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76 4 Long Waves in a ChannelFig. 4.8 Structure of the FORTRAN code for the following exercises4.3 Exercise 5: Long Waves in a Channel4.3.1 AimThe aim of this exercise is to simulate the progression of shallow-water surfacegravity waves in a channel of uniform water depth.4.3.2 InstructionsWe employ the finite-di ference Eqs. (4.17), (4.19) and (4.21) for a one-dimensionalchannel with closed ends under a variety of initial and forcing conditions. In thefollowing applications, the channel has a length of 1000 m being resolved by a gridspacing of Δx = 10 m. We choose 101 grid cells in the x-direction plus anothergrid cell on each end of the channel as boundary grid points. With the choice of anuneven number of grid points, the centre of the channel is define by a single gridcell.Undisturbed water depth is set to 10 m. Dry boundary cells are assigned a waterdepth of zero. Two different forcing scenarios are considered (Fig. 4.9). In Scenario1, we commence the simulation with a 110-m wide region centred in the channel inwhich sea level is initially 1 m higher than elsewhere. This scenario is referred to as“dam-break simulation”.In Scenario 2, we place a wave paddle in the middle of the channel and let the sealevel oscillate with an amplitude of 1 m and a period of 20 s. In both scenarios, thesolutions are explored for different values of the parameter ɛ in the Shapiro filte .The choice of ɛ = 0 switches off this f lter. The time step is set to Δt = 0.1 s, whichsatisfie the CFL stability criterion.