Real and apparent invariants in the transformation of the equations governing wave-motion in the general flow of a general fluid

Abstract

The ten equations are derived that govern, to the first order, the propagation of small general perturbations in the general unsteady flow of a general fluid, in three space-variables and time. The condition that any hypersurface is a wave hypersurface of these equations is obtained, and the envelope of all such wave hypersurfaces that pass through a given point at a given time, i. e. the wave hyperconoid, is determined. These results, which are all invariant under galilean transformation, are progressively specialized, through homentropic flow and irrotational homentropic flow, to steady uniform flow, for which both the convected wave-equation and the standard wave-equation, with their wave hypersurfaces, are finally recovered. A special class of reference-frames is considered, namely those whose origins move with the fluid. It is then shown that, for observers at the origins of all such reference frames, the wave hypersurfaces satisfy specially simple equations locally. These equations are identical with those for waves in a uniform fluid at rest relative to the reference frame, except that the wave speed is not constant but varies with position and time in accordance with the variable mean flow. These specially simple equations appear to be invariant for galilean transformations between all such observers. These results are briefly applied, in reverse order, to Maxwell’s equations, and to equations more general than Maxwell’s, for the electric and magnetic field-strengths.