On vortex waves in compressible fluids. I The hollow-core vortex

In 1880, Kelvin published his analysis of small-amplitude waves carried by a straight-line vortex in an incompressible, inviscid fluid. The most significant of these waves are the 'bending modes', in which the axis of the vortex becomes helical. The corresponding angular wavenumber, m, is 1, but Kelvin found solutions for all M and all axial wavenumbers k. For the hollow-core vortex, the model also studied here, he found two modes for each k and m. For ka much less than 1, where a is the core radius, the waves are of two types. The majority are 'fast', with frequencies W of the order of k/a(2), where k is the vortex circulation. The axisymmetric (m = 0) modes and one bending mode are 'slow'; for these, omega is of the order of kk(2), apart from a logarithmic factor.