Three dimensional application of the complementary mild-slope equation

Yaron Toledo*, Yehuda Agnon

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


The complementary mild-slope equation (CMSE) is a depth-integrated equation, which models refraction and diffraction of linear time-harmonic water waves. For 2D problems, it was shown to give better agreements with exact linear theory compared to other mild-slope (MS) type equations. However, no reference was given to 3D problems. In contrast to other MS-type models, the CMSE is derived in terms of a stream function vector rather than in terms of a velocity potential. For the 3D case, this complicates the governing equation and creates difficulties in formulating an adequate number of boundary conditions. In this paper, the CMSE is re-derived using Hamilton's principle from the Irrotational Green-Naghdi equations with a correction for the 3D case. A parabolic version of it is presented as well. The additional boundary conditions needed for 3D problems are constructed using the irrotationality condition. The CMSE is compared with an analytical solution and wave tank experiments for 3D problems. The results show very good agreement.

Original languageEnglish
Pages (from-to)1-8
Number of pages8
JournalCoastal Engineering
Issue number1
StatePublished - Jan 2011
Externally publishedYes


  • Coastal and offshore engineering
  • Linear waves
  • Mild-slope equation
  • Stream function formulation


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