# Transverse Wave Propagation in a Thin Isotropic Plate Part I
## 2 Materials and Methods
### Analytical Solution of Geometric Models of Thin Isotropic Plates
#### 2.2.1 [PERSON]'s Model
where angles of tangents rotation \(\varphi_{x}\) and \(\varphi_{y}\) \[\varphi_{x}=\tfrac{\partial w}{\partial x}=\tfrac{16}{abc}\tfrac{F_{0}}{\rho h} \sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\tfrac{h_{1}(\gamma_{mn}c)}{ \gamma_{mn}}\tfrac{a_{n}}{\omega_{mn}^{2}}\sin\ \alpha_{n}x_{F}\sin\ \beta_{m}y_{F}\cos\ \alpha_{n}x\ \sin\ \beta_{m}y\ \sin^{2\tfrac{\omega_{mn}}{2}}t\] (33) \[\varphi_{y}=\tfrac{\partial w}{\partial y}=\tfrac{\partial w}{\partial y}= \tfrac{16}{abc}\tfrac{F_{0}}{\rho h}\sum\limits_{m=1}^{\infty}\sum\limits_{n= 1}^{\infty}\tfrac{h_{1}(\gamma_{mn}c)}{\gamma_{mn}}\tfrac{\beta_{m}}{\omega_ {mn}^{2}}\sin\ \alpha_{n}x_{F}\sin\ \beta_{m}y_{F}\sin\ \alpha_{n}x\ \cos\ \beta_{m}y\ \sin^{2\tfrac{\omega_{mn}}{2}}t\] (34) \(\bullet\) The velocities \(i\), \(\dot{v}\), \(\dot{w}\) in the direction of the individual axes are obtained by deriving the displacements in these axes according to time: \[i = \frac{\partial u}{\partial t}\!=-\frac{8z}{abc}\frac{F_{0}}{\rho h }\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\frac{J_{1}(\gamma_{mn}c )}{\gamma_{mn}}\frac{\alpha_{n}}{\omega_{mn}}\sin\ \alpha_{n}x_{F}\sin\ \beta_{m}y_{F}\cos\ \alpha_{n}x\ \sin\ \beta_{m}y\ \sin\ \omega_{mn}t\] (35) \[\dot{v} = \frac{\partial v}{\partial t}\!=-\frac{8z}{abc}\frac{F_{0}}{\rho h }\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\frac{J_{1}(\gamma_{mn} c)}{\gamma_{mn}}\frac{\beta_{m}}{\omega_{mn}}\sin\ \alpha_{n}x_{F}\sin\ \beta_{m}y_{F}\sin\ \alpha_{n}x\ \cos\ \beta_{m}y\ \sin\ \omega_{mn}t\] (36) \[\omega_{mn}=\left(\alpha_{n}^{2}+\beta_{m}^{2}\right)\sqrt{\frac{ D}{\rho\,\hbar}}=\left(\alpha_{n}^{2}+\beta_{m}^{2}\right)\frac{c\,\hbar}{ \sqrt{12}}\ D=\frac{E\,\hbar^{3}}{12\,\left(1-\,\mu^{2}\right)}\]0 (37) \(\bullet\) Due to the deflection, which is caused by the impact of the load body on the plate, bending stresses also arise in the plate \(\sigma_{x}\), \(\sigma_{y}\), defined in form \[\sigma_{x}(x,y,z,t)=\tfrac{F_{2}}{1-\mu^{2}}\tfrac{16}{abc}\tfrac{F_{0}}{\rho h }\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\tfrac{h_{1}(\gamma_{mn} c)}{\gamma_{mn}}\tfrac{a_{n}^{2}+\mu_{0}^{2}}{\omega_{mn}^{2}  
* Displacements \(w\), \(u\), \(v\) in axis \(z\), \(x\), \(y\), respectively, are defined: \[ \begin{array}{c}w=w(x,y,t)=\frac{16z}{abc}\frac{F_{0}}{ph}\sum\limits_{m=1}^{ \infty}\sum\limits_{n=1}^{\infty}\sum\limits_{n=1}^{\infty}\frac{\frac{\hbar( \gamma_{mn}c)}{\gamma_{mn}}\frac{\sin\alpha_{n}x_{Y}\ \cos\ \beta_{m}y_{r}}{\left[\frac{\hbar^{2}}{2}(\alpha_{n}^{2}+\beta_{m}^{2})+1 \right]\omega_{mn}^{2}}\sin\ \alpha_{n}x\ \sin\ \beta_{m}y\cdot\\ \cdot\sin^{2}\frac{\alpha_{mn}}{\sqrt{\frac{\hbar^{2}}{2}}(\alpha_{n}^{2}+ \beta_{m}^{2})+1}\sqrt{\frac{\hbar^{2}}{2}}\ \mathrm{D}=\frac{\frac{\hbar}{2}\hbar^{3}}{12 \ (1-\mu^{-})}\\ \gamma_{mn}=\sqrt{\frac{\hbar^{2}}{2}\mu}\ \alpha_{n}=n\frac{\pi}{\hbar}\ \beta_{m}=m \frac{\pi}{\hbar}\\ \end{array} \] (43) \[ \begin{array}{c}u=-z\varphi_{x}=-\frac{16z}{abc}\frac{F_{0}}{ph}\sum \limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\frac{\frac{\hbar}{1}(\gamma_ {mn}c)\ \alpha_{n}\mathrm{sin\ }\alpha_{n}x_{Y}\ \sin\ \beta_{m}y_{r}}{\gamma_{mn}\left[\frac{\hbar^{2}}{2}(\alpha_{n}^{2}+\beta_{m}^ {2})+1\right]\omega_{mn}^{2}}\cos\ \alpha_{n}x\ \sin\ \beta_{m}y\ \sin^{2}\frac{\omega_{mn}}{2}t\\ v=-z\varphi_{y}=-\frac{16z}{abc}\frac{F_{0}}{ph}\sum\limits_{m=1}^{\infty}\sum \limits_{n=1}^{\infty}\frac{\frac{\hbar}{1}(\gamma_{mn}c)\beta_{m}\mathrm{ sin\ }\alpha_{n}x_{Y}\ \sin\ \beta_{m}y_{r}}{\gamma_{mn}\left[\frac{\hbar^{2}}{2}(\alpha_{n}^{2}+\beta_{m}^ {2})+1\right]\omega_{mn}^{2}}\sin\ \alpha_{n}x\ \cos\ \beta_{m}y\ \sin^{2}\frac{\omega_{mn}}{2}t\\ \end{array} \] (44) where angles of tangents rotation \(\varphi_{x}\) and \(\varphi_{y}\) \[