## 2 Theory  
Figure 2 shows the general concept of a structural-acoustic model, in which the structural part is in contact with a finite cavity by physical boundaries. The enclosed volume would induce the acoustic resonant behavior, which is normally modeled using the well-known wave equation [1; 2; 15; 16; 17]. This wave equation has been used in numerous applied science/engineering problems for the past few decades. The structural part in the model usually is a plate or a shell, which is subject to an external force and coupled with the internal acoustic pressure within the cavity. The structural vibration directly affects the magnitude of the acoustic pressure acting on the structure. This is so-called structural-acoustic interaction. In the past, many researchers just focused on particular linear/nonlinear plate modellings (e.g., [5; 6; 7; 8; 9; 10; 11; 12; 18]) or linear structural-acoustic problems with perfectly rigid acoustic boundaries. In this paper, the focuses include the nonlinear structural-acoustic interaction and non-rigid acoustic boundary. Figure 3 shows the problem considered in this study. The acoustic pressure within the rectangular tube is obtained from the homogeneous wave equation. \[\
abla^{2}P^{h}-\frac{1}{C_{a}^{2}}\frac{\partial^{2}P^{h}}{\partial t^{2}}=0 \tag{1}\]  
where \(P^{h}\) is the \(h\)-th harmonic component of the acoustic pressure within the cavity; \(Ca\) is the speed of sound. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Solution** \\ **Method** \\ \end{tabular} } & \multicolumn{2}{c}{\begin{tabular}{c} **Zero Level (1 Harmonic Term)** \\ \end{tabular} } & \multicolumn{2}{c}{\begin{tabular}{c} **1 st Level (2 Harmonic Terms)** \\ \end{tabular} } & \multicolumn{2}{c}{\begin{tabular}{c} **2 nd Level (3 Harmonic Terms)** \\ \end{tabular} } \\ \cline{2-7}  & \begin{tabular}{c} **No. of nonlinear** \\ **algebraic** \\ **equations** \\ \end{tabular} & \begin{tabular}{c} **No. of linear** \\ **algebraic** \\ **equations** \\ \end{tabular} & \begin{tabular}{c} **No. of linear** \\ **algebraic** \\ **equations** \\ \end{tabular} & \begin{tabular}{c} **No. of nonlinear** \\ **algebraic** \\ **equations** \\ \end{tabular} & \begin{tabular}{c} **No. of linear** \\ **algebraic** \\ **equations** \\ \end{tabular} \\ \hline \
\begin{tabular}{c} Proposed \\ harmonic \\ balance method \\ \end{tabular} & 1 & 0 & 1 & 2 & 1 & 3 \\ \hline
\begin{tabular}{c} Classical \\ harmonic \\ balance method \\ \end{tabular} & 1 & 0 & 2 & 0 & 3 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Numbers of equations for the cubic nonlinear undamped beam problem. In Figure 4, the boundary conditions of the perfectly rigid and fully open cases at \(x=0\) and \(a\); \(y=0\) and \(b\) are given by  
\[\frac{\partial P^{h}}{\partial x}=\frac{\partial P^{h}}{\partial y}=0,\text{ for perfectly rigid} \tag{2a}\] \[P^{h}=0,\text{ for fully open} \tag{2b}\]  
According to Equations (2a) and (2b), the acoustic pressure mode shapes of the perfectly rigid and fully open cases are given by  
\[\varphi_{uv}(x,y)=\cos\Bigl{(}\frac{u\pi}{a}x\Bigr{)}\cos\Bigl{(}\frac{v\pi}{ b}y\Bigr{)} \tag{3a}\] \[\varphi_{uv}(x,y)=\sin\Bigl{(}\frac{(u+1)\pi}{a}x\Bigr{)}\sin\Bigl{(}\frac{(v+ 1)\pi}{b}y\Bigr{)} \tag{3b}\]  
where \(\varphi_{uv}(x,y)\) is the (\(u\),\(v\)) acoustic mode, \(a\) and \(b\) are the cross-sectional dimensions of the duct, and \(u\) and \(v\) are the acoustic mode numbers. Figure 3: Schematic diagram of a rectangular tube with a flexible end and non-rigid acoustic boundaries. Figure 2: General concept of a structural–acoustic model. Hence, the acoustic pressure mode shape of non-rigid case is defined by  
\[\varphi_{uv}(x,y)=\cos\biggl{(}\frac{(u+\eta)\pi}{a}x-\eta\frac{\pi}{2}\biggr{)} \cos\biggl{(}\frac{(v+\eta)\pi}{b}y-\eta\frac{\pi}{2}\biggr{)} \tag{4}\]  
where \(\eta=\frac{\varphi}{90};\theta\) is the phase shift parameter. If \(\eta\) is set as zero in Equation (4), then the acoustic mode shapes become those in the perfectly rigid case.