Refraction of Waves from a Point Source into a Medium of Higher Velocity

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Abstract
A point source is placed in one medium and the fields in a second medium, separated from the first by an infinite plane boundary, are calculated for the case that the wave velocity in the second medium exceeds that in the first medium. Absorption is neglected and the problem is solved both by ray methods and by evaluating, by the method of steepest descents, of an exact solution of the wave equation. The agreement between the solutions by different methods permits a ready and expected interpretation of the wave solution. Rays incident on the boundary at angles exceeding the critical angle are totally reflected; however, directly transmitted energy penetrates to all points in the second medium. At points in the second medium outside the critical angle, near the surface and far from the source, the directly transmitted fields are much smaller than the fields, exponentially decaying from the surface, which result from totally reflected rays. This suggests an experiment to measure quantitatively the penetration of the fields into the second medium in total reflection, easily performable for the case of sound waves penetrating from air into water.