Effect of the Third Approximation in the Analysis of the Evolution of a Nonlinear Elastic P-Wave. Part 2*

Abstract

The propagation of a nonlinearly elastic longitudinal plane displacement wave for a symmetric initial wave profile described by a Gaussian function and an asymmetric initial profile described by a Whittaker function is analyzed. The basic novelty is that the evolution of waves is analyzed by the approximate methods by taking into account the first three approximations. The harmonic wave is analyzed only to compare with the new results for the bell-shaped wave. Some significant differences between the evolutions of waves are shown. Common to these profiles is the distortion of the initial profile due to the nonlinear self-interaction of the wave. The bell-shaped (symmetrical profile) wave retains symmetry when moving in a nonlinearly elastic body. For some initial sets of parameters, this wave does not change the wavelength initially and only shows a tendency towards the formation of two humps instead of one when we take into account the second conditional harmonic (two bell-shaped waves form and adjoin each other and halve wavelength) and the sinking of the left and elevation of the right hump when we take into account the third harmonic. The asymmetric profile described by Whittaker function retains the wavelength and asymmetry when the second approximation of the second harmonic is taken into account, but the amplitude increases rapidly. With additional allowance for the third harmonic, two asymmetric humps are formed, the humps resemble the evolution of the symmetric profile when the second harmonic is taken into account. Common to the evolutions of the symmetric and asymmetric profiles is the distortion of the initial wave profile: the formation of two symmetric humps when the second harmonic is taken into account in the analysis of the Gaussian function and two asymmetric humps when the third harmonic is taken into account in the analysis of the Whittaker function. Taking the third harmonic into account for a symmetric profile makes the two humps asymmetrical for symmetric and asymmetric profiles, only in these two profiles, the asymmetry is different.