TY - JOUR

T1 - A time-dependent nonlinear mild-slope equation for water waves

AU - Beji, S.

AU - Nadaoka, K.

PY - 1997

Y1 - 1997

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0030965930&partnerID=8YFLogxK

U2 - 10.1098/rspa.1997.0018

DO - 10.1098/rspa.1997.0018

M3 - Article

AN - SCOPUS:0030965930

SN - 1364-5021

VL - 453

SP - 319

EP - 332

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

IS - 1957

ER -