Change of the Ray Propagation Mode in Smoothly Irregular Waveguides

We study the ray propagation path in a plane smoothly irregular waveguide. The following two modes of ray propagation are possible: with reflections and without reflections from the waveguide walls. In each of these modes, the problem has an adiabatic invariant. We obtain an asymptotic formula for the value of the adiabatic invariant jump as the propagation mode changes.