Propagation of initially bi-harmonic sound waves in a 1D semi-infinite medium with hysteretic non-linearity.

Materials with hysteretic non-linearity have the property of memorizing specific previous extrema in the stress/strain loading history. Because of this complexity, the analytical theory describing the non-linear evolution of acoustic waves in such materials is currently restricted to simplex wave propagation processes with a single minimum and a single maximum over a wave period. In the present paper a numerical model is presented which is valid for an arbitrary strain wave profile, and the results for the frequency-mixing process in acoustic waves composed initially of two harmonic frequency components are analyzed. The model simulations demonstrate that an initially complex wave transforms into a simplex wave during propagation. In addition, we have studied the mutual influence of the initial frequency components, and we have found regimes of induced absorption and induced transparency.