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.

Minor changes so that it matches the published version. 35 pages, 8 Figures

References in corpus (8)

• Stationary Metrics and Optical Zermelo-Randers-Finsler Geometry
• The geometry of sound rays in a wind
• A general model for wildfire propagation with wind and slope
• Huygens' envelope principle in Finsler spaces and analogue gravity
• Anisotropic conformal invariance of lightlike geodesics in pseudo-Finsler manifolds
• Geodesic sprays and frozen metrics in rheonomic Lagrange manifolds
• A Spacetime Geometry picture of Forest Fire Spreading and of Quantum Navigation
• Anisotropic Wavefronts and Laguerre Geometry

Cited by in corpus (7)

• A general model for wildfire propagation with wind and slope
• Globally hyperbolic spacetimes: slicings, boundaries and counterexamples
• Snell's law revisited and generalized via Finsler Geometry
• An account on links between Finsler and Lorentz Geometries for Riemannian Geometers
• Lightlike hypersurfaces and time-minimizing geodesics in cone structures
• On the application of Lorentz-Finsler geometry to model wave propagation
• A general model for time-minimizing navigation on a mountain slope under gravity