Harmonic generation of longitudinal elastic waves

The propagation of the fundamental and harmonically generated longitudinal elastic waves is treated by means of an asymptotic iterative procedure directly in the governing nonlinear differential equation. Explicit results are obtained for the steady-state, spatial growth of the second and third harmonic and the depletion of the input wave. As expected, the analysis indicates that the amplitude of the Nth harmonic depends on all the elastic constants up to order N+1. However, the forms for the amplitudes obtained in the asymptotic solution reveal that, for the known range of ratios of elastic constants of successively increasing order and propagation distances commonly encountered in harmonic generation experiments, only the quadratic nonlinearity, which depends on the second- and third-order elastic constants, is required to accurately account for the experimental results. In addition, the amplitude dependence of the phase velocity of the fundamental longitudinal wave is determined.