Applications of cone structures to the anisotropic rheonomic Huygens' principle

Abstract

A general framework for the description of classic wave propagation is introduced. This relies on a cone structure $C$ determined by an intrinsic space $\Sigma$ of velocities of propagation (point, direction and time-dependent) and an observers' vector field $\partial_t$ whose integral curves provide both a Zermelo problem for the wave and an auxiliary Lorentz-Finsler metric $G$ compatible with $C$. The PDE for the wavefront is reduced to the ODE for the $t$-parametrized cone geodesics of $C$. Particular cases include time-independence ($\partial_t$ is Killing for $G$), infinitesimally ellipsoidal propagation ($G$ can be replaced by a Lorentz metric) or the case of a medium which moves with respect to $\partial_t$ faster than the wave (the strong wind case of a sound wave), where a conic time-dependent Finsler metric emerges. The specific case of wildfire propagation is revisited.