• Journal of the Korean Society for Marine Environment & Energy
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• v.13 no.3
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• pp.174-180
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• 2010

The inhomogeneous Helmholtz equation is introduced for variable water depth and potential function and separation of variables are introduced for the derivation. Only harmonic wave motions are considered. The governing equation composed of the potential function for irrotational flow is directly applied to the still water level, and the inhomogeneous Helmholtz equation for variable water depth is obtained. By introducing the wave amplitude and wave phase gradient the governing equation with complex potential function is transformed into two equations of real variables. The transformed equations are the first and second-order ordinary differential equations, respectively, and can be solved in a forward marching manner when proper boundary values are supplied, i.e. the wave amplitude, the wave amplitude gradient, and the wave phase gradient at a side boundary. Simple spatially-centered finite difference numerical schemes are adopted to solve the present set of equations. The equation set is applied to two test cases, Booij’ inclined plane slope profile, and Bragg’ wavy bed profile. The present equations set is satisfactorily verified against other theories including the full linear equation, Massel's modified mild-slope equation, and Berkhoff's mild-slope equation etc.