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

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The generation of second and third harmonics by an acoustic wave propagating along one dimension in a weakly nonlinear elastic medium is analyzed by successive approximations starting with the linear case. The medium is loaded harmonically in time with frequency ω0 at a single point in space. It is important to recall two known facts: The first, nonlinear waves have a speed of propagation that depends on their amplitude, a reflection of the fact that nonlinear oscillators have an amplitude-dependent period. The second fact is that although both a free and a loaded medium generate higher harmonics, the second harmonic of the free medium scales like the square of the linear wave, but 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 (“secular”) terms in a successive approximations solution scheme to the homogeneous (i.e., “free”) problem. The result is then used to solve the inhomogeneous (i.e., “loaded”) 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 of wavelength 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. It depends both on the third- and fourth-order elastic constants, as well as on the frequency and amplitude of the loading. A distance scale emerges: it depends on nonlinearities, on the loading frequency and on the amplitude of the linear wave. In this distance scale, at short distances from the source a second harmonic scaling proportional to the amplitude of the linear wave squared, and to the distance from the source, is recovered. It 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 ω0 and the other two, third harmonics, with 3ω0. In each pair, one term scales like the amplitude of the linear wave to the five-thirds, and the other to the seven-thirds.