 Crack pressure in elliptic coordinates

For $h=H_0(1-x^2/a^2{)}^{1/6}$, elliptic coordinates make $h^3/\sin \eta$ constant on the upper crack face. Pressure continuity and harmonicity reduce the boundary condition to $F^{\prime }(0)=\alpha F(0)$ for the cosine mode, while uniform far flow gives $F\sim -e^{\xi }/2$. Solving $F^{\prime \prime }=F$ yields the displayed pressure. The midpoint crack flux is $2aU\alpha /(1+\alpha )$; small $\alpha$ leaves the background pressure gradient nearly unchanged, while large $\alpha$ makes the crack nearly equipotential.