= Solution
Let $J(x)=\int_x^b\kappa(s)\,ds=\eta-I(x)$. The <WKB approximation> in the barrier can be rewritten as
$$
E_{\rm II}=\frac{D_R e^{-J}+G_R e^J}{\sqrt\kappa},\qquad
D_R=G e^\eta,\quad G_R=D e^{-\eta}.
$$
The right <Airy turning-point connection formula> gives
$$
E_{\rm III}=\frac{2D_R\cos(\Psi-\pi/4)-G_R\sin(\Psi-\pi/4)}{\sqrt{k_3}},
$$
so $A_{\rm t}=D_R+iG_R/2$ and $A_{\rm back}=D_R-iG_R/2$. Impose \b[no incident <wave> from the right], $A_{\rm back}=0$. Then
$$
D_R=\frac{iG_R}{2},\qquad
G=\frac{iD}{2}e^{-2\eta},\qquad
A_{\rm t}=iD e^{-\eta},\qquad
A_{\rm in}=D\left(1+\frac14e^{-2\eta}\right).
$$
Consequently, within the leading <Airy turning-point connection formula> calculation,
$$
t=\frac{A_{\rm t}}{A_{\rm in}}=\frac{i e^{-\eta}}{1+e^{-2\eta}/4},\qquad
r=\frac{A_{\rm r}}{A_{\rm in}}=\frac{1-e^{-2\eta}/4}{1+e^{-2\eta}/4}.
$$
For an opaque barrier, $\eta\gg1$,
$$
\boxed{t\sim i e^{-\eta},\qquad T=|t|^2\simeq e^{-2\eta}.}
$$
The <transmission coefficient> is a ratio of <Poynting fluxes>; the $k^{-1/2}$ normalization already includes their local <wavenumber> factors. The displayed denominator is useful for consistent matching and <Wronskian flux> conservation, but does not claim accuracy for subleading terms beyond the <WKB approximation>, or for merging <wave turning points>.
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