Shallow wave equations (First order quasilinear systems)

[gash789]

Homework Statement

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)

Homework Equations

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

The Attempt at a Solution

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

where b is a "sink or source of momentum". This implied to me that it is a rate of change of momentum such as
$P_{t}$
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
$h_{x}=\frac{\partial h}{\partial x}$

 

[lippyka]

u_{x}=\frac{\partial u}{\partial x}u_{t}=\frac{\partial u}{\partial t} P_{x}=\frac{\partial P}{\partial x}