Context: Fundamental properties (behavior) of infinitesimal disturbances to flow. The coordinate system is placed on the bottom of a channel. Basic linearized equations per unit width are denoted as:
$$\frac{\partial h'}{\partial t}+{{h}_{o}}\frac{\partial U'}{\partial x}+{{U}_{o}}\frac{\partial h'}{\partial x}=0 , \tag{1}$$
$$\begin{align} &\frac{\partial {U}'}{\partial t}+{{U}_{o}}\frac{\partial {U}'}{\partial x}+g\frac{\partial {h}'}{\partial x} \\ &\quad = g\left\{ \sin \theta -{{n}^{2}}\left( {{U}_{o}}^{2}+2{{U}_{o}}{U}' \right)\left( \frac{1}{{{h}_{o}}^{{4}/{3}\;}}-\frac{4}{3}\frac{{{h}'}}{{{h}_{o}}^{{7}/{3}\;}} \right) \right\} \\ \tag{2.1}\\ &\quad = g\left\{ \sin \theta -\frac{{{n}^{2}}{{U}_{o}}^{2}}{{{h}_{o}}^{{4}/{3}\;}}-{{n}^{2}}\frac{2{{U}_{o}}U'}{{{h}_{o}}^{{4}/{3}\;}}+{{n}^{2}}\frac{4}{3}{{U}_{o}}^{2}\frac{h'}{{{h}_{o}}^{{7}/{3}\;}} \right\} \tag{2.2} \end{align} $$
and
$$\frac{{{\partial }^{2}}{h}'}{\partial {{t}^{2}}}+2{{U}_{o}}\frac{{{\partial }^{2}}{h}'}{\partial x\partial t}+\left( {{U}_{o}}^{2}-g{{h}_{o}} \right)\frac{{{\partial }^{2}}{h}'}{\partial {{x}^{2}}}+\frac{2g{{S}_{o}}}{{{U}_{o}}}\left( \frac{\partial {h}'}{\partial t}+\frac{5}{3}{{U}_{o}}\frac{\partial {h}'}{\partial x} \right)=0 \tag{3}$$
where
- $U$: cross sectional averaged velocity
- $\theta$ and ${S}_{o}$: channel slope
- $n$: roughness coefficient
- ${U}_{o}$: uniform flow velocity at ${h}_{o}$
- ${h}_{o}$: depth evaluated by Manning's formula $\sin \theta =\frac{{{n}^{2}}{{U}_{o}}^{2}}{{{h}_{o}}^{4/3}}$
Eq. (3) is obtained from (1) and (2.2) but I am not sure how. I would be grateful if someone could help.