# Transverse Wave Propagation in a Thin Isotropic Plate Part I
## 2 Materials and Methods
### General Analytical Solution  
The analytical solution of the plates considers material and geometric properties. We assume three planes of symmetry, i.e., nine independent constants. The individual equations determine the relationship between the stress and strain components, which we can write in the simplified form of the abbreviated indices of the generalized [PERSON]'s law in the form [13]  
\[\sigma_{i}=c_{ij}\cdot\epsilon_{j}\quad\text{ for }i,j=1,2,\ldots,6 \tag{1}\]  
where \(\sigma_{i}\)--stress tensor, \(c_{ij}\)--matrix of elastic coefficients, and \(\epsilon_{j}\)--strain tensor. If the coordinate axes coincide with the material axes (axes of body symmetry), the relationship between the stress components and the deformation components can be used. The relationships between stress and strain components can also be expressed in inverse form  
\[\epsilon_{j}=s_{ij}\cdot\sigma_{i}\quad\text{ for }i,j=1,2,\ldots,6 \tag{2}\]  
where \(s_{ij}\)--matrix of elastic modules. Then, for the coordinate system \(x\), \(y\  
Assuming small deformations, i.e., according to geometric linear theory, the components of the deformation tensor are a function of the displacement vector in the Cartesian coordinate system:  
\[ \begin{array}{c}\varepsilon_{x}=\frac{\partial u}{\partial x},\qquad\qquad \varepsilon_{y}=\frac{\partial v}{\partial y},\qquad\qquad\varepsilon_{z}= \frac{\partial v}{\partial z},\\ \gamma_{xy}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x},\qquad \gamma_{xz}=\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x},\quad \gamma_{yz}=\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}.\end{array}  \tag{7}\]  
Assuming \(\varepsilon_{z}=0\), it follows that the vertical displacement \(w\) of the general point of the median plane is not a function of \(z\). \[w=w(x,y,t). \tag{8}\]  
From the relations (7), the displacement in the \(x\)-direction, or in the \(y\)-direction  
\[u=u(x,y,z,t),\quad v=v(x,y,z,t). \tag{9}\]  
From the last two relations of the system (7) after integration (integration functions are zero under transverse loading) we get  
\[u=-z\frac{\partial w}{\partial x},\qquad\quad v=-z\frac{\partial w}{\partial y}. \tag{10}\]  
The above relations are based on the analysis of the displacement of the general point A of the normal to the median surface at the distance \(z\) (Figure 3). After deformation of the plate due to the excitation load, the tangent to the median surface at point A rotates by an angle \(\varphi\) with respect to both the \(x\)-axis and the \(y\)-axis. Then, the displacements in the direction of the \(x\) and \(y\) axes can be described by relations using the angles of rotation \(\varphi_{x}\) and \(\varphi_{y}\)  
\[u=-z\cdot\varphi_{x},\qquad\quad v=-z\cdot\varphi_{y}. \tag{11}\]  
\[\varphi_{x}=\frac{\partial w}{\partial x},\qquad\quad\varphi_{y}=\frac{ \partial w}{\partial y}. \tag{12}\]  
where the z-axis displacement is a function \(w=w(x,y)\). Then, using Equations (10)-(12), Equation (6) can be rearranged to the form  
\[ \begin{array}{c}\sigma_{x}=\frac{-z\cdot\varepsilon_{x}}{1-\mu_{xy}\mu_{x }}\Big{(}\frac{\partial^{2}w}{\partial x^{2}}+\mu_{yx}\frac{\partial^{2}w}{ \partial y_{z}}\Big{)},\qquad\sigma_{y}=\frac{-z\cdot\varepsilon_{x}}{1-\mu_ {xy}\mu_{x}}\Big{(}\frac{\partial^{2}w}{\partial y^{2}}+\mu_{xy}\frac{ \partial^{2}w}{\partial x^{2}}\Big{)},\\ \tau_{xy}=-2z\cdot G_{xy}\frac{\partial^{2}w}{\partial x\partial y}.\end{array}  \tag{13}\]  
The three degrees of freedom of the plate element, i.e., the displacement \(w\) in the \(z\)-axis direction, the rotation around the \(x\)-axis by the angle \(\varphi_{x}\) and the rotation around the \(y\)-axis by the angle \(\varphi_{y}\), correspond to the three motion equations of forces in the \(z\)-direction and moments to axes parallel to the \(x\) and \(y\) directions. In thin plate theory, these equations of motion are usually formulated in integral form for the plate element \(dx\times dy\times h\), using specific shear forces and bending and torsional moments (Figure 4).