= Power-transformed Taylor–Goldstein energy identity
For a mode with nonreal <phase velocity> $c$, let $V=U-c$, choose a continuous branch of $V^a$, and put $\widehat w=V^aq$. Under impermeable boundary conditions, multiplying the transformed <Taylor–Goldstein equation> by $V^{2a}\overline q$ and applying <integration by parts> gives
$$
\int V^{2a}(|q'|^2+k^2|q|^2)dz=\int\left[\{N^2+a(a-1)(U')^2\}V^{2a-2}+(a-1)U''V^{2a-1}\right]|q|^2dz.
$$
Choosing $a=1/2$ gives the <Miles–Howard theorem>; choosing $a=1$ makes the real <phase velocity> of an unstable mode a weighted mean of $U$.
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