Lagrangian nonlinear progressive water waves on sloping bottoms

Yang Yih Chen, Bin-Da Yang, Hung Chu Hsu, Meng-Syue Li, Kuei Sen Yang

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Citations (Scopus)


In this paper, a new third-order Lagrangian asymptotic solution describing nonlinear water wave propagation on the surface of a uniform sloping bottom is presented. The model is formulated in the Lagrangian variables and we use a two-parameter perturbation method to develop a new mathematical derivation. The particle trajectories, wave pressure and Lagrangian velocity potential are obtained as a function of the nonlinear wave steepness ε and the bottom slope α perturbed to third order. The analytical solution in Lagrangian form satisfies state of the normal pressure at the free surface. The condition of the conservation of mass flux is examined in detail for the first time. The two important properties in Lagrangian coordinates, Lagrangian wave frequency and Lagrangian mean level, are included in the third-order solution. The solution can also be used to estimate the mean return current for waves progressing over the sloping bottom. The Lagrangian solution untangle the description of the features of wave shoaling in the direction of wave propagation from deep to shallow water, as well as the process of successive deformation of a wave profile and water particle trajectories leading to wave breaking. Comparing the theoretical results and experimental data shows good agreement.

Original languageEnglish
Title of host publicationProceedings of the 20th (2010) International Offshore and Polar Engineering Conference, ISOPE-2010
Number of pages9
Publication statusPublished - 2010 Sep 10
Event20th International Offshore and Polar Engineering Conference, ISOPE-2010 - Beijing, China
Duration: 2010 Jun 202010 Jun 25

Publication series

NameProceedings of the International Offshore and Polar Engineering Conference
ISSN (Print)1098-6189
ISSN (Electronic)1555-1792


Other20th International Offshore and Polar Engineering Conference, ISOPE-2010

All Science Journal Classification (ASJC) codes

  • Energy Engineering and Power Technology
  • Ocean Engineering
  • Mechanical Engineering


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