# Transverse Wave Propagation in a Thin Isotropic Plate Part I
## 3 Solution and Discussion
### Comparison of [PERSON]'s and [PERSON]'s Models  
[PERSON]'s geometrical model, unlike [PERSON]'s geometrical model, respects the influence of the cross-section rotation, as mentioned above. Since the same material model ([PERSON]'s) was used for both models in the analytical calculation, it is appropriate to perform a mutual comparison of displacements, velocities and stresses. Figure 14: Normal stress time trends in \(x\) and \(y\) axis direction for points T\({}_{1}\) and T\({}_{2}\) (Rayleigh). Figure 13: Velocity time trend in z axis direction for point T\({}_{1}\) and T\({}_{2}\) (Rayleigh). Figure 15a shows a comparison of the course of the transverse displacement in the direction of the y-axis at the point T\({}_{2}\) at time \(t_{1},t_{2},t_{3}\). From the course of both displacements, it is evident that the differences are relatively significant and range from 55.1 to 107.7% (see Table 5). This is probably due to the use of a improved model (Rayleigh). Similarly, the deviations of the transverse wave in the x-axis (see Figure 15b) are significantly different for both models at different times and range from 6.3 to 90.4%, see Table 6.  
With the displacement in the \(z\)-axis direction (Figure 16), it is evident that for both models, the agreement is very good for longer times (\(t_{2},t_{3}\)), and difference is up to 10.6%. The bigger difference is only in the short time \(t_{1}\), where it is about 58%. At both points, these differences are always almost identical (see Tables 7 and 8). The larger difference in time \(t_{1}\) is probably due to the corrections used in the [PERSON] model. Thus, it can be said that [PERSON]'s model is more accurate--it shows a faster response to impact (the same in both directions). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Mark**} & \multirow{2}{*}{**Time**} & \multicolumn{2}{c}{**Displacement**} & \multicolumn{2}{c}{**Comparison**} \\  & & **[PERSON]** & **[PERSON]** & **Col. 4 – Col. 3** & \multicolumn{2}{c}{Col. 5} \\ \hline
1 & 2 & 3 & 4 & 5 & 6 \\  & \(10^{-6}\) [s] & \(10^{-10}\) [m] & \(10^{-10}\) [m] & \(10^{-10}\) [m] & [\%] \\ \(t_{1}\) & 3.000 & \(0.125\) & \(0.238\) & \(0.110\) & 90.4 \\ \(t_{2}\) & 3.900 & \(-0.708\) & \(-1.246\) & \(-0.538\) & 75.9 \\ \(t_{3}\) & 5.400 & \(4.373\) & \(4.647\) & 0.274 & 6.30 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison \(u\) displacements in \(x\)-axis direction as a function of time for [PERSON]’s and [PERSON]’s model at point T\({}_{2}\). Figure 15: Comparison of displacements \(u\), \(v\) in the direction of the \(x\) and \(y\) axes at the point T\({}_{1}\) (**a**) and T\({}_{2}\) (**b**) as a function of time for the [PERSON] and Rayleigh model. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Mark**} & \multirow{2}{*}{**Time**} & \multicolumn{2}{c}{**Displacement**} & \multicolumn{2}{c}{**Comparison**} \\  & & **[PERSON]** & **[PERSON]** & **Col. 4 – Col. 3** & \multicolumn{2}{c}{Col. 5 -100} \\ \hline
1 & 2 & 3 & 4 & 5 & 6 \\  & \(10^{-6}\) [s] & \(10^{-10}\) [m] & \(10^{-10}\) [m] & \(10^{-10}\) [m] & \([\%]\) \\ \(t1\) & 4.000 & 0.128 & 0.054 & \(-0.074\) & \(57.8\) \\ \(t2\) & 6.500 & \(-0.686\) & \(-0.613\) & \(0.073\) & \(10.6\) \\ \(t3\) & 16.800 & 6.939 & 6.830 & \(-0.109\) & \(1.60\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison of displacement \(w\) in the \(z\) axis as a function of time for [PERSON]’s and [PERSON]’s model at point T\({}_{1}\). Figure 16: Comparison of displacement \(w\) in the direction of the \(z\)-axis at point T\({}_{1}\) (**a**) and T\({}_{2}\) (**b**) as a function of time for [PERSON] and [PERSON] model.