Overhanging and touching waves in constant vorticity flows

Vera Mikyoung Hur, Miles H. Wheeler

Research output: Contribution to journalArticlepeer-review


We show the existence of periodic traveling waves at the free surface of a two dimensional, infinitely deep, and constant vorticity flow, under gravity, whose profiles are overhanging, including one which intersects itself to enclose a bubble of air. Numerical evidence has long suggested such overhanging and touching waves, but a rigorous proof has been elusive. Crapper's celebrated capillary waves in an irrotational flow have recently been shown to yield an exact solution to the problem for zero gravity, and our proof uses the implicit function theorem to construct nearby solutions for weak gravity.

Original languageEnglish (US)
Pages (from-to)572-590
Number of pages19
JournalJournal of Differential Equations
StatePublished - Nov 25 2022


  • Constant vorticity
  • Crapper
  • Free surface waves
  • Overhanging
  • Periodic traveling
  • Touching

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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