Wave Refraction and Wave Height Variation Due to Current

Abstract

This paper deals with the wave refraction due to current by a theoretical treatment and a numerical model. The theory of refraction of deep-water waves traversing a simple horizontal current with vertical axis of shear by Longuet-Higgins and Stewart is extended to the case of shallow-water waves. It is found that the wave direction and wave height of shallow-water waves refracted by the current vary more rapidly than those of deep-water waves, and the variation of wave height is expressed as the product of the similar coefficients to the shoaling and refraction coefficients and a coefficient representing the effect of the radiation stress. A numerical model is presented for the wave refraction due to current. The variation of wave direction is calculated with the wave orthogonal equation along a path propagating in the direction of the sum of the wave velocity and the current, which is derived from the irrotational condition of wave number. The variation of wave height is also calculated with the equation of wave energy conservation in current derived by Longuet-Higgins and Stewart along another path propagating in the direction of the sum of the group velocity and the current. The numerical results agree roughly with the theoretical results, and the validity of the numerical model is roughly confirmed. The effects of the relative wave velocity to the current, the derivatives of wave direction with respect to the horizontal coordinates and the radiation stress on the wave refraction due to current are found to be not negligible.