We consider the diffraction of an incident plane mode from a semi-infinite soft surface duct (\(\vert y\vert < a\), \(x<0\)) which has geometrical representation in Figure 1. We form a two-dimensional pentafurcated structure such that four semi-infinite soft plates are located inside two infinite hard plates. Infinite hard plates are located at \(y=\pm d\).

We introduce the scalar potential function \(\varphi (x, y, t)\) and define acoustic pressure by \(p=-\rho_{o}\frac{\partial \varphi }{ \partial t}\) and velocity vector by \(\vec{u}=\operatorname{grad}(\varphi)\), respectively (where \(\rho_{o}\) indicates the density in equilibrium state). The potential *φ* satisfies the following wave equation:

$$ \triangledown^{2}\varphi =\bigl(1/c^{2}\bigr) \varphi_{tt}. $$

(2.1)

We assume

$$ \varphi (x,y,t)=\operatorname{Re}\bigl[v(x,y)e^{-iwt}\bigr], $$

(2.2)

where *c* is the speed of the sound, *w* is angular frequency, and the wave number is defined as \(k=\frac{w}{c}\). Equation (2.2) identically satisfies equation (2.1) which eventually results into a well-known Helmholtz equation in 2*D* that is within the duct

$$ \bigl(\triangledown^{2}+k^{2}\bigr)v(x,y)=0. $$

(2.3)

We will solve the problem subject to the following boundary conditions:

$$\begin{aligned}& v_{y}(x,y) =0,\quad y=\pm d, -\infty < x< \infty, \end{aligned}$$

(2.4)

$$\begin{aligned}& v(x,y) =0, \quad y=\pm b, -\infty < x< 0, \end{aligned}$$

(2.5)

$$\begin{aligned}& v(x,y) =0,\quad y=\pm a, -\infty < x< 0. \end{aligned}$$

(2.6)

The wave field satisfies the radiation conditions:

$$ v(x, y)-v^{i}(x,y)\quad \mbox{is outgoing and bounded as } x\rightarrow \pm \infty, $$

(2.7)

which simply ensures the boundedness of the obtained solution. In order to ensure the unique solution, different extra conditions, which are termed as edge conditions, are imposed. However, the problem is well posed as mentioned above. Therefore, no extra condition is required for the uniqueness of the solutions. Yet, in the case of higher order boundaries, the use of edge conditions becomes relevant. This statement is well supported through a number of research articles (see, for instance, [15, 23]).

### 2.1 Region I: \(\{-d \leq y \leq -b, x<0\}\)

In the given problem, the potential solution is obtained in each region by applying the method of separation of variables. Then we use the straightforward eigenfunction expansion technique proposed by Mei [24] and Linton & Mclver [25].

The potential solution of equation (2.3) in region I can be written as

$$ v(x,y)=\sum_{n=1}^{\infty }A_{n} e^{-i\hat{\alpha }_{n}x} \biggl( \sqrt{ \frac{2}{(d-b)}}\sin \alpha_{n}(y+b) \biggr), $$

(2.8)

which satisfies equations (2.4), (2.5) and radiation conditions (2.7), where \(A_{n}\) represents transmitted field amplitudes in region I.

The orthonormal relation is defined as

$$\begin{aligned}& \int_{-d}^{-b} \biggl( \sqrt{\frac{2}{(d-b)}}\sin \alpha_{m}(y+b) \biggr) \biggl( \sqrt{\frac{2}{(d-b)}}\sin \alpha_{n}(y+b) \biggr)\,dy \\& \quad =\delta _{mn},\quad m,n=1,2,3, \dots, \end{aligned}$$

(2.9)

where \(\delta_{mn}\) is the Kronecker delta defined by

$$ \delta_{mn}=\textstyle\begin{cases} 0,& m\neq n, \\ 1,& m=n. \end{cases} $$

(2.10)

The associated eigenvalues are

$$\begin{aligned}& \hat{\alpha }_{1}=\sqrt{k^{2}- \biggl( \frac{\pi }{2(d-b)} \biggr) ^{2}}, \end{aligned}$$

(2.11)

$$\begin{aligned}& \hat{\alpha }_{2}= \sqrt{k^{2}- \biggl( \frac{3\pi }{2(d-b)} \biggr) ^{2}} \end{aligned}$$

(2.12)

$$\begin{aligned}& \vdots \end{aligned}$$

(2.13)

$$\begin{aligned}& \hat{\alpha }_{n}=\sqrt{k^{2}- \biggl( \frac{ ( 2n-1) \pi }{2(d-b)} \biggr) ^{2}},\quad n=1,2,3,\dots, \end{aligned}$$

(2.14)

with \(0<\operatorname{Im}\hat{\alpha }_{1}<\operatorname{Im}\hat{\alpha }_{2}\dots \) and \(\operatorname{Re}\hat{\alpha }_{n}>0\).

The eigenvalues \(\alpha_{n}=\frac{(2n-1)\pi }{2(d-b)}\) are the solution of the following relation:

$$ \cos \alpha_{n}(d-b)=0,\quad n=1,2,3,\dots. $$

(2.15)

### 2.2 Region II: \(\{-b \leq y \leq -a, x<0\}\)

The potential solution of equation (2.3) in this region is given by

$$ v(x,y)=\sum_{n=1}^{\infty }B_{n}e^{-i\hat{\beta }_{n}x} \biggl( \sqrt{\frac{2}{ ( b-a) }}\sin \beta_{n}(y+b) \biggr), $$

(2.16)

which satisfies equations (2.5), (2.6) and radiation conditions (2.7), where \(B_{n}\) is the amplitudes of transmitted field in region II.

The eigenfunctions satisfy the orthonormal relation

$$\begin{aligned}& \int_{-b}^{-a} \biggl( \sqrt{\frac{2}{ ( b-a) }} \sin \beta _{m}(y+b) \biggr) \biggl( \sqrt{\frac{2}{ ( b-a) }} \sin \beta _{n}(y+b) \biggr)\,dy \\& \quad =\delta_{mn},\quad m,n=1,2,3,\dots. \end{aligned}$$

(2.17)

The associated eigenvalues are

$$\begin{aligned}& \hat{\beta }_{1}=\sqrt{k^{2}- \biggl( \frac{\pi }{(b-a)} \biggr) ^{2}}, \end{aligned}$$

(2.18)

$$\begin{aligned}& \hat{\beta }_{2}=\sqrt{k^{2}- \biggl( \frac{2\pi }{(b-a)} \biggr) ^{2}} \end{aligned}$$

(2.19)

$$\begin{aligned}& \vdots \end{aligned}$$

(2.20)

$$\begin{aligned}& \hat{\beta }_{n}=\sqrt{k^{2}- \biggl( \frac{n\pi }{(b-a)} \biggr) ^{2}},\quad n=1,2,3,\dots, \end{aligned}$$

(2.21)

with \(0<\operatorname{Im}\hat{\beta }_{1}<\operatorname{Im}\hat{\beta }_{2}\dots \) and \(\operatorname{Re}\hat{\beta }_{n}>0\).

The eigenvalues \(\beta_{n}=\frac{n\pi }{(b-a)}\) satisfy the equation

$$ \sin \beta_{n}(b-a)=0,\quad n=1,2,3,\dots. $$

(2.22)

### 2.3 Region III: \(\{-a \leq y \leq a, x<0\}\)

The potential solution of equation (2.3) in region III is defined as

$$ v(x,y)=\sum_{n=1}^{\infty }C_{n}e^{-i\hat{\gamma }_{n}x} \biggl( \sqrt{ \frac{1}{a}}\sin \gamma_{n}(y+a) \biggr) + e^{\hat{\gamma }_{1}x} \biggl( \sqrt{ \frac{1}{a}}\sin \gamma_{1}(y+a) \biggr), $$

(2.23)

which also satisfies equation (2.6) and radiation conditions (2.7), where the incident wave \(e^{i\hat{\gamma }_{1}x} ( \sqrt{ \frac{1}{a}}\sin \gamma_{1}(y+a)) \) is excited in the lowest mode propagating from \(x=-\infty \).

The eigenfunctions in this region define the orthonormal relation

$$ \int_{-a}^{a} \biggl( \sqrt{\frac{1}{a}}\sin \gamma_{m}(y+a) \biggr) \biggl( \sqrt{\frac{1}{a}}\sin \gamma_{n}(y+a) \biggr)\,dy= \delta_{mn},\quad m,n=1,2,3, \dots. $$

(2.24)

The associated eigenvalues are

$$\begin{aligned}& \hat{\gamma }_{1}=\sqrt{k^{2}- \biggl( \frac{\pi }{2a} \biggr) ^{2}}, \end{aligned}$$

(2.25)

$$\begin{aligned}& \hat{\gamma }_{2}=\sqrt{k^{2}- \biggl( \frac{\pi }{a} \biggr) ^{2}} \end{aligned}$$

(2.26)

$$\begin{aligned}& \vdots \end{aligned}$$

(2.27)

$$\begin{aligned}& \hat{\gamma }_{n}=\sqrt{k^{2}- \biggl( \frac{n\pi }{2a} \biggr) ^{2}},\quad n=1,2,3,\dots, \end{aligned}$$

(2.28)

with \(0<\operatorname{Im}\hat{\gamma }_{1}<\operatorname{Im}\hat{\gamma }_{2}\dots \) and \(\operatorname{Re}\hat{\gamma }_{n}>0\).

The eigenvalues \(\gamma_{n}=\frac{n\pi }{2a}\) are the roots of the equation

$$ \sin (\gamma_{n}2a)=0,\quad n=1,2,3,\dots. $$

(2.29)

### 2.4 Region IV: \(\{a \leq y \leq b, x<0\}\)

The general potential solution of equation (2.3) in region IV is defined as

$$ v(x,y)=\sum_{n=1}^{\infty }D_{n}e^{-i\hat{\beta }_{n}x} \biggl( \sqrt{\frac{2}{ ( b-a) }}\sin \beta_{n}(y-a) \biggr), $$

(2.30)

which satisfies equations (2.5), (2.6) and radiation conditions (2.7), where \(D_{n}\) is the amplitudes of transmitted field in region IV.

The eigenfunctions satisfy the orthonormal relation

$$\begin{aligned}& \int_{a}^{b} \biggl( \sqrt{\frac{2}{ ( b-a) }} \sin \beta_{m}(y-a) \biggr) \biggl( \sqrt{\frac{2}{ ( b-a) }}\sin \beta_{n}(y-a) \biggr)\,dy \\& \quad = \delta_{mn}, \quad m,n=1,2,3, \dots. \end{aligned}$$

(2.31)

### 2.5 Region V: \(\{b \leq y \leq d, x<0\}\)

The general potential solution of equation (2.3) in this region is defined as

$$ v(x,y)=\sum_{n=1}^{\infty }E_{n}e^{-i\hat{\alpha }_{n}x} \biggl( \sqrt{\frac{2}{ ( d-b) }}\sin \alpha_{n}(y-b) \biggr), $$

(2.32)

which satisfies equations (2.4), (2.5) and radiation conditions (2.7), where \(E_{n}\) is the amplitudes of transmitted field in region V.

The eigenfunctions satisfy the orthonormal relation

$$\begin{aligned}& \int_{b}^{d} \biggl( \sqrt{\frac{2}{ ( d-b) }} \sin \alpha_{m}(y-b) \biggr) \biggl( \sqrt{\frac{2}{ ( d-b) }}\sin \alpha_{n}(y-b) \biggr)\,dy \\& \quad = \delta_{mn},\quad m,n=1,2,3, \dots. \end{aligned}$$

(2.33)

### 2.6 Region VI: \(\{-d \leq y \leq d, x>0\}\)

The general potential solution of equation (2.3) in region VI is defined as

$$ v(x,y)=\sum_{n=1}^{\infty }F_{n}e^{i\hat{\lambda }_{n}x} \bar{ \Psi }_{n}^{(\mathrm{VI})}(y)\,dy, $$

(2.34)

which satisfies equation (2.4) and radiation conditions (2.7).

We define the vertical orthonormal eigenfunctions in region VI as

$$\begin{aligned} \bar{\Psi }_{n}^{(\mathrm{VI})}(y)\,dy =\textstyle\begin{cases} \sqrt{\frac{1}{2d}},& \mbox{if } n = 1, \\ \sqrt{\frac{1}{d}}\cos \lambda_{n}(y+d),& \mbox{if } n \neq 1. \end{cases}\displaystyle \end{aligned}$$

(2.35)

The associated eigenvalues are

$$\begin{aligned}& \hat{\lambda }_{2}=\sqrt{k^{2}- \biggl( \frac{\pi }{2d} \biggr) ^{2}}, \end{aligned}$$

(2.36)

$$\begin{aligned}& \hat{\lambda }_{3}=\sqrt{k^{2}- \biggl( \frac{\pi }{d} \biggr) ^{2}} \end{aligned}$$

(2.37)

$$\begin{aligned}& \vdots \end{aligned}$$

(2.38)

$$\begin{aligned}& \hat{\lambda }_{n}=\sqrt{k^{2}- \biggl( \frac{ ( n-1) \pi }{2d} \biggr) ^{2}},\quad n=1,2,3,\dots, \end{aligned}$$

(2.39)

with \(0<\operatorname{Im}\hat{\lambda }_{1}<\operatorname{Im}\hat{\lambda }_{2}\dots \) and \(\operatorname{Re}\hat{\lambda }_{n}>0\).

The eigenvalues \(\lambda_{n}=\frac{ ( n-1) \pi }{2d}\) satisfy the equation

$$ \sin (\lambda_{n}2d)=0,\quad n=1,2,3,\dots. $$

(2.40)