17 - Plane waves at boundaries

Summary

The reflection and transmission processes that occur when a travelling wave meets a discontinuity in a string, and the standing waves set up when incident and reflected waves are superposed, were analyzed in chapter 9. We were able to adapt the results to other situations, such as the reflection of electromagnetic plane waves meeting a dielectric surface at right angles, or the formation of acoustic standing waves in a pipe.

Some of the most interesting ways in which the presence of a boundary between two media can affect the behaviour of waves in those media become apparent only when the waves are travelling in directions other than normal to the boundary. Likewise, standing waves in a region of three-dimensional space have characteristics which we cannot learn about by studying standing waves on strings. In this chapter, therefore, we think about plane waves meeting plane boundaries, not in general at normal incidence.

As a preliminary step, we learn how to handle plane waves in three-dimensional problems.

Reflection and refraction

Although we began by discussing transverse waves on strings, we have seen already that the disturbance in a one-dimensional wave need not be concentrated along a line. In a sinusoidal travelling wave ψ(z, t) for which ψ is a quantity like acoustic pressure or the strength of an electric field, which can have a value at any point in space, the phase angle ωt – kz has the same value at all points which have the same value of z; these points lie on a surface perpendicular to the direction of propagation.