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At a fixed time, the complete spatial derivative isThe first term describes geometric variation along the channel at a fixed surface elevation. The second describes the extra wetted area produced by a varying water level; $S_h$ is the surface width.

Assume a fixed channel, negligible lateral inflow, uniform cross-sectional velocity, incompressible water and the hydrostatic long-wave approximation. Integrating volume conservation over a short channel segment givesInviscid horizontal momentum, with $h$ measured from the fixed datum, is $u_t+u{u}_x+g{h}_x=0$. The coefficient matrix acting on $(h_x,u_x)$ has eigenvaluesThese are the variable-section channel characteristics. Put $D_{\pm }={\partial }_t+(u\pm c){\partial }_x$. Adding the corresponding linear combinations of the two equations givesThe opposite signs in the two combinations cancel both derivative terms in the source calculation.

If $c=c(h)$, then $D_{\pm }c=c_h{D}_{\pm }h$ and $d{c}^2/dh=2c{c}_h$. Taking $\xi =t$ along a characteristic gives the requested formA sufficient geometry assumption is $S(x,h)=b(x)A(h)$, for which $S/S_h=A/A^{\prime }$ is independent of $x$ at fixed height.

For a completely general $S(x,h)$, the printed form omits an explicit wave-speed geometry term. The correct conversion is insteadFor example, a lake at rest with constant $h$ but position-dependent $c$ has $D_{\pm }c=\pm c{c}_x$. The printed left side is then nonzero while its right side is zero. The $h$-based compatibility equation remains valid, including when ${\partial }_h(c^2)=0$ makes the speed-based expression singular.