An asymptotic theory for internal reflection in weakly inhomogeneous elastic waveguides

An asymptotic theory for internal reflection in the plane elastic waveguide, slowly varying along one of the longitudinal directions, is developed by the method of matched asymptotic expansions. The reflection takes place within cross-sections in which the vibration frequencies coincide with the cut-off frequencies. After transforming the original elasticity equations into a Schrödinger type equation, the results are derived in a general form. These results may be applied to other areas of mathematical physics, provided the governing equations allow such a transformation.