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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 givesThus 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.

For a mode with nonreal phase velocity $c$, let $V=U-c$, choose a continuous branch of $V^a$, and put $\hat{w}=V^{a}q$. Under impermeable boundary conditions, multiplying the transformed Taylor–Goldstein equation by $V^{2a}\overline{q}$ and applying integration by parts givesChoosing $a=1/2$ gives the Miles–Howard theorem; choosing $a=1$ makes the real phase velocity of an unstable mode a weighted mean of $U$.

For a stationary internal gravity wave, the Taylor–Goldstein equation gives $m^2=N^2/U^2-U^{\prime \prime }/U-k^2$. In a propagating interval with slowly varying $m$, the WKB approximation has amplitude proportional to $m^{-1/2}$ and phase derivative $m$. The conditions $|m^{\prime }|\ll m^2$ and $|m^{\prime \prime }|\ll |m{|}^3$ fail at a regular turning level. With $U>0$, the upward branch has $m>0$ and negative intrinsic frequency; laboratory group velocity is $U(k^2,km)/(k^2+m^2)$ when the curvature term is negligible.

For a regular real mode between rigid boundaries, multiply the Taylor–Goldstein equation by its vertical structure and integrate to get the stationary quotient $k^2=\int ({\ell }^2{\varphi }^2-{\varphi }_z^2)dz/\int {\varphi }^{2}dz$. Along a differentiable dispersion branch, stationarity removes the variation of the eigenfunction, giving $2k=(\partial I/\partial c)dc/dk$. Since ${\partial }_c{\ell }^2=N^2/(U-c{)}^3+{\ell }^2/(U-c)$, the displayed group velocity follows from $d(kc)/dk$. Critical levels and vanishing branch derivatives require a separate limiting analysis.