 Variable-section channel characteristics

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.