Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 329 / 3 / d / Solution

Let $\phi =\pi /2-\varphi \to 0$ near a shoulder of the barrier. Since $\cos \varphi \sim \phi$, the solution from part c predictsThus the supposedly thin radial region broadens while its large-thickness assumption eventually weakens, and azimuthal derivatives become singular; the approximation cannot remain uniform at the shoulder.

The radial and azimuthal derivative scales of $h^3$ areThey become comparable when $\Delta \sim R\phi$. Using the expression above,soAt this transition,