# Oscillating Nonlinear Acoustic Waves in a Mooney-Rivlin Rod
## 3 Numerical Analysis
### Analysis of Wave Propagation
#### 3.2.2 Waves at Harmonic Excitation  
Fairly large harmonic loadings with frequencies varied in a range \(\omega\in[1,30]\) rad/s. The loading amplitude \(\sigma_{0}=1\) was chosen to ensure the strain amplitude near the left end was \(\varepsilon_{0}\approx 0.3\). The plot in Figure 2d shows that slower moving parts of the wave at a smaller \(\epsilon\) will be overtaken by the faster moving parts, especially those related to large negative strains; according to [24], these effects will eventually lead to the appearance of shock wave fronts. The relative displacements, distances, time, and specific energies appearing in Figure 3 are, respectively, defined as  
\[U*=\frac{U}{\omega c_{0}};\qquad l*=\frac{l}{\omega c_{0}};\qquad\quad t*=\frac {\omega t}{2\pi}\qquad\quad E*=\frac{E}{\rho c_{0}^{2}}, \tag{34}\]  
where \(c_{0}\) is the limiting wave velocity defined by Equation (19) and the relative displacement magnitude is defined as max (\(U*\)) over time.  
Figure 2: (**a**) Tangential modulus; (**b**) tangential modulus in the vicinity of extremum; (**c**) Cauchy stress; (**d**) speed of propagation.  
The performed numerical analysis reveals (i) considerable attenuation of the wave amplitudes with distance from the excitation source, Figure 3a; (ii) a decrease in both specific strain and specific kinetic energy with distance in expense of the rise in specific thermal energy, Figure 3b; and (iii) the appearance of multiple shock wave fronts corresponding to the multiple overtaking occurring when faster moving pulses with negative strain overtake slower moving pulses with positive strain, see Figure 3c. Moreover, the propagating multiple shock wave fronts seen in Figure 3c indicate that all material is involved in the process of the mechanical energy dissipation and the increase in entropy and heat release; thus, in a purely mechanical system obeying the Mooney-Rivlin equation of state, the initially harmonic excitation leads to the attenuation of acoustic waves.  
Another remark concerns the appearance of shock wave fronts in a rod modelled by a neo-Hookean potential, which follows (13) by taking \(C_{01}=0\)[15]. This yields:  
\[W=C_{10}\bigg{(}\lambda^{2}+\frac{2}{\lambda}-3\bigg{)}=C_{10}\bigg{(}(1+ \varepsilon)^{2}+\frac{2}{(1+\varepsilon)}-3\bigg{)} \tag{35}\]  
and  
\[E(\varepsilon)\equiv\partial_{t}\sigma=2C_{10}\frac{3+9\varepsilon+12 \varepsilon^{2}+8\varepsilon^{3}+2\varepsilon^{4}}{(1+\varepsilon)^{3}}, \tag{36}\]  
Figure 3: Mooney–Rivlin potential: (**a**) attenuation of the relative displacement magnitude; (**b**) attenuation of specific strain and kinetic energy and an increase in thermal energy; (**c**) appearing multiple shock wave fronts at overtaking pulses of different signs at a point located at a relative distance of 100 from the left edge.  
from where the speed of sound becomes:  
\[c(\varepsilon)=\sqrt{\frac{2C_{10}}{\rho}}\sqrt{\frac{3+9\varepsilon+12\varepsilon ^{2}+8\varepsilon^{3}+2\varepsilon^{4}}{\left(1+\varepsilon\right)^{3}}} \tag{37}\]  
Now, taking \(C_{10}=0.916\) MPa, as in Equation (33), we can observe shock wave fronts and the associated dissipation of mechanical energy in a rod modelled by the neo-Hookean potential. It is interesting to note that plots for the spatial variation in specific strain, kinetic and thermal energy, along with the displacement field for neo-Hookean potential, are almost identical to the plots shown in Figure 3 for the Mooney-Rivlin potential. This is explained by a small material constant \(C_{01}\) in Equation (33), which, for the neo-Hookean potential, produces visually indistinguishable results. And finally, no dissipation of mechanical energy and shock wave fronts were observed in a linear elastic medium [24; 34].