# Acoustic and Dynamic Response of Unbaffled Plates of Arbitrary Shape
## 2 Models and Methods
### Boundary Element Method  
The integral equation (13) is solved as follows. First, the plate is divided into panels (of trapezoidal/triangular shape), as depicted in Figure 2. The numbers of panels are set as \(I\) in the \(x\)-direction and \(J\) in the \(y\)-direction. Therefore, \(I\cdot J\) is the total number of panels. The pressure in each panel is assumed to be constant and therefore the \"jump\" in pressure at panel \(ij\) is equal to \(\left(\Delta\widetilde{P}_{mn}\right)_{ij}\) due to the normal mode \(W_{mn}\). The integral on the surface of the plate is then evaluated as the sum of all panels, resulting in the following expression:  
\[q_{mn}^{0}\rho_{\infty}\omega^{2}l_{c}W_{mn}(x,y)=-\frac{1}{4\pi}\sum\limits_ {i=1}^{I}\sum\limits_{j=1}^{I}\left(\Delta\widetilde{P}_{vib(mn)}\right)_{ij} \iint\limits_{S_{pij}}\frac{\partial^{2}}{\partial z^{2}}\left(\frac{e^{-ikR} }{R}\right)\Bigg{|}_{z=0}d\xi d\eta \tag{14}\]The integral equation at \(I\cdot J\) control points located at the geometric center of each panel is satisfied to obtain a linear system of \(I\cdot J\) equations for the pressure jump distribution (either \(\Delta\bar{P}_{vib}\)) associated with the mode \(W_{mn}\). Condition (14) may be represented in a matrix form as:  
\[[AIC]\Big{\{}\Delta\bar{P}\Big{\}}_{mn}=-4\pi\rho_{\infty}\omega^{2}l_{c}q^{0} _{mn}\{W_{mn}\} \tag{15}\]  
where the influence coefficients of the matrix \([AIC]\) are defined as follows:  
\[AIC^{rs}_{ij}=\iint\limits_{S_{r\bar{r}ij}}\frac{\partial^{2}}{\partial z^{2}} \left(\frac{e^{-ikR}}{R}\right)\Bigg{|}_{z=0}d\xi d\eta \tag{16}\]  
In the integral expressed in Equation (16), special care must be taken for the singularity that occurs when the element of integration coincides with the element where the boundary condition is satisfied. Let the integrand be established as follows. \[K(x_{rs}-\xi,y_{rs}-\eta)=K(x_{0},y_{0}) \tag{17}\]  
The singular part of function is added and subtracted to prevent numerical instabilities. Therefore, the integral can be expressed as follow:  
\[\iint\limits K(x_{0},y_{0})dx_{0}dy_{0}=\iint\limits\left[K(x_{0},y_{0})-K_{s }(x_{0},y_{0})\right]dx_{0}dy_{0}+F.P.\iint\limits K_{s}(x_{0},y_{0})dx_{0}dy _{0} \tag{18}\]  
The first integral on the right side is no longer singular when \(x_{0}\to 0\) and \(y_{0}\to 0\), and it can be evaluated numerically. The second integral can be analytically evaluated, and the finite part in Hadamard s sense can be used. The robustness of the integration method is validated using the following example. For very low frequencies, function \(K_{s}(x_{0},y_{0})\) can be expanded in the form of Taylor series, as follows:  
\[K_{s}(x_{0},y_{0})\approx-\frac{1}{R^{3}}-\frac{k^{2}}{2R}+\frac{ik^{3}}{3} \tag{19}\]  
Figure 2: Division of plate into elements. If a rectangular element of sides 2a and 2b is considered, the integration of the expanded form of the function \(K_{s}\), that is, Equation (19), after the finite part of the integral of the term, \(\
icefrac{{1}}{{R^{3}}}\), is selected, yields the following expression:  
\[\frac{-4\sqrt{a^{2}+b^{2}}}{ab}-\frac{k^{2}}{2}4b\ln\left|\frac{a+\sqrt{a^{2}+b^ {2}}}{b}\right|-\frac{k^{2}}{2}4a\ln\left|\frac{b+\sqrt{a^{2}+b^{2}}}{ba} \right|+\frac{ik^{3}}{3}(4 ab) \tag{20}\]  
The results obtained using Equation (20) are compared with the numerical procedure described for evaluating the integrals in Equation (18) and Figure 3. Both methods are in excellent agreement, and the slight difference was attributed to the number of terms used in the Taylor series expansion. As the procedure for evaluating the integral in Equation (18) is independent of the nondimensional frequency \(k\), the above results demonstrate the robustness of the integration method. The method described to treat the singularity provides accurate results (see Figure 3). Although other methods, such as the ones given by [PERSON] (1966) and [PERSON] (1997), for example, are equally effective for treating this singularity.