Propagation of nonlinear logitudinal waves in an elastic rod

The nonlinear longitudinal wave propagation in a rod is studied by the homotopy analysis method and obtained analytical solutions which are the same as earlier known results. We also employed the explicit and implicit finite difference methods to solve this nonlinear equation, respectively. The numerical results showed the wave form depends on many factors such as driving frequency, amplitude, nonlinear elastic constant, and wave velocity. We found that compared with the explicit finite difference method, the implicit finite difference method need less amount of the grid but require more computational time to obtain reliable solutions. We also found out that the energy of multiple-frequency components becomes larger with longer traveling distance.