(adsbygoogle = window.adsbygoogle || []).push({}); 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]

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# Shallow wave equations (First order quasilinear systems)

Can you offer guidance or do you also need help?

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