Transmission of Sound Through Finite Multiple-Panel Partition

Abstract

This chapter is organized as three parts: in the first part, the vibroacoustic performance of a rectangular double-panel partition with enclosed air cavity and simply mounted on an infinite acoustic rigid baffle is investigated analytically. The sound velocity potential method rather than the commonly used cavity modal function method is employed, which possesses good expandability and has significant implications for further vibroacoustic investigations. The simply supported boundary condition is accounted for by using the method of modal function, and double Fourier series solutions are obtained to characterize the vibroacoustic behaviors of the structure. Results for sound transmission loss (STL), panel vibration level, and sound pressure level are presented to explore the physical mechanisms of sound energy penetration across the finite double-panel partition. Specifically, focus is placed upon the influence of several key system parameters on sound transmission, including the thickness of air cavity, structural dimensions, and the elevation angle and azimuth angle of the incidence sound. Further extensions of the sound velocity potential method to typical framed double-panel structures are also proposed.

In the second part, the air-borne sound insulation performance of a rectangular double-panel partition clamp mounted on an infinite acoustic rigid baffle is investigated both analytically and experimentally, and compared with that of a simply supported one. With the clamped (or simply supported) boundary accounted for by using the method of modal function, a double series solution for the sound transmission loss (STL) of the structure is obtained by employing the weighted residual (Galerkin) method. Experimental measurements with Al double-panel partitions having air cavity are subsequently carried out to validate the theoretical model for both types of the boundary condition, and good overall agreement is achieved. A consistency check of the two different models (based separately on clamped modal function and simply supported modal function) is performed by extending the panel dimensions to infinite where no boundaries exist. The significant discrepancies between the two different boundary conditions are demonstrated in terms of the STL versus frequency plots as well as the panel deflection mode shapes.

In the third part, an analytical model for sound transmission through a clamped triple-panel partition of finite extent and separated by two impervious air cavities is formulated. The solution derived from the model takes the form of that for a clamp-supported rectangular plate. A set of modal functions (or more strictly speaking, the basic functions) are employed to account for the clamped boundary conditions, and the application of the virtual work principle leads to a set of simultaneous algebraic equations for determining the unknown modal coefficients. The sound transmission loss (STL) of the triple-panel partition as a function of excitation frequency is calculated and compared with that of a double-panel partition. The model predictions are then used to explore the physical mechanisms associated with the various dips on the STL versus frequency curve, including the equivalent “mass-spring” resonance, the standing-wave resonance, and the panel modal resonance. The asymptotic variation of the solution from a finite-sized partition to an infinitely large partition is illustrated in such a way as to demonstrate the influence of the boundary conditions on the soundproofing capability of the partition. In general, a triple-panel partition outperforms a double-panel partition in insulating the incident sound, and the relatively large number of system parameters pertinent to the triple-panel partition in comparison with that of the double-panel partition offers more design space for the former to tailor its noise reduction performance.

Appendices

Appendices

1.1.1 Appendix A

The deflection coefficients of the two panels are

$$ \left\{{\alpha}_{1, kl}\right\}={{\left[\begin{array}{cccccccccc}\hfill {\alpha}_{1,11}\hfill & \hfill {\alpha}_{1,21}\hfill & \hfill \dots \hfill & \hfill {\alpha}_{1, M1}\hfill & \hfill {\alpha}_{1,12}\hfill & \hfill {\alpha}_{1,22}\hfill & \hfill \dots \hfill & \hfill {\alpha}_{1, M2}\hfill & \hfill \dots \hfill & \hfill {\alpha}_{1, M N}\hfill \end{array}\right]}^{\mathrm{T}}}_{MN\times 1} $$

(1.A.1)

$$ \left\{{\alpha}_{2, kl}\right\}={{\left[\begin{array}{cccccccccc}\hfill {\alpha}_{2,11}\hfill & \hfill {\alpha}_{2,21}\hfill & \hfill \dots \hfill & \hfill {\alpha}_{2, M1}\hfill & \hfill {\alpha}_{2,12}\hfill & \hfill {\alpha}_{2,22}\hfill & \hfill \dots \hfill & \hfill {\alpha}_{2, M2}\hfill & \hfill \dots \hfill & \hfill {\alpha}_{2, M N}\hfill \end{array}\right]}^{\mathrm{T}}}_{MN\times 1} $$

(1.A.2)

The left-hand side of Eq. (1.74) represents the generalized force, where

$$ \left\{{F}_{kl}\right\}=2 j\omega {\rho}_0 I{{\left[\begin{array}{cccccccccc}\hfill {f}_{11}\hfill & \hfill {f}_{21}\hfill & \hfill \dots \hfill & \hfill {f}_{M1}\hfill & \hfill {f}_{12}\hfill & \hfill {f}_{22}\hfill & \hfill \dots \hfill & \hfill {f}_{M2}\hfill & \hfill \dots \hfill & \hfill {f}_{M N}\hfill \end{array}\right]}^{\mathrm{T}}}_{M N\times 1} $$

(1.A.3)

$$ {\lambda}_{1, mn}^{*1}=3{\left(\frac{m}{a}\right)}^4+3{\left(\frac{n}{b}\right)}^4+2{\left(\frac{m}{a}\right)}^2{\left(\frac{n}{b}\right)}^2 $$

(1.A.4)

$$ {\Delta}_1^{*1}={\left[\!\arraycolsep4pt\begin{array}{cccccccccc}\hfill {\lambda}_{1,11}^{*1}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill {\lambda}_{1,21}^{*1}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \ddots \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {\lambda}_{1, M1}^{*1}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {\lambda}_{1,12}^{*1}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {\lambda}_{1,22}^{*1}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \ddots \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {\lambda}_{1, M2}^{*1}\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \ddots \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {\lambda}_{1, M N}^{*1}\hfill \end{array}\!\!\right]}_{MN\times MN} $$

(1.A.5)

$$ {\lambda}_{1, n}^{*2}=\frac{2{n}^4}{b^4}{\left[\arraycolsep4pt\begin{array}{ccccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill \cdots \hfill & \hfill 1\hfill \\ {}\hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill \cdots \hfill & \hfill 1\hfill \\ {}\hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill \cdots \hfill \\ {}\hfill \cdots \hfill & \hfill \cdots \hfill & \hfill \cdots \hfill & \hfill \ddots \hfill & \hfill 1\hfill \\ {}\hfill 1\hfill & \hfill 1\hfill & \hfill \cdots \hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]}_{M\times N} $$

(1.A.6)

$$ {\Delta}_1^{*2}={\left[\arraycolsep4pt\begin{array}{cccc}\hfill {\lambda}_{1,1}^{*2}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill {\lambda}_{1,2}^{*2}\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \ddots \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {\lambda}_{1, N}^{*2}\hfill \end{array}\right]}_{MN\times MN} $$

(1.A.7)

$$ {\lambda}_1^{*3}=\frac{2}{a^4}{\left[\begin{array}{cccc}\hfill {1}^4\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill {2}^4\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \ddots \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {M}^4\hfill \end{array}\right]}_{M\times N} $$

(1.A.8)

$$ {\Delta}_1^{*3}={\left[\arraycolsep5pt\begin{array}{lllll} 0 & {\lambda}_1^{*3} & {\lambda}_1^{*3} & \cdots & {\lambda}_1^{*3} \\[3pt] {} {\lambda}_1^{*3} & 0 & {\lambda}_1^{*3} & \cdots & {\lambda}_1^{*3} \\[3pt] {} {\lambda}_1^{*3} & {\lambda}_1^{*3} & 0 & \cdots & \cdots \\[3pt] {} \cdots & \cdots & \cdots & \ddots & {\lambda}_1^{*3} \\[3pt] {} {\lambda}_1^{*3} & {\lambda}_1^{*3} & \cdots & {\lambda}_1^{*3} & 0 \end{array}\right]}_{MN\times MN} $$

(1.A.9)

$$ {\Delta}_2^{*1}=\frac{9 ab}{4}{\left[\begin{array}{cccc}\hfill 1\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill 1\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \ddots \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill 1\hfill \end{array}\right]}_{MN\times MN} $$

(1.A.10)

$$ {\lambda}_2^{*2}=\frac{3 ab}{2}{\left[\arraycolsep4pt\begin{array}{ccccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill \cdots \hfill & \hfill 1\hfill \\ {}\hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill \cdots \hfill & \hfill 1\hfill \\ {}\hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill \cdots \hfill \\ {}\hfill \cdots \hfill & \hfill \cdots \hfill & \hfill \cdots \hfill & \hfill \ddots \hfill & \hfill 1\hfill \\ {}\hfill 1\hfill & \hfill 1\hfill & \hfill \cdots \hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]}_{M\times N} $$

(1.A.11)

$$ {\Delta}_2^{*2}={\left[\begin{array}{cccc}\hfill {\lambda}_2^{*2}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill {\lambda}_2^{*2}\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \ddots \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {\lambda}_2^{*2}\hfill \end{array}\right]}_{MN\times MN} $$

(1.A.12)

$$ {\lambda}_2^{*3}=\frac{3 ab}{2}{\left[\begin{array}{cccc}\hfill 1\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill 1\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \ddots \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill 1\hfill \end{array}\right]}_{M\times N} $$

(1.A.13)

$$ {\Delta}_2^{*3}={\left[\arraycolsep5pt\begin{array}{lllll} 0 & {\lambda}_2^{*3} & {\lambda}_2^{*3} & \cdots & {\lambda}_2^{*3} \\[3pt] {} {\lambda}_2^{*3} & 0 & {\lambda}_2^{*3} & \cdots & {\lambda}_2^{*3} \\[3pt] {} {\lambda}_2^{*3} & {\lambda}_2^{*3} & 0 & \cdots & \cdots \\[3pt] {} \cdots & \cdots & \cdots & \ddots & {\lambda}_2^{*3} \\[3pt] {} {\lambda}_2^{*3} & {\lambda}_2^{*3} & \cdots & {\lambda}_2^{*3} & 0 \end{array}\right]}_{MN\times MN} $$

(1.A.14)

$$ {\lambda}_2^{*4}= ab{\left[\arraycolsep4pt\begin{array}{ccccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill \cdots \hfill & \hfill 1\hfill \\ {}\hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill \cdots \hfill & \hfill 1\hfill \\ {}\hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill \cdots \hfill \\ {}\hfill \cdots \hfill & \hfill \cdots \hfill & \hfill \cdots \hfill & \hfill \ddots \hfill & \hfill 1\hfill \\ {}\hfill 1\hfill & \hfill 1\hfill & \hfill \cdots \hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]}_{M\times N} $$

(1.A.15)

$$ {\Delta}_2^{*4}={\left[\arraycolsep5pt\begin{array}{lllll} 0 & {\lambda}_2^{*4} & {\lambda}_2^{*4} & \cdots & {\lambda}_2^{*4} \\[4pt] {} {\lambda}_2^{*4} & 0 & {\lambda}_2^{*4} & \cdots & {\lambda}_2^{*4} \\[4pt] {} {\lambda}_2^{*4} & {\lambda}_2^{*4} & 0 & \cdots & \cdots \\[4pt] {} \cdots & \cdots & \cdots & \ddots & {\lambda}_2^{*4} \\[4pt] {} {\lambda}_2^{*4} & {\lambda}_2^{*4} & \cdots & {\lambda}_2^{*4} & 0 \end{array}\right]}_{MN\times MN} $$

(1.A.16)

Using the definition of the sub-matrices presented above, one obtains

$$ \begin{aligned} {\left[{T}_{11, kl}\right]}_{MN\times MN}&=4{D}_1{\pi}^4 ab\left({\Delta}_1^{*1}+{\Delta}_1^{*2}+{\Delta}_1^{*3}\right)\\ & \quad -\left({m}_1{\omega}^2{+} j\omega {\rho}_0\frac{2\omega {e}^{2 j{k}_z H}}{k_z\left(1-{e}^{2 j{k}_z H}\right)}\right)\cdot \left({\Delta}_2^{*1}{+}{\Delta}_2^{*2}{+}{\Delta}_2^{*3}+{\Delta}_2^{*4}\right) \end{aligned} $$

(1.A.17)

$$ {\left[{T}_{12, kl}\right]}_{MN\times MN}= j\omega {\rho}_0\frac{2\omega {e}^{j{k}_z H}}{k_z\left(1-{e}^{2 j{k}_z H}\right)}\left({\Delta}_2^{*1}+{\Delta}_2^{*2}+{\Delta}_2^{*3}+{\Delta}_2^{*4}\right) $$

(1.A.18)

$$ {\left[{T}_{21, kl}\right]}_{MN\times MN}= j\omega {\rho}_0\frac{2\omega {e}^{j{k}_z H}}{k_z\left(1-{e}^{2 j{k}_z H}\right)}\left({\Delta}_2^{*1}+{\Delta}_2^{*2}+{\Delta}_2^{*3}+{\Delta}_2^{*4}\right) $$

(1.A.19)

$$ \begin{aligned} {\left[{T}_{22, kl}\right]}_{MN\times MN}&=4{D}_2{\pi}^4 ab\left({\Delta}_1^{*1}+{\Delta}_1^{*2}+{\Delta}_1^{*3}\right)\\ & \quad -\left({m}_2{\omega}^2{+} j\omega {\rho}_0\frac{2\omega {e}^{2 j{k}_z H}}{k_z\left(1-{e}^{2 j{k}_z H}\right)}\right)\cdot \left({\Delta}_2^{*1}{+}{\Delta}_2^{*2}{+}{\Delta}_2^{*3}+{\Delta}_2^{*4}\right) \end{aligned} $$

(1.A.20)

1.1.2 Appendix B

The modal coefficients of the three panels are

$$ \left\{{\boldsymbol{\alpha}}_1\right\}={{\left[\begin{array}{cccccccccc}\hfill {\alpha}_{1,11}\hfill & \hfill {\alpha}_{1,21}\hfill & \hfill \dots \hfill & \hfill {\alpha}_{1, M1}\hfill & \hfill {\alpha}_{1,12}\hfill & \hfill {\alpha}_{1,22}\hfill & \hfill \dots \hfill & \hfill {\alpha}_{1, M2}\hfill & \hfill \dots \hfill & \hfill {\alpha}_{1, M N}\hfill \end{array}\right]}^{\mathtt{T}}}_{MN\times 1} $$

(1.B.1)

$$ \left\{{\boldsymbol{\alpha}}_2\right\}={{\left[\begin{array}{cccccccccc}\hfill {\alpha}_{2,11}\hfill & \hfill {\alpha}_{2,21}\hfill & \hfill \dots \hfill & \hfill {\alpha}_{2, M1}\hfill & \hfill {\alpha}_{2,12}\hfill & \hfill {\alpha}_{2,22}\hfill & \hfill \dots \hfill & \hfill {\alpha}_{2, M2}\hfill & \hfill \dots \hfill & \hfill {\alpha}_{2, M N}\hfill \end{array}\right]}^{\mathrm{T}}}_{MN\times 1} $$

(1.B.2)

$$ \left\{{\boldsymbol{\alpha}}_3\right\}={{\left[\begin{array}{cccccccccc}\hfill {\alpha}_{3,11}\hfill & \hfill {\alpha}_{3,21}\hfill & \hfill \dots \hfill & \hfill {\alpha}_{3, M1}\hfill & \hfill {\alpha}_{3,12}\hfill & \hfill {\alpha}_{3,22}\hfill & \hfill \dots \hfill & \hfill {\alpha}_{3, M2}\hfill & \hfill \dots \hfill & \hfill {\alpha}_{3, M N}\hfill \end{array}\right]}^{\mathrm{T}}}_{MN\times 1} $$

(1.B.3)

The generalized forces can be written in vector form as

$$ \left\{\mathbf{F}\right\}=2 j\omega {\rho}_0 I{{\left[\begin{array}{cccccccccc}\hfill {f}_{11}\hfill & \hfill {f}_{21}\hfill & \hfill \dots \hfill & \hfill {f}_{M1}\hfill & \hfill {f}_{12}\hfill & \hfill {f}_{22}\hfill & \hfill \dots \hfill & \hfill {f}_{M2}\hfill & \hfill \dots \hfill & \hfill {f}_{M N}\hfill \end{array}\right]}^{\mathrm{T}}}_{M N\times 1} $$

(1.B.4)

$$ {\lambda}_{1, mn}^{*1}=3{\left(\frac{m}{a}\right)}^4+3{\left(\frac{n}{b}\right)}^4+2{\left(\frac{m}{a}\right)}^2{\left(\frac{n}{b}\right)}^2 $$

(1.B.5)

$$ {\Delta}_1^{*1}={\left[\begin{array}{cccccccccc}\hfill {\lambda}_{1,11}^{*1}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill {\lambda}_{1,21}^{*1}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \ddots \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {\lambda}_{1, M1}^{*1}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {\lambda}_{1,12}^{*1}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {\lambda}_{1,22}^{*1}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \ddots \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {\lambda}_{1, M2}^{*1}\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \ddots \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {\lambda}_{1, M N}^{*1}\hfill \end{array}\right]}_{MN\times MN} $$

(1.B.6)

$$ {\lambda}_{1, n}^{*2}=\frac{2{n}^4}{b^4}{\left[\arraycolsep4pt\begin{array}{ccccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill \cdots \hfill & \hfill 1\hfill \\ {}\hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill \cdots \hfill & \hfill 1\hfill \\ {}\hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill \cdots \hfill \\ {}\hfill \cdots \hfill & \hfill \cdots \hfill & \hfill \cdots \hfill & \hfill \ddots \hfill & \hfill 1\hfill \\ {}\hfill 1\hfill & \hfill 1\hfill & \hfill \cdots \hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]}_{M\times N} $$

(1.B.7)

$$ {\Delta}_1^{*2}={\left[\arraycolsep4pt\begin{array}{cccc}\hfill {\lambda}_{1,1}^{*2}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill {\lambda}_{1,2}^{*2}\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \ddots \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {\lambda}_{1, N}^{*2}\hfill \end{array}\right]}_{MN\times MN} $$

(1.B.8)

$$ {\lambda}_1^{*3}=\frac{2}{a^4}{\left[\begin{array}{cccc}\hfill {1}^4\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill {2}^4\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \ddots \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {M}^4\hfill \end{array}\right]}_{M\times N} $$

(1.B.9)

$$ {\Delta}_1^{*3}={\left[\arraycolsep5pt\begin{array}{lllll} 0 & {\lambda}_1^{*3} & {\lambda}_1^{*3} & \cdots & {\lambda}_1^{*3} \\[3pt] {} {\lambda}_1^{*3} & 0 & {\lambda}_1^{*3} & \cdots & {\lambda}_1^{*3} \\[3pt] {} {\lambda}_1^{*3} & {\lambda}_1^{*3} & 0 & \cdots & \cdots \\[3pt] {} \cdots & \cdots & \cdots & \ddots & {\lambda}_1^{*3} \\[3pt] {} {\lambda}_1^{*3} & {\lambda}_1^{*3} & \cdots & {\lambda}_1^{*3} & 0 \end{array}\right]}_{MN\times MN} $$

(1.B.10)

$$ {\Delta}_2^{*1}=\frac{9 ab}{4}{\left[\begin{array}{cccc}\hfill 1\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill 1\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \ddots \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill 1\hfill \end{array}\right]}_{MN\times MN} $$

(1.B.11)

$$ {\lambda}_2^{*2}=\frac{3 ab}{2}{\left[\arraycolsep4pt\begin{array}{ccccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill \cdots \hfill & \hfill 1\hfill \\ {}\hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill \cdots \hfill & \hfill 1\hfill \\ {}\hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill \cdots \hfill \\ {}\hfill \cdots \hfill & \hfill \cdots \hfill & \hfill \cdots \hfill & \hfill \ddots \hfill & \hfill 1\hfill \\ {}\hfill 1\hfill & \hfill 1\hfill & \hfill \cdots \hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]}_{M\times N} $$

(1.B.12)

$$ {\Delta}_2^{*2}={\left[\arraycolsep4pt\begin{array}{cccc}\hfill {\lambda}_2^{*2}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill {\lambda}_2^{*2}\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \ddots \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {\lambda}_2^{*2}\hfill \end{array}\right]}_{MN\times MN} $$

(1.B.13)

$$ {\lambda}_2^{*3}=\frac{3 ab}{2}{\left[\begin{array}{cccc}\hfill 1\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill 1\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \ddots \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill 1\hfill \end{array}\right]}_{M\times N} $$

(1.B.14)

$$ {\Delta}_2^{*3}={\left[\arraycolsep5pt\begin{array}{lllll} 0 & {\lambda}_2^{*3} & {\lambda}_2^{*3} & \cdots & {\lambda}_2^{*3} \\[3pt] {} {\lambda}_2^{*3} & 0 & {\lambda}_2^{*3} & \cdots & {\lambda}_2^{*3} \\[3pt] {} {\lambda}_2^{*3} & {\lambda}_2^{*3} & 0 & \cdots & \cdots \\[3pt] {} \cdots & \cdots & \cdots & \ddots & {\lambda}_2^{*3} \\[3pt] {} {\lambda}_2^{*3} & {\lambda}_2^{*3} & \cdots & {\lambda}_2^{*3} & 0 \end{array}\right]}_{MN\times MN} $$

(1.B.15)

$$ {\lambda}_2^{*4}= ab{\left[\arraycolsep4pt\begin{array}{ccccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill \cdots \hfill & \hfill 1\hfill \\ {}\hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill \cdots \hfill & \hfill 1\hfill \\ {}\hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill \cdots \hfill \\ {}\hfill \cdots \hfill & \hfill \cdots \hfill & \hfill \cdots \hfill & \hfill \ddots \hfill & \hfill 1\hfill \\ {}\hfill 1\hfill & \hfill 1\hfill & \hfill \cdots \hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]}_{M\times N} $$

(1.B.16)

$$ {\Delta}_2^{*4}={\left[\arraycolsep5pt \begin{array}{lllll} 0 & {\lambda}_2^{*4} & {\lambda}_2^{*4} & \cdots & {\lambda}_2^{*4} \\[3pt] {} {\lambda}_2^{*4} & 0 & {\lambda}_2^{*4} & \cdots & {\lambda}_2^{*4} \\[3pt] {} {\lambda}_2^{*4} & {\lambda}_2^{*4} & 0 & \cdots & \cdots \\[3pt] {} \cdots & \cdots & \cdots & \ddots & {\lambda}_2^{*4} \\[3pt] {} {\lambda}_2^{*4} & {\lambda}_2^{*4} & \cdots & {\lambda}_2^{*4} & 0 \end{array}\right]}_{MN\times MN} $$

(1.B.17)

In the context of the above sub-matrices, the elemental matrices can be derived as

$$ \begin{aligned}{\left[{\mathbf{T}}_{11}\right]}_{MN\times MN}&=4{D}_1{\pi}^4 ab\left({\Delta}_1^{*1}+{\Delta}_1^{*2}+{\Delta}_1^{*3}\right)\\ {}& \quad - \left({m}_1{\omega}^2+ j\omega {\rho}_0\frac{2\omega {e}^{2 j{k}_z{H}_1}}{k_z\left(1-{e}^{2 j{k}_z{H}_1}\right)}\right)\cdot \left({\Delta}_2^{*1}+{\Delta}_2^{*2}+{\Delta}_2^{*3}+{\Delta}_2^{*4}\right)\end{aligned} $$

(1.B.18)

$$ {\left[{\mathbf{T}}_{12}\right]}_{MN\times MN}= j\omega {\rho}_0\frac{2\omega {e}^{j{k}_z{H}_1}}{k_z\left(1-{e}^{2 j{k}_z{H}_1}\right)}\left({\Delta}_2^{*1}+{\Delta}_2^{*2}+{\Delta}_2^{*3}+{\Delta}_2^{*4}\right) $$

(1.B.19)

$$ {\left[{\mathbf{T}}_{21}\right]}_{MN\times MN}= j\omega {\rho}_0\frac{2\omega {e}^{j{k}_z{H}_1}}{k_z\left(1-{e}^{2 j{k}_z{H}_1}\right)}\left({\Delta}_2^{*1}+{\Delta}_2^{*2}+{\Delta}_2^{*3}+{\Delta}_2^{*4}\right) $$

(1.B.20)

$$ {\fontsize{8.5}{10}\selectfont{\begin{aligned}[b] {\left[{\mathbf{T}}_{22}\right]}_{MN\times MN}&=4{D}_2{\pi}^4 ab\left({\Delta}_1^{*1}+{\Delta}_1^{*2}+{\Delta}_1^{*3}\right)\\ {}& \quad +\left(\!{-}{m}_2{\omega}^2{+}\frac{\omega^2{\rho}_0\left({e}^{j{k}_z\left({H}_1{-}{H}_2\right)}-{e}^{j{k}_z\left({H}_1+{H}_2\right)}\right)}{k_z\left(-1+{e}^{2 j{k}_z{H}_1}\right) \sin \left({k}_z\left({H}_1-{H}_2\right)\right)}\!\right)\left({\Delta}_2^{*1}{+}{\Delta}_2^{*2}{+}{\Delta}_2^{*3}{+}{\Delta}_2^{*4}\right)\end{aligned} }} $$

(1.B.21)

$$ {\left[{\mathbf{T}}_{23}\right]}_{MN\times MN}=\frac{\omega^2{\rho}_0}{k_z \sin \left({k}_z\left({H}_1-{H}_2\right)\right)}\left({\Delta}_2^{*1}+{\Delta}_2^{*2}+{\Delta}_2^{*3}+{\Delta}_2^{*4}\right) $$

(1.B.22)

$$ {\left[{\mathbf{T}}_{32}\right]}_{MN\times MN}=\frac{\omega^2{\rho}_0}{k_z \sin \left({k}_z\left({H}_1-{H}_2\right)\right)}\left({\Delta}_2^{*1}+{\Delta}_2^{*2}+{\Delta}_2^{*3}+{\Delta}_2^{*4}\right) $$

(1.B.23)

$$ \begin{aligned} {\left[{\mathbf{T}}_{33}\right]}_{MN\times MN}&=4{D}_3{\pi}^4 ab\left({\Delta}_1^{*1}+{\Delta}_1^{*2}+{\Delta}_1^{*3}\right)\\ & \quad -\left({m}_3{\omega}^2+\frac{\omega^2{\rho}_0{e}^{- j{k}_z\left({H}_1-{H}_2\right)}}{k_z \sin \left({k}_z\left({H}_1-{H}_2\right)\right)}\right)\left({\Delta}_2^{*1}+{\Delta}_2^{*2}+{\Delta}_2^{*3}+{\Delta}_2^{*4}\right) \end{aligned} \\[-12pt] $$

(1.B.24)