= Phase-speed bound for unstable stratified shear modes
For an unstable <Taylor–Goldstein equation> mode in a finite channel, take $a=1$ in the <power-transformed Taylor–Goldstein energy identity>. Its <imaginary part> gives
$$
c_r=\frac{\int U(|q'|^2+k^2|q|^2)dz}{\int(|q'|^2+k^2|q|^2)dz}.
$$
Thus the real <phase velocity> lies strictly between the extremes of a nonconstant smooth shear profile. Equality would force the regular <eigenfunction> to vanish on an interval, hence everywhere by uniqueness for its <ordinary differential equation>.
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