The breakdown of the hodograph transformation for irrotational compressible fluid flow in two dimensions

Abstract

It has been noticed by various authors that the use of the hodograph transformation for problems of compressible fluid flow leads to flows having singularities where the acceleration becomes infinite, streamlines have cusps and the solution of the equations of motion in the hodograph plane ceases to represent any possible physical flow. In this paper we investigate the nature of the singularities which can occur in the representation of one plane upon another and apply our results to the hodograph transformation. Finally, we discuss the significance for practical purposes of such singularities and in particular their connexion with the occurrence of shock waves.