## 2 Materials and Methods
### Acoustic Radiation Formulation of Plate with Point Masses
Furthermore, if \(\sigma_{\text{mn}}\) is considered as radiation efficiency of the plate, then radiation efficiency [10] is obtained by Equation (8)  
\[\sigma_{\text{mn}}=\frac{S_{\text{mn}}}{\left|i_{\text{mn}}^{2}\right|_{ts}} \Big{|}i_{\text{mn}}^{2}\Big{|}ts=\frac{1}{2\pi(a^{2}-b^{2})}\int_{b}^{a}\int \limits_{0}^{2\pi}U^{2}d\varphi dl \tag{8}\]  
where, \(\left|i_{\text{mn}}^{2}\right|_{ts}\) is considered to be the spatially average r.m.s velocity for the two normal surfaces of the plate. If the plate thickness (h) effect is considered, then from the two normal surfaces of the plate at (\(Z=0.5h\) and \(-0.5h\)), the modal sound power [16] due to the sum of two sound radiations is obtained by Equations (9)-(11)  
\[\text{P}_{\text{mn}}(R,\alpha,\beta)=(1+\cos\alpha)\text{P}_{\text{mn}}^{s}( R,\alpha,\beta)+(1-\cos\alpha)\text{P}_{\text{mn}}^{s}(R,\alpha,\beta) \tag{9}\]  
\[\text{P}_{\text{mn}}^{s}(R,\alpha,\beta)=\frac{\rho_{0}c_{0}B_{\text{mn}}e^{ iB_{\text{mn}}R}}{2R}e^{-iB_{\text{mn}}(\frac{b}{2})\cos\alpha}\cos n\beta(-i)^{n+1}A_ {n}\Big{[}U(l)\Big{]} \tag{10}\]  
\[\text{P}_{\text{mn}}^{0}(R,\alpha,\beta)=\frac{\rho_{0}c_{0}B_{\text{mn}}e^{ iB_{\text{mn}}R}}{2R}e^{-iB_{\text{mn}}(\frac{b}{2})\cos\alpha}\cos n(\beta+\phi)(-i)^{n+1}A _{n}\Big{[}U(l)\Big{]} \tag{11}\]  
where, for (m, n)th mode, \(B_{\text{mn}}\) is considered to be the corresponding acoustic wave number whereas s and \(o\) represent the source side and opposite to source side. \[\mathrm{P}\left(l_{p}\right)=\int_{S_{s}}\left(\frac{\partial q}{\partial f} \left(l_{p},l_{s}\right)P\left(l_{p}\right)-\frac{\partial P}{\partial f} \left(l_{s}\right)q\left(l_{p},l_{s}\right)\right)ds\left(l_{s}\right) \tag{3}\]0 Acoustic radiation due to the vibration modes of unbaffled tapered plate in \(Z\) direction with different combination of attached point masses enclosed in a sphere. elements and nodes for uniform unloaded plate comes out to be 14,680 and 3639, respectively. For Case I plate with one point mass the number of elements and nodes after proper convergence comes out to be 14,662 and 3627, respectively. Further, after proper convergence, for Case I plate with two point masses the number of element and nodes after proper convergence comes out to be 14,644 and 3615 respectively. For Case I plate with four point masses, the number of element and nodes after proper convergence found to be 14,620, and 3605 respectively. For other cases of plate with different point masses combination, again a variation of 1-2% of mesh as that of uniform unloaded plate is taken in this numerical simulation technique. Consider the plate is vibrating in air medium with air density \(\rho_{0}\) = 1.21 kg/m\({}^{3}\). At 20 \({}^{\circ}\)C, the speed of sound \(c_{0}\) of air is assumed as 343 m/s. The plate with structural damping coefficient is taken as 0.01. Rayleigh integral is applied to determine acoustic power calculation and ANSYS is used as a tool for numerical simulation. In this paper, we considered a plate with outer radius 'a' and inner radius 'b' as shown in \[\mathrm{P}\left(l_{p}\right)=\int_{S_{s}}\left(\frac{\partial q}{\partial f} \left(l_{p},l_{s}\right)P\left(l_{p}\right)-\frac{\partial P}{\partial f} \left(l_{s}\right)q\left(l_{p},l_{s}\right)\right)ds\left(l_{s}\right) \tag{3}\]1  
In this numerical simulation processes, a comparison for the effect of natural frequency parameter, effect of sound power level, effect of average radiation efficiency, and effect of peak sound power level for different cases of parabolic tapered plate with different combinations of point masses are obtained. For all cases of tapered plate with point masses, the mass of the plate is kept constant. The out of plane (m, n)th modes are considered and the different taper ratios are varied in the range of 0.00-0.75 for this process of simulation. The plate is clamped at inner and free at the outer boundary.