= Airy wall-layer matching determinant
{c}
{title2=$k^{4/3}I_0(C)+iq\operatorname{Ai}\prime(-qC)=0$}
With the <Airy displacement integrals>, the lower pressure-to-velocity ratio is $-iq\operatorname{Ai}\prime(-qC)/I_0(C)$. The decaying upper mode gives that ratio as $\alpha/\delta$. Hence for $\alpha=kR^{-1/4}$ the <dispersion relation> is $k^{4/3}I_0(C)+iq\operatorname{Ai}\prime(-qC)=0$. Writing it without division remains meaningful if an individual moment vanishes. The wavespeed is $c=\delta C$ and a positive wavenumber grows temporally when $\operatorname{Im}C>0$.
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