Singularities in the complex physical plane for deep water waves

Authors:
Chao Xie;Gregory Baker
Affiliations:
The Ohio State University, Department of Mathematics, Columbus, OH;The Ohio State University, Department of Mathematics, Columbus, OH
Venue:
AMERICAN-MATH'10 Proceedings of the 2010 American conference on Applied mathematics
Year:
2010

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Abstract

A boundary integral technique is used to simulate deep water wave motion. A spectral procedure is used to form-fit the Fourier spectrum of the curvature of the wave profile to a prescribed asymptotic expression. The fit provides information on the power and location of the closest curvature singularity to the real axis of the complex arclength plane. This singularity proves to be a pole singularity in the complex arclength plane, and is not an artifact of the parametrization. It approaches the real axis when a plunging breaker occurs and wanders above some level in the complex arclength plane for non-breaking waves. This singularity is found theoretically equivalent to Tanveer's result. When the surface elevation is viewed as a function of horizontal distance x, a square root type singularity occurs in the complex x plane. It corresponds to a breaking wave when it reaches the real axis of the horizontal coordinate.