## 2 Materials and Methods  
The research model in this paper is presented in Figure 1. The governing equation of the Rayleigh-Love theory [12] for wave propagation in rods is as follows:  
\[\frac{\partial^{2}u_{1}}{\partial t^{2}}=c_{01}^{2}\frac{\partial^{2}u_{1}}{ \partial x^{2}}+v_{1}^{2}\kappa_{1}^{2}\frac{\partial^{4}u_{1}}{\partial x^{ 2}\partial t^{2}},\quad-L\leq x\leq 0, \tag{1}\]  
\[\frac{\partial^{2}u_{2}}{\partial t^{2}}=c_{02}^{2}\frac{\partial^{2}u_{2}}{ \partial x^{2}}+v_{2}^{2}\kappa_{2}^{2}\frac{\partial^{4}u_{2}}{\partial x^{ 2}\partial t^{2}},\quad 0\leq x<\infty, \tag{2}\]  
where \(c_{01}=\sqrt{E_{1}/\rho_{1}}\) and \(c_{02}=\sqrt{E_{2}/\rho_{2}}\) are the constant wave velocities without considering [PERSON]'s effect in the striker rod and the the semi-infinite rod, respectively. \(E_{1}\), \(E_{2}\) are the Young's moduli values of the striker rod and the semi-infinite rod, respectively. \(\rho_{1}\) and \(\rho_{2}\) are the mass densities; \(v_{1}\) and \(v_{2}\) are the Poisson ratios; and \(\kappa_{1}\), \(\kappa_{2}\) are the radii of gyration of the cross-section of the striker rod and the semi-infinite rod, respectively. The impedances are \(Z_{1}=A_{1}\rho_{1}c_{1}\) and \(Z_{2}=A_{2}\rho_{2}c_{2}\) for the striker rod and the semi-infinite rod, respectively. \(c_{1}\) and \(c_{2}\) are the constant wave velocities considering [PERSON]'s effect [16] in the striker rod and the semi-infinite rod, respectively, and are determined as follows:  
\[c_{1}=\sqrt{\frac{(1-v_{1})E_{1}}{(1+v_{1})(1-2v_{1})\rho_{1}}}, \tag{3}\]  
\[c_{2}=\sqrt{\frac{(1-v_{2})E_{2}}{(1+v_{2})(1-2v_{2})\rho_{2}}}. \tag{4}\]  
The initial conditions in terms of displacement \(u\) can be written in a piecewise manner:  
\[u_{1}(x,0)=0,\quad\frac{\partial u_{1}(x,0)}{\partial t}=2v_{0},\quad-L\leq x \leq 0. \tag{5}\]  
\[u_{2}(x,0)=0,\quad\frac{\partial u_{2}(x,0)}{\partial t}=0,\quad 0\leq x<\infty, \tag{6}\]  
The boundary conditions and continuity conditions at \(x=-L\), \(x=0\), and \(x=\infty\) are given by:  
\[E_{1}A_{1}\frac{\partial u_{1}}{\partial x}(-L,t)=0,\quad x=-L, \tag{7}\]  
\[u_{1}(0^{-},t)=u_{2}(0^{+},t),\quad x=0, \tag{8}\]To make the Laplace transform process more convenient, we can convert the quantities to dimensionless quantities for substitution into Equations (1) and (2). This is achieved by introducing the following dimensionless variables and parameters:  
\[ \begin{array}{l}\overline{u}_{1}=\frac{u_{1}}{D_{2}},\ \overline{u}_{2}=\frac{u_{2}}{D_{2}}, \ \overline{x}=\frac{x}{D_{2}},\ \overline{t}=\overline{t}_{2}=\frac{c_{2}t}{D_{2}},\ \overline{t}_{1}=\frac{c_{1}t}{D_{2}}= \frac{c_{2}}{c_{2}}\overline{t},\\ \overline{L}=\frac{L}{D_{2}},\ \overline{\kappa}_{1}=\frac{\kappa_{1}}{D_{2}},\ \overline{\kappa}_{2}=\frac{\kappa_{2}}{D_{2}},\ b_{1}=v_{1}\overline{\kappa}_{1 },\ b_{2}=v_{2}\overline{\kappa}_{2},\ \overline{v}_{0}=\frac{v_{0}}{c_{1}},\end{array}  \tag{11}\]  
where \(D_{2}\) is the diameter of the semi-infinite rod. Substituting Equation (11) into Equations (1) and (2), we obtain:  
\[\frac{\partial^{2}\overline{u}_{1}}{\partial\overline{t}^{2}}=m^{2}\frac{ \partial^{2}\overline{u}_{1}}{\partial\overline{x}^{2}}+b_{1}^{2}\frac{ \partial^{4}\overline{u}_{1}}{\partial\overline{x}^{2}\partial\overline{t}^ {2}},\ \ \ -\overline{L}\leq\overline{x}\leq 0 \tag{12}\]  
\[\frac{\partial^{2}\overline{u}_{2}}{\partial\overline{t}^{2}}=\frac{ \partial^{2}\overline{u}_{2}}{\partial\overline{x}^{2}}+b_{2}^{2}\frac{ \partial^{4}\overline{u}_{2}}{\partial\overline{x}^{2}},\ \ \ 0\leq\overline{x}<\infty \tag{13}\]  
where \(m=\frac{c_{1}}{c_{2}}\). Apply Laplace transformation to find the displacement \(U_{1}(\overline{x},s)\) and \(U_{2}(\overline{x},s)\) in \(s\) domain (see in Appendix A) as follows:  
\[