A Lagrangian Finite Element Analysis of Gravity Waves in Water of Variable Depth

The paper is concerned with the problem of gravitational wave propagation in water of variable depth. The problem is formulated in the Lagrangian description, and the ensuing equations are solved numerically by a finite element method. In computations a convecting mesh that follows the material fluid particles is used. As illustrations, results of numerical simulations carried out for plane gravity waves propagating over bottoms of simple geometry are presented. For parameters typical of a laboratory flume, the transformation of a transient wave, generated by a single movement of a piston-like wave maker, is investigated. The results show the evolution of the free-surface elevation, displaying steepening of the wave over sloping beds and its gradual attenuation in regions of uniform depth.