Variational group velocity of a Taylor-Goldstein mode ($c_g=c+\frac{2k^2\int {\varphi }^{2}dz}{\int [N^2/(U-c{)}^3+{\ell }^2/(U-c)]{\varphi }^{2}dz}$)

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.