## 3 Results and Discussion
### Validation of Natural Frequency Parameter and Acoustic Power Calculation
However, for a sound power level up to 60 dB, we get a broader range of frequency denoted as B and C for all taper ratios as reported in Figure 15.  
It may be inferred from Figures 10-15 that a plate with different combinations of rectangular and concentric stiffener patches plays a significant role in sound power reduction in different frequency bands. A plate with 4 rectangular stiffener patches combination causes maximum sound power level reduction in comparison to 1 rectangular stiffener patch and 2 rectangular stiffener patch combinations for a Case I plate; whereas, for a plate with 4 concentric stiffener patches the lowest sound power is observed in comparison to other combinations. However, the stiffness contribution due to various taper ratios has a very limited impact on sound power level reduction in comparison to that of modes and excitation locations of plate with different combinations of rectangular and concentric stiffener patches. Furthermore, from Figures 10-15, it is observed that for an excitation frequency  
Figure 14: Numerical comparison of sound power level for annular plate attached with 2 concentric stiffener patches and having parabolic decreasing thickness variations (Case I) for different taper ratios \(\mathrm{T_{x}}\). Figure 15: Numerical comparison of sound power level for annular plate attached with 4 concentric stiffener patches and having parabolic decreasing thickness variations (Case I) for different taper ratios \(\mathrm{T_{x}}\). up to 2000 HZ, the effect of different combinations of rectangular and concentric stiffener patches and stiffness variation due to different taper ratios do not have a significant effect on sound power radiation for clamped-free boundary condition. However, when the excitation frequency increases beyond 2000 HZ and up to the first peak, the sound power level is higher for only higher taper ratios for a Case I plate with both 1 rectangular stiffener and 1 concentric stiffener patch, and variation of sound power level due to variation of peaks for different taper ratios is observed for a plate with both 2 rectangular stiffener and 2 concentric stiffener patches and for 4 rectangular stiffener and 4 concentric stiffener patch combinations. For a Case II plate, beyond a forcing frequency 2000 HZ, the highest sound power level is associated with plate for 2 rectangular stiffener patches combinations; while for a Case III plate, the sound power level is seen to decrease for all combinations of rectangular stiffener patches with increasing taper ratio. Similar effect is observed for plate with concentric stiffener patches where plate with 2 concentric stiffener patches is seen to have highest radiation power. Again for case III plate the sound power is seen to be decreased for all combination of concentric stiffener patches. Furthermore, different modes do influence the sound power peaks as evident from Figures 10-15. Sound power level peak obtained for different modes (0, 0) and (0, 1) almost remain same for different taper ratios. However, no such sound power similarity of modes (0, 0) and (0, 1) is observed for plates with concentric stiffener patches. The sound power level does shift towards a lower frequency with increasing taper ratio for all combinations of rectangular and concentric stiffener patches. For a higher frequency beyond 4000 HZ, it is observed that different taper ratios alter its stiffness at higher forcing frequency for different cases of thickness variation. It is noteworthy that for a higher frequency beyond 4000 HZ and up to 8000 HZ, a plate with different combination of rectangular and concentric stiffener patches alters its stiffness at higher forcing frequency and the acoustic power curve tends to intersect each other at this high forcing region. Table 7 compares the peak sound power level of a plate having different parabolically varying thickness with different combinations of rectangular and concentric stiffener patches for a taper ratio T\({}_{\text{x}}\) = 0.75.