Ray-based description of normal mode amplitudes in a range-dependent   waveguide
  An analogue of the geometrical optics for description of the modal structure of a wave field in a range-dependent waveguide is considered. In the scope of this approach the mode amplitude is expressed through solutions of the ray equations. This analytical description accounts for mode coupling and remains valid in a nonadiabatic environment. It has been used to investigate the applicability condition of the adiabatic approximation. An applicability criterion is formulated as a restriction on variations of the action variable of the ray. 
A.L. Virovlyansky, A.Yu. Kazarova, L.Ya. Lyubavin