## 3 Results and Discussion
### Validation of Natural Frequency Parameter and Acoustic Power Calculation
It is interesting to note that a plate with 4 rectangular stiffener patches combination shows the lowest peak sound power level among all cases of thickness variations, and the lowest peak sound of 77 dB is obtained for Case III plate whereas the highest peak sound power level of 84 dB is obtained for a Case II plate with 2 rectangular stiffener patches combination. Similar effect is again observed for plate with concentric stiffener patch combination. The lowest sound power of 76 dB is observed for the plate with the 4 concentric stiffener patches combination, and the highest power of 83 dB is observed for plate with 1 concentric stiffener patch combination. Figures 16-21 shows the numerical comparison of sound power levels for Case I, Case II and Case III plates for different combinations of rectangular and concentric stiffener patches for taper ratio T\({}_{\text{x}}\) = 0.75. From Figures 16-21, it is observed that for all cases of thickness variation and for excitation frequency up to 2000 HZ, different parabolic thickness variation does not have any significant effect on sound power radiation. Furthermore, from Figures 16-18 it is seen that beyond excitation frequency of 2000 HZ and up to the first peak, a Case II plate with 2 rectangular stiffener patches shows the highest radiation power of 84 dB in comparison to a radiation power of 82 dB for a Case I plate with 1 rectangular stiffener patch combination. However, at this forcing frequency of 2000 HZ case III plate remains unaffected and shows the lowest peak sound level for all cases of thickness variations and so it is suggested that Case III plate is the lowest sound power radiator among all cases of thickness variation with different combinations of rectangular stiffener patches. Again, a similar effect is observed for plate with the concentric stiffener patches combination. Beyond 200 HZ, Case II plate with 2 concentric stiffener patches is a very good sound radiator of sound power 83 dB in comparison to 82 dB of plate with 1 stiffener patch combination. A Case III plate with all combination of concentric stiffener patches is found to be a poor sound radiator. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Type of**} & \multirow{2}{*}{**Plate Thickness**} & \multirow{2}{*}{**Unloaded Plate Thickness**} & \multicolumn{2}{c}{**Plate with 1**} & \multicolumn{2}{c}{**Plate with 2**} & \multicolumn{2}{c}{**Plate with 4**} \\
**Stiffener** & & **Variation for** & **Variation for** & **Target Ratio T\({}_{\star}\) = 0.75** & **Rectangular/Concentric** & **Rectangular/Concentric** & **Rectangular/Concentric** \\
**Patches** & **Target Ratio** & **Target Ratio** & **Sound Power** & **Radiation** & **Sound Power** & **Radiation** & **Sound Power** & **Radiation** & **Sound Power** & **Radiation** \\
**T\({}_{\star}\)** & **0.75** & **Level (dB)** & **Efficacy (\({}_{\text{min}}\))** & **Level (dB)** & **Efficacy (\({}_{\text{min}}\))** & **Level (dB)** & **Efficacy (\({}_{\text{min}}\))** & **Level (dB)** & **Efficacy (\({}_{\text{min}}\))** \\ \hline Rectangular & Case I & 83 & 1.079 & 82 & 1.058 & 81 & 1.048 & 79 & 1.015 \\ Stiffener & Case II & 85 & 1.135 & 81 & 1.045 & 84 & 1.094 & 78 & 1.002 \\ Case II & 79 & 1.020 & 79 & 1.020 & 78 & 1.007 & 77 & 0.994 \\ \hline Concentric & Case I & 83 & 1.079 & 82 & 1.058 & 81 & 1.048 & 77 & 0.994 \\ Stiffener & Case II & 85 & 1.135 & 81 & 1.045 & 83 & 1.079 & 79 & 1.020 \\ Case III & 79 & 1.020 & 78 & 1.020 & 78 & 1.007 & 76 & 0.935 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Numerical comparison of peak sound power level and radiation efficiency for annular plate having different parabolically thickness variations with different attachments of rectangular and concentric stiffer patches for T\({}_{\star}\) = 0.75. Figure 16: Numerical comparison of sound power level for annular plate having parabolic decreasing thickness variation (case I) for different attachments of rectangular stiffener patches for taper ratio T\({}_{\star}\) = 0.75.