## 6 Examples  
### Love-Wave Case  
First, a two-layer model (Model Nov71 in Table 4) is chosen to demonstrate the validity of the proposed algorithms and of the developed code for the case of Love waves. Figure 2 shows the computational results for Model Nov71. Although the package is capable of modeling multiple modes, we only show the results of the fundamental mode for comparison with the published studies [30; 31]. It is noteworthy that, in Figure 2 we plot curves of \(10\times\partial c/\partial h\) and \(10\times\partial U/\partial h\) instead of their original values due to their large differences in quantitative values compared to the other derivatives, so to make the changes of these two curves recognizable. In addition, in the work of [PERSON] [31], to facilitate the derivation of the analytical formulism specifically for the case of two layers, he introduced adimensionless quantity \(\rho=\rho_{2}/\rho_{1}\) and plotted the curves of \(\partial c/\partial\rho\) and \(\partial U/\partial\rho\). Therefore, to obtain the same curves, we have made use of the chain rule:  
\[\partial c/\partial\rho=\frac{\partial c}{\partial\rho_{2}}\frac{\partial\rho _{2}}{\partial\rho}=\rho_{1}\frac{\partial c}{\partial\rho_{2}}\text{ and }\partial U/\partial\rho=\frac{\partial U}{\partial\rho_{2}}\frac{ \partial\rho_{2}}{\partial\rho}=\rho_{1}\frac{\partial U}{\partial\rho_{2}}. \tag{17}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Layer** & **S-Wave Velocity (m-s\({}^{-1}\))** & **Density (kg\(\cdot\)m\({}^{-3}\))** & **Thickness (m)** \\ \hline \multicolumn{4}{c}{**Model Nov71**} \\ \multicolumn{4}{c}{**Two layered model in Novotny (1971) [31]**} \\ \(1\) & 3500 & 2700 & 35,000 \\ \(2\) & 4500 & 3300 & \(\infty\) \\ \hline \multicolumn{4}{c}{**Model Nov70**} \\ \multicolumn{4}{c}{**The model of the Canadian shield CANSD in Novotny (1970) [30]**} \\ \(1\) & 3470 & 2700 & 6000 \\ \(2\) & 3640 & 2800 & 10,500 \\ \(3\) & 3850 & 2850 & 18,700 \\ \(4\) & 4720 & 3300 & 80,000 \\ \(5\) & 4540 & 3440 & 100,000 \\ \(6\) & 4510 & 3530 & 100,000 \\ \(7\) & 4760 & 3600 & 80,000 \\ \(8\) & 5120 & 3760 & \(\infty\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Two models chosen for validation of the package. Figure 2: _Cont._  
Obviously, the computational results in Figure 2 are rather consistent with the work of [PERSON] [31] (see Figures 1-3 of his paper). To verify the correctness of these results more clearly, the relevant quantities calculated at 20, 30, and 40 s are shown in Table 5 (related to phase velocity) and in Table 6 (related to group velocity), respectively. As shown in the Tables, they are in good agreement with the outcomes from [PERSON] [31] (see Tables 1 and 2 of his paper). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**T (s)** & \multicolumn{2}{c}{\(U\) (m-s\({}^{-1}\))} & \multicolumn{2}{c}{\(\partial U\)/\(\partial\beta_{1}\)} & \multicolumn{2}{c}{\(\partial U\)/\(\partial\beta_{2}\)} \\ \hline  & **[PERSON]** & **This paper** & **[PERSON]** & **This paper** & **[PERSON]** & **This paper** \\ \hline
20 & 3384.38 & 3384.3839 & 1.16642 & 1.1664245 & \(-\)0.14784 & \(-\)0.1478377 \\
30 & 3489.61 & 3489.6075 & 1.42265 & 1.4226530 & \(-\)0.20843 & \(-\)0.2084253 \\
40 & 3706.66 & 3706.6645 & 1.36767 & 1.3676722 & \(-\)0.03879 & \(-\)0.0387899 \\ \hline  & & & \(\partial U\)/\(\partial h\) & \multicolumn{2}{c}{\(\partial U\)/\(\partial\rho\)} \\ \cline{3-6}  & & **[PERSON]** & **This paper** & **[PERSON]** & **This paper** \\ \hline
20 & & \(-\)0.00094 & \(-\)0.0009381 & \(-\)0.08530 & \(-\)0.0853015 \\
30 & & \(-\)0.01576 & \(-\)0.0157647 & 0.02334 & 0.0233376 \\
40 & & \(-\)0.02588 & \(-\)0.0258753 & 0.25547 & 0.2554651 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Computational results relevant to group velocities for Model Nov71.