= Coastal adjustment of an elevated strip
{title2=$a=\sqrt{gH}/|f|$}
For a resting shallow-water strip $-L<x<0$ adjacent to an impermeable coast, conserved linear <potential vorticity> gives $a^2\eta_{xx}-\eta=-\eta_{\rm initial}$. Decay offshore, matching and conserved volume give $\eta=\eta_0e^{x/a}\sinh(L/a)$ outside the strip and $\eta=\eta_0[1-e^{-L/a}\cosh(x/a)]$ inside it. The adjusted velocity is $v=g\eta_x/f$, with zero tangential wall velocity inherited from initial rest and alongshore uniformity.
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