Airy wall-layer matching determinant ($k^{4/3}{I}_0(C)+iq\mathrm{Ai}\prime (-qC)=0$)

With the Airy displacement integrals, the lower pressure-to-velocity ratio is $-iq\mathrm{Ai}\prime (-qC)/I_0(C)$. The decaying upper mode gives that ratio as $\alpha /\delta$. Hence for $\alpha =k{R}^{-1/4}$ the dispersion relation is $k^{4/3}{I}_0(C)+iq\mathrm{Ai}\prime (-qC)=0$. Writing it without division remains meaningful if an individual moment vanishes. The wavespeed is $c=\delta C$ and a positive wavenumber grows temporally when $\mathrm{Im}C>0$.