Direction-Independent Algorithm for Simulating Nonlinear Pressure Waves

Abstract

This study formulates a frequency-domain computational scheme for simulating nonlinear wave propagation in a homogeneous medium governed by the Westervelt equation. The need for such numerical treatment arises in both engineering and medical imaging applications, where finite-amplitude pressure waves trigger nonlinear effects that may critically affect the sensory data. The primary advantage of the proposed approach over commonly used approximations, which account for nonlinear effects via the Burgers’ equation, lies in its ability to handle nonlinearities due to arbitrarily inclined incident waves, which becomes especially important for focused sound beams with large apertures, i.e., wide ranges of inclination angles. The proposed direction-independent algorithm has a direct mathematical connection with the Westervelt equation, as opposed to the Burger’s equation (that relies on the plane-wave hypothesis), and has computational efficiency that is comparable to that of the traditional approach. The developments are illustrated by numerical examples that verify the method against an analytical solution and highlight the significance of accurately modeling nonlinear waves.