= Cosine-ridge shallow-water geostrophic response
{title2=$\phi''-L_R^{-2}\phi=-f\widehat b/h_0$}
For a uniform upstream current whose free-surface and bottom slopes compensate, steady <quasi-geostrophic approximation> flow with uniform incoming <potential vorticity> obeys the displayed inversion equation. Let $\widehat b=\epsilon\cos(x/L)$ on $|x|<a=\pi L/2$ and zero outside. With $C=(f\epsilon/h_0)/(L^{-2}+L_R^{-2})$, the decaying even response is $\phi=C[\cos(x/L)+(L_R/L)e^{-a/L_R}\cosh(x/L_R)]$ inside and $\phi=C(L_R/L)e^{-a/L_R}\cosh(a/L_R)e^{-(|x|-a)/L_R}$ outside. Continuity of $\phi$ and $\phi'$ fixes the coefficients. Its streamlines satisfy $y=y_\infty+\phi(x)/U$, so the response extends beyond the compact ridge through exponentially decaying balanced tails.
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