= Solution
Take $\alpha>0$; a real disturbance with negative <wavenumber> is recovered by <complex conjugation>. To obtain the <pressure equation for a shear-flow normal mode>, apply $i\alpha$ to the streamwise momentum equation and differentiate the normal momentum equation. The <mass conservation> equation cancels the viscous divergence and the terms proportional to $U-c$, leaving $2i\alpha U'v=\alpha^2p-p''$. Thus \b[the <fluid pressure> equation and wall value are]
$$
\boxed{p''-\alpha^2p=-2i\alpha U'v,\qquad
p(0)=\frac{u''(0)}{i\alpha R}=\frac{v^{(3)}(0)}{\alpha^2R}.}
$$
The wall value follows from streamwise momentum with <no-slip boundary condition>. Normal momentum also gives $p'(0)=v''(0)/R$.
In the lower <viscous boundary layer>, <mass conservation> gives $v=O(\alpha\delta)$ if $u=O(1)$. Since $U\sim y$, convection and shear have size $\alpha\delta$ when $c=O(\delta)$; matching these with transverse viscous diffusion gives $R^{-1}\delta^{-2}\sim\alpha\delta$. The <pressure gradient> balance then gives $p=O(\delta)$. Thus \b[the lower scalings are]
$$
\boxed{\delta=(\alpha R)^{-1/3},\quad c=\delta C,\quad
v=\alpha\delta V,\quad p=\delta P.}
$$
The neglected streamwise viscous term is smaller by $(\alpha\delta)^2$. In these variables the leading equations are
$$
V_Y=-iu,\qquad i(Y-C)u+V=-iP+u_{YY},\qquad P_Y=0.
$$
Differentiating the second and using the first yields \b[the Airy reduction] $\boxed{u_{YYY}=i(Y-C)u_Y}$.
Put $q=e^{i\pi/6}$, so $q^3=i$. The bounded matching branch of the <Airy function> gives $u_Y=A\operatorname{Ai}(q(Y-C))$; the other independent Airy solution grows along the ray and is excluded. Integrating with the two wall conditions gives
$$
u(Y)=A\int_0^Y\operatorname{Ai}(q(s-C))\,ds,\qquad
V(Y)=-iA\int_0^Y(Y-s)\operatorname{Ai}(q(s-C))\,ds,
$$
while the wall momentum equation gives the constant <fluid pressure>
$$
P_0=-iAq\operatorname{Ai}'(-qC).
$$
Define the convergent <Airy displacement integrals>
$$
I_0(C)=\int_0^\infty\operatorname{Ai}(q(s-C))\,ds,\qquad
I_1(C)=\int_0^\infty s\operatorname{Ai}(q(s-C))\,ds.
$$
The <Airy function> equation and its decaying <derivative> imply $I_1-CI_0=iq\operatorname{Ai}'(-qC)$. Consequently, as $Y\to\infty$,
$$
\boxed{u\to B=AI_0,\qquad P\to P_0=A(CI_0-I_1),\qquad
V=-iBY+iAI_1+o(1).}
$$
Both differentiated lower momentum and the constant of integration are satisfied; retaining that constant is essential for the final matching.
For the middle region introduce a perturbation <stream function> $\Phi$ with $u=\Phi'$ and $v=-i\alpha\Phi$. The leading <inviscid flow> streamwise equation is
$$
(U-c)\Phi'-U'\Phi=-p,\qquad
\left(\frac{\Phi}{U-c}\right)'=-\frac{p}{(U-c)^2}.
$$
For a <shear flow>, this is the integrated form of the <Rayleigh equation for inviscid shear flow>. The leading matched solution for fixed $y=O(1)$ is $\Phi\sim BU$, so $u\sim BU'$ and $v\sim-i\alpha BU$. In the overlap with the lower <viscous boundary layer>, the first constant correction is
$$
\Phi\sim B(y-c)+\delta P_0=By-\delta AI_1,
$$
which agrees with the integrated lower solution. For a regular smooth profile, the normal momentum equation gives $p'\sim-\alpha^2(U-c)\Phi$. Thus <fluid pressure> variation across a fixed-width middle region is $O(\alpha^2B)$; in the final distinguished scaling this is smaller than its $O(\delta)$ value, so $p\sim\delta P_0$ throughout that region. Since $U\to1$, \b[the middle <velocity> tends to] $\boxed{v\sim-i\alpha B}$, which does not decay and requires another region.
The <fluid pressure> equation becomes $p''-\alpha^2p=0$ when $U\sim1$. Therefore the upper scale is $\Delta=1/\alpha$, with $z=\alpha y$. The inviscid decaying upper solution is
$$
\boxed{p=P_\infty e^{-z},\quad
v=-\frac{iP_\infty}{1-c}e^{-z},\quad
u=-\frac{P_\infty}{1-c}e^{-z}.}
$$
Matching <velocity> as $z\downarrow0$ gives $P_\infty\sim\alpha B$; matching <fluid pressure> to the middle region gives $\delta P_0\sim\alpha B$. This is <three-layer long-wave shear-flow matching>, expressed as a <matched asymptotic expansion>.
Under $\alpha=kR^{-1/4}$ with fixed positive $k$, $\delta=k^{-1/3}R^{-1/4}$, and $\alpha/\delta=k^{4/3}$. Eliminating the nonzero amplitude $A$ gives \b[the <Airy wall-layer matching determinant> and <dispersion relation>]
$$
\boxed{k^{4/3}I_0(C)+iq\operatorname{Ai}'(-qC)=0.}
$$
Equivalently, with $\mathcal T(C)=\int_{-qC}^{\infty e^{i\pi/6}}\operatorname{Ai}(\zeta)\,d\zeta=qI_0(C)$,
$$
\boxed{k^{4/3}\mathcal T(C)=e^{-i\pi/6}\operatorname{Ai}'(-qC).}
$$
The determinant form does not divide by a possibly vanishing $I_0$. The <wavespeed> is $c=\delta C$; temporal growth has sign $\operatorname{Im}C$ for positive $\alpha$.
There is a genuine conflict in the printed assumptions: fixed-$k$ $\alpha=kR^{-1/4}$ cannot satisfy $R^{-1/7}\ll\alpha$ as $R\to\infty$. For order-one lower matching amplitudes, constant leading middle <fluid pressure> requires $\alpha^2\ll\delta$, equivalently $\alpha\ll R^{-1/7}$; this is the reverse inequality and is satisfied by the final distinguished scaling. Thinness also requires $\alpha R\gg1$. The <dispersion relation> above is therefore derived under the final distinguished limit, for which the complete consistent window is $R^{-1}\ll\alpha\ll R^{-1/7}$. Both printed limits cannot be imposed simultaneously.
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