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Edge waves: theories past and present.

R S Johnson1

  • 1School of Mathematics and Statistics, University of Newcastle upon Tyne, Newcastle upon Tyne, UK. r.s.johnson@ncl.ac.uk

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|March 16, 2007
PubMed
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This study explores classical water-wave theories for edge waves, detailing linear and nonlinear models. It presents exact solutions and discusses their application to wave run-up patterns.

Area of Science:

  • Fluid Dynamics
  • Oceanography
  • Wave Theory

Background:

  • Edge waves are a phenomenon in classical water-wave theory.
  • Existing theories for edge waves are reviewed, focusing on traveling waves.

Purpose of the Study:

  • To formulate governing equations and boundary conditions for edge waves.
  • To present Stokes' and Ursell's discoveries on traveling edge waves.
  • To explore linear, long-wave approximations, and weakly nonlinear extensions.

Main Methods:

  • Formulation of governing equations and boundary conditions.
  • Construction of the linear problem and its spectrum.
  • Application of Whitham's nonlinear extension and Gerstner's exact solution.
  • Multiple-scale analysis for edge waves over varying depths.

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Main Results:

  • Comparison of results from different approximations and flow types (irrotational vs. not).
  • An exact solution for edge waves derived from Gerstner's solution.
  • A shallow-water equation model with an exact edge wave solution.
  • Theoretical predictions for edge wave run-up patterns.

Conclusions:

  • The study provides a unified framework for comparing various edge wave theories.
  • Exact solutions and nonlinear models are presented for further study.
  • The research offers insights into edge wave behavior and mode interactions.