Critical times of generating shocks on smooth one-dimensional pressure wakes in inviscid fluids

The balance equations which govern the unsteady one-dimensional motion of an inviscid compressible fluid are considered, taking into account pressure wakes which spread through compressible media. Multiple-valued solutions arise in connection with the nonlinearity of the balance equations. The considered motion represents the flow in an infinite tube caused by a small periodical movement of a piston at the end of the tube. Previous studies concerning the one-dimensional equation describing the flow have only provided a rough estimate of the time t(0) when the classical solution breaks down. In the present investigation, an exact analytical formula for t(0) is derived.