= Solution
Let the original <area> be $A_0>0$ and the increased upstream <area> be $A_1>A_0$. At $\epsilon=0$, 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 speed
$$
c_s=\frac{A_1^{5/4}-A_0^{5/4}}{A_1-A_0}.
$$
At $x=1$ 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 $\epsilon$, 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 $c=\tfrac54A_0^{1/4}$ and <diffusivity> $D=\epsilon A_0^{3/4}/4$. 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.
\Image[/past-exam-of-the-mathematics-and-the-environment-and-mathematical-physiology-course-of-the-university-of-oxford/2010-river-response.png]
{title=Sharp and diffused downstream hydrographs, with subcritical and supercritical bed-surface phases}
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