Wave refraction at an interface: Snell’s law versus Chapman’s law.

Abstract

Wave energy streamlines determine the paths along which energy is transported from a wave source. The energy streamlines provide insights into mechanisms of wave propagation and scattering and are often utilized to visualize wave fields. In contrast to rays, which are essentially an asymptotic, short‐wave concept, energy streamlines adequately represent arbitrary wave fields. However, usefulness of energy streamlines in studies of wave fields is limited by the fact that, unlike rays, no general laws governing energy streamline refraction are known. Here a simple refraction law is derived for energy streamlines of acoustic and linearly polarized electromagnetic waves. The refraction law is an extension of a result recently established by D. M. F. Chapman for sound refraction at plane interfaces of homogeneous media. While Snell's law of ray refraction at an interface is formulated in terms of the ratio of sound speeds in media in contact in the case of acoustic waves, Chapman's law implies that refraction of acoustic energy streamlines is controlled by the ratio of mass densities. Similarities and differences between Snell's and Chapman's refraction laws are discussed. It is shown that analysis of energy streamlines provides a new, helpful perspective on wave transmission through an interface.