Shallow water equation



The study of wawe motion in shallow water leads to a system of conservation laws. To derive the 1D equations, we consider fluid in channel and assume that the vertical velocity of the fluid is negligible and the horizontal velocity v(x,t) is roughly constant through any vertical cross section. This is true if we consider small amplitude wawes in a fluid that is shallow relative to the wawelength.
We now assume the fluid is incompressible , so the density  is constant. Instead the height h(x,t) varies, and so the total mass in [x1; x2] at time t is

The momentum at each point is pv(x,t) and integrating this vertically gives the mass flux to be .The constant p drops out of the conservation of mass equation, whixh taken the familar form

The conservation of mementum equation also takes the same form as in the Euler equation

but now the pressure p is determined from hydrostatic law. This gives after  cancelling

Finally the explicit dependence on g can be eliminated by introducing the variable . The system for the 1D shallow water equations the become :


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