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The Birkhoff-Rott integral expresses the fluid velocity on a vortex sheet.This integral converges if certain quantities decay at horizontal infinity, butcan also be summed over periodic images in the horizontally periodic case.However, non-decaying, non-periodic cases are also of interest, such as theinteraction of periodic wavetrains with non-commensurate periods (i.e.spatially quasiperiodic solutions), or non-periodic disturbances to periodicwavetrains. We therefore develop a more general single formula for theBirkhoff-Rott integral, which unifies and extends the cases of decay andperiodicity. We verify that under some reasonable conditions this new versionof the Birkhoff-Rott integral is the restriction to the vortex sheet of anincompressible, irrotational velocity field, with continuous normal componentbut with a jump in tangential velocity across the vortex sheet. We give anumber of examples of non-decaying, non-periodic sheet positions and sheetstrengths for which our assumptions may be verified. While we develop this inthe case of two-dimensional fluids, the methodology applies equally well tothree-dimensional fluids.

The velocity field and Birkhoff-Rott integral for non-decaying, non-periodic vortex sheets