Multiple scales expansions near critical rays

The theory of geometrical optics breaks down whenever a ray is excited tangentially to a boundary. Examples of this include the total internal reflection of a wave incident upon the boundary between two media and the excitation of a surface wave. Model problems for both phenomena are considered, taking the incident fields to be a modulation of an appropriate plane wave. Using a multiple scales analysis, a description of the local field near the boundary point where the tangential ray is excited is obtained in each case. These analyses clearly indicate the mechanisms by which further rays are radiated from the boundary (forming the head wave and acoustic surface wave, respectively) and predict rapidly varying, nonspecular features in the total reflected field.