Vortex sound: An alternative derivation of Mohring's formulation

摘要

In the 1964 theory of vortex sound by Powell the source strengths are the volume integrals of (ζ∧u), [u=flow velocity, ζ=▽∧u=vorticity, ( )≡( )/t], for dipole sound and of yx(ζ∧u)x" for quadrupole sound (y=local coordinate, yx=y · x/x with x=far-field observation point). M?hring solved the same inhomogeneous wave equation by a Green's function technique that transformed these to the volume integrals of 1/2(y)x and of 1/3yx(y)x, respectively. By considering fixed vortex rings of variable strength the first of the latter pair was given in 1964 by a simple physical argument, apart from a trivial geometrical equivalence. By considering that the radial flow velocity at a spherical bounding surface due to a system of vortex rings (as given by the so-called Biot–Savart analogy for incompressible flow), fixed but of variable strength, drives the contiguous acoustic fields, simple physical arguments are presented that yield the results for dipole and quadrupole vortex sound in M?hring's form.