On the transformation of the linearized equation of unsteady supersonic flow

Abstract

The linearized potential equation for unsteady motion in frictionless, supersonic flow is transformed from the classical wave equation to the canonical form ϕ x x − ϕ y y − ϕ z z = ϕ τ τ {\phi _{xx}} - {\phi _{yy}} - {\phi _{zz}} = {\phi _{\tau \tau }} with the aid of a modified Lorentz transformation. Possible invariant transformations of the latter, including the classical Lorentz transformation, are discussed. Eleven coordinate systems (each of which has its counterpart in the classical theory of the wave equation) permitting separation of variables are set forth, their derivation being based on the analogy between the hyperbolic metric defined by ( d s ) 2 = ( d x ) 2 − ( d y ) 2 − ( d z ) 2 {\left ( {ds} \right )^2} = {\left ( {dx} \right )^2} - {\left ( {dy} \right )^2} - {\left ( {dz} \right )^2} and the Euclidean (Cartesian) metric. A few practical applications are indicated.