 Coastal adjustment of an elevated strip

For a resting shallow-water strip $-L<x<0$ adjacent to an impermeable coast, conserved linear potential vorticity gives $a^2{\eta }_{xx}-\eta =-{\eta }_{\mathrm{initial}}$. Decay offshore, matching and conserved volume give $\eta ={\eta }_0{e}^{x/a}\sinh (L/a)$ 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.