Inferences that can be drawn about the structure of the upper mantle from observations of surface‐reflected longitudinal body phases are discussed. In particular, some aspects of previous observational, statistical, and theoretical work on the phasepPare reviewed. Recently, mathematical work has thrown more light on the properties of the cusps of the surface‐reflected longitudinal phases. Seismological experiments are discussed that are designed to determine the actual behavior of the wave pulses near the cusps. Some preliminary attention has been given to this problem using a seismic array with linear dimension about 10 times greater than the observed wavelength. Computer analysis of the pulses observed at the array elements leads to estimates of the gradients of the travel‐time curves. Weighting must be arranged to take into account the measured lateral refraction and crustal delays at each element. Analysis of the deep‐focus earthquake of April 2, 1963, is presented as an