# Transverse Wave Propagation in a Thin Isotropic Plate Part I
## 3 Solution and Discussion
### Kirchhoff's Model  
_Results of the Solution_  
The results of the transverse wave displacements in the direction of the \(x\), \(y\), \(z\) axes--see Figures 7 and 8.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Name** & **Notation** & **Unit** & **Value** \\ \hline Young’s module & \(E\) & Pa & \(7.1\times 10^{10}\) \\ Poisson’s ratio & \(\mu\) & - & 0.34 \\ Density & \(\rho\) & kg\(\cdot\)m\({}^{-3}\) & 2699 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material parameters of aluminum plate.  
Figure 6: Solved model. \(C_{R}\)—impact center, T\({}_{1}\), T\({}_{2}\)—observed points, \(a\)—plate length, \(b\)—plate width, \(h\)—plate thickness.  
Figure 7 and Table 2 show that the difference in wave propagation in the \(x\) and \(y\) axis directions is insignificant (T\({}_{1}\) and T\({}_{2}\)). The course in both directions shows a very good agreement, as shown in Figure 7. The board material has identical properties in both directions (it is isotropic). Local extremes were reached at times \(t_{1}\), \(t_{2}\), \(t_{3}\)--see Table 2, where the difference in individual times is max. 3.1%.  
The propagation of the wave in the \(z\)-axis shows a high agreement when viewed from the direction of the \(x\) and \(y\) axes, see Table 3. The extremes that were achieved at point T\({}_{1}\) and T\({}_{2}\) are max. 0.4%.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Mark**} & \multirow{2}{*}{**Time**} & \multicolumn{2}{c}{**Displacement**} & \multicolumn{2}{c}{**Comparison**} \\  & & \(u\) **(T\({}_{2}\))** & \(v\) **(T\({}_{1}\))** & **Col. 4 – Col. 3** & \multicolumn{2}{c}{Col. 5\(\cdot 100\)} \\ \hline
1 & 2 & 3 & 4 & 5 & 6 \\  & [s] & [m] & [m] & [m] & [\%] \\ \(t_{1}\) & 3.000 \(\times\)\(10^{-6}\) & 2.383 \(\times\)\(10^{-11}\) & 2.362 \(\times\)\(10^{-11}\) & \(-\)0.021 \(\times\)\(10^{-11}\) & 0.9 \\ \(t_{2}\) & 3.900 \(\times\)\(10^{-6}\) & \(-\)1.336 \(\times\)\(10^{-11}\) & \(-\)1.294 \(\times\)\(10^{-11}\) & 0.042 \(\times\)\(10^{-11}\) & 3.1 \\ \(t_{3}\) & 5.400 \(\times\)\(10^{-6}\) & 4.647 \(\times\)\(10^{-11}\) & 4.716 \(\times\)\(10^{-11}\) & 0.069 \(\times\)\(10^{-11}\) & 1.5 \\ \hline \end{tabular}
\end{table}
Table 2: Displacement of wave in \(x\) (\(u\)) and \(y\) (\(v\)) axis direction in points T\({}_{1}\), T\({}_{2}\) ([PERSON]).  
Figure 8: Displacement in the direction of the \(z\) axes for point T\({}_{1}\) and T\({}_{2}\) as a function of time (Kirchhoff).  
Figure 7: Displacement in the direction of the \(x\) and \(y\) axes for point T\({}_{1}\) and T\({}_{2}\) as a function of time (Kirchhoff).  
Similar results were obtained for the wave propagation velocity, see Figure 9. After reaching the reverse side of the plate, the wave reflected from the edge, manifested in the oscillation of the velocity trend after time \(t_{3}\).  
The normal stress trend is also smooth until the reflected wave is captured. It is obvious that the normal stress shows a very good agreement in both directions, see Figure 10. Small inaccuracies before time \(t_{1}\) are probably caused by inaccuracies in the Bessel function calculation.