RELAXATION PHENOMENA IN NONLINEAR ACOUSTICS

The effects of relaxation on the propagation of acoustic waves in a one-dimensional thermoviscous medium are considered through a modified Westervelt equation that includes either a memory term or non local effects that depend exponentially on time, and result in an integrodifferential equation. This equation has been transformed into coupled nonlinear partial differential equations that have been discretized in time with a second order accurate method, the nonlinear terms have been linearized with respect to the previous time level, and the resulting system of linear ordinary differential equations at each time level has been discretized by means of either second order accurate or a three point, compact, fourth-order accurate finite difference method. It is shown that, for the lossless Westervelt equation without memory terms and a sinusoidal pressure variation at the upstream boundary, an acceleration wave is formed; the steepness of this wave increases as time increases and eventually results in a shock wave. When relaxation is included, an acceleration wave is still produced, but this wave does not result in a shock wave although its front is very steep and is analogous to that observed in the viscous Burgers equation. When sinusoidal pressure conditions of different amplitude, frequency and phase are imposed at both the upstream and the downstream boundaries, no shock waves are observed, although complex interactions between acceleration waves and steep moving pressure fronts are observed. These interactions depend on the relaxation times, the nonlinearity parameter, the sound absorption coefficient and the amplitude, frequency and phase of the acoustic pressure at the boundaries, are periodic in time, and exhibit relative maxima and minima which are, in absolute value, higher than the amplitude of the pressure at the boundaries.