Dynamical space-time ray tracing and modified horizontal ray method

TL;DR

The paper addresses acoustic wave propagation in shallow-water environments by unifying vertical-mode (Sturm-Liouville) representations with dynamical space-time ray tracing. It treats time as an additional spatial coordinate to describe frequency-modulated signals in a dispersive medium, deriving a Hamiltonian framework with the eikonal equation $|\mathbf{k}|^2 - q^2(w,\mathbf{r}) = 0$ and a principal-amplitude transport law along space-time rays. Central contributions include the introduction of adiabatic vertical modes with eigenvalues $q_l$, a variational (Jacobi) analysis for ray tubes, explicit expressions for phase and amplitude evolution, and a formalism to compute observable fronts and gradients in space-time. The framework enables efficient prediction of modulation, time compression, front tilts, and caustics, and provides a foundation for extending to multi-mode coupling in 3D, with broad applicability to dispersive media beyond acoustics.

Abstract

The 'vertical modes and horizontal rays' method, commonly applied for simulating acoustic wave propagation in shallow water is advanced in this research. Our approach to this method involves the use of the so-called space-time rays, which are constructed by decomposing the time-dependent sound field into adiabatic vertical modes, the solutions to the Sturm-Liouville problem. The introduction of the time coordinate, while still considering it as an additional space coordinate instead of merely a parameter along the ray, allows us to describe the propagation of frequency-modulated signals in an effectively frequency-dispersive medium. The consideration of the extension of Hamiltonian ray-tracing methods (also used for the description of Gaussian beams and so-called quasiphotons) leads to a simple description of observable effects such as changes in modulation, time compression, differences between angles of phase and amplitude fronts, space-time caustics, etc., in dynamics - on the moving line or at some point of observation while having the general form of the source (for example, also a moving one).