Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 82 / 4 / a / Solution

Consider a nonreal phase velocity $c$, so $U-c$ never vanishes. Substitute $\varphi =(U-c)\psi$ into the Rayleigh equation for inviscid shear flow. Differentiating cancels the two terms involving $U^{\prime \prime }$, leavingFor $k>0$, the wall condition on normal velocity gives $\varphi =0$, hence $\psi =0$, at both walls. Multiply by $\overline{\psi }$ and use integration by parts to obtain the weighted identity for Howard's semicircle theorem:For a nonzero normal mode, $N=\int Wdy>0$. The imaginary and real parts of this identity give, respectively,Since $c_i\ne 0$, these implyThe first equation says that $c_r$ is a positive weighted mean of $U$, and consequently $U_{\min }\le c_r\le U_{\max }$. The division by $c_i$ is essential: the weighted-mean assertion concerns nonreal modes. A neutral mode may have a critical level where $U=c$ and the substitution for $\psi$ is singular.

Pointwise, $(U-U_{\max })(U-U_{\min })\le 0$. Integrating against the nonnegative weight $W$ and substituting both identities yieldsCompleting the square proves Howard's semicircle theorem:Thus all nonreal phase velocities lie in the closed disk with the real interval $[U_{\min },U_{\max }]$ as diameter. For the convention $e^{ik(x-ct)}$ and $k>0$, exponentially growing normal modes occupy its upper semicircle, while decaying modes occupy its lower semicircle.