Accuracy of the slow scale approximation for predicting harmonic generation in plane shear wave propagation

Nonlinear shear wave propagation is typically modeled using the slow scale approximation, where it is assumed that nonlinear effects occur over the course of many wavelengths of propagation, i.e., that the nonlinear length z sh is larger than the wavelength λ. However, Catheline et al. [Phys. Rev. Lett. 91, 164301 (2003)] reported shock formation in an initially sinusoidal plane shear wave within one wavelength of propagation from the source in gelatin. Here, we investigate limits of the slow scale approximation for cubically nonlinear plane shear waves by comparing with the exact solution of the nonlinear wave equation. The dimensionless parameter Λ = λ/z sh quantifies the source amplitude limits before the slow scale approximation breaks down. As Λ (amplitude) increases, the slow scale approximation overpredicts phase speed, leading to an underestimation of shock formation distance and overestimation of waveform distortion and harmonic generation. For the case of Catheline et al., Λ = 1.07 and we find an 11% overprediction of the third harmonic amplitude at the shock formation distance. This analysis provides criteria on source amplitude for when the slow scale approximation may be invalid in the modeling of nonlinear shear wave propagation in soft tissues. [Work supported by the ARL:UT McKinney Fellowship in Acoustics.]