= Exact linear-drag cosine basin circulation
{title2=$r\Psi''+\beta\Psi'-rm^2\Psi=F_0$}
For <streamfunction> $\psi=\Psi(x)\cos(my)$ with impermeable walls at $x=0,L$, let $A=-F_0/(rm^2)$ and positive exponents satisfy $\lambda-\gamma=\beta/r$, $\lambda\gamma=m^2$. The exact profile is $\Psi=A[1-C_We^{-\lambda x}-C_Ee^{\gamma(x-L)}]$, where $C_W=(1-e^{-\gamma L})/(1-e^{-(\lambda+\gamma)L})$ and $C_E=(1-e^{-\lambda L})/(1-e^{-(\lambda+\gamma)L})$. Substitution proves the <ordinary differential equation> and both boundary conditions. The usual narrow <western boundary current> limit requires $\lambda L\gg1$ and $\gamma L\ll1$, with $\lambda\simeq\beta/r$ and $\gamma\simeq rm^2/\beta$; the opposite first inequality does not describe weak drag.
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