Parabolic approximation in ray coordinates for high-frequency nonlinear waves in a inhomogeneous and high speed moving fluid

A new theoretical formulation for long-range propagation of nonlinear high-frequency acoustical waves is developed, that combines the advantages of both the parabolic and the ray approximations. The resulting scalar equation for the acoustical pressure is valid for any three dimensional, inhomogeneous and moving (subsonic) medium. Instead of selecting a preferred direction, the paraxial approximation is performed around the local directions of propagation given by geometrical acoustics, which are self-adapted to the medium and source characteristics. The parabolic approximation reintroduces diffraction effects neglected in geometrical acoustics but known to be dominant in regions where this one gets singular, such as caustics or shadow zones. This is expected to provide an innovative and efficient way to model and simulate nonlinear, high-frequency and transient wave propagation in complex environments. To further demonstrate the validity and versatility of the formulation, numerous equations from the literature in linear and nonlinear acoustics are obtained very straightforwardly from the general formulation simply as particular cases. This includes nonlinear ray theory, standard parabolic approximation for an inhomogeneous and slowly moving fluid, nonlinear propagation into shadow zones (produced by convex objects or temperature gradients), echo-gallery waves, wave focusing at caustics and nonlinear finite-amplitude beams along curved rays.