Applications of cone structures to the anisotropic rheonomic Huygens' principle

TL;DR

This paper introduces a comprehensive framework for wave propagation using cone structures and Lorentz-Finsler metrics, applicable to various media including wildfire spread, unifying different propagation scenarios.

Contribution

It develops a general geometric approach to wavefront evolution, incorporating time-dependent and anisotropic effects through cone structures and Finsler metrics, extending classical models.

Findings

Unified geometric framework for wave propagation.

Application to wildfire spread modeling.

Reduction of PDEs to ODEs for cone geodesics.

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.