The Velocity Field and Birkhoff–Rott Integral for Nondecaying, Nonperiodic Vortex Sheets

The Birkhoff--Rott integral expresses the fluid velocity on a vortex sheet. This integral converges if certain quantities decay at horizontal infinity, but can also be summed over periodic images in the horizontally periodic case. However, nondecaying, nonperiodic cases are also of interest, such as the interaction of periodic wave trains with noncommensurate periods (i.e., spatially quasiperiodic solutions), or nonperiodic disturbances to periodic wave trains. We therefore develop a more general single formula for the Birkhoff--Rott integral, which unifies and extends the cases of decay and periodicity. We verify that under some reasonable conditions this new version of the Birkhoff--Rott integral is the restriction to the vortex sheet of an incompressible, irrotational velocity field, with continuous normal component but with a jump in tangential velocity across the vortex sheet. We give a number of examples of nondecaying, nonperiodic sheet positions and sheet strengths for which our assumptions may be verified. While we develop this in the case of two-dimensional fluids, we expect the methodology will apply equally well to three-dimensional fluids.