The Asymptotic Behaviour Near the Crest of Waves of Extreme Form.
Abstract
This paper concerns waves of permanent form on the free surface of an ideal liquid which is in two-dimensional, irrotational motion under the action of gravity. It considers only extreme waves, often called waves of greatest heights; each of these is the end-member of a one-parameter family of waves and is distinguished from other smaller members of the family by a sharp crest. Although this corner is physically unrealistic, oceanographers have given such idealised, extreme waves a great deal of attention since Stokes postulated their existence iln 1880. This paper is a contribution to the strict mathematical theory of extreme waves, which has emerged only since 1978. An asymptotic series is known that describes the flow near the crest, but it has never been proved whether the leading term in this expansion has a non-zero coefficient or not (and so whether it is in fact the leading term or not). The author shows that the coefficient is non-zero and determines its sign. The result should play a useful part in numerical computation of extreme waves. Keywords: Nonlinear integral equations; Analytic functions; Water waves.
Document Details
- Document Type
- Technical Report
- Publication Date
- Aug 01, 1985
- Accession Number
- ADA161064
Entities
People
- J. B. Mcleod
Organizations
- University of Wisconsin–Madison