Numerical analysis of the limits of effective medium theory for nonlinearly propagating dispersive waves

Abstract

Acoustic metameterials are an increasingly popular approach to control both linear and nonlinear sound propagation. In this approach, heterogeneous structures are engineered to behave as predetermined continuous media. A key requirement for a system to be accurately approximated as a homogeneous material for wave motion is that the smallest length scale of the waveform be much larger than any length scale associated with the microscale structure of the system. However, the fact that the characteristic length scales of a finite-amplitude waveform can change due to nonlinear effects means that care must be taken when analyzing such signals in an acoustic metamaterial. This talk presents recent work toward understanding the breakdown of effective medium theory as finite-amplitude waves propagate in a heterogeneous system and weak shocks or solitons begin to form. A finite-difference time-domain code was used to model the propagation of a pulse of sound in a one-dimensional propagation domain using both a full description of the system and an effective-medium description accounting for effective mass density, compressibility, coefficient of nonlinearity, and dispersion. The difference between the solutions of the two models provides a method by which one can estimate the error associated with using effective properties.