Fermat's and Huygens' principles, and hyperbolic equations and their equivalence in wavefront construction

Abstract

Consider propagation of a wavefront in a medium. According to Fermat's principle a ray, travelling from one point P₀ to another point P₁ in space, chooses a path such that the time of transit is stationary. Given initial position of a wavefront Ω₀, we can use rays to construct the wavefront Ω_t at any time t. Huygens' method states that all points of a wavefront Ω₀ at t = 0 can be considered as point sources of spherical secondary wavelets and after time t the new position Ω_t of the wavefront is an envelope of these secondary wavelets. The equivalence of the two methods of construction of a wavefront Ω_t in a medium governed by a general hyperbolic system of equations does not seem to have been proved. Hyperbolic equations have their own method of construction of a wavefront. We shall discuss this still open (as far as I know) problem for a general hyperbolic system and briefly sketch the relation between the three methods for a particular case when the medium is governed by Euler equations of a polytropic gas in free space [16].