 Airy scaling at a variable-speed wave turning point

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 $O({\epsilon }^{2/3})$.