The assumption that the compressional waves of earthquakes follow the ordinary laws of refraction, the energy within any pencil of rays remaining permanently within that pencil, has been found to lead to too small amplitudes for the indirect waves from near earthquakes. In this paper a system consisting of two superposed compressible, but non-rigid, media is considered; the lower is supposed to transmit compressional waves with the greater velocity. It is found that ah explosion within the upper medium produces a disturbance at the upper surface involving the direct wave and all the reflected waves that might be expected; but in addition a wave is found that appears to have travelled along the interface with the velocity of sound in the lower medium. This indirect wave would have zero amplitude on the simple laws of refraction with plane boundaries. The variation of its amplitude with distance from the focus is in reasonable accordance with seismological observation, and its time of arrival agrees with that inferred from the laws of refraction for boundaries with slight curvature. But if the direct wave begins with a finite velocity, the indirect one will begin with a finite acceleration; in seismological language an iPg will be associated with an ePn. The indirect wave will also take longer than the direct one to give its maximum displacement.