The vibrations of an infinite system of vortex rings

The vibrations that may be set up and maintained in the central filament of a single vortex ring of small but finite section have been investi­gated by Thomson and others, but no corresponding investigation appears to have been undertaken for a system of parallel rings, although the matter is of some importance in connection with the state of motion behind a moving body. In a previous paper the authors have examined the stability of an infinite system of equal vortex rings situated in parallel planes with their centres evenly spaced along an infinite line and with their planes at right angles to that line. Instability was there found to occur for disturbances confined to displacements of the centre of each ring along the central axis, the filament of each ring still remaining circular. In the present paper the investigation is extended to deformation of the vortex filaments, and some interesting conclusions are drawn regarding natural modes of vibration of the infinite system of vortex rings, such as may occur without the longitudinal instability referred to in the previous paper becoming apparent. It is found, for example, that for any given ratio of radius of ring section to radius of ring there exists a critical ratio of ring spacing to radius, separating the region of stable oscillation from that of instability, a result in some respects closely analogous to that found by Kármán for the stability of two infinite parallel rows of rectilinear vortices.