 Solution

The real Airy function asymptotics used for the Airy turning-point connection formula are, for $z,s\to +\infty$,These real-axis asymptotics fix both normalization and phase.^[ref] Near $a$, $I\sim 2z^{3/2}/3$, $\Phi \sim 2(-z{)}^{3/2}/3$, and ${\kappa }^{-1/2}\sim {\ell }^{1/2}{z}^{-1/4}$. Matching to the constants of c gives $C_A=2\sqrt{\pi }{\ell }^{1/2}D$ and $C_B=\sqrt{\pi }{\ell }^{1/2}G$. ThereforeIn particular, the decaying component matches the cosine with the requested $-\pi /4$ shift, with a factor two in its oscillatory coefficient. The growing component matches a sine with the indicated minus sign. Expanding into travelling waves givesThe factor two is essential when converting the real oscillatory components to incident and reflected travelling waves.