ABSTRACT

The multigrid technique IS used to solve a variant of the mild-slope equation which takes into account wave-current Interaction The different versions of wave-current Interaction equations are first compared with each other and then the governing equation IS recast Into a form suitable for treatment by the multigrid technique Experimental data are used to test the model Results from the present model as well as from a hyperbolic solution of the same governing equation are presented for the case examined by LIU (1983)

INTRODUCTION

The mild-slope equatlon (Berkhoff, 1972) is valid over the complete range of water depths for pure wave problem subject to slope limitations This means that the effects of bottom inhomogeneties are included in the equation The effect of a current IS neglected In practical coastal engineering problems significant tidal currents can occur near a river inlet or a harbour entrance When currents and waves are In the same direction, this results In the lengthening of waves and reduction of wave heights However, waves are shortened and steepened by opposing currents, often to the extent of inducing breaking Therefore, a valuable contribution would be to add terms to the mild slope equation representing the effect of a current

For the purpose of computing practical cases, where slow variations of depth and current can be both taken into account. Booij (1981) has generalized the original mild slope equation (Berkhoff, 1972) using Lagrangian theory As he pointed out, the derivation IS only valid for current fields with zero rotation However. In practical applications these conditions rarely occur and rotational current fields should be considered Booij also compared his wave-current equation with the wave-action equation which is defined as the basic equation of pure wave refraction By neglecting higher order terms, his equation can be split into two equations corresponding to real part and imaginary part The Imaginary part is the same as the wave-action equation In order to apply his equation to large area, he derived a parabolic equation from the original wave-current interaction equation

Liu (1983) derived a similar equation to Boolj's model for wave-current interaction A parabolic method was then used to solve the equation To demonstrate the application of the parabolic approximation to wave-current interactions, numerical calculations were performed for a problem that was originally studied by Arthur (1950) The case IS that of a current system existing on a uniform beach with a slope of 1/50

Kirby (1984) presented a careful study to wave-current interactions He derived a new equation by using the Lagrangian formulation which was also used before by Booij (1981) His investigation demonstrated that Booij's equation, after neglecting the second order terms, represents the correct mild-slope equation for linear wave current Interaction motion, although the dynamic free surface boundary condition which Booij applied to derive the equation was incorrect Due to limitations in the procedure used to derive the equation, Liu's model is not a complete version as also pointed out by Kirby

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