## 3 Results and Discussion
### Validation of Natural Frequency Parameter and Acoustic Power Calculation
Acoustic Response Solution of Tapered Annular Circular Plate with Different Combination of Rectangular and Concentric Stiffener Patches with Different Taper Ratios  
In this paper, the sound power level (dB, reference = 10\({}^{-12}\) watts) of an annular circular plate with a different attachment of rectangular and concentric stiffener patches is estimated. The plate for the sound power level is analyzed for all cases of different parabolic thickness variation due to transverse vibration. The taper ratio is maintained from a range (0.00-0.75). The sound power level is investigated by applying 1 N concentrated load under time-varying harmonic excitations at different excitation locations at different nodes, and a harmonic frequency range of (0-8000) HZ is taken to determine the sound radiation characteristic. The Case I plate with parabolic decreasing thickness variation is taken as a convergence study. Figures 8 and 9 compare the sound power level for a Case I plate obtained analytically and numerically for the taper ratio T\({}_{\rm x}\) = 0.75 for 4 rectangular stiffener patches and 4 concentric stiffener patches, respectively, for different modes. A good agreement of computed results is seen in the comparison of sound power as depicted from Figures 8 and 9. Figures 10-12 shows the numerical comparison of sound power level for Case I plate with different combinations of rectangular stiffener patches for different taper ratios and for different modes under forced excitation. From Figure 10, a sound power level up to 30 dB is seen, and we do not get any broad range of frequencies for different taper ratios for plate with 1 rectangular stiffener patch. However, for a sound power level up  
Figure 10: Numerical comparison of sound power level for annular plate attached with 1 rectangular stiffener patch and having parabolic decreasing thickness variations (Case I) for different taper ratios \(\rm T_{x}\). Figure 9: Comparison of sound power level analytically and numerically for annular plate attached with 4 concentric stiffener patches and having parabolic decreasing thickness variations (Case I) for taper ratio \(\rm T_{x}\) = 0.75.  
bands A only with all taper ratios, T\({}_{\rm x}\) = 0.00, 0.25, 0.50 and 0.75 and, therefore, this is the available design alternative. However, for a sound power level up to 40 dB, we get more design options for the sound power level in different frequency bands, i.e., B, C and D as reported in Figure 12. Furthermore, Figures 13-15 show the numerical comparison of the sound power level for a Case I plate with different combinations of concentric stiffener patches for different taper ratios. From Figure 13, it is seen that for a sound power level up to 30 dB, we do not get any design options for different taper ratios for the plate with both 1 concentric stiffener patch and 4 concentric stiffener patches; and for 2 concentric stiffener patches, we do not find any sound power level upto 10 dB. However, for a sound power level up to 40 dB, we get all taper ratios, T\({}_{\rm x}\) = 0.00, 0.25, 0.50 and 0.75 as design options in frequency bands A and B for plate with 1 concentric stiffener patch combinations reported in Figure 13. It is noteworthy that for a sound power level up to 50 dB, we get more design options for the sound power level in different frequency bands, i.e., C, D and E as reported in Figure 13. From Figure 14, it is apparent that for a sound power level up to 20 dB, then in frequency band A only taper ratio T\({}_{\rm x}\) = 0.00, 0.25, 0.50 and 0.75 are available design alternatives for a plate with the 2 concentric stiffener patches combination. But for sound power level up to 30 dB, we get wider frequency bands, B, C and D for different taper ratios as reported in Figure 14. From Figure 15, it is seen that for a sound power level up to 40 dB is possible only in frequency bands A only with all taper ratios, T\({}_{\rm x}\) = 0.00, 0.25, 0.50 and 0.75 and, therefore, this is the available design alternative for plate with 4 concentric stiffener patches combination.