## 3 Experimental and Numerical Methods
### Finite Element Model  
In a Cartesian coordinate system in two dimensions, we can express Equation (25), as follows. \[\frac{\partial^{2}\xi_{x}}{\partial t^{2}}+\sigma_{0}(\
abla\cdot \boldsymbol{v})\frac{\partial\xi_{x}}{\partial t}-\frac{G}{\rho}\left(\frac{ \partial^{2}\xi_{x}}{\partial x^{2}}+\frac{\partial^{2}\xi_{x}}{\partial y^{2} }\right)=-\frac{G}{\rho}\frac{\partial}{\partial x}\left(\frac{\partial\xi_{x} }{\partial x}+\frac{\partial\xi_{y}}{\partial y}\right)+\alpha\frac{(\lambda+2 G)}{\rho}\frac{\partial}{\partial x}\left(\frac{\partial\xi_{x}}{\partial x}+\frac{ \partial\xi_{y}}{\partial y}\right) \tag{32}\] \[\frac{\partial^{2}\xi_{y}}{\partial t^{2}}+\sigma_{0}(\
abla\cdot \boldsymbol{v})\frac{\partial\xi_{y}}{\partial t}-\frac{G}{\rho}\left(\frac{ \partial^{2}\xi_{y}}{\partial x^{2}}+\frac{\partial^{2}\xi_{y}}{\partial y^{2} }\right)=-\frac{G}{\rho}\frac{\partial}{\partial y}\left(\frac{\partial\xi_{ x}}{\partial x}+\frac{\partial\xi_{y}}{\partial y}\right)+\alpha\frac{(\lambda+2 G)}{\rho}\frac{\partial}{\partial y}\left(\frac{\partial\xi_{x}}{\partial x}+\frac{ \partial\xi_{y}}{\partial y}\right) \tag{33}\]  
Using [PERSON]'s ratio \(\
u\), we can rewrite the volume expansion term appearing on the right-hand side in the following form. Here, \(y\) is chosen to be the axis along which an external load is applied to the specimen. \[\frac{\partial\xi_{x}}{\partial x}+\frac{\partial\xi_{y}}{\partial y}=-\
u \frac{\partial\xi_{y}}{\partial y}+\frac{\partial\xi_{y}}{\partial y} \tag{34}\]  
Using Equation (34), canceling common terms from the left and right-hand sides and further simplifying several terms, we can rewrite Equations (32) and (33), as follows. \[\frac{\partial^{2}\xi_{x}}{\partial t^{2}}+\sigma_{0}(\
abla\cdot \boldsymbol{v})\frac{\partial\xi_{x}}{\partial t}-\frac{G}{\rho}\frac{ \partial^{2}\xi_{x}}{\partial y^{2}}=-\frac{G}{\rho}\frac{\partial^{2}\xi_{y} }{\partial x\partial y}+\alpha\frac{(\lambda+2G)(1-\
u)}{\rho}\frac{\partial^ {2}\xi_{y}}{\partial x\partial y} \tag{35}\] \[\frac{\partial^{2}\xi_{y}}{\partial t^{2}}+\sigma_{0}(\
abla \cdot\boldsymbol{v})\frac{\partial\xi_{y}}{\partial t}-\frac{G}{\rho}\frac{ \partial^{2}\xi_{y}}{\partial x^{2}}=\frac{G\
u}{\rho}\frac{\partial^{2}\xi_{ y}}{\partial y^{2}}+\alpha\frac{(\lambda+2G)(1-\
u)}{\rho}\frac{\partial^{2}\xi_{y}}{ \partial y^{2}} \tag{36}\]  
Equation (36) indicates that the longitudinal displacement component travels as longitudinal and transverse waves. The transverse wave characteristics comes from the \(\partial^{2}\xi_{y}/\partial x^{2}\) term on the left-hand side of the equation. The longitudinal wave characteristics comes from the first term on the right-hand side. It is important to note that the quantities multiplied to the secondary spatial derivative (\(\partial^{2}\xi_{y}/\partial y^{2}\)) term on the right-hand side represents the phase velocity of the longitudinal wave. It depends on the shear modulus \(G\), the Poisson's ratio \(\
u\), the density \(\rho\), the degree of the elastic force contribution \(\alpha\), and the Lame's parameter \(\lambda\). The phase velocity of the transverse wave is determined by the quantity multiplied to \(\partial^{2}\xi_{y}/\partial x^{2}\) term on the left-hand side. Hence, the transverse wave velocity only depends on \(G\) and \(\rho\). The transverse component \(\xi_{x}\) is in a different situation according to Equation (35). While the secondary spatial derivative (\(\partial^{2}\xi_{x}/\partial y^{2}\)) term on the left-hand side generates transverse wave characteristics, the right-hand side of this equation does not cause wave dynamics, as it does not contain the secondary spatial derivative of \(\xi_{x}\). Instead, the entire right-hand side is a function of the  
Figure 3: In-plane sensitive Electronic Speckle-Pattern Interferometric setup. secondary spatial derivative of \(\xi_{y}\). This indicates that the spatiotemporal behavior of the transverse component depends on that of the longitudinal component.