{\infty }{D}_n{R}_{\mathrm{mn}}$$

(30)

where

$$R_{\mathrm{mn}}=\{\begin{array}{cc}2a{\delta }_{\mathrm{mn}}\hfill & n=0,m=0,1,2,\dots \hfill \\ \multicolumn{1}{c}{\frac{{\gamma }_n}{{\tau }_m^2-{\gamma }_n^2}{E}_{\mathrm{mn}}}& n\ne 0,m=0,1,2,\dots \hfill \end{array}$$

(31)

$$E_{\mathrm{mn}}={(-1)}^{m+1}\sin \left[(a+h){\gamma }_n\right]-\sin \left[(a-h){\gamma }_n\right]$$

(32)

On using equations (17) and (26) into equation (10), we get

$$\sum _{n=0}^{\infty }{B}_n\cos \left[{\xi }_n(y+a)\right]=\sum _{n=0}^{\infty }{D}_n\cos \left[{\gamma }_n(y+h)\right]$$

(33)

On multiplying equation (33) with $\cos [{\xi }_m(y+a)]$ , integrating from $-b$ to $-a$ and then using the orthogonality relation (equation (20)), we get

$$B_m=\frac{2}{(b-a)\Theta \left({\xi }_m\right)}\sum _{n=0}^{\infty }{D}_n{P}_{\mathrm{mn}}$$

(34)

For all rigid–rigid cases

$$P_{\mathrm{mn}}=\{\begin{array}{cc}{\delta }_{\mathrm{mn}}(b-a)\hfill & n=0,m=0,1,2,\dots \hfill \\ \multicolumn{1}{c}{\frac{{\gamma }_n}{\frac{m^2{\pi }^2}{{(b-a)}^2}-{\gamma }_n^2}{F}_{\mathrm{mn}}}& n\ne 0,m=0,1,2,\dots \hfill \end{array}$$

(35)

where

$$F_{\mathrm{mn}}={(-1)}^{m+1}\sin \left[(h-a){\gamma }_n\right]+\sin \left[(h-b){\gamma }_n\right]$$

(36)

For all rigid–soft cases

$$P_{\mathrm{mn}}=\{\begin{array}{cc}\frac{2(b-a)}{\pi }\hfill & n=0,m=0,1,2,\dots \hfill \\ \multicolumn{1}{c}{\frac{1}{\frac{{(2m+1)}^2{\pi }^2}{4{(b-a)}^2}-{\gamma }_n^2}{G}_{\mathrm{mn}}}& n\ne 0,m=0,1,2,\dots \hfill \end{array}$$

(37)

where

$$\begin{array}{c}\multicolumn{1}{c}{G_{\mathrm{mn}}={(-1)}^{m+1}{\gamma }_n\sin \left[(h-a){\gamma }_n\right]}\\ \multicolumn{1}{c}{+\frac{(2m+1)\pi }{2(b-a)}\cos \left[(h-b){\gamma }_n\right]}\end{array}$$

(38)

On using equations (22) and (26) into equation (12), we get

$$\sum _{n=0}^{\infty }{C}_n\cos \left[{\lambda }_n(y-a)\right]=\sum _{n=0}^{\infty }{D}_n\cos \left[{\gamma }_n(y+h)\right]$$

(39)

On multiplying equation (39) with $\cos [{\lambda }_m(y-a)]$ , integrating from a to b and then using the orthogonality relation (equation (25)), we get

$$C_m=\frac{2}{(b-a)\Theta \left({\lambda }_m\right)}\sum _{n=0}^{\infty }{D}_n{Q}_{\mathrm{mn}}$$

(40)

For all rigid–rigid cases

$$Q_{\mathrm{mn}}=\{\begin{array}{cc}{\delta }_{\mathrm{mn}}(b-a)\hfill & n=0,m=0,1,2,\dots \hfill \\ \multicolumn{1}{c}{\frac{{\gamma }_n}{\frac{m^2{\pi }^2}{{(b-a)}^2}-{\gamma }_n^2}{H}_{\mathrm{mn}}}& n\ne 0,m=0,1,2,\dots \hfill \end{array}$$

(41)

where

$$H_{\mathrm{mn}}={(-1)}^{m+1}\sin \left[(b+h){\gamma }_n\right]+\sin \left[(a+h){\gamma }_n\right]$$

(42)

For all rigid–soft cases

$$Q_{\mathrm{mn}}=\{\begin{array}{cc}\frac{2(b-a)}{\pi }\hfill & n=0,m=0,1,2,\dots \hfill \\ \multicolumn{1}{c}{\frac{1}{\frac{{(2m+1)}^2{\pi }^2}{4{(b-a)}^2}-{\gamma }_n^2}{I}_{\mathrm{mn}}}& n\ne 0,m=0,1,2,\dots \hfill \end{array}$$

(43)

where

$$\begin{array}{c}\multicolumn{1}{c}{I_{\mathrm{mn}}={(-1)}^{m+1}\frac{(2m+1)\pi }{2(b-a)}\cos \left[(b+h){\gamma }_n\right]}\\ \multicolumn{1}{c}{+{\gamma }_n\sin \left[(a+h){\gamma }_n\right]}\end{array}$$

(44)

Now, to determine the transmitted mode coefficients, we use equations (17), (14), (22), and (26) into equation (13) to obtain

$$\sum _{n=0}^{\infty }{D}_n{s}_n\cos \left[{\gamma }_n(y+h)\right]=\{\begin{array}{cc}0\hfill & -h\le y<-b\hfill \\ \multicolumn{1}{c}{-\sum _{n=0}^{\infty }{B}_n{\upsilon }_n\cos \left[{\xi }_n(y+a)\right]}& -b\le y<-a\hfill \\ \multicolumn{1}{c}{1-\sum _{n=0}^{\infty }{A}_n{\eta }_n\cos \left[{\tau }_n(y+a)\right],}& -a\le y<a\hfill \\ \multicolumn{1}{c}{-\sum _{n=0}^{\infty }{C}_n{\varkappa }_n\cos \left[{\lambda }_n(y-a)\right],}& a\le y<b\hfill \\ \multicolumn{1}{c}{0}& b\le y\le h\hfill \end{array}$$

(45)

On multiplying with $\cos [{\gamma }_m(y+h)]$ , then integrating from $-h$ to h and then using the orthogonality relation (equation (45)), we get

$$\begin{array}{c}\multicolumn{1}{c}{D_m=\frac{R_{0m}}{s_m{ϵ}_{m}h}-\frac{1}{s_m{ϵ}_{m}h}\sum _{n=0}^{\infty }{A}_n{\eta }_n{R}_{\mathrm{nm}}-\frac{1}{s_m{ϵ}_{m}h}\sum _{n=0}^{\infty }{B}_n{\upsilon }_n{P}_{\mathrm{nm}}}\\ \multicolumn{1}{c}{-\frac{1}{s_m{ϵ}_{m}h}\sum _{n=0}^{\infty }{C}_n{\varkappa }_n{Q}_{\mathrm{nm}}}\end{array}$$

(46)

Now, by means of equations (30), (34), and (40), it is straightforward to obtain equation (46) in terms of unknown $D_m,m=0,1,2,\dots ,$ that is

$$\begin{array}{c}{D}_m{s}_m{ϵ}_{m}h+\sum _{n=0}^{\infty }\sum _{p=0}^{\infty }\frac{D_p{\eta }_n{R}_{\mathrm{np}}{R}_{\mathrm{nm}}}{{ϵ}_{n}a}\hfill \\ \multicolumn{1}{c}{+\sum _{n=0}^{\infty }\sum _{q=0}^{\infty }\frac{2D_q{\upsilon }_n{P}_{\mathrm{nq}}{P}_{\mathrm{nm}}}{(b-a)\Theta \left({\xi }_n\right)}+\sum _{n=0}^{\infty }\sum _{l=0}^{\infty }\frac{2D_l{\varkappa }_n{Q}_{\mathrm{nm}}{Q}_{\mathrm{nl}}}{(b-a)\Theta \left({\lambda }_m\right)}=2R_{0m}}\end{array}$$

(47)

In this way, equation (47) leads to a system of infinite equations in which $D_m$ , $m=0,1,2,\dots$ , are unknowns. These are truncated and inverted to determine the unknown model coefficients. Once obtained, $D_m$ , $m=0,1,2,\dots$ , and the quantities $\{A_m,B_m,C_m\}$ , $m=0,1,2,\dots$ , are found easily by using equations (30), (34), and (40).

Energy balance

The energy flux/power inside the duct regions in terms of non-dimensional time harmonic fluid velocity potential is defined by

$$\frac{\partial \mathcal{E}}{\partial t}=\frac{1}{2}\mathrm{Re}\left\{i\underset{\Omega }{\int }\psi {\left(\frac{\partial \psi }{\partial x}\right)}^{*}\mathrm{dy}\right\}$$

(48)

where superscript asterisk (*) denotes the complex conjugate. From the definition of energy flux/power, the incident power is found to be ${\mathcal{P}}_{\mathrm{inc}}=a$ . Likewise, the power/energy flux components in duct region ${\mathcal{R}}_j$ , $j=1,2,3,4$ , are

$${\mathcal{P}}_1=\frac{1}{2}\mathrm{Re}\{\sum _{n=o}^{\infty }|A_n{|}^2{\eta }_n{ϵ}_n\}$$

(49)

$${\mathcal{P}}_2=\frac{b-a}{4a}\mathrm{Re}\{\sum _{n=o}^{\infty }|B_n{|}^2{\upsilon }_n\Theta \left({\xi }_n\right)\}$$

(50)

$${\mathcal{P}}_3=\frac{b-a}{4a}\mathrm{Re}\{\sum _{n=o}^{\infty }|C_n{|}^2{\varkappa }_n\Theta \left({\lambda }_n\right)\}$$

(51)

and

$${\mathcal{P}}_4=\frac{h}{2a}\mathrm{Re}\{\sum _{n=o}^{\infty }|D_n{|}^2{s}_n{ϵ}_n\}$$

(52)

Now, it is important to note that the power fed into the system will be equal to the sum of scattering powers in different duct regions, that is

$${\mathcal{P}}_{\mathrm{inc}}={\mathcal{P}}_1+{\mathcal{P}}_2+{\mathcal{P}}_3+{\mathcal{P}}_4$$

(53)

which is conserved power identity. For analysis purpose, we may scale the incident power at unity. For this, we divide equation (53) by a to get ${\mathcal{E}}_T=\sum _{j=1}^4{\mathcal{E}}_j$ , where ${\mathcal{E}}_j={\mathcal{P}}_j/a$ , $j=1,2,3,4$ , denote the power/energy flux components in duct regions $R_j$ , $j=1,2,3,4$ , for which the incident power is being scaled at unity.

Numerical results and discussion

Here, we truncate the system of equations defined by equation (47) up to N terms and then solve the retained system numerically to discuss the radiated energy flux/power in duct regions and to reconstruct the matching conditions at interface. Here, two types of bounding wall conditions for regions $R_j$ , $j=1,2,3,4$ , are assumed: (a) all are acoustically rigid (Figures 2–5(a)), and (b) rigid–soft for $R_2$ and $R_3$ only (Figures 2–5(b)). For numerical computations, the compressible fluid of density (air) $\rho =1.2043\mathrm{kg}{\mathrm{m}}^{-3}$ and the speed of sound $c=343\mathrm{m}{\mathrm{s}}^{-1}$ are taken from Kaye and Laby. ²⁴

Figure 2.

The energy flux/power components against frequency in discontinuous waveguide, with (a) rigid–rigid and (b) rigid–soft boundaries, where $\overline{a}=0.05\mathrm{m}$ , $\overline{b}=0.1\mathrm{m}$ , $\overline{h}=0.15\mathrm{m}$ , and $N=10$ .

Figure 3.

The energy flux/power components against frequency in planer waveguide, with (a) rigid–rigid and (b) rigid–soft boundaries, where $\overline{a}=0.05\mathrm{m}$ , $\overline{b}=0.1\mathrm{m}$ , $\overline{h}=0.15\mathrm{m}$ , and $N=10$ .

Figure 4.

The energy flux/power components against height discontinuities h, for (a) rigid–rigid and (b) rigid–soft boundaries, where $\overline{a}=0.05\mathrm{m}$ , $\overline{b}=0.1\mathrm{m}$ , and $N=10.$

Figure 5.

The energy flux/power components against b, for (a) rigid–rigid and (b) rigid–soft boundaries, where $\overline{a}=0.05\mathrm{m}$ , $\overline{h}=\overline{b}$ , and $N=10$ .

The radiated power in different duct regions against frequency from $1$ to $3000\mathrm{Hz}$ is shown in Figure 2. The considered configuration comprises structural discontinuities at interface and fixed dimensions: $\overline{a}=0.05\mathrm{m}$ , $\overline{b}=0.1\mathrm{m}$ , and $\overline{h}=0.15\mathrm{m}$ .

Note that the radiated energy against frequency depicted in Figure 2 contains sharp edges and the variation in radiated energy is more noticeable before and after these edges. These edges basically specify the cut-on position of duct modes. For the structural discontinuity case along with rigid bounding characteristics (Figure 2(a)), the two duct modes are cut-on at $f=1441\mathrm{Hz}$ and $f=2281\mathrm{Hz}$ . However, for the rigid–soft case (Figure 2(b)), three duct modes are cut-on at $f=1441\mathrm{Hz}$ , $f=1711\mathrm{Hz}$ , and $f=2281\mathrm{Hz}$ . The more manifest radiated energy is of region $R_1$ and $R_4$ . For rigid–rigid case (Figure 2(a)), by increasing frequency, the energy flux in $R_4$ decreases, which goes to its maximum at cut-on limit of the next duct mode. But once the next, and so on higher modes start propagating, this power varies inversely to its highest and, vice versa for energy flux in $R_1$ . However, in later rigid–soft case (Figure 2(b)), this behavior is opposite up to the first cut-on point, while remains similar to previous case for higher order modes.

In the next figure (Figure 3), we assume the planar waveguide configuration (without structural discontinuity) by taking $\overline{a}=0.05\mathrm{m}$ and $\overline{b}=\overline{h}=0.1\mathrm{m}$ . Note that only the first duct mode is cut-on at $f=1711\mathrm{Hz}$ in given regime for both rigid (Figure 3(a)) and rigid–soft (Figure 3(b)) bounding properties. In rigid–rigid case, the radiated power in different duct regions remains constant against frequency up to their first cut-on and then goes to its largest (see Figure 3(a)). But in the rigid–soft case, the energy radiated in regions $R_1$ and $R_4$ varies inversely.

The effect of variation in symmetric height discontinuity on radiated energy is plotted in Figure 4. For the fixed frequency $f=1000\mathrm{Hz}$ and dimensions $\overline{a}=0.05\mathrm{m}$ and $\overline{b}=0.1\mathrm{m}$ , the non-dimensional height discontinuity $h=k\times \overline{h}$ changed symmetrically from $\overline{h}=\overline{b}$ to $\overline{h}=0.2\mathrm{m}$ . The first cut-on duct mode occurs at $k\times \overline{h}\approx 3.11$ , while the other cut-on modes are beyond the chosen domain. From Figure 4, it is noted that the variation in height discontinuity significantly affects the reflection in $R_1$ and transmission in $R_4$ , for both rigid–rigid case and rigid–soft case. There is no radiated power in $R_2$ and $R_3$ for rigid–soft setting, which is consistent with the transmission loss in these regions.

Figure 5 shows the radiated energy in the planar waveguide against the size of the regions $R_2$ , $R_3$ , and $R_4.$ In symmetrical setting, the vertical dimensions of the regions containing incident wave are fixed at $\overline{a}=0.05\mathrm{m}$ and $\overline{b}=\overline{h}$ . Now, on changing the non-dimensional height $b=k\times \overline{b}$ , for $0.1\mathrm{m}\le \overline{b}\le 0.15\mathrm{m}$ , the size of regions $R_2,$ $R_3$ , and $R_4$ is changed while frequency remains fixed at $f=1000\mathrm{Hz}$ . The cut-on duct mode for the rigid–rigid case occurs at $k\times \overline{b}\approx 3.13$ (see Figure 5(a)), while for the rigid–soft case, two cut-on duct modes exist at $k\times \overline{b}\approx 2.18$ and $k\times \overline{b}\approx 3.13$ (see Figure 5(b)). From Figure 5, it is clear that the change in size of the duct regions significantly affects the radiated energy of these regions.

By comparing all these four graphs, it is clear that the geometric discontinuity and the change of bounding material properties greatly affect the radiated energy duct modes, especially in low-frequency regime $5-1200\mathrm{Hz}$ .

Now, the continuity conditions of pressure and normal velocity are pieced together at interface to validate the truncated solution. Only the results for all rigid setting of waveguide along with height discontinuities are shown in Figures 6 and 7. The vertical dimensions of symmetric waveguide are fixed at $\overline{a}=0.05\mathrm{m}$ , $\overline{b}=0.1\mathrm{m}$ , and $\overline{h}=0.15\mathrm{m}$ . At frequency $f=1000\mathrm{Hz}$ , the real $(\Re )$ and imaginary $(\Im )$ parts of non-dimensional pressure and normal velocity are depicted in Figures 6 and 7. It can be seen that the real and imaginary parts of pressures ${\psi }_j\mathbf{(}0,y\mathbf{)}$ , $j=1,2,3$ , in their respective region match exactly to the pressure ${\psi }_4\mathbf{(}0,y\mathbf{)}$ , $-b<y<b$ . Likewise, the real and imaginary parts of normal velocities ${\psi }_{\mathrm{jx}}\mathbf{(}0,y\mathbf{)}$ , $j=1,2,3$ , in their respective regions match exactly to the normal velocity ${\psi }_{4x}\mathbf{(}0,y\mathbf{)}$ , $-b<y<b$ . However, the real and imaginary parts of ${\psi }_{4x}\mathbf{(}0,y\mathbf{)}$ approach to zero in the regime $-h<y<-b$ and $b<y<h$ , which is exactly the condition considered in equation (13).

Figure 6.

The (a) real and (b) imaginary parts of pressures against duct heights, at interface, with step discontinuities and rigid–rigid boundaries, where $\overline{a}=0.05\mathrm{m}$ , $\overline{b}=0.1\mathrm{m}$ , $\overline{h}=0.15\mathrm{m}$ , and $N=80$ .

Figure 7.

The (a) real and (b) imaginary parts of normal velocities against duct heights, at interface, with step discontinuities and rigid–rigid boundaries, where $\overline{a}=0.05\mathrm{m}$ , $\overline{b}=0.1\mathrm{m}$ , $\overline{h}=0.15\mathrm{m}$ , and $N=80$ .

In this way, the truncated solution reconstructs successfully the matching conditions of pressures and velocities at interface (see equations (11)–(13)). However, there appear some oscillations in normal velocity curves, which lessen on increasing the number of modes. One more physical check on accuracy of the truncated solution is the validation of conserved power identity ${\mathcal{E}}_T=1$ while the inside of the waveguide contains compressible fluid. It holds that even a few number of duct modes are allowed to propagate in their respective regions (see dotted line in Figures 2–5).

Conclusion

In this study, we have discussed the acoustic scattering in a trifurcated waveguide involving structural discontinuity at interface. The inside of the duct regions contains compressible fluid while the bounding wall conditions are chosen to be acoustically rigid and/or soft. By varying inside and bounding characteristics of the regions, two physical problems are formulated together. The MM technique has been explored to solve the envisaged boundary value problems. The eigenfunctions along with the appropriate orthogonality characteristics in their respective regions enable to recast the differential system into linear algebraic system. This system is then truncated and solved numerically. The provided solution remains valid for both discontinuous and planar waveguide structures. It also helps us to discuss the radiated energy against the variation of the geometric discontinuity and the size of the regions. The opposite numerical results are depicted in Figures 2 to 5. This problem can be regarded as a prototype that provides a benchmark scheme to model and solve a range of bifurcated or trifurcated waveguide problems involving geometric discontinuities and various material properties of medium and bounding walls.

Here, it is interesting to notice that the conservation of energy flux across the duct regions not only confirm the propagation of cut-on duct modes in various duct sections but also provide a useful check on the accuracy of algebra. Besides this, the truncated solution has also been used to reconstruct the matching of pressure and normal velocity modes at interface (see Figures 6 and 7). The validation of matching condition at interface along with the conservation of energy flux across the duct regions substantiate the MM solution altogether.