Response of a filamentary vortex to sound

The motion of a test vortex filament induced by a sound wave of frequency high in comparison with the vortex time scales, at low Mach numbers, is studied. To leading order the filament moves with the local speed of the sound wave. The small but finite compressibility introduces retardation effects that modify this behavior: for example, there is a self-induction effect for a point vortex. A differential equation, depending on one cutoff parameter that quantifies this assertion both in two and three dimensions, is derived in a localized induction approximation. This is done by use of a variational principle that governs inviscid irrotational adiabatic flow. The velocity and pressure, generated by the vortex, are expressed in terms of a multivalued velocity potential, as convolutions of a Green’s function and a source localized on the filament. Substitution of these expressions into the action, whose extrema give the equations of fluid flow, renders it a functional of the vortex filament history. Its extrema, with respect to variations of this history, determine the equation of vortex evolution. The multivalued velocity potential is used to recast the usual theory of vortex sound. Incompressible vortex dynamics is used as an example to introduce and illustrate these methods.