Midpoint correction for two quadratic-dispersion waves ($m\bar{\omega }={\bar{k}}^2+(\delta k{)}^2$)

For the dispersion relation $m\omega =k^2$, take two plane waves with wavenumbers $\bar{k}\pm \delta k$ and corresponding angular frequencies $\bar{\omega }\pm \delta \omega$. Adding their two relations proves the title identity, while subtracting gives $\delta \omega /\delta k=2\bar{k}/m$. Thus the exact two-tone envelope speed differs from twice the midpoint carrier phase velocity by a second-order bandwidth correction. In the narrow-band limit the derivative group velocity is exactly twice the phase velocity on the central branch.