Edge-Layer Theory for Shallow-Water Waves over a Step —Reflection and Transmission of a Soliton

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  • Sugimoto Nobumasa
    Department of Mechanical Engineering, Faculty of Engineering Science, Osaka University
  • Nakajima Noriyuki
    Department of Mechanical Engineering, Faculty of Engineering Science, Osaka University
  • Kakutani Tsunehiko
    Department of Mechanical Engineering, Faculty of Engineering Science, Osaka University

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  • Edge-Layer Theory for Shallow-Water Waves over a Step--Reflection and Transmission of a Soliton
  • Edge Layer Theory for Shallow Water Wav

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Abstract

This paper develops an ‘edge-layer’ theory for weakly nonlinear shallow-water waves propagating over a step bottom. Introducing an ‘edge-layer’ in the vicinity of the step, the matched-asymptotic expansion method is applied to establish ‘reduced’ boundary conditions relevant to Boussinesq equations in the uniform shallow-water regions extending on the outside of the ‘edge-layer’. From the matching conditions, two ‘reduced’ boundary conditions are derived which incorporate the higher-order corrections to the well-known Lamb’s conditions. On applying the conditions thus derived, a problem of reflection and transmission of a single soliton incident upon the step is discussed. It is found that for a moderate depth-ratio, Lamb’s conditions still provide correct boundary-values except the phase shift at the step.

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