## 3 Results and Discussion
### Validation of Natural Frequency Parameter and Acoustic Power Calculation
Figure 19: Numerical comparison of sound power level for annular plate having parabolic decreasing thickness variations (Case I) for different combinations of concentric stiffener patches for taper ratio \(\rm T_{x}=0.75\). Figure 17: Numerical comparison of sound power level for annular plate having parabolic decreasing increasing thickness variation (Case II) for different attachments of rectangular stiffener patches for taper ratio \(\rm T_{x}=0.75\). Figure 18: Numerical comparison of sound power level for annular plate having parabolic increasing decreasing thickness variations (Case III) for different combinations of rectangular stiffener patches for taper ratio \(\rm T_{x}=0.75\). that both a plate with 4 rectangular and 4 concentric stiffener patches with all cases of thickness variation (Cases I, II and III) shows the least radiation efficiency for all combinations of rectangular stiffener patches, as evident from Table 7.Therefore, it is interesting to mention that a Case III plate shows the lowest radiation efficiency (\(\sigma_{\rm mn}\)) for all cases of thickness variations and is a poor radiation emitter among all the thickness variations with different combinations of rectangular and concentric stiffener patches. Figure 26 shows the numerical comparison of radiation efficiency for a plate with both 4 rectangular and 4 concentric stiffener patches for taper ratio T\({}_{\rm x}\) = 0.75. It is found that the plate shows almost the same radiation efficiency as depicted from Figure 26. Figure 27 shows the numerical comparison of the sound power level for a plate with 4 rectangular and 4 concentric stiffener patches for taper ratio T\({}_{\rm x}\) = 0.75. It is observed that both the plates show almost the same peak for taper ratio T\({}_{\rm x}\) = 0.75. Hence, the effect of stiffness variation along with the modes has negligible effect for both the combinations. Figure 23: Comparison of radiation efficiency (\(\sigma_{\rm mn}\)) analytically and numerically for annular plate attached with 4 concentric stiffener patches and having parabolic decreasing thickness variations (Case I) for taper ratio T\({}_{\rm x}\) = 0.75. Figure 22: Comparison of radiation efficiency (\(\sigma_{\rm mn}\)) analytically and numerically for annular plate attached with 4 rectangular stiffener patches and having parabolic decreasing thickness variations (Case I) for taper ratio T\({}_{\rm x}\) = 0.75. Figure 24: Numerical comparison of radiation efficiency (\(\sigma_{\rm mn}\)) for annular plate having parabolically decreasing thickness variations (Case I) with different attachment of (a) 1 rectangular stiffener patch (b) 2 rectangular stiffener patches (c) 4 rectangular stiffener patches for taper ratio T\({}_{\rm x}\) =0.75. Figure 16:0 Numerical comparison of radiation efficiency (\(\sigma_{\rm mm}\)) for annular plate having parabolically decreasing thickness variations (Case I) with different attachment of (a) 1 concentric patch (b) 2 concentric patches and (c) 4 concentric patches for taper ratio T\({}_{x}\) = 0.75.  
concentric stiffener patches attached to a plate is reported at the first peak which corresponds to (0, 0) mode of the plate. From Figure 28 it is seen that for a Case I plate with 1 rectangular stiffener patch combination, peak sound power level increases for increasing value of taper ratio whereas for 2 rectangular stiffener patches and 4 rectangular stiffener patches combinations, variations of peak sound power levels are observed for an increasing value of taper ratio. For a Case I plate, the maximum peak sound power level is obtained for taper ratio \(T_{x}\) = 0.75 for plate with 1 rectangular stiffener patch combination. Furthermore, it is seen that peak is minimum for taper ratio, \(T_{x}\) = 0.25 and maximum for taper ratio, \(T_{x}\) = 0.50 for plate with 4 rectangular stiffener patches combination, whereas for a plate with 2 rectangular stiffener patches combination, the peak is at a minimum for taper ratio \(T_{x}\) = 0.25 and maximum for taper ratio \(T_{x}\) = 0.75.