 Airy reduction of the Orr-Sommerfeld equation in constant shear

For nondimensional constant shear $U=y$ and streamwise wavenumber one, put $W=(D^2-1)\hat{v}$. The Orr-Sommerfeld equation gives $W^{\prime \prime }=[1+iRe(y-c)]W$. The displayed substitution transforms it into the Airy ordinary differential equation, because $(e^{i\pi /6}{)}^3=i$. The scale $R{e}^{-1/3}$ balances shear advection against viscous diffusion.