# The Effects of Stress on Second Harmonics in Plate-Like Structures
## 3 Analytical Method of Second Harmonics Under Stress Change
### Analysis of Second Harmonics in Plate
Compared with the data in Table 2, it can be found that, in the same propagation distance, except for mode pairs 8 and 10, other mode pairs meet the rule that the larger the value of \(f_{n}{}^{surf}+f_{n}{}^{vol}\) is, the larger the amplitude of second harmonic  
Figure 4: Group velocity dispersion curves of isotropic steel plate. Figure 3: Phase velocity dispersion curves of isotropic steel plate. mode is. From the perspective of energy transfer, \(f_{n}\,{}^{surf}+f_{n}\,{}^{vol}\) represents the amount of energy transferred from the primary excitation mode to second harmonic mode. The larger \(f_{n}\,{}^{surf}+f_{n}\,{}^{vol}\) is, the greater the energy transferred is, and the larger the amplitude of second harmonic mode should be. The value of \(f_{n}\,{}^{surf}+f_{n}\,{}^{vol}\) of mode pair 10 is not the largest, but the amplitude of second harmonic mode is the largest, thus testifying that the amplitude of the second harmonic mode is not completely dependent on the value of \(f_{n}\,{}^{surf}+f_{n}\,{}^{vol}\). In fact, the amplitude of second harmonic mode is related to the values of \(f_{n}\,{}^{surf}+f_{n}\,{}^{vol}\) and \(P_{mn}\), so the optimal mode pair cannot be selected according to the value of \(f_{n}\,{}^{surf}+f_{n}\,{}^{vol}\) alone. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \begin{tabular}{c} **Number of** \\ **Mode Pair** \\ \end{tabular} & \begin{tabular}{c} **Frequency-Thickness** \\ **_fd(MHz-mm)_** \\ \end{tabular} & \begin{tabular}{c} **Phase Velocity** \\ **(m/s)** \\ \end{tabular} & \begin{tabular}{c} **Group Velocity** \\ **(m/s)** \\ \end{tabular} & \
\begin{tabular}{c} **Modal** \\ **Shape** \\ \end{tabular} &
\begin{tabular}{c} \(f_{n}\,{}^{surf}\)**+\(f_{n}\,{}^{vol}\)**(\(\times\)10\({}^{25}\)) \\ \end{tabular} \\ \hline \multirow{2}{*}{1} & 37.5 & 3144 & 3013 & S & 0 \\ \cline{2-6}  & 75 & 3141 & 2771 & A & \\ \hline \multirow{2}{*}{2} & 37.5 & 3144 & 3013 & S & 0 \\ \cline{2-6}  & 75 & 3144 & 3505 & A & \\ \hline \multirow{2}{*}{3} & 37.5 & 3154 & 1372 & A & 0 \\ \cline{2-6}  & 75 & 3160 & 2741 & A & \\ \hline \multirow{2}{*}{4} & 37.5 & 3171 & 1294 & S & 0 \\ \cline{2-6}  & 75 & 3180 & 2617 & A & \\ \hline \multirow{2}{*}{5} & 37.5 & 3195 & 1361 & S & 0 \\ \cline{2-6}  & 75 & 3206 & 2706 & A & \\ \hline \multirow{2}{*}{6} & 37.5 & 5904 & 2132 & A & 0 \\ \cline{2-6}  & 75 & 5908 & 4917 & A & \\ \hline \multirow{2}{*}{7} & 37.5 & 3171 & 1294 & S & 10.91 \\ \cline{2-6}  & 75 & 3169 & 2744 & S & \\ \hline \multirow{2}{*}{8} & 37.5 & 3144 & 3013 & S & 12.97 \\ \cline{2-6}  & 75 & 3147 & 2765 & S & \\ \hline \multirow{2}{*}{9} & 37.5 & 3195 & 1361 & S & 145.63 \\ \cline{2-6}  & 75 & 3192 & 3142 & S & \\ \hline \multirow{2}{*}{10} & 37.5 & 3933 & 1088 & A & 195.04 \\ \cline{2-6}  & 75 & 3920 & 2492 & S & \\ \hline \multirow{2}{*}{11} & 37.5 & 3448 & 1271 & A & 262.33 \\ \cline{2-6}  & 75 & 3458 & 2331 & S & \\ \hline \multirow{2}{*}{12} & 37.5 & 3227 & 1343 & A & 288.42 \\ \cline{2-6}  & 75 & 3223 & 2728 & S & \\ \hline \multirow{2}{*}{13} & 37.5 & 3267 & 3013 & S & 479.75 \\ \cline{2-6}  & 75 & 3263 & 2644 & S & \\ \hline \multirow{2}{*}{14} & 37.5 & 3316 & 1303 & A & 695.08 \\ \cline{2-6}  & 75 & 3315 & 2581 & S & \\ \hline \multirow{2}{*}{15} & 37.5 & 3376 & 1275 & S & 834.27 \\ \cline{2-6}  & 75 & 3382 & 2488 & S & \\ \hline \multirow{2}{*}{16} & 50 & 3170 & 1828 & S & 12.19 \\ \cline{2-6}  & 75 & 3169 & 2744 & S & \\ \hline \multirow{2}{*}{17} & 37.5 & 3535 & 1349 & S & 27.32 \\ \cline{2-6}  & 75 & 3263 & 2644 & S & \\ \hline \multirow{2}{*}{18} & 37.5 & 3641 & 1176 & A & 40.87 \\ \cline{2-6}  & 75 & 3315 & 2581 & S & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of mode pairs. In Figure 6, the values of \(f_{n}\mathit{surf}+f_{n}\mathit{vol}\) of mode pairs 16-18 are not zero, so the energy transferred from the primary excitation modes to second harmonic modes are not zero.