Perturbation analysis of nonlinear partial reflected wave on a sloping bottom

Meng Syue Li, Yang Yih Chen, Qingping Zou, Hung Chu Hsu

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In this paper, we use the perturbation method to develop a new mathematical derivation to describe nonlinear partial standing wave over uniformly sloping bottoms. In the Lagrangian coordinate system, the particle trajectories are obtained as a function of the nonlinear order parameter ɛ and the bottom slope α to the second order of the perturbation. The setups and mean return flow for an arbitrary bottom in the Lagrangian framework are also found as part of the solutions. The direct influences of reflection coefficient, wave steepness and bottom slope on the surface profiles of partially reflected waves are also derived. This nonlinear analytical solution is verified by reduction to the classical Stokes solution of progressive waves in both the deep water and constant depth limit. This solution allows to describe the successive deformation of partial reflected wave profiles and water particle trajectories prior to wave breaking to be described. The dynamic properties including mass transport, and Lagrangian mean level and radiation stress for nonlinear partially reflected waves on various sloping bottoms are investigated.

Original languageEnglish
Article number103885
JournalNonlinear Analysis: Real World Applications
Early online date28 Mar 2023
Publication statusPublished - Oct 2023


  • Lagrangian
  • Nonlinear waves
  • Partial reflected wave
  • Particle trajectory
  • Radiation stress
  • Sloping bottom

ASJC Scopus subject areas

  • Analysis
  • General Engineering
  • Economics, Econometrics and Finance(all)
  • Computational Mathematics
  • Applied Mathematics


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