Second and third harmonic generation of acoustic waves in a nonlinear elastic solid in one space dimension

TL;DR

This paper analyzes the generation of second and third harmonics in acoustic waves within a nonlinear elastic solid, revealing how nonlinearities and external loading influence harmonic amplitudes and wave speeds.

Contribution

It provides a detailed analytical framework for understanding harmonic generation and wave speed shifts in one-dimensional nonlinear elastic media, including effects of external loading.

Findings

Second harmonic amplitude scales with the linear wave amplitude to the four-thirds.

Third harmonic waves include four amplitude-modulated components with distinct scaling laws.

Wave speed shifts depend on nonlinearities and are used to prevent resonance in solutions.

Abstract

The generation of second and third harmonics by an acoustic wave propagating along one dimension in a weakly nonlinear elastic medium that is loaded harmonically in time with frequency $\omega_0$ at a single point in space, is analyzed by successive approximations starting with the linear case. It is noted that nonlinear waves have a speed of propagation that depends on their amplitude. It is also noted that both a free medium as well as a loaded medium generate higher harmonics, but that although the second harmonic of the free medium scales like the square of the linear wave, this is no longer the case when the medium is externally loaded. The shift in speed of propagation due to the nonlinearities is determined imposing that there be no resonant terms in a successive approximation solution scheme to the homogeneous problem. The result is then used to solve the inhomogeneous case also by successive approximations, up to the third order. At second order, the result is a second harmonic wave whose amplitude is modulated by a long wave, whose wavelength is inversely proportional to the shift in the speed of propagation of the linear wave due to nonlinearities. The amplitude of the long modulating wave scales like the amplitude of the linear wave to the four thirds. At short distances from the source a scaling proportional to the amplitude of the linear wave squared is recovered, as is a second harmonic amplitude that grows linearly with distance from the source and depends on the third-order elastic constant only. The third order solution is the sum of four amplitude-modulated waves, two of them oscillate with frequency $\omega_0$ and the other two, third harmonics, with $3\omega_0$. In each pair, one term scales like the amplitude of the linear wave to the five-thirds, and the other to the seven-thirds.