= Variational group velocity of a Taylor-Goldstein mode
{title2=$c_g=c+\frac{2k^2\int\phi^2dz}{\int[N^2/(U-c)^3+\ell^2/(U-c)]\phi^2dz}$}
For a regular real mode between rigid boundaries, multiply the <Taylor–Goldstein equation> by its vertical structure and integrate to get the stationary quotient $k^2=\int(\ell^2\phi^2-\phi_z^2)dz/\int\phi^2dz$. 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.
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