# Transverse Wave Propagation in a Thin Isotropic Plate Part I
## 2 Materials and Methods
### General Analytical Solution
The specific shear forces in the \(z\) direction act in a plane perpendicular to the axis \(x\) and \(y\). \[q_{xz}=\int_{-\frac{1}{2}}^{+\frac{1}{2}}\tau_{xz}\cdot dz,\qquad q_{yz}=\int_{- \frac{1}{2}}^{+\frac{1}{2}}\tau_{yz}\cdot dz. \tag{14}\]  
Specific bending moment (\(m_{x}\)) acting in a plane parallel to the \(yz\) plane whose vector is parallel to the \(y\)-axis, specific bending moment (\(m_{y}\)) acting in a plane parallel to the \(xz\) plane, the vector parallel to the \(x\)-axis and specific torques (\(m_{xy}=m_{yx}\)) acting in planes of the element parallel to the planes \(xz\) and \(yz\), whose vectors are perpendicular to these planes are defined:  
\[m_{x}=\int_{-\frac{1}{2}}^{+\frac{1}{2}}\sigma_{x}\cdot z\cdot dz,\qquad m_{y}= \int_{-\frac{1}{2}}^{+\frac{1}{2}}\sigma_{y}\cdot z\cdot dz,\qquad m_{xy} =m_{yx}=\int_{-\frac{1}{2}}^{+\frac{1}{2}}\tau_{xy}\cdot z\cdot dz. \tag{15}\]  
These specific moments can be expressed with (13) as a function of the displacements  
\[ \begin{array}{ll}m_{x}{=}-D_{x}\left(\frac{\partial^{2}w}{\partial x^{2}}+ \mu_{yx}\frac{\partial^{2}w}{\partial y^{2}}\right),&\text{where}&D_{x}=\frac{ E_{x}h^{3}}{12\left(1-\mu_{xy}\mu_{yx}\right)},\\ m_{y}=-D_{y}\left(\frac{\partial^{2}w}{\partial y^{2}}+\mu_{xy}\frac{ \partial^{2}w}{\partial x^{2}}\right),&\text{where}&D_{y}=\frac{E_{y}h^{3}}{12 \left(1-\mu_{xy}\mu_{yx}\right)},\\ m_{xy}{=}-2D_{xy}\frac{\partial^{2}w}{\partial x\partial y},&\text{where}&D_{ xy}=\frac{G_{xy}h^{3}}{12}.\end{array}  \tag{16}\]  
where \(D_{x}\), \(D_{y}\)--stiffness modules in the direction \(x\) and \(y\), \(D_{xy}\)--torsional stiffness modulus. After further modifications, three equations of motion can be formulated for the forces and moments acting on the plate element \(dxdyth\). \[ \begin{array}{ll}\frac{\partial u_{xz}}{\partial x}+\frac{\partial u_{xz}} {\partial y}+p(x,y,t)=\rho h\frac{\partial^{2}w}{\partial t^{2}},\\ q_{xy}{=}-\frac{\partial u_{xz}}{\partial x}-\frac{\partial u_{xy}}{ \partial y}-\frac{\partial^{2}w}{\partial t^{2}},&\text{where}&J_{p}{=}\rho \frac{h^{3}}{12}=\frac{d^{2}J_{x}}{dxdyd}=\frac{d^{2}J_{y}}{dxdyd^{\prime}},\\ q_{yz}{=}-\frac{\partial u_{yz}}{\partial y}-\frac{\partial u_{yz}}{\partial x }{=}J_{p}\frac{\partial^{2}w}{\partial t^{2}}.\end{array}  \tag{17}\]  
where \(J_{p}\)--inertia material moment of the element \(dxdyth\) in axis \(x\), or \(y\), passing through the center of the element provided \(dx<h\), \(dy<h\), \(h\)--plate thickness, \(p\)--the function of the excitation load, \(\rho\)--plate density. For moment equations, the right side respects the effect of the rotation of the plate element around the \(x\) and \(y\) axis--[PERSON]'s model. For \(J_{p}=0\). These equations have the character of moment-static equilibrium, when the effect of inertial moments, or cross-section rotation are neglected, which corresponds to the original [PERSON] model. By further arrangement (17) and using (12), (16) we get for [PERSON]'s model (\(J_{p}\
eq 0\))  
\[ \begin{array}{ll}q_{xz}=\frac{\partial}{\partial x}\left[I_{p}\frac{ \partial^{2}w}{\partial x^{2}}-D_{x}\frac{\partial^{2}w}{\partial x^{2}}-D \frac{\partial^{2}w}{\partial y^{2}}\right],&\text{where}&D=D_{x}\mu_{yx}+2D_{ xy}{=}D_{y}\mu_{xy}+2D_{xy},\\ q_{yz}=\frac{\partial}{\partial y}\left[I_{p}\frac{\partial^{2}w}{\partial x^{2}} -D_{y}\frac{\partial^{2}w}{\partial y^{2}}-D\frac{\partial^{2}w}{\partial x^{2 }}\right].\end{array}  \tag{18}\]  
Figure 4: Shear forces and moments acting on the plate element. For the Kirchhoff model \(J_{p}=0\) and relations for \(q_{xx}\) and \(q_{yz}\) are clear.