A time-dependent nonlinear mild-slope equation for water waves

S. Beji*, K. Nadaoka

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

40 Citations (Scopus)


A weakly nonlinear and dispersive water wave equation, which in linearized form yields a new version of the time-dependent mild-slope equation of Smith & Sprinks (1975), is derived. The applicable spectral width of the new wave equation for random waves is found to be more satisfactory than that of Smith and Sprinks (1975). For very shallow depths the equation reduces to the combined form of Airy's nonlinear non-dispersive wave equations; if the lowest-order dispersion is retained it produces the combined form of Boussinesq's equations. In the deep-water limit the equation admits the second-order Stokes waves as analytical solutions. Furthermore, by introducing a right-moving coordinate transformation, the equation is recast into a unidirectional form, rendering the KdV equation in one limit while reproducing the second-order Stokes waves in the other.

Original languageEnglish
Pages (from-to)319-332
Number of pages14
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number1957
Publication statusPublished - 1997


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