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For a Fourier mode $e^{i(kx-\omega t)}$, the wave equation givesTaking the positive-frequency branch,For $k>0$, the phase velocity and group velocity areThe frequency rises from zero and approaches $1$; $c_p$ reaches its maximum at $k=1$, whileshows that the front velocity isThus the wave at the front hasMoreover $c_g/c_p=2/(1+k^2)$, so the packet outruns its crests for $0<k<1$, the speeds agree at $k=1$, and the crests outrun the packet for $k>1$.

The initial conditions give the exact superpositionThe condition $A^{\ast }(-k)=A(k)$ makes this solution real. For $V=x/t$ with $0<V<V_m$, the equationhas two positive roots $0<{\alpha }_1<1/\sqrt{3}<{\alpha }_2$. The second exponential has the corresponding stationary points $-{\alpha }_1,-{\alpha }_2$. Sincethe stationary phase method and pairing of complex-conjugate contributions giveFor $-V_m<V<0$, the same picture is reflected and the stationary points have negative wavenumber. For $|V|>V_m$ there are no real stationary points, so a smooth localized spectrum gives a much smaller nonstationary tail outside the wave front.

Without self-gravity and for $k>0$,The phase velocity and group velocity areso

With self-gravity restored,The product changes sign atFor $k>k_{\mathrm{crit}}$, crests and a localized wave packet travel in the same radial direction. For $k<k_{\mathrm{crit}}$, the group velocity and phase velocity have opposite signs: the envelope and its wave energy propagate opposite to the individual crests. At $k_{\mathrm{crit}}$ the group velocity vanishes and neighboring wave components cause the packet to spread.

The vertical group velocity is the $m$ derivative of the local dispersion relation:At release,Initial upward propagation is possible only when $m_0<0$. In that case $\dot{z}(0)>0$ precisely when

From the dispersion relation, the phase velocity and group velocity areBoth are even functions of $k$, both attain their minimum value $U$ at $k=0$, and both increase like $|k{|}^8$. For every $k\ne 0$,The graph of $c_g$ therefore lies above the graph of $c_p$ away from their common minimum.

For $k>0$, the phase velocity and group velocity areAs $k$ increases from zero, $c_p$ decreases from $+\infty$ toward $c$, while $c_g$ increases from zero toward $c$.

A wave packet is a localized superposition of nearby wavenumbers. Its envelope propagates at the group velocity to leading order.