## 2 Mathematical Modeling and Analysis
### Analytical and Numerical Formulation for Acoustic Radiation from Tapered Annular Circular Plate
Considering the plate thickness (h) effect, the sum of sound radiations [16] from two normal surfaces of the plate at (Z = 0.5 h and \(-\)0.5 h) will represent the modal sound power, which can be given by Equations (9)-(11):  
\[P_{mn}(R,\alpha,\beta)=(1+\cos\alpha)P_{mn}^{s}(R,\alpha,\beta)+(1-\cos\alpha )P_{mn}^{o}(R,\alpha,\beta) \tag{9}\]  
\[P_{mn}^{s}(R,\alpha,\beta)=\frac{\rho_{0}c_{0}B_{mn}e^{iB_{mn}R}}{2R}e^{-iB_{ mn}(\frac{b}{2})\cos\alpha}\cos n\beta(-i)^{n+1}A_{n}\bigg{[}U(l)\bigg{]} \tag{10}\]  
\[\text{P}\big{(}l_{\text{P}}\big{)}=\int_{S_{\text{s}}}\left(\frac{\partial q} {\partial f}\big{(}l_{p},l_{s}\big{)}P\big{(}l_{\text{P}}\big{)}-\frac{ \partial P}{\partial f}(l_{s})q\big{(}l_{p},l_{s}\big{)}\right)ds(l_{s}) \tag{3}\]0  
where, the corresponding acoustic wave number of the (m, n)th mode is represented by \(\mathrm{B}_{m,n}\), s and o in Equations (10) and (11) represent the source side and the opposite to the source side. For numerical analysis, we have used ANSYS (ANSYS, Inc., Canonsburg, PA, USA) as a tool. The plates with rectangular and concentric patches are modeled in ANSYS with Plane 185 with 8 brick nodes and having three degrees of freedom at each node. The mesh is not exactly equal to all the cases of different thickness variation and stiffener. The number of element and nodes for uniform unloaded plates ends up being 5883 and 1664, respectively. For plates with different cases of thickness variation with different stiffener we tried to keep the mesh as close to the mesh of the unloaded plate. For vibration analysis and for a Case I plates with 1 rectangular stiffener, the modal structure consists of 5685 elements with 1638 nodes whereas for Case I plates with 1 concentric stiffener, it has 5524 elements and 1618 nodes. With other combinations of rectangular and concentric stiffener with different parabolic thickness, a variation of 5% of the mesh from that of the unloaded plate is considered. The numerical results obtained using FEM are compared with the existing literature. The structure is modeled as such that the total volume of the plate plus stiffeners is equal to the total volume of the uniform unloaded plate. As a result the whole mass of the plate plus stiffener is equal to the whole mass of the uniform unloaded plate. So for all the cases of plate, the mass will be constant. FLUID 30 and FLUID130 elements are used to create the acoustic medium environment around the plate. For fluid-structure interaction FLUID 30 is used. For the surface on outer sphere, FLUID 130 elements are created by imposing a condition of infinite space around the source and to prevent the back reflection of sound waves to the source. For acoustic analysis, the number of element and nodes for a uniform unloaded plate ends up being 14,680 and 3639, respectively. For a Case I plate with 1 rectangular stiffener, and after proper convergence of modeling, the numbers of elements and nodes ends up being 14,124 and 3465 respectively while for Case I plate with 1 concentric stiffener, the numbers of elements and nodes found to be 13,934 and 3345 respectively. Again, for other combination of rectangular and concentric stiffeners with different parabolic thickness, a variation of 5% of mesh from that of the unloaded plate is taken. Consider the air medium where the plate is vibrating with air density \(\rho_{0}=1.21\) kg/m\({}^{3}\). At 20 \({}^{\circ}\)C, the speed of sound \(c_{0}\) of air is taken as 343 m/s. The structural damping coefficient of the plate is assumed as 0.01.