Linearising shallow-water wave equations We are given the equations
$$\frac{\partial{u}}{\partial{t}}+u\frac{\partial{u}}{\partial{x}}+g\frac{\partial{h}}{\partial{x}}=0$$
and 
$$\frac{\partial{h}}{\partial{t}}+\frac{\partial{(hu)}}{\partial{x}} = 0$$
We are then asked

By linearising these equations about a uniform mean flow of speed $u_{0}$ and uniform thickness $h_0$, derive expressions for the phase and group speeds of linear shallow water waves. Are these waves dispersive?

Having attempted $u=u_0 + u'$ and $h=h_0 + h'$ I arrived at
$$\frac{\partial{u'}}{\partial{t}}+u_0\frac{\partial{u'}}{\partial{x}}+g\frac{\partial{h'}}{\partial{x}}=0$$
and 
$$\frac{\partial{h'}}{\partial{t}}+u_0\frac{\partial{h'}}{\partial{x}}+h_0\frac{\partial{u'}}{\partial{x}}=0$$
Which I don't see how they come out to be proper wave equations from which I can get the velocities.

Differentiating each equation with respect to $t$ and $x$ we have 
\begin{align}
\frac{\partial^2 u'}{\partial t^2} + u_0\frac{\partial^2 u'}{\partial t \partial x} + g \frac{\partial^2 h'}{\partial t \partial x} = 0 \label{1}\tag{1}\\
\frac{\partial^2 u'}{\partial t \partial x} + u_0\frac{\partial^2 u'}{\partial x^2} + g \frac{\partial^2 h'}{\partial x^2} = 0 \label{2}\tag{2}\\
\frac{\partial^2 h'}{\partial t^2} + u_0\frac{\partial^2 h'}{\partial t \partial x} + h_0 \frac{\partial^2 u'}{\partial t \partial x} = 0 \label{3}\tag{3}\\
\frac{\partial^2 h'}{\partial t \partial x} + u_0\frac{\partial^2 h'}{\partial x^2} + h_0 \frac{\partial^2 u'}{\partial x^2} = 0 \label{4}\tag{4}
\end{align}
To find the wave equation for $u'$, rearrange \eqref{4} to make $\frac{\partial^2 h'}{\partial t \partial x}$ the subject, and substitute into \eqref{1}:
$$
\frac{\partial^2 u'}{\partial t^2} + u_0\frac{\partial^2 u'}{\partial t \partial x} - g \left[u_0\frac{\partial^2 h'}{\partial x^2} + h_0 \frac{\partial^2 u'}{\partial x^2}\right] = 0
$$
Then rearrange \eqref{2} to make $\frac{\partial^2 h'}{\partial x^2}$ the subject and substitute this in as well. The resulting equation is
\begin{align}
0 &= \frac{\partial^2 u'}{\partial t^2} + u_0\frac{\partial^2 u'}{\partial t \partial x} - g h_0 \frac{\partial^2 u'}{\partial x^2} + u_0\left[\frac{\partial^2 u'}{\partial t \partial x} + u_0\frac{\partial^2 u'}{\partial x^2}\right]\\
&=\frac{\partial^2 u'}{\partial t^2} + 2u_0 \frac{\partial^2 u'}{\partial t \partial x} + (u_0^2 - gh_0) \frac{\partial^2 u'}{\partial x^2}
\end{align}
This is an inhomogeneous wave equation:
$$
\frac{\partial^2 u'}{\partial t^2} - (gh_0 - u_0^2) \frac{\partial^2 u'}{\partial x^2} = -2u_0 \frac{\partial^2 u'}{\partial t \partial x},
$$
note that it can be 'factored' into the form
$$
\left[\frac{\partial}{\partial t} + (u_0 + \sqrt{gh_0})\frac{\partial}{\partial x}\right]\left[\frac{\partial}{\partial t} + (u_0 - \sqrt{gh_0})\frac{\partial}{\partial x}\right] u' = 0
$$
A similar form can be found for $h'$ by substituting \eqref{2} and then \eqref{4} into \eqref{3}.
Using the wavelike ansatz $$u' = Ae^{-i(\omega t - kx)} $$ we find the dispersion relation $$-\omega^2 + 2u_0 \omega k - k^2(u_0^2 - gh_0) = 0$$ or (solving as a quadratic in $\omega$) $$\omega = \left(u_0 \pm \sqrt{gh_0}\right)k$$
