## 3 Results and Discussion
### Validation of Natural Frequency Parameter and Acoustic Power Calculation
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & & & & & & & \\ \cline{3-8}
**Case** & **Mode** & \multicolumn{3}{c}{\(\mathbf{T_{x}=0.00}\)} & \multicolumn{3}{c}{\(\mathbf{T_{x}=0.25}\)} & \multicolumn{3}{c}{\(\mathbf{T_{x}=0.50}\)} & \multicolumn{3}{c}{\(\mathbf{T_{x}=0.75}\)} \\ \cline{3-8}  & & **Rectangular** & **Concentric** & **Rectangular** & **Concentric** & **Rectangular** & **Concentric** & **Rectangular** & **Concentric** \\  & & **[PERSON]** & **[PERSON]** & **[PERSON]** & **[PERSON]** & **[PERSON]** & **[PERSON]** & **[PERSON]** \\ \hline
1 & (0,0) & 13.4819 & 13.4223 & 12.9912 & 12.9212 & 12.4711 & 12.2552 & 11.9311 & 11.8025 \\  & (0,1) & 13.4957 & 13.4842 & 12.9759 & 13.0540 & 12.4449 & 12.7821 & 11.8929 & 12.1634 \\  & (0,2) & 14.1148 & 14.0015 & 13.6148 & 13.8225 & 13.0942 & 13.3244 & 12.5513 & 13.0512 \\  & (0,3) & 16.6638 & 16.6014 & 16.0741 & 16.1024 & 15.4619 & 15.6012 & 14.8262 & 15.1020 \\ \hline II & (0,0) & 13.4819 & 13.4223 & 12.8960 & 12.8021 & 12.2729 & 12.3230 & 11.6210 & 11.6254 \\  & (0,1) & 13.4957 & 13.4842 & 12.8799 & 12.9523 & 12.2424 & 12.5422 & 11.5764 & 11.7651 \\  & (0,2) & 14.1148 & 14.0015 & 13.5519 & 13.5535 & 12.8951 & 12.8641 & 12.2392 & 12.6821 \\  & (0,3) & 16.6638 & 16.6014 & 15.9622 & 16.0022 & 15.2286 & 15.4021 & 14.4611 & 14.9248 \\ \hline III & (0,0) & 13.4819 & 13.4223 & 13.4891 & 13.3844 & 13.4917 & 13.4012 & 13.4914 & 13.4030 \\  & (0,1) & 13.4957 & 13.4842 & 13.4791 & 13.4214 & 13.4816 & 13.4414 & 13.4342 \\  & (0,2) & 14.1148 & 14.0015 & 14.1116 & 14.0025 & 14.1142 & 14.0521 & 14.1139 & 14.0614 \\  & (0,3) & 16.6638 & 16.6014 & 16.6601 & 16.5254 & 16.6632 & 16.5528 & 16.6629 & 16.5403 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Numerical comparison of natural frequency parameter \(\lambda^{2}\) of a plate with 1 rectangular and 1 concentric stiffener patch for different thickness variations and for different taper ratios \(\mathbf{T_{x}}\). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & \multicolumn{6}{c}{**Natural Frequency Parameter, \(\lambda^{2}\)**} \\ \cline{3-10}
**Case** & **Mode** & \multicolumn{2}{c}{\(\mathbf{T_{x}=0.00}\)} & \multicolumn{2}{c}{\(\mathbf{T_{x}=0.25}\)} & \multicolumn{2}{c}{\(\mathbf{T_{x}=0.50}\)} & \multicolumn{2}{c}{\(\mathbf{T_{x}=0.75}\)} \\ \cline{3-10}  & & **Rectangular** & **Concentric** & **Rectangular** & **Concentric** & **Rectangular** & **Concentric** & **Rectangular** & **Concentric** \\  & & **Stiffener** & **Stiffener** & **Stiffener** & **Stiffener** & **Stiffener** & **Stiffener** & **Stiffener** & **Stiffener** \\ \hline
**1** & (0,0) & 13.4862 & 13.3625 & 12.9886 & 12.8938 & 12.4699 & 12.4829 & 11.9583 & 12.0021 \\  & (0,1) & 13.4937 & 13.4421 & 12.9782 & 12.9436 & 12.4463 & 12.5335 & 11.9217 & 12.0554 \\  & (0,2) & 14.1099 & 14.0275 & 13.6108 & 13.9637 & 13.0904 & 13.2724 & 12.5774 & 13.0086 \\  & (0,3) & 16.6638 & 16.5028 & 16.0744 & 15.9829 & 15.4625 & 15.5421 & 14.8610 & 15.0227 \\ \hline
**II** & (0,0) & 13.4862 & 13.3665 & 12.8937 & 12.7717 & 12.2720 & 12.2929 & 11.6478 & 11.7631 \\  & (0,1) & 13.4937 & 13.4485 & 12.8810 & 12.8342 & 12.2435 & 12.3227 & 11.6042 & 11.9229 \\  & (0,2) & 14.1099 & 14.0267 & 13.5156 & 13.8732 & 12.9216 & 12.7811 & 12.2648 & 12.6456 \\  & (0,3) & 16.6638 & 16.5057 & 15.9625 & 15.9018 & 15.2646 & 15.2882 & 14.4954 & 14.7824 \\ \hline
**III** & (0,0) & 13.4862 & 13.3619 & 13.4831 & 13.3228 & 13.5170 & 13.3415 & 13.5245 & 13.3238 \\  & (0,1) & 13.4937 & 13.4467 & 13.4860 & 13.4039 & 13.5248 & 13.4262 & 13.5168 & 13.402472 \\  & (0,2) & 14.1099 & 14.0228 & 14.1070 & 13.9442 & 14.1421 & 13.9594 & 14.1421 & 13.9684 \\  & (0,3) & 16.6638 & 16.5025 & 16.6603 & 16.4311 & 16.7020 & 16.4435 & 16.7017 & 16.4224 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Numerical comparison of natural frequency parameter \(\lambda^{2}\) of plate with 2 rectangular and 2 concentric stiffener patches for different thickness variations and for different taper ratios \(\mathbf{T_{x}}\).