# Dictionary:Group velocity

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1. The velocity with which the energy in a wavetrain travels. In dispersive media where velocity varies with frequency, the wavetrain changes shape as it progresses so that individual wave crests travel at different velocities (phase velocities, V) than does the envelope of the wavetrain. The velocity of the envelope is the group velocity U:

${\displaystyle U=V-\lambda \left({\frac {dV}{d\lambda }}\right)=V+f\left({\frac {dV}{df}}\right)}$,

where λ=wavelength and f=frequency. Normal mode propagation (q.v.) results in dispersion and thus different values for group and phase velocities. See also Figure D-16 and compare phase velocity.

FIG. D-16. Dispersion. (a) Change in wavelet shape because of dispersion in a Voigt solid. Amplitudes have been normalized. (b) Change of waveshape because of energy shifting to later cycles. The axes of time and offset could be interchanged on either graph. (From Balch and Smolka, 1970.)[1]

2. Angular dispersion; in anisotropic media, group velocity is the velocity of energy transport radially outward from a point source. Also called ray velocity; see Figure A-14.

FIG. A-14. Anisotropy. (a) Application of Huygens’ principle to anisotropic velocity illustrates why phase and ray velocities may differ in both direction and magnitude. (b) The application of Fermat’s principle to anisotropic velocity illustrates why the angle of incidence for a reflection for a coincident source and receiver may not make a right angle with the reflector. (c) SH-wavefronts in transversely isotropic media are elliptical but P- and SV-wavefronts are not.

3. When frequency dispersion and angular dispersion occur together, the velocity of the wave envelope (group velocity) and energy velocity (the ratio of the time-average Poynting vector to the time-average stored energy density) are not the same.

## References

1. Balch, A. H. and Smolka, F. R., 1970, Plane and spherical Voigt waves: Geophysics, 35, 745–761.