On the total reflection of plane waves

Abstract

The reflection and refraction of transverse plane waves at an interface parallel to the direction of polarization is considered when the incident wave is of arbitrary shape and the angle of incidence exceeds the critical angle. It is shown that the solution of this problem depends on the determination of a plane harmonic function h(ε η) satisfying the condition. (∂h∂ξ)η=0-λ(∂h∂η)η=0=2f′(ξ),where λ is a known constant and f(ε) a given function. By using the half-plane analogue of Poisson's formula, h(ε, η) can be expressed in terms of f'(ε).The results show that the reflected and transmitted disturbances exist everywhere at all times even when the incident wave has a well-defined front, and that the transmitted disturbance due to an incident simple pulse is of the order of the reciprocal of the distance from the interface, when this distance is large.It is pointed out that the same analysis can be applied to the treatment of the total reflection of electromagnetic waves of arbitrary shape.Finally, the propagation of waves of arbitrary shape over the surface of a semi-infinite elastic solid is considered and shown to be possible when the velocity of propagation is that of Rayleigh waves. © 1948 Oxford University Press.