= Boundary-driven shallow-water wave steepening
{title2=$x_b=2gH/(3u_b'(0))$}
For an outgoing <simple wave> into water initially at rest at depth $H$, the incoming <Riemann invariant> is $u-2\sqrt{gh}=-2\sqrt{gH}$. Thus $\sqrt{gh}=\sqrt{gH}+u/2$ and the outgoing <characteristic speed> is $v=\sqrt{gH}+3u/2$. A boundary characteristic emitted at time $s$ is $x=v(s)(t-s)$. Its envelope satisfies $t=s+v/v'$ and $x=v^2/v'$. At a zero-velocity onset with positive acceleration, the leading-edge intersection is the displayed distance. Later <characteristic curves> can form a shock earlier elsewhere in a sufficiently strong waveform.
Back to article page
