This paper considers the problem of the propagation of a shock wave down a non-uniform tube. Linearized solutions to the problem (Chester 1954) do not hold when the velocity behind the shock is near or at the sonic speed. By retaining appropriate non-linear terms of the flow equations, a solution is obtained which holds for all conditions behind the shock, and reduces to the linearized solution for conditions away from sonic.

The behaviour of supersonic or subsonic flow entering regions of expanding or contracting area changes is discussed. It is found that subsidiary shocks may be formed; these can be located and described using the present solution. Explicit solutions are given for the cases of supersonic or subsonic flow entering a region of linearly expanding or contracting area. The point of shock formation as well as the path of the subsidiary shock is obtained for the case in which the area contracts.