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
[x_{1}; x_{2}] 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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