= Midpoint correction for two quadratic-dispersion waves
{title2=$m\bar\omega=\bar k^2+(\delta k)^2$}
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.
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