Comparative measurements at large and small distances have been made on the pressure wave from a small underwater explosion. It is shown that: (1) there is a small departure from the $\frac{1}{r}$ law for the peak pressure, (2) there is no detectable change in shape or "spreading" of the pressure wave with propagation, and (3) the shock front is discontinuous within the limitations of the apparatus. A theory is developed which is in quantitative agreement with (1) and (3) but in disagreement with (2). The theory predicts a small spreading of the profile of the wave which is large enough to have been detected. The theory shows that a spherically diverging wave of any amplitude always becomes discontinuous eventually. In practice the distance over which it must propagate before becoming discontinuous may be enormous.