 Solution

Normalize $U$ and $c$ by $U_{\max }$ and write $h={\zeta }_L\in (0,1)$, $\alpha =kL>0$. In each open region $U^{\prime \prime }=0$, so the Rayleigh equation for inviscid shear flow reduces to ${\hat{w}}^{\prime \prime }-{\alpha }^2\hat{w}=0$ for the nonreal modes of interest. Incorporating the wall conditions from the start, the bounded piecewise-linear shear layer hasThese are four independent regional amplitudes before matching. The same expressions extend to noncritical neutral modes and their limits.