Variable-section channel characteristics ($c^2=gS/S_h$)

For cross-sectional area $S(x,h)$ and cross-sectionally uniform velocity, volume conservation gives $S_h(h_t+u{h}_x)+S{u}_x+u{S}_x=0$, while hydrostatic inviscid momentum gives $u_t+u{u}_x+g{h}_x=0$. Their characteristic speeds are $u\pm c$. With $D_{\pm }={\partial }_t+(u\pm c){\partial }_x$, the robust compatibility equation is $(g/c)D_{\pm }h\pm D_{\pm }u=-uc{S}_x/S$. Replacing $D_{\pm }h$ by $D_{\pm }c/c_h$ requires $c_x=0$ at fixed height; otherwise an explicit geometry term remains.

For constant long-wave speed and $S_0\propto e^{\beta x}$, a standing free-surface amplitude obeys ${\eta }^{\prime \prime }+\beta {\eta }^{\prime }+k^2\eta =0$. The transformation $\eta =e^{-\beta x/2}Y$ gives $Y^{\prime \prime }+(k^2-{\beta }^2/4)Y=0$. Impermeable ends at $0,L$ impose ${\eta }^{\prime }=0$, yielding the displayed nonzero-frequency eigenvalues and ${\eta }_n\propto e^{-\beta x/2}[\cos (n\pi x/L)+\beta L\sin (n\pi x/L)/(2n\pi )]$. A separate zero-frequency mode changes the uniform water level.