# Acoustic and Dynamic Response of Unbaffled Plates of Arbitrary Shape
## 2 Models and Methods  
A flat, isotropic, homogeneous, thin plate of arbitrary shape and under any support condition is considered. The plate is in the \(z=0\)_plane_ and one of its sides is selected as the \(x\)-axis (Figure 1).  
The differential equation that governs the transverse displacement \(w\) (\(x\), \(y\), \(t\)) of the plate with constant thickness and without structural damping can be expressed as follows ([PERSON], [15]):  
\[D\
abla^{4}w+\rho_{M}h\frac{\partial^{2}w}{\partial t^{2}}=\Delta\widetilde{P }(x,y,t)+f(x,y,t) \tag{1}\]  
where \(D\) is the flexural rigidity, \(\rho_{M}\) is the material density, \(h\) is the plate thickness, \(\Delta\widetilde{P}\) is the acoustic pressure jump across the plate, surface, and \(f(x,y,t)\) represents the external forces other than the forces exerted by the fluid. The equations for the plate are completed based on the boundary conditions at the edge of the plate.  
The pressure jump \(\Delta\widetilde{P}\) can be expressed as follows:  
\[\Delta\widetilde{P}(x,y,t)=\Delta\widetilde{P}_{\mathit{P}ib}(x,y,t)+\Delta \widetilde{P}_{\mathit{wave}}(x,y,t) \tag{2}\]  
Figure 1: Example of plate.  
where \(\Delta\widetilde{P}_{vib}\) is the pressure jump owing to the plate radiation, and \(\Delta\widetilde{P}_{wave}\) is the pressure jump owing to the load induced by the incident plane wave over its surface when the plate is rigid. It is assumed that small perturbations occur in a compressible, inviscid and irrotational fluid, and these pressure distributions can be calculated by solving the wave equation for the fluid domain, expressed as follows:  
\[\
abla^{2}p-\frac{1}{a_{\infty}{}^{2}}\frac{\partial^{2}p}{\partial t^{2}}=0 \tag{3}\]  
where \(a_{\infty}\) is the speed of sound in the fluid at rest.  
The application of the momentum equation at the plate surface yields the following boundary condition:  
\[\frac{\partial p}{\partial z}=-\rho_{\infty}\frac{\partial^{2}w}{\partial t^{ 2}}\quad\quad\text{at}\;z=\pm 0 \tag{4}\]  
where \(\rho_{\infty}\) is the fluid density.  
At an extended distance from the plate, the Sommerfeld radiation condition must be satisfied. It is assumed that the motion is harmonic for both the fluid and the structure to solve the system of Equations (1), (3) and (4). Therefore, the transverse displacement \(w(x,y,t)\) and pressure \(p(x,y,z,t)\) can be written as follows.  
\[w(x,y,t) =W(x,y)e^{-i\omega t}\] \[p(x,y,z,t) =P(x,y,z)e^{-i\omega t} \tag{5}\]  
The deformation of the plate \(W(x,y)\) can be expressed as a linear combination of the normal modes of the plate in vacuum, which are determined using the FEM model. Let these modes be \(W_{m}(x_{j},y_{j})\), where \((x_{j},y_{j})\) are the coordinates of the FEM model nodes. An analytical expression valid for all \((x,y)\) for the plate is obtained through curve fitting in a Lagrange polynomial form. The deformation of the plate is then derived as follows:  
\[w(x,y,t)=\sum_{m=1}^{M}\sum_{n=1}^{N}q_{nm}^{0}W_{nm}(x,y)e^{-i\omega t} \tag{6}\]  
where the \(q_{nm}^{0}\) are unknown coefficients to be determined when the coupled fluid-structure equations are solved, and \(W_{nm}(x,y)\) are the curve-fitted modes. The modes of the plate in vacuum are chosen as a basis, and the number of degrees of freedom can be significantly reduced.  
Next, the independent variables \(x\), \(y\), and \(z\) are nondimensionalized using the plate characteristic length \(l_{c}\), which is defined as \(l_{c}=\sqrt{S_{p}}\), where \(S_{p}\) is the plate area.