Total Reflection, Phase Changes, and Caustics

Abstract

The simple theory and pertinent acoustic evidence, relating to the classic frequency-independent phase changes upon total reflection, are briefly reviewed. Of particular interest are experimental results showing the effect of such phase changes in modifying the shape of propagating pulses. Thus, it is well known that, in the plane wave approximation, if a narrow pressure pulse is totally reflected off the plane boundary of a high velocity medium, the reflected pulse may develop a tail. This is interpreted physically by the fact that a frequency independent phase change φ corresponds to a delay τ=φ/ω, i.e., a delay inversely proportional to the angular frequency ω. The mechanism of tail formation is thus a delay. In the process of total reflection, it takes time for the energy to penetrate into the high-velocity medium and to be restored back into the medium of incidence. Of particular interest in both ocean and atmospheric acoustics is the case of total internal reflection in a medium of continuously increasing velocity. Standard asymptotic approximations, identical to those used in quantum theory, show that when certain conditions are fulfilled, an effective phase change of π/2 takes place. This of course implies the deformation of pulses upon total reflection, appearance of tails, etc. Some remarkably clear experimental evidence showing this effect has been obtained in recent years. It has sometimes been proposed that similar effects are predicted by the so-called phase changes at a caustic. This is not so. The subject of phase shifts at caustics and loci is well known to students of optics and diffraction theory. The theory shows that factors of eiπ/2 (caustics) or eiπ (foci) make their appearance in the solutions describing the local field in the simple harmonic case. However, the mechanism by which a caustic or focus is produced is important in defining precisely the significance and limitations of the eiπ/2 or eiπ factors in any given case, especially when applied to the spectra of broad-band disturbances traveling through the system. Thus, in the latter case, if one accepts the π/2 caustic phase shift at face value and applies it to a narrow pulse, one runs afoul of the causality principle. On the other hand, it is obvious that a caustic is simply a geometrical construct: The only thing that happens there is an interference between arrivals along neighboring rays, a phenomenon which cannot, in a linear approximation, leave any permanent imprint upon a pulse traveling through such a region. Thus, these “phase shifts” are localized effects, which cannot be applied indiscriminately to the spectra of pulses traveling along the edges of caustics or through foci. [Hudson Lab. of Columbia Univ. Informal Documentation No. 144. This work was supported by the U. S. Office of Naval Research.]