The Total Internal Reflection of SH Waves

In the first part of this paper we discuss the reflection of simple harmonic plane SH waves at angles of incidence greater than the critical angle at the interface between two semi-infinite elastic media. The change of phase which occurs on reflection is related to the rate at which energy crosses the interface. We then calculate the rate at which energy travels parallel to the interface in the inhomogeneous wave and the effect that this has on the group velocity of Love waves. The total reflection of a plane SH pulse is studied in the latter part of the paper. The expression for the reflected pulse involves the use of allied functions as defined by Titchmarsh (1937 § 5.1). General results concerning the allied function are drawn in the appendix from a few conditions imposed on an otherwise general function. A plane pulse has physical meaning as an approximation to a bounded plane pulse or a spherical pulse. We show here that the approximation is no longer valid for incident angles near π/2 or near the critical angle. The pulse which is usually known as the head wave is not given by this approximation but we are able to study the displacements which are forerunners of the reflected pulse and which arrive ahead of the time predicted for the reflected pulse by ray theory. A few special cases are given and deductions are made about the reflected wave when the incident pulse is of symmetric form with a central maximum.