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Applications of cone structures to the anisotropic rheonomic Huygens’ principle. (English) Zbl 1466.35004

Summary: A general framework for the description of classic wave propagation is introduced. This relies on a cone structure \(\mathcal{C}\) determined by an intrinsic space \(\Sigma\) of velocities of propagation (point, direction and time-dependent) and an observers’ vector field \(\partial/\partial t\) whose integral curves provide both a Zermelo problem for the wave and an auxiliary Lorentz-Finsler metric \(G\) compatible with \(\mathcal{C}\). The PDE for the wavefront is reduced to the ODE for the \(t\)-parametrized cone geodesics of \(\mathcal{C}\). Particular cases include time-independence \((\partial/\partial t\) 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 \(\partial/\partial t\) 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.

MSC:

35A18 Wave front sets in context of PDEs
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
83C75 Space-time singularities, cosmic censorship, etc.
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