Impact on the surface of a compressible liquid. 2. Focusing of a weak, near-cylindrical shock wave

The propagation of a weak shock wave produced by an abrupt impact on a rigid boundary of a slightly compressible liquid is considered. Initially the liquid is at rest and occupies the region enclosed by a circular cylindrical surface. At some instant of time the points of this surface obtain velocities directed radially inwards. The velocity value depends on both the time and the angular coordinate and is small compared with the sound velocity. The liquid flow is assumed to be plane. A weak shock wave is formed under the impact and propagates to the centre of the liquid region. Initially the shock front is circular and bends later on due to non-linear effects. The form of the front is very complicated near the centre and depends on the distribution of the initial impact velocity. The evolution of the shock front at this stage is described by the geometrical acoustic theory. As a result a local focusing of separate parts of the shock wave is observed. The position of the shock wave and the liquid flow behind it are described within the framework of the transonic approximation within those zones. The structure of the liquid flow close to the focusing points is shown to be unaffected by any details of the external action and to be described by the solution of the non-linear boundary problem free from parameters.