TOPOGRAPHICAL SCATTERING OF GRAVITY-WAVES

A systematic hierarchy of partial differential equations for linear gr avity waves in water of variable depth is developed through the expans ion of the average Lagrangian in powers of \del h\ (h = depth, Vh = sl ope). The first and second members of this hierarchy, the Helmholtz an d conventional mild-slope equations, are second order. The third membe r is fourth order but may be approximated by Chamberlain & Porter's (1 995) 'modified mild-slope' equation, which is second order and compris es terms in del(2)h and (del h)(2) that are absent from the mild-slope equation. Approximate solutions of the mild-slope and modified mild-s lope equations for topographical scattering are determined through an iterative sequence, starling from a geometrical-optics approximation ( which neglects reflection), then a quasi-geometrical-optics approximat ion, and on to higher-order results. The resulting reflection coeffici ent for a ramp of uniform slope is compared with the results of numeri cal integrations of each of the mild-slope equation (Booij 1983), the modified mild-slope equation (Porter & Staziker 1995), and the full li near equations (Booij 1983). Also considered is a sequence of sinusoid al sandbars, for which Bragg resonance may yield rather strong reflect ion and for which the modified mild-slope approximation is in close ag reement with Mel's (1985) asymptotic approximation.