1. Compressible and incompressible flow. Small perturbations in an otherwise uniform stream of compressible fluid moving at supersonic speed are described by the approximate linearized equation for the velocity potential. When the stream flows in the z-direction, the equation assumes the form

where M is the Mach number of the flow and α2 is positive. In cylindrical polar coordinates (r, z), the equation may be written as

which is Laplace's equation in coordinates (iαr, z). We may therefore relate potentials of incompressible flow which are solutions of (1·2) to potentials of compressible flow which are solutions of (1·1).