2.1. Dynamic effective density and scattered sound pressure

As shown in figure 1(a), metamaterial plates with active feedback control are applied to a submarine hull for avoiding external sonar detection. The theoretical model is illustrated in figure 1(b), which consists of two infinite plates connected by periodic four-link mechanisms. The plate spacing is H, and the length and width of a unit cell are $L_{x}$ and $L_{y}$ , respectively. A Cartesian coordinate system (x, y, z) is established, and there is a baffle between the upper and lower plates at the position x = 0. The baffle is rigid with respect to the medium between the plates, but it does not affect the transverse motion of two plates. As shown in figures 1(c) and 1(d), each four-link mechanism has a pair of lateral and two vertical local resonators. The horizontal and vertical distances between the nodes on the link rod are $ h/2 $ and $ l/2 $ .

Figure 1.

Metamaterial plates with active feedback control: (a) application to submarines; (b) structure with rigid baffle subjected to incident sound wave; (c) four-link mechanism in a unit cell and (d) its mechanical model.

The lateral resonator m _l is connected to the four-link by the spring k _l . The vertical resonator $m_{v1}$ is attached to the upper and lower plates by the spring $k_{v1}$ together with the four-link. The vertical resonators $m_{v1}$ and $m_{v2}$ are connected by a pair of springs with stiffness $k_{v2}$ . The actuator includes a thin coil attached to resonator $m_{v1}$ and a cylindrical magnet in resonator $m_{v2}$ . The voltage load applied to the coil is tuned based on the received acceleration signal of resonator $m_{v1}$ . Then a pair of control magnetic force $F_{\kern-1pt c}$ is generated between resonators $m_{v1}$ and $m_{v2}$ . Displacements of the lateral resonator and link rod node are u _l and $ v $ . The displacements of vertical resonators $m_{v1}$ and $m_{v2}$ are $u_{v1}$ and $u_{v2}$ , and displacements of the upper and lower plates are $w_{1}$ and $w_{2}$ .

Based on the Kirchhoff plate theory, the governing equation of metamaterial plates is

(2.1) \begin{align}D_{\!j}{\nabla} ^{4}w_{\!j}+\rho _{pj}h_{\!j}\frac{\partial ^{2}w_{\!j}}{\partial t^{2}}=q_{\!j}\quad (j=1,2),\end{align}

where ∇⁴ = (∂²/∂x ² + ∂²/∂y ²)², D _j = $ E_{\!j} $ h ³ _j /[12(1 – μ ² _j )] is the bending stiffness, $ \rho_{pj} $ , $ E_{\!j} $ , $ \mu_{\!j} $ and $ h_{\!j} $ are the density, Young’s modulus, Poisson’s ratio and thickness, $ q_{\!j} $ is the internal lumped force generated by the four-link mechanism, and j = 1,2 represent the upper and lower plates, respectively.

The governing equations of vertical and lateral local resonators can be expressed as

(2.2a) \begin{align}& m_{v1}\frac{\partial ^{2}u_{v1}}{\partial t^{2}}=k_{v1}\left(w_{1}-u_{v1}\right)+k_{v1}\left(w_{2}-u_{v1}\right)+2k_{v2}\left(u_{v2}-u_{v1}\right)-F_{c}, \end{align}

(2.2b,c) \begin{align}&\qquad m_{v2}\frac{\partial ^{2}u_{v2}}{\partial t^{2}}=2k_{v2}\left(u_{v1}-u_{v2}\right)+F_{c},\quad m_{l}\frac{\partial ^{2}u_{l}}{\partial t^{2}}=k_{l}\left(v-u_{l}\right)\!, \end{align}

where $F_{\kern-1pt c}$ = B _L I _c , and B _L is the parameter related to the size of the coil and magnet, and the axial distance between them (Guo & Wang Reference Guo and Wang2024).

The governing equation for the current I _c in the coil can be written as

(2.3) \begin{align}R_{e}I_{c}+L_{e}\frac{\partial I_{c}}{\partial t}=U_{\textit{load}}-B_{L}\frac{\partial }{\partial t}\left(u_{v1}-u_{v2}\right)\!,\end{align}

where R _e and L _e are the resistance and inductance, U _loud = [G ₁(∂²/∂t ²) + G ₂] $u_{v1}$ is the external voltage load, and G ₁ (V s² m⁻¹) and G ₂ (V m⁻¹) are the acceleration and displacement feedback coefficients.

Based on the Bloch–Floquet theory of periodic systems, and considering the harmonic motion with time, the displacements of the metamaterial plates are

(2.4a,b) \begin{align}w_{1}=W_{1}\,{\rm e}^{-{\rm i}({k_{x}}x+{k_{y}}y+\omega t)},\quad w_{2}=W_{2}\,{\rm e}^{-{\rm i}({k_{x}}x+{k_{y}}y+\omega t)},\end{align}

where W ₁ and W ₂ are the displacement amplitudes, k _x and k _y are the wavenumbers along the x- and y-axes, ω is the wave frequency, and ${\rm i}=\sqrt{-1}$ .

When the lattice constant of the metamaterial plates is subwavelength, the internal lumped forces are considered uniformly applied across the area of a unit cell, and can be expressed as

(2.5a,b) \begin{align}q_{1}=Bw_{1}+Cw_{2},\quad q_{2}=Cw_{1}+Bw_{2},\end{align}

where

(2.6a,b) \begin{align}B=\frac{1}{L_{x}L_{y}}\left(\frac{k_{v1}^{2}}{A_{2}}-k_{v1}+\frac{k_{l}m_{l}\omega ^{2}}{k_{l}-m_{l}\omega ^{2}}\frac{h^{2}}{2l^{2}}\right)\!,\quad C=\frac{1}{L_{x}L_{y}}\left(\frac{k_{v1}^{2}}{A_{2}}-\frac{k_{l}m_{l}\omega ^{2}}{k_{l}-m_{l}\omega ^{2}}\frac{h^{2}}{2l^{2}}\right)\!.\end{align}

The explicit derivation is presented in Appendix A.

Substituting (2.4) and (2.5) into (2.1), and omitting the time-harmonic factor, we can obtain

(2.7) \begin{align}\left[\!\begin{array}{cc} D_{1}K^{4}-\omega ^{2}\rho _{p1}h_{1}-B & -C\\ -C & D_{2}K^{4}-\omega ^{2}\rho _{p2}h_{2}-B \end{array}\!\right]\left\{\!\!\begin{array}{c} W_{1}\\ W_{2} \end{array}\!\!\right\}=\mathbf{0},\end{align}

where $K=\sqrt{k_{x}^{2}+k_{y}^{2}}$ .

In order to derive the dynamic effective density, an equivalent single plate is considered with identical parameters for the upper plate, and $K^{2}=\omega \sqrt{\rho _{\textit{eff}}h_{1}/D_{1}}$ is obtained. Considering the non-trivial solution for (2.7), the dynamic effective density can be derived as

(2.8) \begin{align}\rho _{\textit{eff}1,2}=\frac{-b_{0}\pm \sqrt{b_{0}^{2}-4a_{0}c_{0}}}{2a_{0}},\end{align}

where

(2.9a) \begin{align}& a_{0}=\omega ^{4}h_{1}^{2}\frac{D_{2}}{D_{1}}, \\[-9pt] \nonumber \end{align}

(2.9b) \begin{align}& b_{0}=-\omega ^{2}h_{1}\left(1+\frac{D_{2}}{D_{1}}\right)B-\omega ^{4}h_{1}^{2}\frac{D_{2}}{D_{1}}\rho _{p1}-\omega ^{4}h_{1}h_{2}\rho _{p2}, \\[-9pt] \nonumber \end{align}

(2.9c) \begin{align}& c_{0}=\omega ^{4}h_{1}h_{2}\rho _{p1}\rho _{p2}+\big(\omega ^{2}h_{1}\rho _{p1}+\omega ^{2}h_{2}\rho _{p2}\big)B+B^{2}-C^{2}. \\[9pt] \nonumber \end{align}

Compared with the scale of sound field, the thickness of the upper and lower plates can be negligible, and their position coordinates are regarded as z = 0 and z = −H. The space is divided into three regions by the metamaterial plates, each of which is occupied by a stationary fluid medium. The density and sound speed of the medium are ρ _η and c _η , in which η = 1, 2, 3 represent the mediums in the $z \gt 0$ , −H < $z \lt 0$ and $z \lt -H$ regions, respectively. Interaction between the fluid medium and mechanical metamaterial is considered, and the whole system is excited by a plane sound wave p _in incident on the upper plate. The propagating direction of the incident wave is parallel to the $x{-}z$ plane, and the angle from the positive x-axis is θ _in . The incident sound pressure is independent of the y-coordinate.

The incident sound pressure and specular reflection pressure can be written as

(2.10a,b) \begin{align}p_{\textit{in}}(x,z)={\rm e}^{-{\rm i}{k_{1}}\left(x\cos \theta _{\textit{in}}+z\sin\theta _{\textit{in}}\right)},\quad p_{sr}(x,z)={\rm e}^{-{\rm i}{k_{1}}(x\cos {\theta _{\textit{in}}}-z\sin\theta _{\textit{in}})},\end{align}

where k ₁ = $ \omega/c_{1} $ is the wavenumber in medium 1, and the harmonic time term e^–iωt is omitted during the following derivations.

Based on the principle of superposition, the total sound pressure $p^{T}_{\eta}$ (x, z) in medium η, and the total displacement $ w^{T}_{\!j} (x) $ of the upper and lower plates, can be expressed by

(2.11a) \begin{align}&\qquad p_{1}^{T}(x,z)=p_{\textit{in}}(x,z)+p_{sr}(x,z)+p_{1}^{e}(x,z)+p_{1}^{r}(x,z), \\[-9pt] \nonumber \end{align}

(2.11b,c) \begin{align}& p_{2}^{T}(x,z)=p_{2}^{e}(x,z)+p_{2}^{r}(x,z),\quad p_{3}^{T}(x,z)=p_{3}^{e}(x,z)+p_{3}^{r}(x,z), \\[-9pt] \nonumber \end{align}

(2.11d,e) \begin{align}&\qquad w_{1}^{T}(x)=w_{1}^{e}(x)+w_{1}^{r}(x),\quad w_{2}^{T}(x)=w_{2}^{e}(x)+w_{2}^{r}(x), \\[9pt] \nonumber \end{align}

where $p^{e}_{\eta}$ (x, z) and $ w^{e}_{\!j}(x) $ are the pressure and displacement components generated by the elastic motion without the rigid baffle, $p^{r}_{\eta}$ (x, z) and $w^{r}_{\!j}$ (x) are the corresponding components from the rigid baffle, and p _sr (x, z), $p^{e}_{1}$ (x, z) and $ p^{r}_{1}(x, z) $ constitute the total scattered sound pressure.

2.2. Scattered sound pressure of metamaterial plates without rigid baffle

According to the ideal fluid assumption (Batcheler Reference Batcheler1967; Lighthill Reference Lighthill1978; Landau & Lifshitz Reference Landau and Lifshitz1987), the sound pressure without rigid baffle in each medium region satisfies the Helmholtz equation

(2.12) \begin{align}\frac{\partial ^{2}p_{\eta }^{e}(x,z)}{\partial x^{2}}+\frac{\partial ^{2}p_{\eta }^{e}(x,z)}{\partial z^{2}}+k_{\eta }^{2}\,p_{\eta }^{e}(x,z)=0\quad (\eta = 1, 2, 3),\end{align}

where $ k_{2} = \omega/c_{2}$ and $ k_{3} = \omega/c_{3}$ are the wavenumbers in mediums 2 and 3.

The Fourier transform with respect to the x-coordinate is applied to partial differential equations, and the transform of the pressure $p^{e}_{\eta}$ (x, z) is defined as

(2.13a) \begin{align}\overline{p}_{\eta }^{e}(\alpha ,z)&=\int _{-\infty }^{\infty }p_{\eta }^{e}(x,z)\,{\rm e}^{-{\rm i}\alpha x}\,\mathrm{d}x, \\[-9pt] \nonumber \end{align}

(2.13b) \begin{align}p_{\eta }^{e}(x,z)&=\frac{1}{2\pi }\int _{-\infty }^{\infty }\overline{p}_{\eta }^{e}(\alpha ,z)\,{\rm e}^{{\rm i}\alpha x}\,\mathrm{d}x, \\[9pt] \nonumber \end{align}

where α is a complex variable.

Then the Fourier transform of (2.12) becomes

(2.14) \begin{align}\frac{\partial ^{2}\overline{p}_{\eta }^{e}(\alpha ,z)}{\partial z^{2}}+\gamma _{\eta }^{2}\,\overline{p}_{\eta }^{e}(\alpha ,z)=0\quad (\eta = 1, 2, 3).\end{align}

The general solutions of (2.14) are

(2.15) \begin{align}\overline{p}_{\eta }^{e}(\alpha ,z)=A_{\eta }^{e}(\alpha)\,{\rm e}^{{\rm i}{\gamma _{\eta }}z}+B_{\eta }^{e}(\alpha)\,{\rm e}^{-{\rm i}{\gamma _{\eta }}z},\end{align}

where $ \gamma^{2}_{\eta} = k^{2}_{\eta} - \alpha^{2} $ , $ A^{e}_{\eta}(\alpha) $ and $ B^{e}_{\eta}(\alpha) $ are the functions ofα to be determined.

For real values of α, the function γ _η (α) is defined as

(2.16) \begin{align}\gamma _{\eta }(\alpha )=\begin{cases} \sqrt{k_{\eta }^{2}-\alpha ^{2}}, & k_{\eta }^{2}\geq \alpha ^{2}, \\ {\rm i}\sqrt{\alpha ^{2}-k_{\eta }^{2}},& k_{\eta }^{2}\lt \alpha ^{2}, \end{cases}\end{align}

and for complex values of α, the branch cut is chosen to ensure Im(γ _η ) $\geq$ 0.

The total pressure $ p^{t}_{1} $ (x, z) in medium 1 is

(2.17) \begin{align}p_{1}^{t}(x,z)=p_{\textit{in}}(x,z)+p_{sr}(x,z)+p_{1}^{e}(x,z).\end{align}

Substituting (2.10)–(2.17) and applying the Fourier transform gives

(2.18) \begin{align}\overline{p}_{1}^{t}(\alpha ,z)=2\pi \delta \left(\alpha +k_{1}\cos \theta _{\textit{in}}\right)\big({\rm e}^{-{\rm i}{k_{1}}z\sin \theta _{\textit{in}}}+{\rm e}^{{\rm i}{k_{1}}z\sin \theta _{\textit{in}}}\big)+\overline{p}_{1}^{e}(\alpha ,z).\end{align}

The continuous velocity conditions about the pressure at fluid–structure interfaces are (Xin & Lu Reference Xin and Lu2010; He et al. Reference He, Wang and Wang2021)

(2.19a,b) \begin{align}&\,\,\,\,\left.\frac{\partial \overline{p}_{1}^{e}(\alpha ,z)}{\partial z}\right| _{z=0}=\omega ^{2}\rho _{1}\,\overline{w}_{1}^{e}(\alpha ),\quad \left.\frac{\partial \overline{p}_{2}^{e}(\alpha ,z)}{\partial z}\right| _{z=0}=\omega ^{2}\rho _{2}\,\overline{w}_{1}^{e}(\alpha ),\\[-9pt] \nonumber \end{align}

(2.19c,d) \begin{align}&\left.\frac{\partial \overline{p}_{2}^{e}(\alpha ,z)}{\partial z}\right| _{z=-H}=\omega ^{2}\rho _{2}\,\overline{w}_{2}^{e}(\alpha ),\quad \left.\frac{\partial \overline{p}_{3}^{e}(\alpha ,z)}{\partial z}\right| _{z=-H}=\omega ^{2}\rho _{3}\,\overline{w}_{2}^{e}(\alpha ). \\[9pt] \nonumber \end{align}

Considering the radiation condition for outgoing waves in mediums 1 and 3, $ B^{e}_{1} = 0 $ and $ A^{e}_{3} = 0 $ can be obtained. Substituting (2.15) into (2.19), $ A^{e}_{\eta} (\alpha) $ and $ B^{e}_{\eta} (\alpha) $ can be derived as

(2.20a) \begin{align} A_{1}^{e}(\alpha )&=\frac{-{\rm i}\omega ^{2}\rho _{1}}{\gamma _{1}}\,\overline{w}_{1}^{e}(\alpha ), \\[-9pt] \nonumber \end{align}

(2.20b) \begin{align}A_{2}^{e}(\alpha )&=\frac{-\omega ^{2}\rho _{2}}{2\gamma _{2}\sin \gamma _{2}H}\big[\overline{w}_{1}^{e}(\alpha )\,{\rm e}^{{\rm i}{\gamma _{2}}H}-\overline{w}_{2}^{e}(\alpha )\big], \\[-9pt] \nonumber \end{align}

(2.20c) \begin{align}B_{2}^{e}(\alpha )&=\frac{-\omega ^{2}\rho _{2}}{2\gamma _{2}\sin \gamma _{2}H}\big[\overline{w}_{1}^{e}(\alpha )\,{\rm e}^{-{\rm i}{\gamma _{2}}H}-\overline{w}_{2}^{e}(\alpha )\big], \\[-9pt] \nonumber\end{align}

(2.20d) \begin{align}B_{3}^{e}(\alpha )&=\frac{{\rm i}\omega ^{2}\rho _{3}}{\gamma _{3}}\,\overline{w}_{2}^{e}(\alpha )\,{\rm e}^{-{\rm i}{\gamma _{3}}H}. \\[9pt] \nonumber \end{align}

The sound pressure at the coupling interface is expressed as

(2.21a) \begin{align}\overline{p}_{1}^{e}(\alpha ,0)&=\frac{-{\rm i}\omega ^{2}\rho _{1}}{\gamma _{1}}\,\overline{w}_{1}^{e}(\alpha ), \\[-9pt] \nonumber \end{align}

(2.21b) \begin{align}\overline{p}_{2}^{e}(\alpha ,0)&=\frac{-\omega ^{2}\rho _{2}}{\gamma _{2}\sin \gamma _{2}H}\left[\overline{w}_{1}^{e}(\alpha )\cos \gamma _{2}H-\overline{w}_{2}^{e}(\alpha )\right]\!, \\[-9pt] \nonumber \end{align}

(2.21c) \begin{align}\overline{p}_{2}^{e}(\alpha ,-H)&=\frac{-\omega ^{2}\rho _{2}}{\gamma _{2}\sin \gamma _{2}H}\left[\overline{w}_{1}^{e}(\alpha )-\overline{w}_{2}^{e}(\alpha )\cos \gamma _{2}H\right]\!, \\[-9pt] \nonumber \end{align}

(2.21d) \begin{align}\overline{p}_{3}^{e}(\alpha ,-H)&=\frac{{\rm i}\omega ^{2}\rho _{3}}{\gamma _{3}}\,\overline{w}_{2}^{e}(\alpha ). \\[9pt] \nonumber \end{align}

Governing equations of the metamaterial plates with fluid load are (Xin & Lu Reference Xin and Lu2010; He et al. Reference He, Wang and Wang2021)

(2.22a) \begin{align} D_{1}\frac{\partial ^{4}w_{1}^{e}(x)}{\partial x^{4}}-\omega ^{2}M_{1}\,w_{1}^{e}(x)&=-p_{1}^{t}\left(x,0\right)+p_{2}^{e}\left(x,0\right)+\!\sum _{n=-\infty }^{\infty }\!F_{1}^{e}(x)\,\delta \left[x-\left(\frac{1}{2}+n\!\right)\!L_{x}\!\right]\!, \\[-12pt] \nonumber \end{align}

(2.22b) \begin{align} D_{2}\frac{\partial ^{4}w_{2}^{e}(x)}{\partial x^{4}}-\omega ^{2}M_{2}\,w_{2}^{e}(x)&=-p_{2}^{e}\left(x,-H\right)+p_{3}^{e}\left(x,-H\right)\nonumber\\&\quad +\sum _{n=-\infty }^{\infty }F_{2}^{e}(x)\,\delta \left[x-\left(\frac{1}{2}+n\right)L_{x}\right]\!, \\[9pt] \nonumber\end{align}

where $ M_{\!j} $ = $ \rho_{pj} $ $ h_{\!j} $ is the mass per unit area, $ F^{e}_{1} (x) $ and $ F^{e}_{2} (x) $ are the internal lumped forces, and $ \delta (\cdot) $ is the Dirac delta function.

The Fourier transform of (2.22) is

(2.23a) \begin{align} \big(D_{1}\alpha ^{4}-\omega ^{2}M_{1}\big)\overline{w}_{1}^{e}(\alpha )&=-\overline{p}_{1}^{t}(\alpha ,0)+\overline{p}_{2}^{e}(\alpha ,0)+\sum _{n=-\infty }^{\infty }F_{1}^{e}\left(x_{n}\right){\rm e}^{-{\rm i}\alpha {x_{n}}}, \end{align}

(2.23b) \begin{align} \big(D_{2}\alpha ^{4}-\omega ^{2}M_{2}\big)\overline{w}_{2}^{e}(\alpha )&=-\overline{p}_{2}^{e}(\alpha ,-H)+\overline{p}_{3}^{e}(\alpha ,-H)+\sum _{n=-\infty }^{\infty }F_{2}^{e}\left(x_{n}\right){\rm e}^{-{\rm i}\alpha {x_{n}}}, \end{align}

where x _n = x ₀ + nL _x , with x ₀ = $L_{x}$ /2.

Based on (2.5) and (2.6), the internal lumped forces can be expressed as

(2.24a,b) \begin{align}F_{1}^{e}(x)=K_{1}\,w_{1}^{e}(x)+K_{2}\,w_{2}^{e}(x),\quad F_{2}^{e}(x)=K_{2}\,w_{1}^{e}(x)+K_{1}\,w_{2}^{e}(x),\end{align}

where K ₁ = BL _x and K ₂ = CL _x .

According to the property of an incident sound wave and the periodicity of structure, the displacement of two plates can be written as

(2.25a,b) \begin{align}w_{1}^{e}\left(x+nL_{x}\right)=w_{1}^{e}(x){\rm e}^{-{\rm i}{k_{1}}n{L_{x}}\cos \theta _{\textit{in}}},\quad w_{2}^{e}\left(x+nL_{x}\right)=w_{2}^{e}(x){\rm e}^{-{\rm i}{k_{1}}n{L_{x}}\cos \theta _{\textit{in}}}.\end{align}

Substituting (2.25) into (2.24), we can derive

(2.26a,b) \begin{align}F_{1}^{e}\left(x_{n}\right)=F_{1}^{e}\left(x_{0}\right){\rm e}^{-{\rm i}{k_{1}}n{L_{x}}\,{\cos }\, \theta _{\textit{in}}},\quad F_{2}^{e}(x_{n})=F_{2}^{e}(x_{0}){\rm e}^{-{\rm i}{k_{1}}n{L_{x}}\,{\cos }\, \theta _{\textit{in}}}.\end{align}

During the following derivation, we consider $ F^{e}_{1} $ and $ F^{e}_{2} $ instead of $ F^{e}_{1}(x_{0})$ and $ F^{e}_{2}(x_{0})$ , respectively. Based on (2.18), (2.21) and (2.26), (2.23) can be derived as

(27a) \begin{align} &\left(D_{1}\alpha ^{4}-\omega ^{2}M_{1}-\frac{{\rm i}\omega ^{2}\rho _{1}}{\gamma _{1}}+\frac{\omega ^{2}\rho _{2}\cos \gamma _{2}H}{\gamma _{2}\,{\sin }\, \gamma _{2}H}\right)\,\overline{w}_{1}^{e}(\alpha )-\frac{\omega ^{2}\rho _{2}}{\gamma _{2}\,{\sin }\, \gamma _{2}H}\overline{w}_{2}^{e}(\alpha )\nonumber\\&\quad =-4\pi \delta \left(\alpha +k_{1}\,{\cos }\, \theta _{\textit{in}}\right)+F_{1}^{e}{\rm e}^{-{\rm i}\alpha {x_{0}}}\sum _{n=-\infty }^{\infty }{\rm e}^{-{\rm i}n{L_{x}}\left(\alpha +k_{1}\,{\cos }\, \theta _{\textit{in}}\right)}, \end{align}

(2.27b) \begin{align} &\frac{-\omega ^{2}\rho _{2}}{\gamma _{2}\,{\sin }\, \gamma _{2}H}\,\overline{w}_{1}^{e}(\alpha )+\left(D_{2}\alpha ^{4}-\omega ^{2}M_{2}-\frac{{\rm i}\omega ^{2}\rho _{3}}{\gamma _{3}}+\frac{\omega ^{2}\rho _{2}\cos \gamma _{2}H}{\gamma _{2}\,{\sin }\, \gamma _{2}H}\right)\overline{w}_{2}^{e}(\alpha )\nonumber\\&\quad =F_{2}^{e}\,{\rm e}^{-{\rm i}\alpha {x_{0}}}\sum _{n=-\infty }^{\infty }{\rm e}^{-{\rm i}n{L_{x}}\left(\alpha +k_{1}\,{\cos }\, \theta _{\textit{in}}\right)}. \end{align}

The matrix form of (2.27) is

(2.28) \begin{align} \boldsymbol{S}(\alpha )\left\{\!\!\begin{array}{c} \overline{w}_{1}^{e}(\alpha )\\[5pt] \overline{w}_{2}^{e}(\alpha )\end{array}\!\!\right\}=-4\pi\left\{\!\!\begin{array}{c} \delta(\alpha +k_1 \cos \theta_{\textit{in}} )\\ 0\end{array}\!\!\right\}+\left\{\!\!\begin{array}{c} F_1^e\\[5pt] F_2^e\end{array}\!\!\right\} {\rm e}^{-{\rm i}\alpha {x_{0}}}\sum _{n=-\infty }^{\infty }{\rm e}^{-{\rm i}n{L_{x}}\left(\alpha +k_{1}\,{\cos }\, \theta _{\textit{in}}\right)} \end{align}

where S(α) represents the dynamic stiffness matrix with fluid damping effect (Springer, Scheit & Becker Reference Springer, Scheit and Becker2017).

The elements of matrix S(α) are

(2.29a) \begin{align} & S_{11}(\alpha )=D_{1}\alpha ^{4}-\omega ^{2}M_{1}-\frac{{\rm i}\omega ^{2}\rho _{1}}{\gamma _{1}}+\frac{\omega ^{2}\rho _{2}\cos \gamma _{2}H}{\gamma _{2}\,{\sin }\, \gamma _{2}H}, \end{align}

(2.29b) \begin{align} & S_{12}(\alpha )=S_{21}(\alpha )=-\frac{\omega ^{2}\rho _{2}}{\gamma _{2}\,{\sin }\, \gamma _{2}H}, \end{align}

(2.29c) \begin{align} & S_{22}(\alpha )=D_{2}\alpha ^{4}-\omega ^{2}M_{2}-\frac{{\rm i}\omega ^{2}\rho _{3}}{\gamma _{3}}+\frac{\omega ^{2}\rho _{2}\cos \gamma _{2}H}{\gamma _{2}\,{\sin }\, \gamma _{2}H}. \\[9pt] \nonumber \end{align}

According to the Poisson summation formula, we can obtain

(2.30) \begin{align}\sum _{n=-\infty }^{\infty }{\rm e}^{-{\rm i}n{L_{x}}(\alpha +{k_{1}}\,{\cos }\, \theta _{\textit{in}})}=\frac{2\pi }{L_{x}}\sum _{n=-\infty }^{\infty }\delta \left(\alpha +k_{1}\,{\cos }\, \theta _{\textit{in}}-\frac{2n\pi }{L_{x}}\right)\!.\end{align}

Based on (2.28) and (2.30), the following expressions are obtained:

(2.31a) \begin{align} & \overline{w}_{1}^{e}(\alpha )=\frac{-4\pi S_{22}(\alpha )}{{\varDelta }(\alpha )}\delta \!\left(\alpha -\kappa \right)+\frac{2\pi }{L_{x}{\varDelta }(\alpha )}\left[F_{1}^{e}S_{22}(\alpha )-F_{2}^{e}S_{12}(\alpha )\!\right]{\rm e}^{-{\rm i}\alpha {x_{0}}}\!\sum _{n=-\infty }^{\infty }\!\delta \!\left(\alpha -\kappa _{n}\right)\!, \end{align}

(2.31b) \begin{align} & \overline{w}_{2}^{e}(\alpha )=\frac{4\pi S_{12}(\alpha )}{{\varDelta }(\alpha )}\delta \left(\alpha -\kappa \right)-\frac{2\pi }{L_{x}{\varDelta }(\alpha )}\left[F_{1}^{e}S_{12}(\alpha )-F_{2}^{e}S_{11}(\alpha )\right]{\rm e}^{-{\rm i}\alpha {x_{0}}}\sum _{n=-\infty }^{\infty }\delta \!\left(\alpha -\kappa _{n}\right)\!, \end{align}

where κ = –k ₁cos θ _in , κ _n = 2nπ/ $L_{x}$ – k ₁cos θ _in and Δ(α) = S ₁₁(α)S ₂₂(α) – S ₁₂(α)S ₂₁(α).

By Fourier inverse transform in (2.31), the displacement $ w^{e}_{\!j}(x) $ can be derived as

(2.32a) \begin{align} & w_{1}^{e}(x)=\frac{-2S_{22}(\kappa){\rm e}^{{\rm i}\kappa x}}{{\varDelta }(\kappa)}+\frac{Q_{1}(x)F_{1}^{e}+Q_{3}(x)F_{2}^{e}}{L_{x}}, \end{align}

(2.32b) \begin{align} & w_{2}^{e}(x)=\frac{2S_{12}(\kappa){\rm e}^{{\rm i}\kappa x}}{{\varDelta }(\kappa)}+\frac{Q_{3}(x)F_{1}^{e}+Q_{2}(x)F_{2}^{e}}{L_{x}}, \end{align}

where

(2.33a–c) \begin{align}&Q_{1}(x)=\sum _{n=-\infty }^{\infty }\frac{S_{22}(\kappa _{n})}{{\varDelta }(\kappa _{n})}{\rm e}^{{\rm i}{\kappa _{n}}\left(x-x_{0}\right)}, \qquad Q_{2}(x)=\sum _{n=-\infty }^{\infty }\frac{S_{11}(\kappa _{n})}{{\varDelta }(\kappa _{n})}{\rm e}^{{\rm i}{\kappa _{n}}\left(x-x_{0}\right)},\nonumber\\& Q_{3}(x)=\sum _{n=-\infty }^{\infty }\frac{-S_{12}(\kappa _{n})}{{\varDelta }(\kappa _{n})}{\rm e}^{{\rm i}{\kappa _{n}}(x-{x_{0}})}.\end{align}

It is obvious that as $|n|$ increases, the term in the summation symbol will quickly approach zero. As a result, the sum can be evaluated by truncation. And Q ₁ = Q ₁(x ₀), Q ₂ = Q ₂(x ₀) and Q ₃ = Q ₃(x ₀) are used during the following derivation. Based on (2.24) and (2.32), the internal lumped forces generated by the four-link mechanism at x = x ₀ can be derived as

(2.34a) \begin{align} & F_{1}^{e}=\frac{-2L_{x}{\rm e}^{{\rm i}\kappa {x_{0}}}}{T_{1}\Delta (\kappa )}\left[T_{2}S_{22}(\kappa )-T_{3}S_{12}(\kappa )\right]\!, \end{align}

(2.34b) \begin{align} & F_{2}^{e}=\frac{-2L_{x}{\rm e}^{{\rm i}\kappa {x_{0}}}}{T_{1}\Delta (\kappa )}\left[T_{3}S_{22}(\kappa )-T_{4}S_{12}(\kappa )\right]\!, \end{align}

where

(2.35a) \begin{align} & T_{1}=L_{x}^{2}-L_{x}K_{1}(Q_{1}+Q_{2})-2L_{x}K_{2}Q_{3}+\big(K_{1}^{2}-K_{2}^{2}\big)\big(Q_{1}Q_{2}-Q_{3}^{2}\big), \end{align}

(2.35b) \begin{align} & T_{2}=L_{x}K_{1}-\big(K_{1}^{2}-K_{2}^{2}\big)Q_{2}, \end{align}

(2.35c) \begin{align} & T_{3}=L_{x}K_{2}+\big(K_{1}^{2}-K_{2}^{2}\big)Q_{3}, \end{align}

(2.35d) \begin{align} & T_{4}=L_{x}K_{1}-\big(K_{1}^{2}-K_{2}^{2}\big)Q_{1}. \end{align}

According to (2.15), (2.20) and (2.31), the sound pressures in mediums 1 and 2 can be expressed as

(2.36a) \begin{align} &\overline{p}_{1}^{e}(\alpha ,z)=\frac{4\pi {\rm i}\omega ^{2}\rho _{1}S_{22}(\alpha ){\rm e}^{{\rm i}{\gamma _{1}}z}}{{\Delta }(\alpha )\gamma _{1}}\delta \left(\alpha -\kappa \right)\nonumber\\&\qquad\qquad\quad-\frac{2\pi {\rm i}\omega ^{2}\rho _{1}{\rm e}^{{\rm i}{\gamma _{1}}z}}{L_{x}{\Delta }(\alpha )\gamma _{1}}\left[F_{1}^{e}S_{22}(\alpha )-F_{2}^{e}S_{12}(\alpha )\right]{\rm e}^{-{\rm i}\alpha {x_{0}}}\sum _{n=-\infty }^{\infty }\delta\! \left(\alpha -\kappa _{n}\right)\!, \end{align}

(2.36b) \begin{align} &\overline{p}_{2}^{e}(\alpha ,z)=\frac{\omega ^{2}\rho _{2}\,{\cos }\, \gamma _{2}(z+H)}{{\Delta }(\alpha )\gamma _{2}\sin \gamma _{2}H}\nonumber\\&\quad\times\left\{4\pi \delta \left(\alpha -\kappa \right)S_{22}(\alpha )-\frac{2\pi }{L_{x}}\left[F_{1}^{e}S_{22}(\alpha )-F_{2}^{e}S_{12}(\alpha )\right]{\rm e}^{-{\rm i}\alpha {x_{0}}}\sum _{n=-\infty }^{\infty }\delta\! \left(\alpha -\kappa _{n}\right)\right\}\nonumber\\&\quad+\frac{\omega ^{2}\rho _{2}\,{\cos }\, \gamma _{2}z}{{\Delta }(\alpha )\gamma _{2}\sin \gamma _{2}H}\left\{4\pi \delta \left(\alpha -\kappa \right)S_{12}(\alpha )-\frac{2\pi }{L_{x}}\left[F_{1}^{e}S_{12}(\alpha )-F_{2}^{e}S_{11}(\alpha )\right]{\rm e}^{-{\rm i}\alpha {x_{0}}}\right.\nonumber\\&\quad\times\left. \sum _{n=-\infty }^{\infty }\delta\! \left(\alpha -\kappa _{n}\right)\right\}\!. \end{align}

The inverse Fourier transform of (2.36) can be derived as

(2.37a) \begin{align}& p_{1}^{e}(x,z)=\frac{2{\rm i}\omega ^{2}\rho _{1}S_{22}(\kappa)}{{\Delta }(\kappa)k_{1}\,{\sin }\, \theta _{\textit{in}}}{\rm e}^{-{\rm i}{k_{1}}\left(x\,{\cos }\, \theta _{\textit{in}}-z\,{\sin }\, \theta _{\textit{in}}\right)}\nonumber\\&\qquad\qquad-\frac{{\rm i}\omega ^{2}\rho _{1}}{L_{x}}\sum _{n=-\infty }^{\infty }\frac{F_{1}^{e}S_{22}(\kappa _{n})-F_{2}^{e}S_{12}(\kappa _{n})}{{\Delta }(\kappa _{n})\gamma _{1}(\kappa _{n})}{\rm e}^{{\rm i}\left[\kappa _{n}\left(x-x_{0}\right)+\gamma _{1}(\kappa _{n})z\right]}, \end{align}

(2.37b) \begin{align}& p_{2}^{e}(x,z)=\frac{2\omega ^{2}\rho _{2}\left[S_{22}(\kappa){\cos }\, \gamma _{2}(\kappa)(z+H)+S_{12}(\kappa){\cos }\, \gamma _{2}(\kappa)z\right]{\rm e}^{{\rm i}\kappa x}}{{\Delta }(\kappa)\gamma _{2}(\kappa)\sin \gamma _{2}(\kappa)H}-\frac{\omega ^{2}\rho _{2}}{L_{x}}\sum _{n=-\infty }^{\infty }\nonumber\\&\times\frac{\left[\!F_{1}^{e}S_{22}(\kappa _{n})-F_{2}^{e}S_{12}(\kappa _{n})\!\right]{\cos }\, \gamma _{2}(\kappa _{n})(z+H)+\left[\!F_{1}^{e}S_{12}(\kappa _{n})-F_{2}^{e}S_{11}(\kappa _{n})\!\right]{\cos }\, \gamma _{2}(\kappa _{n})z}{{\Delta }(\kappa _{n})\gamma _{2}(\kappa _{n})\sin \gamma _{2}(\kappa _{n})H}\nonumber\\&\times {\rm e}^{{\rm i}{\kappa _{n}}\left(x-x_{0}\right)}. \end{align}

2.3. Scattered sound pressure resulting from rigid baffle

When the rigid baffle exists, $ p^{r}_{1}(x, z) $ and $ p^{r}_{3} (x,z) $ still satisfy (2.14). However, $ p^{r}_{2} (x,z) $ is no longer continuous for the x-coordinate, and its governing equation becomes

(2.38) \begin{align}\frac{\partial ^{2}p_{2}^{r}(x,z)}{\partial x^{2}}+\frac{\partial ^{2}p_{2}^{r}(x,z)}{\partial z^{2}}+k_{2}^{2}p_{2}^{r}(x,z)=0\quad (x \neq 0).\end{align}

The normal velocity of fluid on the rigid baffle at x = 0 should be zero, and the corresponding equation can be expressed as (Ayton Reference Ayton2016; Huang Reference Huang2017)

(2.39) \begin{align}\left.\frac{\partial p_{2}^{T}(x,z)}{\partial x}\right| _{x=0}=\left.\frac{\partial p_{2}^{e}(x,z)}{\partial x}\right| _{x=0}+\left.\frac{\partial p_{2}^{r}(x,z)}{\partial x}\right| _{x=0}=0.\end{align}

According to (2.37b ), we can further obtain

(2.40) \begin{align}\left.\frac{\partial p_{2}^{r}(x,z)}{\partial x}\right| _{x=0}=X{\rm e}^{{{\rm i}\gamma _{2}}(\kappa )z}+Y{\rm e}^{-{\rm i}{\gamma _{2}}(\kappa )z}+\sum _{n=-\infty }^{\infty }\big[X_{n}{\rm e}^{{\rm i}{\gamma _{2}}({\kappa _{n}})z}+Y_{n}{\rm e}^{-{\rm i}{\gamma _{2}}({\kappa _{n}})z}\big],\end{align}

where

(2.41a) \begin{align} X&=\frac{-{\rm i}\omega ^{2}\rho _{2}\kappa \left[{\rm e}^{{\rm i}{\gamma _{2}}(\kappa )H}S_{22}(\kappa )+S_{12}(\kappa )\right]}{\Delta (\kappa )\gamma _{2}(\kappa )\sin \gamma _{2}(\kappa )H}, \\[-9pt] \nonumber \end{align}

(2.41b) \begin{align} Y&=\frac{-{\rm i}\omega ^{2}\rho _{2}\kappa \left[{\rm e}^{-{\rm i}{\gamma _{2}}(\kappa )H}S_{22}(\kappa )+S_{12}(\kappa )\right]}{\Delta (\kappa )\gamma _{2}(\kappa )\sin \gamma _{2}(\kappa )H}, \\[-9pt] \nonumber \end{align}

(2.42a) \begin{align} X_{n}&=\frac{{\rm i}\omega ^{2}\rho _{2}\kappa _{n}{\rm e}^{-{\rm i}{\kappa _{n}}{x_{0}}}}{2L_{x}\Delta (\kappa _{n})\gamma _{2}(\kappa _{n})\sin \gamma _{2}(\kappa _{n})H}\Big\{F_{1}^{e}\big[{\rm e}^{{\rm i}{\gamma _{2}}({\kappa _{n}})H}S_{22}(\kappa _{n})+S_{12}(\kappa _{n})\big]\Big.\nonumber\\[4pt]&\quad\Big. -F_{2}^{e}\big[{\rm e}^{{\rm i}{\gamma _{2}}({\kappa _{n}})H}S_{12}(\kappa _{n})+S_{11}(\kappa _{n})\big]\Big\}, \\[-9pt] \nonumber \end{align}

(2.42b) \begin{align} Y_{n}&=\frac{{\rm i}\omega ^{2}\rho _{2}\kappa _{n}{\rm e}^{-{\rm i}{\kappa _{n}}{x_{0}}}}{2L_{x}\Delta (\kappa _{n})\gamma _{2}(\kappa _{n})\sin \gamma _{2}(\kappa _{n})H}\Big\{F_{1}^{e}\big[{\rm e}^{-{\rm i}{\gamma _{2}}({\kappa _{n}})H}S_{22}(\kappa _{n})+S_{12}(\kappa _{n})\big]\Big.\nonumber\\[4pt]&\quad\Big. -F_{2}^{e}\big[{\rm e}^{-{\rm i}{\gamma _{2}}({\kappa _{n}})H}S_{12}(\kappa _{n})+S_{11}(\kappa _{n})\big]\Big\}. \\[9pt] \nonumber \end{align}

According to the geometric characteristic of metamaterial plates and the loading condition, the mechanical field generated by the rigid baffle is symmetric with respect to x = 0. It can be obtained from (2.40) that $ \partial p^{r}_{2} (x, z) / \partial x $ and $ p^{r}_{2} (x,z) $ are even and odd functions of x, respectively. Then we can see that $ p^{r}_{1}(x, z) $ , $ p^{r}_{3} (x,z) $ , $ w^{r}_{1} (x) $ and $ w^{r}_{2} (x) $ are odd functions.

The Wiener–Hopf method is applied to the sound pressure scattered by the rigid baffle, and the half-range Fourier transforms are defined as

(2.43a) \begin{align}\overline{p}_{2+}^{r}(\alpha ,z)&=\int _{-\infty }^{0}p_{2}^{r}(x,z){\rm e}^{-{\rm i}\alpha x}\,\mathrm{d}x, \\[-9pt] \nonumber \end{align}

(2.43b) \begin{align}\overline{p}_{2-}^{r}(\alpha ,z)&=\int _{0}^{\infty }p_{2}^{r}(x,z){\rm e}^{-{\rm i}\alpha x}\,\mathrm{d}x, \end{align}

where the former integral function is analytic for all complex α in the upper half-plane Im(α) $\geq$ 0, and the latter is analytic in the lower half-plane Im(α) $\leq$ 0. In addition, both functions are algebraically bounded as $|\alpha|$ approaches infinity.

Based on (2.43a ) with integration by parts, the Fourier transform of (2.38) can be derived as

(2.44) \begin{align}\frac{\partial ^{2}\overline{p}_{2+}^{r}(\alpha ,z)}{\partial z^{2}}+\gamma _{2}^{2}\overline{p}_{2+}^{r}(\alpha ,z)+\left.\frac{\partial p_{2}^{r}(x,z)}{\partial x}\right| _{x=0}+{\rm i}\alpha p_{2}^{r}(0,z)=0.\end{align}

Adding the corresponding equation evaluated at –α to (2.44), and substituting (2.40), we can obtain

(2.45) \begin{align}&\left(\frac{\partial ^{2}}{\partial z^{2}}+\gamma _{2}^{2}\right)\left[\overline{p}_{2+}^{r}(\alpha ,z)+\overline{p}_{2+}^{r}(-\alpha ,z)\right]=-2\big[X{\rm e}^{{\rm i}{\gamma _{2}}(\kappa)z}+Y{\rm e}^{-{\rm i}{\gamma _{2}}(\kappa)z}\big]\nonumber\\&\quad -2\sum _{n=-\infty }^{\infty }\big[X_{n}{\rm e}^{{\rm i}{\gamma _{2}}(\kappa _{n})z}+Y_{n}{\rm e}^{-{\rm i}{\gamma _{2}}(\kappa _{n})z}\big].\end{align}

The solution to this equation can be directly derived as the superposition of the general and particular solutions as

(2.46) \begin{align}\overline{p}_{2+}^{r}(\alpha ,z)+\overline{p}_{2+}^{r}\left(-\alpha ,z\right)&=A_{2}^{r}(\alpha ){\rm e}^{{\rm i}{\gamma _{2}}z}+B_{2}^{r}(\alpha ){\rm e}^{-{\rm i}{\gamma _{2}}z}+2\frac{X{\rm e}^{{\rm i}{\gamma _{2}}(\kappa)z}+Y{\rm e}^{-{\rm i}{\gamma _{2}}(\kappa)z}}{\alpha ^{2}-\kappa ^{2}}\nonumber\\&\quad+2\sum _{n=-\infty }^{\infty }\frac{X_{n}{\rm e}^{{\rm i}{\gamma _{2}}(\kappa _{n})z}+Y_{n}{\rm e}^{-{\rm i}{\gamma _{2}}(\kappa _{n})z}}{\alpha ^{2}-\kappa _{n}^{2}}.\end{align}

Then, setting z = 0 and z = −H, we obtain

(2.47a) \begin{align}\overline{p}_{2+}^{r}(\alpha ,0)+\overline{p}_{2+}^{r}\left(-\alpha ,0\right)&=A_{2}^{r}(\alpha )+B_{2}^{r}(\alpha )+2\frac{X+Y}{\alpha ^{2}-\kappa ^{2}}+2\sum _{n=-\infty }^{\infty }\frac{X_{n}+Y_{n}}{\alpha ^{2}-\kappa _{n}^{2}}, \end{align}

(2.47b) \begin{align} \overline{p}_{2+}^{r}(\alpha ,-H)+\overline{p}_{2+}^{r}(-\alpha ,-H)&=A_{2}^{r}(\alpha ){\rm e}^{-{\rm i}{\gamma _{2}}H}\!+\!B_{2}^{r}(\alpha ){\rm e}^{{\rm i}{\gamma _{2}}H}\!+\!2\frac{X{\rm e}^{-{\rm i}{\gamma _{2}}(\kappa)H}+Y{\rm e}^{{\rm i}{\gamma _{2}}(\kappa)H}}{\alpha ^{2}-\kappa ^{2}}\nonumber\\&\quad+2\sum _{n=-\infty }^{\infty }\frac{X_{n}{\rm e}^{-{\rm i}{\gamma _{2}}(\kappa_{n})H}+Y_{n}{\rm e}^{{\rm i}{\gamma _{2}}(\kappa_{n})H}}{\alpha ^{2}-\kappa _{n}^{2}}. \end{align}

Similar to (2.19b ) and (2.19c ), the boundary conditions can be written as

(2.48a,b) \begin{align} \left.\frac{\partial \overline{p}_{2+}^{r}(\alpha ,z)}{\partial z}\right| _{z=0}=\omega ^{2}\rho _{2}\overline{w}_{1+}^{r}(\alpha ),\quad \left.\frac{\partial \overline{p}_{2+}^{r}(\alpha ,z)}{\partial z}\right| _{z=-H}=\omega ^{2}\rho _{2}\overline{w}_{2+}^{r}(\alpha ).\end{align}

Substituting (2.46) into (2.48), we can derive that

(2.49a) \begin{align} \omega ^{2}\rho _{2}\left[\overline{w}_{1+}^{r}(\alpha )+\overline{w}_{1+}^{r}(-\alpha )\right]&={\rm i}\gamma _{2}\left[A_{2}^{r}(\alpha )-B_{2}^{r}(\alpha )\right]+\frac{2{\rm i}\gamma _{2}(\kappa )}{\alpha ^{2}-\kappa ^{2}}\left(X-Y\right)\nonumber\\&\quad+\sum _{n=-\infty }^{\infty }\frac{2{\rm i}\gamma _{2}(\kappa _{n})}{\alpha ^{2}-\kappa _{n}^{2}}\left(X_{n}-Y_{n}\right)\!, \end{align}

(2.49b) \begin{align}\omega ^{2}\rho _{2}\left[\overline{w}_{2+}^{r}(\alpha )+\overline{w}_{2+}^{r}(-\alpha )\right]&={\rm i}\gamma _{2}\big[A_{2}^{r}(\alpha ){\rm e}^{-{\rm i}{\gamma _{2}}H}-B_{2}^{r}(\alpha ){\rm e}^{{\rm i}{\gamma _{2}}H}\big]\nonumber\\[4pt]&\quad+\frac{2{\rm i}\gamma _{2}(\kappa )}{\alpha ^{2}-\kappa ^{2}}\big[X{\rm e}^{-{\rm i}{\gamma _{2}}(\kappa )H}-Y{\rm e}^{{\rm i}{\gamma _{2}}(\kappa )H}\big]\nonumber\\[4pt]&\quad +\sum _{n=-\infty }^{\infty }\frac{2{\rm i}\gamma _{2}(\kappa _{n})}{\alpha ^{2}-\kappa _{n}^{2}}\big[X_{n}{\rm e}^{-{\rm i}{\gamma _{2}}(\kappa_{n} )H}-Y_{n}{\rm e}^{{\rm i}{\gamma _{2}}(\kappa _{n})H}\big]. \end{align}

The unknown functions $ A^{r}_{2} (\alpha) $ and $ B^{r}_{2} (\alpha) $ can be eliminated by (2.47) and (2.49), and we can obtain

(2.50a) \begin{align}\overline{p}_{2+}^{r}(\alpha ,0)+\overline{p}_{2+}^{r}\left(-\alpha ,0\right)&=\frac{-\omega ^{2}\rho _{2}}{\gamma _{2}\,{\sin }\, \gamma _{2}H}\left\{\left[\overline{w}_{1+}^{r}(\alpha )+\overline{w}_{1+}^{r}\left(-\alpha \right)\right]{\cos }\, \gamma _{2}H\right.\nonumber\\[4pt]&\quad\left.-\left[\overline{w}_{2+}^{r}(\alpha )+\overline{w}_{2+}^{r}\left(-\alpha \right)\right]\right\}-F(\alpha ), \\[-9pt] \nonumber \end{align}

(2.50b) \begin{align}\overline{p}_{2+}^{r}(\alpha ,-H)+\overline{p}_{2+}^{r}(-\alpha ,-H)&=\frac{-\omega ^{2}\rho _{2}}{\gamma _{2}\,{\sin }\, \gamma _{2}H}\left\{\left[\overline{w}_{1+}^{r}(\alpha )+\overline{w}_{1+}^{r}\left(-\alpha \right)\right]\right.\nonumber\\[4pt]&\quad\left.-\left[\overline{w}_{2+}^{r}(\alpha )+\overline{w}_{2+}^{r}\left(-\alpha \right)\right]{\cos }\, \gamma _{2}H\right\}-G(\alpha ), \\[12pt] \nonumber \end{align}

where expressions for $ F(\alpha) $ and $ G(\alpha) $ are

(2.51a) \begin{align}F(\alpha )&=\frac{-2}{\alpha ^{2}-\kappa ^{2}}\left[X+Y+\frac{{\rm i}\gamma _{2}(\kappa)}{\gamma _{2}\,{\sin }\, \gamma _{2}H}\big(\tilde{X}\,{\cos }\, \gamma _{2}H-\tilde{Y}\big)\right]\nonumber\\&\quad+\sum _{n=-\infty }^{\infty }\frac{-2}{\alpha ^{2}-\kappa _{n}^{2}}\left[X_{n}+Y_{n}+\frac{{\rm i}\gamma _{2}(\kappa _{n})}{\gamma _{2}\,{\sin }\, \gamma _{2}H}\big(\tilde{X}_{n}\,{\cos }\, \gamma _{2}H-\tilde{Y}_{n}\big)\right]\!, \\[-9pt] \nonumber \end{align}

(2.51b) \begin{align}G(\alpha )&=\frac{-2}{\alpha ^{2}-\kappa ^{2}}\left\{X{\rm e}^{-{\rm i}{\gamma _{2}}(\kappa)H}+Y{\rm e}^{{\rm i}{\gamma _{2}}(\kappa)H}+\frac{{\rm i}\gamma _{2}(\kappa)}{\gamma _{2}\,{\sin }\, \gamma _{2}H}\big(\tilde{X}-\tilde{Y}\,{\cos }\, \gamma _{2}H\big)\right\}\nonumber\\&\quad+\!\sum _{n=-\infty }^{\infty }\frac{-2}{\alpha ^{2}-\kappa _{n}^{2}}\left\{X_{n}{\rm e}^{-{\rm i}{\gamma _{2}}(\kappa _{n})H}+Y_{n}{\rm e}^{{\rm i}{\gamma _{2}}(\kappa _{n})H}+\frac{{\rm i}\gamma _{2}(\kappa _{n})}{\gamma _{2}\,{\sin }\, \gamma _{2}H}\big(\tilde{X}_{n}-\tilde{Y}_{n}\,{\cos }\, \gamma _{2}H\big)\right\}\!. \\[12pt] \nonumber \end{align}

and

(2.52a,b) \begin{align}\tilde{X}&=\frac{2\omega ^{2}\rho _{2}\kappa S_{22}(\kappa )}{\Delta (\kappa )\gamma _{2}(\kappa )}, \qquad \tilde{Y}=-\frac{2\omega ^{2}\rho _{2}\kappa S_{12}(\kappa)}{\Delta (\kappa)\gamma _{2}(\kappa)}, \\[-9pt] \nonumber \end{align}

(2.53a) \begin{align}\tilde{X}_{n}&=\frac{-\omega ^{2}\rho _{2}\kappa _{n}{\rm e}^{-{\rm i}{\kappa _{n}}{x_{0}}}}{L_{x}\Delta (\kappa _{n})\gamma _{2}(\kappa _{n})}\left[F_{1}^{e}S_{22}(\kappa _{n})-F_{2}^{e}S_{12}(\kappa _{n})\right]\!, \\[-9pt] \nonumber \end{align}

(2.53b) \begin{align}\tilde{Y}_{n}&=\frac{\omega ^{2}\rho _{2}\kappa _{n}{\rm e}^{-{\rm i}{\kappa _{n}}{x_{0}}}}{L_{x}\Delta (\kappa _{n})\gamma _{2}(\kappa _{n})}\left[F_{1}^{e}S_{12}(\kappa _{n})-F_{2}^{e}S_{11}(\kappa _{n})\right]\!. \\[12pt] \nonumber \end{align}

Based on the odd functions and half-range Fourier transform in (2.43), the following equations can be obtained:

(2.54a–c) \begin{align}\overline{p}_{2+}^{r}\left(-\alpha ,z\right)=-\overline{p}_{2-}^{r}(\alpha ,z), \qquad \overline{w}_{1+}^{r}\left(-\alpha \right)=-\overline{w}_{1-}^{r}(\alpha ), \qquad \overline{w}_{2+}^{r}\left(-\alpha \right)=-\overline{w}_{2-}^{r}(\alpha ).\end{align}

Then (2.50) becomes

(2.55a) \begin{align}\overline{p}_{2+}^{r}(\alpha ,0)-\overline{p}_{2-}^{r}(\alpha ,0)&=\frac{-\omega ^{2}\rho _{2}}{\gamma _{2}\,{\sin }\, \gamma _{2}H}\left\{\left[\overline{w}_{1+}^{r}(\alpha )-\overline{w}_{1-}^{r}(\alpha )\right]{\cos }\, \gamma _{2}H\right.\nonumber\\&\quad\left.-\left[\overline{w}_{2+}^{r}(\alpha )-\overline{w}_{2-}^{r}(\alpha )\right]\right\}-F(\alpha ), \\[-9pt] \nonumber \end{align}

(2.55b) \begin{align}\overline{p}_{2+}^{r}(\alpha ,-H)-\overline{p}_{2-}^{r}(\alpha ,-H)&=\frac{-\omega ^{2}\rho _{2}}{\gamma _{2}\,{\sin }\, \gamma _{2}H}\left\{\left[\overline{w}_{1+}^{r}(\alpha )-\overline{w}_{1-}^{r}(\alpha )\right]\right.\nonumber\\&\quad\left.-\left[\overline{w}_{2+}^{r}(\alpha )-\overline{w}_{2-}^{r}\left(-\alpha \right)\right]{\cos }\, \gamma _{2}H\right\}-G(\alpha ). \\[12pt] \nonumber\end{align}

Although both γ ₂ and sinγ ₂ H are multivalued functions, we know through analysis that the functions $1/\gamma _{2}\sin \gamma _{2}H$ and $\cos \gamma _{2}H/\gamma _{2}\sin \gamma _{2}H$ in (2.55) have no branch points and are algebraically bounded as $|\alpha|$ approaches infinity. Their only singularities are an infinite number of poles located at $\alpha =\pm \sqrt{k_{2}^{2}-(m\pi /H)^{2}}=\pm \alpha _{m}$ (m = 0, 1, 2, …). Therefore, the singularities in (2.55) are ±κ, ±κ _n and ±α _m .

The Wiener–Hopf additive decomposition can be applied to obtain

(2.56a) \begin{align}\overline{p}_{2+}^{r}(\alpha ,0)&=\frac{1}{2\pi {\rm i}}\int _{{C_{1}}}\frac{\overline{p}_{2+}^{r}\left(\xi ,0\right)-\overline{p}_{2-}^{r}\left(\xi ,0\right)}{\xi -\alpha }\,\mathrm{d}\xi , \\[-9pt] \nonumber \end{align}

(2.56b) \begin{align}\overline{p}_{2-}^{r}(\alpha ,0)&=\frac{1}{2\pi {\rm i}}\int _{{C_{2}}}\frac{\overline{p}_{2+}^{r}\left(\xi ,0\right)-\overline{p}_{2-}^{r}\left(\xi ,0\right)}{\xi -\alpha }\,\mathrm{d}\xi . \\[12pt] \nonumber \end{align}

The integral contours C ₁ and C ₂ in (2.56) are shown in figure 2. Based on (2.55), the singularities in the integrand of (2.56) are α, ±κ, ±κ _n and ±α _m . The path on real axis is deformed to lie above the negative singularities and below the positive ones when approaching them. It can be seen from residue formulas that all singularities are simple poles. The integral contour C ₁ encloses the poles at −κ, −κ _n and −α _m , while contour C ₂ encloses the poles at κ, κ _n and α _m . According to the Cauchy residue theory, the integral in (2.56) is transformed into the residue calculation as the following forms:

(2.57a) \begin{align} \overline{p}_{2+}^{r}(\alpha ,0)&=\frac{-\omega ^{2}\rho _{2}}{\gamma _{2}\,{\sin }\, \gamma _{2}H}\left[\overline{w}_{1+}^{r}(\alpha )\,{\cos }\, \gamma _{2}H-\overline{w}_{2+}^{r}(\alpha )\right]\nonumber\\&\quad-\frac{\omega ^{2}\rho _{2}\alpha }{H}\sum _{m=0}^{\infty }\frac{\varepsilon _{m}\left[\overline{w}_{1+}^{r}(\alpha _{m})-(-1)^{m}\overline{w}_{2+}^{r}(\alpha _{m})\right]}{\alpha _{m}\!\left(\alpha ^{2}-\alpha _{m}^{2}\right)}-F_{+}(\alpha ), \end{align}

(2.57b) \begin{align} \overline{p}_{2-}^{r}(\alpha ,0)&=\frac{-\omega ^{2}\rho _{2}}{\gamma _{2}\,{\sin }\, \gamma _{2}H}\left[\overline{w}_{1-}^{r}(\alpha )\,{\cos }\, \gamma _{2}H-\overline{w}_{2-}^{r}(\alpha )\right]\nonumber\\&\quad-\frac{\omega ^{2}\rho _{2}\alpha }{H}\sum _{m=0}^{\infty }\frac{\varepsilon _{m}\left[\overline{w}_{1+}^{r}(\alpha _{m})-(-1)^{m}\overline{w}_{2+}^{r}(\alpha _{m})\right]}{\alpha _{m}\!\left(\alpha ^{2}-\alpha _{m}^{2}\right)}+F_{-}(\alpha ), \end{align}

(2.58a) \begin{align} \overline{p}_{2+}^{r}(\alpha ,-H)&=\frac{-\omega ^{2}\rho _{2}}{\gamma _{2}\,{\sin }\, \gamma _{2}H}\left[\overline{w}_{1+}^{r}(\alpha )-\overline{w}_{2+}^{r}(\alpha )\,{\cos }\, \gamma _{2}H\right]\nonumber\\&\quad-\frac{\omega ^{2}\rho _{2}\alpha }{H}\sum _{m=0}^{\infty }\frac{\varepsilon _{m}\left[(-1)^{m}\overline{w}_{1+}^{r}(\alpha _{m})-\overline{w}_{2+}^{r}(\alpha _{m})\right]}{\alpha _{m}\!\left(\alpha ^{2}-\alpha _{m}^{2}\right)}-G_{+}(\alpha ), \end{align}

(2.58b) \begin{align} \overline{p}_{2-}^{r}(\alpha ,-H)&=\frac{-\omega ^{2}\rho _{2}}{\gamma _{2}\,{\sin}\,\gamma _{2}H}\left[\overline{w}_{1-}^{r}(\alpha )-\overline{w}_{2-}^{r}(\alpha )\,{\cos }\, \gamma _{2}H\right]\nonumber\\&\quad-\frac{\omega ^{2}\rho _{2}\alpha }{H}\sum _{m=0}^{\infty }\frac{\varepsilon _{m}\left[(-1)^{m}\overline{w}_{1+}^{r}(\alpha _{m})-\overline{w}_{2+}^{r}(\alpha _{m})\right]}{\alpha _{m}\!\left(\alpha ^{2}-\alpha _{m}^{2}\right)}+G_{-}(\alpha ), \end{align}

Figure 2.

Integration contour of Wiener–Hopf additive decomposition for (a) the lower half-plane and (b) the upper half-plane.

where

(2.59) \begin{align}\varepsilon _{m}&=\begin{cases}1, & m=0,\\2, & m=1,2,3,\ldots, \end{cases} \end{align}

(2.60a) \begin{align}F_{+}(\alpha )&=\sum _{m=0}^{\infty }\frac{{\rm i}\gamma _{2}(\kappa )\varepsilon _{m}\big[\tilde{X}-(-1)^{m}\tilde{Y}\big]}{H\alpha _{m}\big(\kappa ^{2}-\alpha _{m}^{2}\big)(\alpha +\alpha _{m})}+\sum _{n=-\infty }^{\infty }\sum _{m=0}^{\infty }\frac{{\rm i}\gamma _{2}(\kappa _{n})\varepsilon _{m}\big[\tilde{X}_{n}-(-1)^{m}\tilde{Y}_{n}\big]}{H\alpha _{m}\big(\kappa _{n}^{2}-\alpha _{m}^{2}\big)(\alpha +\alpha _{m})}, \\[-9pt] \nonumber \end{align}

(2.60b) \begin{align}F_{-}(\alpha )&=-\sum _{m=0}^{\infty }\frac{{\rm i}\gamma _{2}(\kappa)\varepsilon _{m}\big[\tilde{X}-(-1)^{m}\tilde{Y}\big]}{H\alpha _{m}\big(\kappa ^{2}-\alpha _{m}^{2}\big)(\alpha -\alpha _{m})}-\sum _{n=-\infty }^{\infty }\sum _{m=0}^{\infty }\frac{{\rm i}\gamma _{2}(\kappa _{n})\varepsilon _{m}\big[\tilde{X}_{n}-(-1)^{m}\tilde{Y}_{n}\big]}{H\alpha _{m}\big(\kappa _{n}^{2}-\alpha _{m}^{2}\big)(\alpha -\alpha _{m})}, \end{align}

(2.61a) \begin{align}G_{+}(\alpha )&=\sum _{m=0}^{\infty }\frac{{\rm i}\gamma _{2}(\kappa )\varepsilon _{m}\big[(-1)^{m}\tilde{X}-\tilde{Y}\big]}{H\alpha _{m}\big(\kappa ^{2}-\alpha _{m}^{2}\big)(\alpha +\alpha _{m})}+\sum _{n=-\infty }^{\infty }\sum _{m=0}^{\infty }\frac{{\rm i}\gamma _{2}(\kappa _{n})\varepsilon _{m}\big[(-1)^{m}\tilde{X}_{n}-\tilde{Y}_{n}\big]}{H\alpha _{m}\big(\kappa _{n}^{2}-\alpha _{m}^{2}\big)(\alpha +\alpha _{m})}, \\[-9pt] \nonumber \end{align}

(2.61b) \begin{align}G_{-}(\alpha )&=-\sum _{m=0}^{\infty }\frac{{\rm i}\gamma _{2}(\kappa )\varepsilon _{m}\big[(-1)^{m}\tilde{X}-\tilde{Y}\big]}{H\alpha _{m}\big(\kappa ^{2}-\alpha _{m}^{2}\big)(\alpha -\alpha _{m})}-\sum _{n=-\infty }^{\infty }\sum _{m=0}^{\infty }\frac{{\rm i}\gamma _{2}(\kappa _{n})\varepsilon _{m}\big[(-1)^{m}\tilde{X}_{n}-\tilde{Y}_{n}\big]}{H\alpha _{m}\big(\kappa _{n}^{2}-\alpha _{m}^{2}\big)(\alpha -\alpha _{m})}. \end{align}

According to (2.57) and (2.58), we can obtain

(2.62a) \begin{align}&\overline{p}_{2+}^{r}(\alpha ,0)+\overline{p}_{2-}^{r}(\alpha ,0)=\frac{-\omega ^{2}\rho _{2}}{\gamma _{2}\,{\sin }\, \gamma _{2}H}\left\{\left[\overline{w}_{1+}^{r}(\alpha )\!+\overline{w}_{1-}^{r}(\alpha )\!\right]{\cos }\, \gamma _{2}H\!-\!\left[\overline{w}_{2+}^{r}(\alpha )+\overline{w}_{2-}^{r}(\alpha )\!\right]\right\}\nonumber\\&\quad-\frac{2\omega ^{2}\rho _{2}\alpha }{H}\sum _{m=0}^{\infty }\frac{\varepsilon _{m}\left[\overline{w}_{1+}^{r}(\alpha _{m})-(-1)^{m}\overline{w}_{2+}^{r}(\alpha _{m})\right]}{\alpha _{m}\!\left(\alpha ^{2}-\alpha _{m}^{2}\right)}-F_{+}(\alpha )+F_{-}(\alpha ), \end{align}

(2.62b) \begin{align}&\overline{p}_{2+}^{r}(\alpha ,-H)+\overline{p}_{2-}^{r}(\alpha ,-H)=\frac{-\omega ^{2}\rho _{2}}{\gamma _{2}\,{\sin }\, \gamma _{2}H}\left\{\left[\overline{w}_{1+}^{r}(\alpha )-\overline{w}_{1-}^{r}(\alpha )\right]\right.\nonumber\\&\quad\left.-\left[\overline{w}_{2+}^{r}(\alpha )+\overline{w}_{2-}^{r}(\alpha )\right]{\cos }\, \gamma _{2}H\right\}-\frac{2\omega ^{2}\rho _{2}\alpha }{H}\sum _{m=0}^{\infty }\frac{\varepsilon _{m}\left[(-1)^{m}\overline{w}_{1+}^{r}(\alpha _{m})-\overline{w}_{2+}^{r}(\alpha _{m})\right]}{\alpha _{m}\!\left(\alpha ^{2}-\alpha _{m}^{2}\right)}\nonumber\\&\quad-G_{+}(\alpha )+G_{-}(\alpha ). \end{align}

The governing equations of the corresponding displacement components can be written as

(2.63a) \begin{align}&D_{1}\frac{\partial ^{4}w_{1}^{r}(x)}{\partial x^{4}}-\omega ^{2}M_{1}w_{1}^{r}(x)=-p_{1}^{r}\left(x,0\right)+p_{2}^{r}\left(x,0\right)+F_{1}^{r}(x), \\[-9pt] \nonumber \end{align}

(2.63b) \begin{align}&D_{2}\frac{\partial ^{4}w_{2}^{r}(x)}{\partial x^{4}}-\omega ^{2}M_{2}w_{2}^{r}(x)=-p_{2}^{r}\left(x,-H\right)+p_{3}^{r}\left(x,-H\right)+F_{2}^{r}(x), \end{align}

where

(2.64a,b) \begin{align}F_{1}^{r}(x)=Bw_{1}^{r}(x)+Cw_{2}^{r}(x), \qquad F_{2}^{r}(x)=Cw_{1}^{r}(x)+Bw_{2}^{r}(x).\end{align}

Based on (2.21) and (2.43), the Fourier transform of (2.63) can be expressed as

(2.65a) \begin{align}&\left(D_{1}\alpha ^{4}-\omega ^{2}M_{1}-\frac{{\rm i}\omega ^{2}\rho _{1}}{\gamma _{1}}-B\right)\left[\overline{w}_{1+}^{r}(\alpha )+\overline{w}_{1-}^{r}(\alpha )\right]-C\left[\overline{w}_{2+}^{r}(\alpha )+\overline{w}_{2-}^{r}(\alpha )\right]\nonumber\\&\quad=\overline{p}_{2+}^{r}(\alpha ,0)+\overline{p}_{2-}^{r}(\alpha ,0), \\[-9pt] \nonumber \end{align}

(2.65b) \begin{align}&\left(D_{2}\alpha ^{4}-\omega ^{2}M_{2}-\frac{{\rm i}\omega ^{2}\rho _{3}}{\gamma _{3}}-B\right)\left[\overline{w}_{2+}^{r}(\alpha )+\overline{w}_{2-}^{r}(\alpha )\right]-C\left[\overline{w}_{1+}^{r}(\alpha )+\overline{w}_{1-}^{r}(\alpha )\right]\nonumber\\&\quad=-\left[\overline{p}_{2+}^{r}(\alpha ,-H)+\overline{p}_{2-}^{r}(\alpha ,-H)\right]\!. \end{align}

Substituting (2.62) into (2.65), the governing equations of displacement can be derived in the form

(2.66) \begin{align}&\hat{\boldsymbol{S}}(\alpha )\left\{\!\begin{array}{c} \overline{w}_{1+}^{r}(\alpha )+\overline{w}_{1-}^{r}(\alpha )\\ \overline{w}_{2+}^{r}(\alpha )+\overline{w}_{2-}^{r}(\alpha ) \end{array}\!\right\}=\sum _{m=0}^{\infty }\frac{2\mathrm{i}\gamma _{2}(\kappa)\alpha \varepsilon _{m}}{H\alpha _{m}\!\left(\kappa ^{2}-\alpha _{m}^{2}\right)\left(\alpha ^{2}-\alpha _{m}^{2}\right)}\left\{\!\!\begin{array}{ll} -\tilde{X}+(-1)^{m}\tilde{Y}\\ (-1)^{m}\tilde{X}-\tilde{Y} \end{array}\!\!\right\}\nonumber\\&\quad+\sum _{n=-\infty }^{\infty }\sum _{m=0}^{\infty }\frac{2{\rm i}\gamma _{2}(\kappa _{n})\alpha \varepsilon _{m}}{H\alpha _{m}\!\left(\kappa _{n}^{2}-\alpha _{m}^{2}\right)\left(\alpha ^{2}-\alpha _{m}^{2}\right)}\left\{\!\!\begin{array}{ll} -\tilde{X}_{n}+(-1)^{m}\tilde{Y}_{n}\\ (-1)^{m}\tilde{X}_{n}-\tilde{Y}_{n} \end{array}\!\!\right\}\nonumber\\&\quad+\frac{2\omega ^{2}\rho _{2}\alpha }{H}\sum _{m=0}^{\infty }\frac{\varepsilon _{m}\!\left[\overline{w}_{1+}^{r}(\alpha _{m})-(-1)^{m}\overline{w}_{2+}^{r}(\alpha _{m})\right]}{\alpha _{m}\!\left(\alpha ^{2}-\alpha _{m}^{2}\right)}\left\{\!\!\begin{array}{ll} -1\\ (-1)^{m} \end{array}\!\!\right\}\!,\end{align}

where the elements of matrix $\hat{\boldsymbol{S}}(\alpha )$ are

(2.67a) \begin{align}&S_{11}^{r}(\alpha )=D_{1}\alpha ^{4}-\omega ^{2}M_{1}-\frac{{\rm i}\omega ^{2}\rho _{1}}{\gamma _{1}}+\frac{\omega ^{2}\rho _{2}\cos \gamma _{2}H}{\gamma _{2}\,{\sin }\, \gamma _{2}H}-B, \end{align}

(2.67b) \begin{align}&S_{12}^{r}(\alpha )=S_{21}^{r}(\alpha )=-\frac{\omega ^{2}\rho _{2}}{\gamma _{2}\,{\sin }\, \gamma _{2}H}-C, \end{align}

(2.67c) \begin{align}&S_{22}^{r}(\alpha )=D_{2}\alpha ^{4}-\omega ^{2}M_{2}-\frac{{\rm i}\omega ^{2}\rho _{3}}{\gamma _{3}}+\frac{\omega ^{2}\rho _{2}\cos \gamma _{2}H}{\gamma _{2}\,{\sin }\, \gamma _{2}H}-B. \end{align}

It is convenient to write

(2.68) \begin{align}\hat{\boldsymbol{S}}^{-1}(\alpha )=\boldsymbol{K}(\alpha )=\boldsymbol{K}_{+}(\alpha )+\boldsymbol{K}_{-}(\alpha ),\end{align}

where

(2.69a) \begin{align}\boldsymbol{K}_{+}(\alpha )&=\frac{1}{2\pi {\rm i}}\int _{\underset{C_{3}}{-\infty }}^{\infty }\frac{\boldsymbol{K}(\xi){\rm d} \xi }{\xi -\alpha }, \end{align}

(2.69b) \begin{align}\boldsymbol{K}_{-}(\alpha )&=-\frac{1}{2\pi {\rm i}}\int _{\underset{C_{4}}{-\infty }}^{\infty }\frac{\boldsymbol{K}(\xi){\rm d} \xi }{\xi -\alpha }. \end{align}

The C ₃ and C ₄ represent the integral contours with negative and positive infinity as the upper and lower limits. And the contours C ₃ and C ₄ can be deformed away from the real axis to lie below and above the pole at ξ = α, respectively.

Since the elements of $\hat{\boldsymbol{S}}(\alpha )$ given by (2.67) are even functions of α, (2.69) can be derived as

(2.70a) \begin{align}\boldsymbol{K}_{+}(\alpha )&=\frac{\alpha }{\pi {\rm i}}\int _{0}^{\infty }\frac{\boldsymbol{K}(\xi){\rm d} \xi }{\xi ^{2}-\alpha ^{2}}, \end{align}

(2.70b) \begin{align}\boldsymbol{K}_{-}(\alpha )&=-\frac{\alpha }{\pi {\rm i}}\int _{0}^{\infty }\frac{\boldsymbol{K}(\xi){\rm d} \xi }{\xi ^{2}-\alpha ^{2}}, \end{align}

and the derivatives are

(2.71a) \begin{align}\boldsymbol{K}_{+}^{\prime}(\alpha )&=\frac{\boldsymbol{K}_{+}(\alpha )}{\alpha }+\frac{2\alpha ^{2}}{\pi {\rm i}}\int _{0}^{\infty }\frac{\boldsymbol{K}(\xi){\rm d} \xi }{(\xi ^{2}-\alpha ^{2})^{2}}, \end{align}

(2.71b) \begin{align}\boldsymbol{K}_{-}^{\prime}(\alpha )&=\frac{\boldsymbol{K}_{-}(\alpha )}{\alpha }-\frac{2\alpha ^{2}}{\pi {\rm i}}\int _{0}^{\infty }\frac{\boldsymbol{K}(\xi){\rm d} \xi }{(\xi ^{2}-\alpha ^{2})^{2}}. \end{align}

During the numerical calculation, the integration contour is deformed to lie slightly below the real axis, and the range that meets the error requirements is truncated.

The following Wiener–Hopf additive decomposition is performed:

(2.72) \begin{align}\frac{2\alpha \boldsymbol{K}(\alpha )}{\alpha ^{2}-\alpha _{m}^{2}}&=\left[\frac{\boldsymbol{K}_{+}(\alpha )+\boldsymbol{K}_{+}(\alpha _{m})}{\alpha +\alpha _{m}}+\frac{\boldsymbol{K}_{+}(\alpha )-\boldsymbol{K}_{+}(\alpha _{m})}{\alpha -\alpha _{m}}\right]\nonumber\\&\quad+\left[\frac{\boldsymbol{K}_{-}(\alpha )-\boldsymbol{K}_{+}(\alpha _{m})}{\alpha +\alpha _{m}}+\frac{\boldsymbol{K}_{-}(\alpha )+\boldsymbol{K}_{+}(\alpha _{m})}{\alpha -\alpha _{m}}\right]\!,\end{align}

where the first term on the right-hand side is analytic in the upper complex half-plane, and the second term is analytic in the lower one.

Multiplying both sides of (2.66) by K(α) and considering (2.72), the Wiener–Hopf additive decomposition can be applied to obtain

(2.73a) \begin{align}&\left\{\!\!\begin{array}{ll} \overline{w}_{1+}^{r}(\alpha )\\ \overline{w}_{2+}^{r}(\alpha ) \end{array}\!\!\right\}=\sum _{m=0}^{\infty }\frac{{\rm i}\gamma _{2}(\kappa )\varepsilon _{m}}{H\alpha _{m}(\kappa ^{2}-\alpha _{m}^{2})}\left[\frac{\boldsymbol{K}_{+}(\alpha )+\boldsymbol{K}_{+}(\alpha _{m})}{\alpha +\alpha _{m}}+\frac{\boldsymbol{K}_{+}(\alpha )-\boldsymbol{K}_{+}(\alpha _{m})}{\alpha -\alpha _{m}}\right]\nonumber\\&\quad\times\left\{\!\begin{array}{ll} -\tilde{X}+(-1)^{m}\tilde{Y}\\ (-1)^{m}\tilde{X}-\tilde{Y} \end{array}\!\right\}+\sum _{n=-\infty }^{\infty }\sum _{m=0}^{\infty }\frac{{\rm i}\gamma _{2}(\kappa _{n})\varepsilon _{m}}{H\alpha _{m}(\kappa _{n}^{2}-\alpha _{m}^{2})}\left[\frac{\boldsymbol{K}_{+}(\alpha )+\boldsymbol{K}_{+}(\alpha _{m})}{\alpha +\alpha _{m}}\right.\nonumber\\&\quad\left.+\frac{\boldsymbol{K}_{+}(\alpha )-\boldsymbol{K}_{+}(\alpha _{m})}{\alpha -\alpha _{m}}\right]\left\{\!\begin{array}{ll} -\tilde{X}_{n}+(-1)^{m}\tilde{Y}_{n}\\ (-1)^{m}\tilde{X}_{n}-\tilde{Y}_{n} \end{array}\!\right\}\nonumber\\&\quad+\frac{\omega ^{2}\rho _{2}}{H}\sum _{m=0}^{\infty }\frac{\varepsilon _{m}\left[\overline{w}_{1+}^{r}(\alpha _{m})-(-1)^{m}\overline{w}_{2+}^{r}(\alpha _{m})\right]}{\alpha _{m}}\left[\frac{\boldsymbol{K}_{+}(\alpha )+\boldsymbol{K}_{+}(\alpha _{m})}{\alpha +\alpha _{m}}\right.\nonumber\\&\quad\left.+\frac{\boldsymbol{K}_{+}(\alpha )-\boldsymbol{K}_{+}(\alpha _{m})}{\alpha -\alpha _{m}}\right]\left\{\!\!\begin{array}{ll} -1\\ (-1)^{m} \end{array}\!\!\right\}\!, \\[-9pt] \nonumber \end{align}

(2.73b) \begin{align}&\left\{\!\!\begin{array}{ll} \overline{w}_{1-}^{r}(\alpha )\\ \overline{w}_{2-}^{r}(\alpha ) \end{array}\!\!\right\}=\sum _{m=0}^{\infty }\frac{{\rm i}\gamma _{2}(\kappa )\varepsilon _{m}}{H\alpha _{m}(\kappa ^{2}-\alpha _{m}^{2})}\left[\frac{\boldsymbol{K}_{-}(\alpha )-\boldsymbol{K}_{+}(\alpha _{m})}{\alpha +\alpha _{m}}+\frac{\boldsymbol{K}_{-}(\alpha )+\boldsymbol{K}_{+}(\alpha _{m})}{\alpha -\alpha _{m}}\right]\nonumber\\&\quad\times\left\{\!\begin{array}{ll} -\tilde{X}+(-1)^{m}\tilde{Y}\\ (-1)^{m}\tilde{X}-\tilde{Y} \end{array}\!\right\}+\sum _{n=-\infty }^{\infty }\sum _{m=0}^{\infty }\frac{{\rm i}\gamma _{2}(\kappa _{n})\varepsilon _{m}}{H\alpha _{m}(\kappa _{n}^{2}-\alpha _{m}^{2})}\left[\frac{\boldsymbol{K}_{-}(\alpha )-\boldsymbol{K}_{+}(\alpha _{m})}{\alpha +\alpha _{m}}\right.\nonumber\\&\quad\left.+\frac{\boldsymbol{K}_{-}(\alpha )+\boldsymbol{K}_{+}(\alpha _{m})}{\alpha -\alpha _{m}}\right]\left\{\!\begin{array}{ll} -\tilde{X}_{n}+(-1)^{m}\tilde{Y}_{n}\\ (-1)^{m}\tilde{X}_{n}-\tilde{Y}_{n} \end{array}\!\right\}\nonumber\\&\quad+\frac{\omega ^{2}\rho _{2}}{H}\sum _{m=0}^{\infty }\frac{\varepsilon _{m}\left[\overline{w}_{1+}^{r}(\alpha _{m})-(-1)^{m}\overline{w}_{2+}^{r}(\alpha _{m})\right]}{\alpha _{m}}\left[\frac{\boldsymbol{K}_{-}(\alpha )-\boldsymbol{K}_{+}(\alpha _{m})}{\alpha +\alpha _{m}}\right.\nonumber\\&\quad\left.+\frac{\boldsymbol{K}_{-}(\alpha )+\boldsymbol{K}_{+}(\alpha _{m})}{\alpha -\alpha _{m}}\right]\left\{\!\!\begin{array}{ll} -1\\ (-1)^{m} \end{array}\!\!\right\}\!. \end{align}

Setting α to α ₀, α ₁, …, α _{M – 1} in (2.73a ), and considering the truncation at $|n|$ = N, the following matrix equation can be derived:

(2.74) \begin{align}\boldsymbol{J}\tilde{\boldsymbol{w}}=\tilde{\boldsymbol{F}},\end{align}

where the elements of vector $\tilde{\boldsymbol{w}}$ are

(2.75) \begin{align}\tilde{w}_{\zeta }=\overline{w}_{1+}^{r}\big(\alpha _{\zeta -1}\big)-(-1 )^{\zeta -1}\overline{w}_{2+}^{r}\big(\alpha _{\zeta -1}\big)\quad (\zeta = 1, 2, \ldots , M),\end{align}

the elements of matrix J include

(2.76a) \begin{align}J_{\zeta \zeta }&=1+\frac{\omega ^{2}\rho _{2}\varepsilon _{\zeta -1}}{H\alpha _{\zeta -1}^{2}}\left[K_{11+}(\alpha _{\zeta -1})+K_{22+}(\alpha _{\zeta -1})+2(-1)^{\zeta }K_{12+}(\alpha _{\zeta -1})\right]\nonumber\\&\quad+\frac{\omega ^{2}\rho _{2}\varepsilon _{\zeta -1}}{H\alpha _{\zeta -1}}\left[K_{11+}^{\prime}(\alpha _{\zeta -1})+K_{22+}^{\prime}(\alpha _{\zeta -1})+2(-1)^{\zeta }K_{12+}^{\prime}(\alpha _{\zeta -1})\right]\!, \\[-9pt] \nonumber \end{align}

(2.76b) \begin{align}J_{\zeta \eta }&=\frac{\omega ^{2}\rho _{2}\varepsilon _{\eta -1}}{H\alpha _{\eta -1}(\alpha _{\zeta -1}+\alpha _{\eta -1})}\left\{K_{11+}(\alpha _{\zeta -1})+K_{11+}(\alpha _{\eta -1})+(-1)^{\zeta }\left[K_{12+}(\alpha _{\zeta -1})\right.\right.\nonumber\\&\quad\left.+K_{12+}(\alpha _{\eta -1})\right]\left.+(-1 )^{\eta }\left[K_{12+}(\alpha _{\zeta -1})+K_{12+}(\alpha _{\eta -1})\right]+(-1)^{\zeta +\eta }\left[K_{22+}(\alpha _{\zeta -1})\right.\right.\nonumber\\&\quad\left.\left.+K_{22+}(\alpha _{\eta -1})\right]\right\}+\frac{\omega ^{2}\rho _{2}\varepsilon _{\eta -1}}{H\alpha _{\eta -1}(\alpha _{\zeta -1}-\alpha _{\eta -1})}\left\{K_{11+}(\alpha _{\zeta -1})-K_{11+}(\alpha _{\eta -1})+(-1)^{\zeta }\right.\nonumber\\&\quad\times\left[K_{12+}(\alpha _{\zeta -1})-K_{12+}(\alpha _{\eta -1})\right]+(-1)^{\eta }\left[K_{12+}(\alpha _{\zeta -1})-K_{12+}(\alpha _{\eta -1})\right]+(-1)^{\zeta +\eta }\nonumber\\&\quad\times\left.\left[K_{22+}(\alpha _{\zeta -1})-K_{22+}(\alpha _{\eta -1})\right]\right\},\quad\eta \neq \zeta ). \end{align}

and the elements of vector $\tilde{\boldsymbol{F}}$ can be expressed as

\begin{align*}&\tilde{F}_{\zeta }=\sum _{\begin{array}{cc} & \eta =1\\& \eta \neq \zeta \end{array}}^{M}\frac{-{\rm i}\gamma _{2}(\kappa )\varepsilon _{\eta -1}}{H\alpha _{\eta -1}(\kappa ^{2}-\alpha _{\eta -1}^{2})}\left\{2\alpha _{\zeta -1}\left[K_{11+}(\alpha _{\zeta -1})+(-1)^{\zeta }K_{12+}(\alpha _{\zeta -1})\right.\right.\nonumber\\&\quad\left.\left.+(-1 )^{\eta }K_{12+}(\alpha _{\zeta -1})+(-1)^{\zeta +\eta }K_{22+}(\alpha _{\zeta -1})\right]-2\alpha _{\eta -1}\left[K_{11+}(\alpha _{\eta -1})\right.\right.\nonumber\\&\quad\left.\left.+(-1)^{\zeta }K_{12+}(\alpha _{\eta -1})+(-1 )^{\eta }K_{12+}(\alpha _{\eta -1})+(-1)^{\zeta +\eta }K_{22+}(\alpha _{\eta -1})\right]\right\}\frac{\tilde{X}-(-1)^{\eta -1}\tilde{Y}}{\big(\alpha _{\zeta -1}^{2}-\alpha _{\eta -1}^{2}\big)}\nonumber\\&\quad-\frac{{\rm i}\gamma _{2}(\kappa )\varepsilon _{\zeta -1}}{H\alpha _{\zeta -1}\big(\kappa ^{2}-\alpha _{\zeta -1}^{2}\big)}\left[\frac{K_{11+}(\alpha _{\zeta -1})+2(-1)^{\zeta }K_{12+}(\alpha _{\zeta -1})+K_{22+}(\alpha _{\zeta -1})}{\alpha _{\zeta -1}}\right.\nonumber\\&\quad\left.+K_{11+}^{\prime}(\alpha _{\zeta -1})+2(-1)^{\zeta }K_{12+}^{\prime}(\alpha _{\zeta -1})+K_{22+}^{\prime}(\alpha _{\zeta -1})\right]\big[\tilde{X}-(-1)^{\zeta -1}\tilde{Y}\big]\end{align*}

\begin{align*}&\quad-\sum _{n=-N}^{N}\sum _{\begin{array}{cc} \eta =1\\ \eta \neq \zeta \end{array}}^{M}\frac{{\rm i}\gamma _{2}(\kappa _{n})\varepsilon _{\eta -1}}{H\alpha _{\eta -1}\big(\kappa _{n}^{2}-\alpha _{\eta -1}^{2}\big)}\left\{2\alpha _{\zeta -1}\left[K_{11+}(\alpha _{\zeta -1})+(-1)^{\zeta }K_{12+}(\alpha _{\zeta -1})\right.\right.\nonumber\\&\quad\left.\left.+(-1 )^{\eta }K_{12+}(\alpha _{\zeta -1})+(-1)^{\zeta +\eta }K_{22+}(\alpha _{\zeta -1})\right]\right.\left.-2\alpha _{\eta -1}\left[K_{11+}(\alpha _{\eta -1})\right.\right.\nonumber\\&\quad\left.\left.+(-1)^{\zeta }K_{12+}(\alpha _{\eta -1})+(-1 )^{\eta }K_{12+}(\alpha _{\eta -1})+(-1)^{\zeta +\eta }K_{22+}(\alpha _{\eta -1})\right]\right\}\frac{\tilde{X}_{n}-(-1)^{\eta -1}\tilde{Y}_{n}}{\big(\alpha _{\zeta -1}^{2}-\alpha _{\eta -1}^{2}\big)}\nonumber\\\end{align*}

(2.77) \begin{align}&\quad-\sum _{n=-N}^{N}\frac{{\rm i}\gamma _{2}(\kappa _{n})\varepsilon _{\zeta -1}}{H\alpha _{\zeta -1}\big(\kappa _{n}^{2}-\alpha _{\zeta -1}^{2}\big)}\left[\frac{K_{11+}(\alpha _{\zeta -1})+2(-1)^{\zeta }K_{12+}(\alpha _{\zeta -1})+K_{22+}(\alpha _{\zeta -1})}{\alpha _{\zeta -1}}\right.\nonumber\\&\quad\left.+K_{11+}^{\prime}(\alpha _{\zeta -1})+2(-1)^{\zeta }K_{12+}^{\prime}(\alpha _{\zeta -1})+K_{22+}^{\prime}(\alpha _{\zeta -1})\right]\big[\tilde{X}_{n}-(-1)^{\zeta -1}\tilde{Y}_{n}\big].\end{align}

Considering the above matrix equation, the approximate values of $\tilde{w}_{\zeta }$ are calculated and substituted into (2.66) to obtain the displacement $\overline{w}_{1}^{r}(\alpha )$ of the upper plate. With reference to (2.14) and (2.20), the pressure in medium 1 can be expressed as

(2.78) \begin{align}\overline{p}_{1}^{r}(\alpha ,z)=\frac{-{\rm i}\omega ^{2}\rho _{1}}{\gamma _{1}}\overline{w}_{1}^{r}(\alpha ) {\rm e}^{{\rm i}{\gamma _{1}}z}.\end{align}

Then the inverse Fourier transform is performed to obtain

(2.79) \begin{align}p_{1}^{r}\left(R,\theta \right)=\frac{\omega ^{2}\rho _{1}}{2\pi {\rm i}}\int _{-\infty }^{\infty }\frac{\overline{w}_{1}^{r}(\alpha )}{\gamma _{1}}\,{\rm e}^{{\rm i}R\left(\gamma _{1}\,{\sin }\, \theta +\alpha\, {\cos }\, \theta \right)}\,\mathrm{d}\alpha ,\end{align}

where

(2.80a,b) \begin{align}x=R\cos \theta , \qquad z=R\sin \theta .\end{align}

According to the stationary phase method, the far-field scattered sound pressure resulting from the rigid baffle can be expressed as

(2.81) \begin{align}p_{1}^{r}\left(R,\theta \right)=-\frac{\omega ^{2}\rho _{1}\overline{w}_{1}^{r}\left(k_{1}\,{\cos }\, \theta \right)}{\sqrt{2\pi k_{1}R}}\,{\rm e}^{{\rm i}\left(k_{1}R+{\pi }/{4}\right)}.\end{align}