Fermat's Principle and Huygens' Wavefront Construction

According to Fermat’s principle a ray, in wave propagation in a medium, going 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 ?0 , we can use rays to construct the wavefront ?1 at any time t1 . Huygens’ method states that all points of a wavefront ?0 at t = 0 can be considered as point sources of spherical secondary wavelets and after time t1 the new position ?1 of the wavefront is an envelope of these secondary wavelets. The equivalence of the two methods of construction of a wavefront ? in a medium governed by a general hyperbolic system of equations does not seem to have been proved. We shall discuss this still open (as far as I know) problem for a general hyperbolic system and prove the equivalence for a  particular case when the medium is governed by Euler equations of a polytropic gas.According to Fermat’s principle a ray, in wave propagation in a medium, going 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 ?0 , we can use rays to construct the wavefront ?1 at any time t1 . Huygens’ method states that all points of a wavefront ?0 at t = 0 can be considered as point sources of spherical secondary wavelets and after time t1 the new position ?1 of the wavefront is an envelope of these secondary wavelets. The equivalence of the two methods of construction of a wavefront ? in a medium governed by a general hyperbolic system of equations does not seem to have been proved. We shall discuss this still open (as far as I know) problem for a general hyperbolic system and prove the equivalence for a particular case when the medium is governed by Euler equations of a polytropic gas.