## 6 Travel Time History  
In Figures 15-17, points P\({}_{1}\) to P\({}_{5\text{L}}\) are positions located in the striker rod and P\({}_{5\text{R}}\) to P\({}_{10}\) are positions located in the semi-infinite rod. These illustrate the stress wave travel time histories at positions P\({}_{1}\) to P\({}_{10}\) of two rods with various cross-sectional areas (\(A_{1}=A_{2}\)  
Figure 14: Comparison of stress wave propagation in the semi-infinite rod observed at \(x=0.5\) m under the impact of a striker rod with the same impedance by varying the cross-sectional area and material properties (\(Z_{1}=Z_{2}=32,\)087.204 kg/s, \(A_{1}=0.5A_{2}=0.5\times 7.069\times 10^{-4}\) m\({}^{2}\), \(E_{2}=195\) GPa, \(v_{1}=v_{2}=0.3\), \(\rho_{2}=7850\) kg/m\({}^{3}\), \(2v_{0}=11.6\) m/s, \(L=0.2\) m). The results of no Poisson’s effect are determined based on the research by [PERSON] and [PERSON] [11].  
From the results, it is clear that the amplitude of the stress waves differs in each case, with the wave amplitude increasing as the cross-sectional area \(A_{1}\) increases. This demonstrates the sensitivity of wave transmission to the cross-sectional area; specifically, when \(A_{1}\) is larger than A\({}_{2}\), the amplitude of the stress wave across the rods increases, implying that a larger cross-sectional area facilitates the transmission of more impact energy. In contrast, when \(A_{1}\) is smaller than \(A_{2}\), the amplitude decreases, which may indicate increased reflection rather than transmission of wave energy at the interface between the rods. Positions P\({}_{1}\) to P\({}_{\text{SL}}\) in the striker rod have pre-existing wave energy at the interface between the rods. In the initial state, the striker rod moves with a velocity of \(2v_{0}\), creating a stress in the semi-infinite rod, which will cause a reflection back to the striker rod. The waveform shape at points P\({}_{\text{5R}}\) to P\({}_{10}\) is the same in form, but the timing of the wave initiation differs among the positions, and the amplitude of wave propagation is different in all cases, reflecting the impact of the varying cross-sectional areas on the amplitude of wave propagation at these positions. This phenomenon underscores the critical role of the cross-sectional area in the modulation of wave energy transmission, where a larger \(A_{1}\) facilitates the conveyance of a greater amount of wave energy across the rods.  
Due to the traction-free boundary condition at the left end, the striker stress is zero at any instance at this point. The stress at each point reaches its maximum value when a stress wave travels through and then gradually decreases to zero. The duration of the stress pulse \(T_{j}\) in the striker rod decays linearly in each cycle as \(T_{j}=\frac{2L}{c_{1}}\big{(}1+\frac{x}{c_{1}}\big{)}\), where \(-L\leq x\leq 0\), and the duration of the stress pulse \(T_{1}\) in the semi-infinite rod is \(T_{i}=\frac{2L}{c_{1}}\), where \(x\geq 0\). The wave velocity \(c_{1}\) in all of the cases presented here is the same, because only the cross-sectional area of the striker rod is changed in the three cases, while all of the material properties are the same. In the case where \(Z_{1}=Z_{2}\), as shown in Figure 15, the stress wave propagation occurs in a single cycle, resulting in a clear signal after cycle 1. In the case where \(Z_{1}\
eq Z_{2}\), as shown in Figures 16 and 17, the wave propagation occurs over multiple consecutive cycles, creating a complex wave signal. This indicates that in order to easily identify the stress wave signal, using the condition \(Z_{1}=Z_{2}\) at the material interface is the most appropriate. Specifically, Figure 16 shows that when \(Z_{1}>Z_{2}\), a series of compression waves are displayed over consecutive cycles, causing the two rods to remain joined. Conversely, Figure 17 demonstrates that when \(Z_{1}<Z_{2}\), a tensile wave appears after the first cycle, leading to the separation of the two rods after the impact.