## 4 Result and Discussion
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{**Mode**} & \multicolumn{4}{c}{**Natural Frequency Parameter, \(\lambda^{2}\)**} \\ \cline{3-6}  & & \(\mathbf{T_{x}=0.00}\) & \(\mathbf{T_{x}=0.25}\) & \(\mathbf{T_{x}=0.50}\) & \(\mathbf{T_{x}=0.75}\) \\ \hline \multirow{4}{*}{I} & (0,0) & 13.4852 & 12.9907 & 12.4701 & 11.9301 \\  & (0,1) & 13.4902 & 12.9765 & 12.4443 & 11.8937 \\  & (0,2) & 14.1025 & 13.6040 & 13.0838 & 12.5412 \\  & (0,3) & 16.6635 & 16.0712 & 15.4619 & 14.8258 \\ \hline \multirow{4}{*}{II} & (0,0) & 13.4842 & 12.8924 & 12.2715 & 11.6202 \\  & (0,1) & 13.4902 & 12.8805 & 12.2418 & 11.5762 \\  & (0,2) & 14.1035 & 13.5102 & 12.8858 & 12.2304 \\  & (0,3) & 16.6638 & 15.9618 & 15.2302 & 14.4612 \\ \hline \multirow{4}{*}{III} & (0,0) & 13.4852 & 13.4820 & 13.4850 & 13.4852 \\  & (0,1) & 13.4906 & 13.4853 & 13.4902 & 13.4902 \\ \cline{1-1}  & (0,2) & 14.1037 & 14.1008 & 14.1022 & 14.1032 \\ \cline{1-1}  & (0,3) & 16.6638 & 16.6612 & 16.6639 & 16.6635 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Numerical comparison of different frequency parameter \(\lambda^{2}\) with different modes of plate with two point masses combinations for different parabolic thickness variations and for different taper parameters (T\({}_{x}\)). Acoustic Radiation of Tapered Annular Circular Plate with Different Attachment of Point Masses with Different Taper Ratios  
In this numerical simulation processes, the sound power level (dB, reference = 10\({}^{-12}\) watts) of annular circular plate with different combinations of point masses is considered. The plate with different parabolic thickness variation is analyzed due to transverse vibration. The different taper ratios are taken as range from (0.00-0.25). A concentrated load of 1N is considered under time-varying harmonic excitations which are acted at different excitation location at different nodes. A harmonic frequency range of 0-8000 Hz is taken to determine the sound radiation characteristic. We considered the Case I plate with parabolic decreasing thickness variation as a convergence study. Figure 7 compares analytically and numerically the sound power level for Case I plate with four point masses combination for taper ratio, T\({}_{\mathrm{x}}\) = 0.75 and for different modes. On comparison of sound power, we observed a good agreement of computed results as depicted from Figure 7. In this numerical simulation, the numerical comparison of sound power level for Case I plate with different combinations of point masses for different taper ratios are reported in Figures 8-10. From Figures 8-10, it is investigated that for sound power level up to 20 dB, we do not get any design options for different taper ratios for plate with both one point mass and for two point masses combinations. However, for four point masses combination, we do not find any sound power level upto 30 dB. However, for sound power level up to 30 dB, we get all taper ratios, T\({}_{\mathrm{x}}\) = 0.00, 0.25, 0.50, and 0.75 as design options in frequency bands A and B for plate with one point mass combinationas reported in Figure 8. It is noteworthy that, for sound power level up to 50 dB, we get more design options for sound power levels in different frequency bands, i.e., C, D, and E as reported in Figure 8. From Figure 9, it is apparent that for sound power level up to 30 dB, then in frequency band A only taper ratio T\({}_{\mathrm{x}}\) = 0.00, 0.25, 0.50, and 0.75 are available design alternative for plate with two point mass combination. However, for sound power level up to 50 dB, we get wider frequency bands, B, C, and D for different taper ratios as reported in Figure 9. From Figure 10, it is investigated that for sound power level up to 40 dB is possible only in frequency bands A, B and C only with all taper ratios T\({}_{\mathrm{x}}\) = 0.00, 0.25, 0.50, and 0.75 and therefore is the available design alternative for plate with four point mass combination.