Impact on the surface of a compressible liquid. 1. Propagation of a weak, nearly plane shock wave

The evolution of a weak, nearly plane shock wave produced by the impact on the plane boundary of a compressible liquid is considered. At the initial moment the liquid is at rest and occupies the lower half-plane. Then the points of its boundary get instantly velocities directed into the liquid domain. This leads to the formation of a shock wave the intensity of which is non-uniform due to a non-uniform distribution of the impact velocities. Initially the shock wave is plane but then it bends due to the non-linear effects and can later be focused. To analyze the liquid flow, the method of matched asymptotic expansions is used. For finite times the flow and the evolution of the shock wave are described within the framework of the acoustic approximation. For large times the flow becomes non-linear, and the form of the shock front depends essentially on the characteristics of the liquid flow behind it. If the non-uniformity of the impact velocity distribution is slight then the focusing of the shock wave is shown not to occur. The influence of viscosity of the liquid on the structure of its motion is discussed.