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4.2 Long Surface Gravity Waves 71Fig. 4.4 Snapshot of a shallow-water wave of 1 m in amplitude and 500 m in wavelength in a 20 mdeep ocean. Shown are sea-level elevation and vertical displacements of flui parcels at selecteddepths4.2.7 Numerical GridWe use a spatial grid of constant grid spacing in which velocity grid points arelocated halfway between adjacent sea-level grid points (Fig. 4.5).Fig. 4.5 The staggered grid. The cell index k refers to a certain grid cell. Water depth is calculatedat sea-level grid points4.2.8 Finite-Difference SchemeOn the basis of the staggered grid (see Fig. 4.5), the momentum equation (4.12) canbe written in finite-di ference form as:u n+1k= u n k − Δt g( η n k+1 − ηn k)/Δx (4.17)where n is time level, k is grid index, Δt is time step, and Δx is grid spacing. Acontrol volume (Fig. 4.6) is used to discretise equation (4.3). Accordingly, we canwrite this equation as:η ∗ k = ηn k − Δt ( u n+1kh e − u n+1k−1 h w)/Δx (4.18)