Separation of Variables Applied to the Wave Equation in Laterally Inhomogeneous Media

The new eigenfunction expansion formula based upon the method of separation of variables is derived. The method gives the exact transport equation and the generalized eikonal equation without the need of asymptotic series expansions. The generalized eikonal equation extends the classical eikonal equation to a rapidly varying medium. It governs the wave propagation with dispersive phase velocity depending upon the amplitude. Both equations are separated by making use of conventional phase-ray coordinates. The solution is expressed in terms of the elementary Sturm-Liouville eigenfunctions or modes for the vibrating string. The generalized eikonal equation is solved by employing the modified Hamiltonian formalism. This results in frequency-dependent phase-ray trajectories associated with the energy flux vector field.