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Shallow wave equations (First order quasilinear systems)

  1. Dec 6, 2011 #1
    1. The problem statement, all variables and given/known data

    I am trying to show the shallow wave equations (Pg 35 Ockendon). For a shallow channel of water, with one free surface h(x,t) that runs parallel to the x-axis, the fluid has a constant density and the pressure is nearly hydrostatic and equal to P(x,t)=ρg(h-y)

    2. Relevant equations
    I have already shown that
    [itex]
    h_{t}+[hu]_{x}=0
    [/itex]
    But I am struggling to show
    [itex]
    \rho\left(u_{t}+u u_{x}\right)=-P_{x}=-\rho g h_{x}
    [/itex]


    3. The attempt at a solution

    I have begun by trying to show using the Navier Stokes equation that
    [itex]
    \rho\left(u_{t}+u u_{x}\right)=b
    [/itex]

    where b is a "sink or source of momentum". This implied to me that it is a rate of change of momentum such as
    [itex]
    P_{t}
    [/itex]
    of which could be solved using the original equation for the pressure, but the equation the book has suggests it should be a spatial derivative. This does not make sense to me?

    Just to be clear
    [itex]h_{x}=\frac{\partial h}{\partial x} [/itex]
     
  2. jcsd
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