Body waves that form an internal caustic attain a±π/2 phase shift in the vicinity of the point where the ray touches the caustic. This phase shift distorts a pulse propagating along the ray into its allied function. Asymptotic and numerically exact solutions for a particular model illustrate this property for waves on the back branch of a travel-time curve and the class of multiply reflected waves analogous to PdP waves in the Earth. Numerically exact solutions show that the π/2 phase shift is a high-frequency effect and that the phase shift tends toward zero for very low frequencies. The frequency dependence of the phase shift acts to maintain causality for broad-band pulses. Properties of the phase shift associated with internal caustics can be used to predict wave forms in a variety of applications; multiply reflected waves in a sphere (pP, PP, PPP) provide a simple example.