 Solution

At $a=-x_T$, defineThe first-order Taylor expansion is $k^2(x)=-\alpha (x-a)+O((x-a{)}^2)$, so the local wave equation becomesThis is the Airy ordinary differential equation. Its $z<0$ side is oscillatory and its $z>0$ side is evanescent. A matching overlap requires $\ell \ll |x-a|\ll x_T$, hence well-separated wave turning points with $\ell \ll x_T$. At $b=x_T$, the coordinate $z_R=(b-x)/\ell$ gives the same Airy ordinary differential equation, with its evanescent side again at positive argument.