The hodograph transformation in trans-sonic flow. I. Symmetrical channels

It seems likely that any general theory of compressible flow applicable to problems with regions both of sub- and supersonic flow (such problems have been called ‘trans-sonic’) must be based on the ‘hodograph transformation’ (due originally to Molenbroek 1890 and Chaplygin 1904). This is because in the hodograph plane, in which the independent variables are the magnitude and direction of the velocity, the equations of motion are linear; while in the physical plane they are not even approximately linear for trans-sonic problems. But the hodograph transformation presents difficulties quite apart from those of applying suitable boundary conditions. It has in fact singularities, notably near the sonic speed and the velocity at infinity. This fact considerably elaborates its use. In the present paper a study is initiated of the application of the hodograph transformation to trans-sonic problems by considering the steady plane adiabatic flow of a gas in symmetrical channels in which the velocity rises from zero at infinity on the left to a supersonic value at infinity on the right; this is a problem easier to begin on than those with a body inside the field of flow, which are known in practice to involve shock-waves and hence regions of non-adiabatic flow.