缓坡方程的推广

为了描述水波和强烈的环境流在非平整海底上的相互作用，运用无旋运动的Lagrangian变分原理，对经典的Berkhoff缓坡方程进行了改进。假定水流沿水深方向基本上保持均匀性，这正如潮流运动的特征。海底地形由慢变、快变两个分量叠加构成；慢变分量满足缓坡逼近假定，快变分量的波长与表面波波长为同一量级，但其振幅小于表面波的振幅。在以上假定条件下，得到了适用于非平整海底的推广型浅水方程和应用性更加广泛的波－流－非平整海底相互作用的一般缓坡方程，并且从理论上证明一般缓坡方程包含了以下3种著名的缓坡型方程：经典的Berkhoff缓坡方程；波－流相互作用的Kirby缓坡方程、Dingemans关于沙纹海底的缓坡方程。最后，通过与Bragg反射实验数据的比较，表明该模型可以准确地反映快变海底的典型地貌特征。

EXTENSIONS OF THE MILD-SLOPE EQUATION

A modification of the classic mild-slope equation of Berkhoff is developed for wa-ter waves propagating over strong ambient currents and an uneven bottom, using a Lagrangian formulation for irrotational motions. The current is essentially uniform with depth, as would be the case in many tidal flows. The bottom topography consists of two components: the slowly varying component which satisfies the mild-slope approximation, and the fast varying component with wavelengths on the order of the surface wavelength but amplitudes smaller than that of the surface wavelength. The extended shallow-water equations on uneven bottoms and a more widely applicable mild-slope equation involving wave-current-uneven bottom interactions are thus derived, and the mild-slope equation is verified against other theoretical results related to the mild-slope approximation by showing that it contains as special cases the classical mild-slope equation of Berkhoff, Kirby's extended mild-slope equation with current and Dingemans's mild-slope equation for rippled bed. Finally, the laboratory data from the Bragg reflection can be used as a verification of the model for describing accurately the typical features of geomorphy over rapidly varying topography.