Second and Third Harmonic Generation of Elastic Waves in a Nonlinear Solid Under Shear Loading

The problem of nonlinear generation of a second harmonic longitudinal—acoustic—wave propagating along one direction, “”, but with an external transverse—shear—monochromatic localized load of amplitude , is solved to second and third order in the nonlinearities through a successive approximation scheme. The longitudinal wave so generated, of amplitude , is the superposition of two monochromatic waves of frequency , where is the loading frequency. One of the waves travels with the speed of linear longitudinal waves and the other with the speed of linear shear waves. An alternative description is to say that the result is an amplitude‐modulated wave of frequency , with the carrier wave traveling at a speed that is twice the harmonic mean of the longitudinal and transverse velocities, and the modulating wave has a wavelength , where (resp. ) is the longitudinal (resp. transverse) wavelength of the linear wave. The third‐order solution has a much richer structure: there are shear waves of frequency as well as , and wave vectors , , and , where (resp ) is the longitudinal (resp transverse) wave vector of the linear wave, that scale as . There are additional amplitude‐modulated waves that oscillate with frequencies and as well, but scale as . Their modulating wavelength depends on a third‐order elastic coefficient. Fourier analysis in is used, with the consequence that the solution is valid for all values of the distance from the loading point. It also allows for the computation of the correction to the amplitude and speed of propagation of the linear wave due to nonlinearities.