Acoustic shock wave propagation in a heterogeneous medium: A numerical simulation beyond the parabolic approximation

Abstract

Numerical simulation of nonlinear acoustics and shock waves in a weakly heterogeneous and lossless medium is considered. The wave equation is formulated so as to separate homogeneous diffraction, heterogeneous effects, and nonlinearities. A numerical method called heterogeneous one-way approximation for resolution of diffraction (HOWARD) is developed, that solves the homogeneous part of the equation in the spectral domain (both in time and space) through a one-way approximation neglecting backscattering. A second-order parabolic approximation is performed but only on the small, heterogeneous part. So the resulting equation is more precise than the usual standard or wide-angle parabolic approximation. It has the same dispersion equation as the exact wave equation for all forward propagating waves, including evanescent waves. Finally, nonlinear terms are treated through an analytical, shock-fitting method. Several validation tests are performed through comparisons with analytical solutions in the linear case and outputs of the standard or wide-angle parabolic approximation in the nonlinear case. Numerical convergence tests and physical analysis are finally performed in the fully heterogeneous and nonlinear case of shock wave focusing through an acoustical lens.

Extending the Utility of the Parabolic Approximation in Medical Ultrasound Using Wide-Angle Diffraction Modeling

“…The equation has also been applied to therapeutic ultrasound to model hyperthermia using a linear array [15], was compared against other models in the context of time-domain analysis of shock propagation in heterogenous media [16], and was used to model nonlinear ultrasound fields in air [17]. WAPE models employ high-order approximations of the diffraction operator by retaining more terms in its Taylor expansion [13], approximation by rational function [11], [12], [14], [15], or more advanced methods [18].…”

Rapid quantitative imaging of high intensity ultrasonic pressure fields

“…To generate the FUS-blurred background images used to train the reconstructor, spatially- and temporally-resolved steady-state FUS pressure fields with nonlinearity were simulated using a modified angular spectrum method 48 with frequency domain attenuation and dispersion, absorbing boundary layers, and an adaptive propagation step size. An operator splitting term was used to separate the terms in the retarded-time formulation of the nonlinear angular spectrum equation 49 . The attenuation and dispersion term was solved directly in the frequency domain using a filtering approach.…”

High-intensity ultrasonic waves in fluids

“…This is the so-called one-way approximation (Dagrau et al, 2011). Then the numerical solution of the Westervelt equation becomes doable for many practical situations (Jing et al, 2011;Kreider et al, 2013;Yuldashev and Khokhlova, 2011).…”