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River discharge is volume flux across a channel section. For approximately uniform cross-sectional velocity it equals flow area times mean velocity. Its time dependence at a fixed station is a hydrograph.

A hydrograph records discharge against time at a fixed location. A nonlinear kinematic wave can steepen an upstream increase into a sharp front; longitudinal diffusion or additional storage rounds the resulting rise.

First neglect inertia relative to the $O(ϵ)$ pressure correction, requiring $F\ll 1$ for smooth fields on these scales. The positive velocity isMultiplying by area gives the discharge $Q=A^{5/4}-(ϵ/4)A^{3/4}{A}_x+O({ϵ}^2)$. HenceThe leading equation is a nonlinear kinematic wave equation. The pressure correction has positive diffusion coefficient for $A>0$ and smooths steep gradients. Dropped terms are $O({ϵ}^2)$ and inertial terms $O(ϵF)$ on the assumed scales; sufficiently thin fronts require a separate balance rather than an unrestricted Taylor expansion.

Let the original area be $A_0>0$ and the increased upstream area be $A_1>A_0$. At $ϵ=0$, the conservation law flux $q(A)=A^{5/4}$ is strictly convex. Characteristics behind the increase travel faster than those ahead, forming an admissible shock wave. Rankine-Hugoniot conditions give its speedAt $x=1$ the idealized hydrograph jumps from $Q_0=A_0^{5/4}$ to $Q_1=A_1^{5/4}$ at $t=1/c_s$. With positive $ϵ$, the diffusion term spreads the rise over a finite interval. The hydrograph is rounded, with an earlier small response and a later approach to the new discharge, rather than an instantaneous jump.

The physical cause is the longitudinal pressure gradient: a local increase in depth influences neighbouring sections and redistributes water before pure downstream transport would deliver a discontinuity. For a small increase, linearization about $A_0$ has speed $c=\frac{5}{4}{A}_0^{1/4}$ and diffusivity $D=ϵA_0^{3/4}/4$. The smooth curve in the figure is the exact half-line step response for this linearized advection-diffusion equation; it illustrates the shape without claiming to solve a large nonlinear jump exactly.