Generation of the second, fourth, and eighth harmonics by a hyperelastic longitudinal planewave: numerical simulation

The generation of the first, second, fourth, and eighth harmonics of a harmonic longitudinal plane wave propagating in quadratic nonlinear materials described by the classical Murnaghan model is numerically modeled. This situation was also analyzed analytically. The materials are fibrous composites with micro- and nanoscale structure. Using such materials introduces additional real restrictions on the main parameters and allows simulating various real situations: from the generation of all harmonics for feasible distances and times to the generation of the second harmonics for feasible distances and times. The contribution of each approximation to the overall wave pattern is analyzed. It is shown that these approximations affect the prediction of the evolution of the initial wave profile: first there is a tendency to the generation of the second harmonic which then transforms into a tendency to the generation of the fourth and eight harmonics