# Transverse Wave Propagation in a Thin Isotropic Plate Part I
## 3 Solution and Discussion
### [PERSON]'s Model  
_Results of the Solution_  
As mentioned above, [PERSON] extended [PERSON]'s geometric model of the plate by the effect of the rotation of the cross sections of the plate. In the analytical solution of this model, the simplest material model was again used, i.e., [PERSON]'s model. Figure 11 shows the displacement in transverse wave in the x and y axis directions for point T\({}_{1}\) and T\({}_{2}\). \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**} \\  & & \(w\) (T\({}_{2}\)) & \(w\) (T\({}_{1}\)) & **Col. 4 – Col. 3** & \multicolumn{2}{c}{Col. 5 100} \\ \hline
1 & 2 & 3 & 4 & 5 & 6 \\  & [s] & [m] & [m] & [m] & [?o] \\ \(t_{1}\) & \(4.000\times 10^{-6}\) & \(1.284\times 10^{-10}\) & \(1.284\times 10^{-10}\) & \(0.000\times 10^{-10}\) & 0.0 \\ \(t_{2}\) & \(6.500\times 10^{-6}\) & \(-6.864\times 10^{-10}\) & \(-6.859\times 10^{-10}\) & \(0.005\times 10^{-10}\) & 0.1 \\ \(t_{3}\) & \(1.680\times 10^{-5}\) & \(6.966\times 10^{-10}\) & \(6.939\times 10^{-10}\) & \(-0.027\times 10^{-10}\) & 0.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Trend of wave in \(z\) (\(w\)) axis direction in point T\({}_{1}\), T\({}_{2}\) (Kirchhoff). Figure 10: Normal stress time trends in \(x\) and \(y\) axis direction for points T\({}_{1}\) and T\({}_{2}\) (Kirchhoff). Figure 9: Velocity time trend in \(z\) axis direction for point T\({}_{1}\) and T\({}_{2}\) (Kirchhoff). Figure 12 shows the course of the transverse displacement in the z-axis direction at points T\({}_{1}\) and T\({}_{2}\). In Table 4, the displacement by wave propagation in the z-axis direction at point T\({}_{1}\) and T\({}_{2}\) for times \(t_{1}\), \(t_{2}\) and \(t_{3}\) is evaluated numerically. Figure 12 shows both waveforms simultaneously. It follows from Table 4 that the difference between the two points is the largest at time \(t_{1}\), namely 3.4%. In the other two times, it is only 0.1%, which is a very good agreement. The differences can be attributed to the inaccuracies of the Bessel function calculation. In the Figure 13, z-axis velocity profile at the T\({}_{1}\) and T\({}_{2}\) points is presented. From Figure 13, it can be seen that after passing the transverse wave through the thickness of the plate, it is reflected \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**} \\  & & \(w\) (T\({}_{2}\)) & \(w\) (T\({}_{1}\)) & **Col. 4 – Col. 3** & \multicolumn{2}{c}{Col. 5} \\ \hline \multirow{2}{*}{\(1\)} & 2 & 3 & 4 & 5 & 6 \\  & [s] & [m] & [m] & [m] & [\%] \\ \(t_{1}\) & \(4.000\times 10^{-6}\) & \(5.259\times 10^{-11}\) & \(5.437\times 10^{-10}\) & \(0.178\times 10^{-10}\) & \(3.40\) \\ \(t_{2}\) & \(6.500\times 10^{-6}\) & \(-6.132\times 10^{-10}\) & \(-6.133\times 10^{-10}\) & \(-0.001\times 10^{-10}\) & \(0.1\) \\ \(t_{3}\) & \(1.680\times 10^{-5}\) & \(6.825\times 10^{-9}\) & \(6.830\times 10^{-9}\) & \(0.005\times 10^{-9}\) & \(0.1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Displacement by wave propagation in \(z\) (\(w\)) axis direction for points T\({}_{1}\) and T\({}_{2}\) (Rayleigh). Figure 11: Displacement time trends in the direction of the \(x\) and \(y\) axes for point T\({}_{1}\) and T\({}_{2}\) (Rayleigh). Figure 12: Displacement time trends in the direction of the \(z\) axis for point T\({}_{1}\) and T\({}_{2}\) (Rayleigh). from the back and thus the time trend is deformed. Figure 13 again shows a very good agreement of the velocity across the plate at point T\({}_{1}\) and T\({}_{2}\). Figure 14 shows the trend of normal stress in the \(x\) and \(y\) axis, i.e., \(\sigma_{y}\) at point T\({}_{1}\) (\(\sigma_{x}\) = 0) and \(\sigma_{x}\) at point T\({}_{2}\) (\(\sigma_{y}\) = 0). Figure 14 then compares stresses. The above comparison shows that the transverse wave propagates from the point of impact along circular paths.