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For a plane wave with amplitude $\alpha$, wavenumber $k>0$ and angular frequency $\omega >0$, the dispersion relation $\omega =\omega (k)$ specifies which frequencies the governing equations permit. A point of fixed wave phase obeys $kx-\omega t=\text{constant}$, so the phase speed isIt is the speed of individual crests. For a narrow wave packet, two nearby plane waves with differences $\Delta k$ and $\Delta \omega$ have a slowly varying wave envelope depending on $\Delta kx-\Delta \omega t$. Taking the limit gives the group velocityThis is the envelope's propagation speed and, in the present linear lossless wave model, the energy transport speed for a narrow packet.

A dispersive wave has phase speed dependent on wavenumber, so different components can change their relative phases and alter the packet's shape. A nondispersive wave on the chosen propagation branch has constant phase speed over the relevant band:Indeed $d{c}_p/dk=(c_g-c_p)/k$, so equality throughout a band is equivalent to constant phase speed there. Equality at an isolated wavenumber does not make the entire wave family nondispersive.

A dispersive wave has phase speed depending on frequency. Its phase velocity is $v_p=\omega /k$, whereas the group velocity $v_g=d\omega /dk$ describes a narrow wave packet in a lossless single branch. Examples are deep-water gravity waves, ${\omega }^2=gk$, and electromagnetic waves in a cold plasma, ${\omega }^2={\omega }_p^2+c^2{k}^2$.

The fixed-edge membrane waveguide has transverse profiles $\sin (n\pi y/b)$ withThere is no nonzero n=0 displacement satisfying both fixed-edge conditions. Even though the unconfined membrane is nondispersive, transverse confinement makes propagation along x dispersive. The membrane-guide phase and energy speeds are, for a propagating mode,For positive $k_x$, the phase speed exceeds v while group speed is below v. At cutoff $k_x=0$ the formal phase speed diverges, but group speed vanishes; no finite longitudinal energy flux is implied.

With the supplied width and speed, $f_{c,n}=10n\mathrm{kHz}$. At15kHz only n=1 propagates:The negative value represents travel in the opposite direction; higher transverse modes are evanescent.

A physicist and author of teaching texts on classical mechanics, electromagnetism, quantum mechanics and statistical mechanics. His treatment of dispersive waves gives a useful example of the texts’ approach: a wave packet is built by adding sinusoidal waves, then expanding its dispersion relation to distinguish group velocity from phase velocity. This explains why the envelope of a water-wave packet can travel at a different speed from its crests.^[ref]