## 1 Introduction  
The study of stress wave propagation is a fundamental aspect of materials science and engineering, with implications for the design, analysis, and application of mechanical systems subjected to dynamic loading conditions [1; 2; 3; 4; 5; 6; 7; 8; 9]. In particular, the interaction of stress waves at the interfaces of materials with differing impedances is critical for understanding the behavior of waves under impact loading. The traditional models often consider idealized conditions, neglecting factors such as lateral inertia and the Poisson effect, which can significantly influence the accuracy of the predictions in real-world situations [10; 11]. The Rayleigh-Love rod theory [12; 13; 14], an advanced theoretical framework, incorporates these aspects to provide a more comprehensive understanding of one-dimensional stress wave propagation behavior.  
This paper delves into the complex dynamics of stress wave propagation in a system composed of a moving striker rod and an initially stationary semi-infinite Rayleigh-Love rod. The focus is primarily on the role of different general impedances in modifying the wave's response during its propagation through the system. The general impedance \(Z=A\rho c\), where \(A\) is the cross-sectional area, \(\rho\) is the mass density, and \(c\) is the wave velocity of a rod, is a parameter that embodies the mechanical resistance against wave transmission at an interface, and it is a crucial factor in determining the wave's behavior upon encountering a discontinuity of material properties. This research utilizes the Rayleigh-Love rod theory, which accounts for lateral inertia and the Poisson effect, to provide a more detailed insight into stress wave propagation in rods.  
The novelty of this paper lies in addressing the challenges associated with the inversion of Laplace transforms for analyzing stress wave propagation, taking into account lateral inertia and Poisson's effects. This represents a significant advancement over previous research [13; 14], which only discussed impacts between a striker rod and a semi-infinite rod with the same impedance. To solve complex wave equations derived from the Rayleigh-Love rod theory with varying general impedances, we employ the approximate formula for numerical inversion of Laplace transforms introduced by [PERSON] et al. [15]. The precision achieved by this method can be very high, even in cases in which the transformations are irrational or transcendental functions of the complex variable. The algorithms can be easily and efficiently implemented using programming packages such as MATLAB or Mathematica. Based on the results obtained from the proposed method, this study demonstrates that alterations in general impedances at the interface between the striker rod and the semi-infinite rod lead to distinct reflection and transmission behaviors of the stress waves. These interactions can result in the amplification or attenuation of the wave, underscoring the significant impact that impedance variations can have on the propagation of stress waves.  
Through a thorough analysis and comparison with the results of previous research, this paper confirms the accuracy of the findings and contributes to the body of knowledge by providing detailed insights into the effects of different general impedances on stress wave propagation in rods. The implications of this research are vast, offering valuable guidance for the design and evaluation of mechanical systems and materials subjected to dynamic impacts, ensuring their integrity and performance in practical applications.