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 Σ of velocities of propagation (point, direction and time-dependent) and an observers’ vector field ∂/∂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 (∂/∂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 ∂/∂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.