The shallow water equations are:
\begin{aligned} &\frac{du}{dt} = -g\frac{\partial h}{\partial x} + fv\\ &\frac{dv}{dt} = -g\frac{\partial h}{\partial y} - fu\\ &\frac{dh}{dt} = -h(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}) \end{aligned} where h is the height (or depth) of the shallow water, $u$ is the eastward velocity, $v$ is the northward velocity, g is the acceleration of gravity and f is the coriolis parameter.
my problem is how to linearize these equations. From the linearization theorem, I know that I should take \begin{aligned} &u = \overline{u} + u'\\ &v = \overline{v} + v'\\ &h = \overline{h} + h' \end{aligned} then take them into the original equations, but I just do not know how. From my understanding, take the equation of $h$ as an example, the result should be \begin{equation} \frac{\partial}{\partial t}(\overline{h} + h') + (\overline{u} + u')\frac{\partial}{\partial x}(\overline{h} + h') + (\overline{v} + v')\frac{\partial}{\partial y}(\overline{h} + h') = -(\overline{h} + h')(\frac{\partial \overline{u}}{\partial x} + \frac{\partial \overline{v}}{\partial y} + \frac{\partial u'}{\partial x} + \frac{\partial v'}{\partial y}) \end{equation} by using the equation \begin{aligned} \frac{d\overline{h}}{dt} = -\overline{h}(\frac{\partial \overline{u}}{\partial x}+\frac{\partial \overline{v}}{\partial y}) \end{aligned} and ignore the terms like $u'\frac{\partial h'}{\partial x}$ and $v'\frac{\partial h'}{\partial y}$, the result I obtained is \begin{equation} \frac{\partial h'}{\partial t} + \overline{u}\frac{\partial h'}{\partial x} + \overline{v}\frac{\partial h'}{\partial y} = -\overline{h}(\frac{\partial u'}{\partial x} + \frac{\partial v'}{\partial y}) - h'(\frac{\partial \overline{u}}{\partial x} + \frac{\partial \overline{v}}{\partial y}) \end{equation} but in my textbook, the result is just \begin{equation} \frac{\partial h'}{\partial t} = -\overline{h}(\frac{\partial u'}{\partial x} + \frac{\partial v'}{\partial y}) \end{equation} so I want to ask that where did I make a mistake? Actually, I got the same problem in the linearization of the equations of $u$ and $v$.