## 4 Longitudinal waves in homogeneous and heterogeneous rods
### Analytical solution  
Let \(\Omega\) be a slender body, such that only the propagation of longitudinal waves is of interest. According to the schematic of Figure 4, the body is assumed to be loaded by a constant pressure \(\bar{\tau}\), acting over the time interval \(\Delta t_{\tau}\). To get insights into the effect of inner boundaries on wave reflection and dispersion, a two-phase body in considered, with phases A and B (of length \(L\) in the direction of wave propagation) respectively featuring [PERSON]'s moduli \(E_{A}\) and \(E_{B}\), and mass densities \(\varrho_{A}\) and \(\varrho_{B}\); the relevant longitudinal wave speeds thus are \(c_{A}=\sqrt{\frac{E_{A}}{\varrho_{A}}}\) and \(c_{B}=\sqrt{\frac{E_{B}}{\varrho_{B}}}\).  
In view of the elastic response of the bulk, a tensile wave propagating inside material A turns out to be characterized by the following relationship between the stress amplitude \(\sigma_{\text{in}}\) and the particle velocity \(v_{\text{in}}\):  
\[\sigma_{\text{in}}=-\varrho_{A}c_{A}v_{\text{in}} \tag{33}\]  
Figure 6: Wave propagating along a homogeneous rod discretized with an unstructured mesh: stress field at \(t=1.8\cdot 10^{-2}\) s. Blue solid line: FE solution; black solid line: standard FE-FCT solution; orange dashed line: newly proposed FE-FCT solution.  
After impinging upon the interface \(\Gamma\) at \(t=\bar{t}\), the incident wave induces reflected and transmitted waves, respectively characterized by relations [17]:  
\[\sigma^{-}= +\varrho_{A}c_{A}v^{-} \tag{34}\] \[\sigma^{+}= -\varrho_{B}c_{B}v^{+} \tag{35}\]  
where \(\sigma^{-}\) and \(\sigma^{+}\) are the amplitudes of the reflected and transmitted stress waves, and \(v^{-}\) and \(v^{+}\) are the relevant particle velocities. Equilibrium across \(\Gamma\) requires:  
\[\sigma_{\text{in}}+\sigma^{-}=\sigma^{+} \tag{36}\]  
while compatibility reads:  
\[[\dot{u}]=v^{+}-\left(v_{\text{in}}+v^{-}\right) \tag{37}\]  
\([\dot{u}]\) being the opening velocity on \(\Gamma\). Account taken of the interface constitutive law (10), written for mode I loadings as \(\tau\equiv\sigma^{+}=E_{\Gamma}[u]\), one arrives at the differential equation:  
\[\dot{\sigma}^{+}=\frac{1}{t_{c}}\left(\sigma_{\text{in}}-\frac{ \gamma+1}{2\gamma}\sigma^{+}\right) \tag{38}\]  
where: \(t_{c}=\frac{\varrho_{A}c_{A}}{2E_{\Gamma}}\) is the time-scale of the opening process along interface \(\Gamma\) (sometimes called characteristic relaxation time [17]); \(\gamma=\frac{\varrho_{B}\,c_{B}}{\varrho_{A}c_{A}}\). Upon integration of (38), the solution in terms of time histories  
Figure 7: Wave propagating along a homogeneous rod discretized with an unstructured mesh: stress field at \(t=1.8\cdot 10^{-2}\) s. Effect of parameters \(\eta_{D}=\eta_{A}\) on the capability of the newly proposed FE-FCT scheme to accurately track the stress discontinuity across the wave fronts.  
of the reflected and transmitted waves reads:  
\[\sigma^{-} = \left(\frac{\gamma-1}{2\gamma}-\exp\left[-\frac{\gamma+1}{2\gamma} \frac{t-\bar{t}}{t_{c}}\,\right]\right)\frac{2\gamma}{\gamma+1}\sigma_{\text{in}} \tag{39}\] \[\sigma^{+} = \left(1-\exp\left[-\frac{\gamma+1}{2\gamma}\frac{t-\bar{t}}{t_{c }}\,\right]\right)\frac{2\gamma}{\gamma+1}\sigma_{\text{in}} \tag{40}\]  
In case of perfect adhesion between the two phases, i.e. in case of \(E_{\Gamma}\rightarrow\infty\), the exponential terms in (39) and (40) drop to zero. The time-dependent feature of the solution is thus lost in view of the non-evolving microstructure along the interface.