Airy scaling at a variable-speed wave turning point ($\delta =({\epsilon }^2/q{)}^{1/3}$)

In ${\epsilon }^2{\Psi }^{\prime \prime }+2{\epsilon }^2(c^{\prime }/c){\Psi }^{\prime }+Q(Y)\Psi =0$, with $Q\sim -q(Y-Y_s)$ and $q>0$, set $Y-Y_s=({\epsilon }^2/q{)}^{1/3}\eta$. The leading equation is ${\Psi }_{\eta \eta }-\eta \Psi =0$, the Airy ordinary differential equation. The first derivative and nonlinear coefficient corrections are smaller by $O({\epsilon }^{2/3})$.

The local inner solution is a combination of Airy functions $\mathrm{Ai},\mathrm{Bi}$. Decay in the forbidden region removes $\mathrm{Bi}$. The remaining Airy turning-point connection formula is oscillatory on the allowed side and represents an incident/reflected pair, not a single upward branch. Without the forbidden-side condition the local differential equation alone leaves both coefficients free.

A simple ray turning point has outer amplitude proportional to $|Y-Y_s{|}^{-1/4}$. At the Airy scaling at a variable-speed wave turning point, $|Y-Y_s|=O({\epsilon }^{2/3})$, this gives an inner wavefield of size $O({\epsilon }^{-1/6})$ when the incident action normalization is order one. The enhancement is finite for nonzero $\epsilon$ and is resolved by the Airy function.