三维缓变流场上波浪折射-绕射的缓坡方程

THE MILD-SLOPE EQUATION FOR REFRACTION- DIFFRACTION OF SURFACE WAVES ON THREE DIMENSIONAL SLOWLY VARYING CURRENTS

运用Ｌｕｋｅ变分原理，建立了波浪在三维缓变流场中和缓变海底上折射、绕射的一般缓 坡方程．据此给出了在几何－光学逼近（△Ｓ）２＝ｋ２有效时，波浪、环境流和海底坡度必须满 足的若干条件．对一般缓坡方程进行了分类，在一种特定流场结构的假定下，得到了方程的行 波解．

When deep ocean waves enter shoaling waters, their height and direction undergo significant modification due to interaction with bottom topographical variations and strong ambient currents, exerting a direct and important influence on sediment transport, ship navigation, and forces on coastal structures. Using Luke's variational principle, the general time-dependent mild- slope equation for refraction-diffraction of surface waves over slowly varying media of topography and three dimensional currents is derived, which can be reduced to Kirby's mild-slope equation for wave-current interaction when neglecting the vertical effect of ambient currents. The linear approach to the boundary value problem involving the wave propagating over the slowly changing bottom and the slowly varying three dimensional currents gives a particular representation for the potential φ and the free surface displacement , in which a nonlinear scale parameter ε is introduced. Based on the general two dimensional solution of Laplace equation on the horizontal bottom, the vertical structure of the velocity field is adopted and supposed to be hold locally for the mild-slope bottom. An alternative form of the mild-slope equation for time-harmonic waves is also obtained, which is suitable for parabolic wave model, and valid for large current velocities,that is for velocities |U| which are compaxable to the wave celerity C and the group velocity CO. For the geometric- optic approximation (△S)2 = k2 to be valid, a number of conditions concerning water wave, ambient current and bottom slope should be met. It is shown that the time-harmonic mild-slope equation can be divided into three types depending on the following conditions: hyperbolic, if |U|2 - CCg > 0, parabolic, if |U|2 - CCg = 0; elliptic, if |U|2 - CCg < 0. Finally, assuming a steady state current structure, all solutions of travelling waves are founded.