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A superposition of 1D plane waves (blue) each traveling at a different phase velocity (traced by blue dots) results in a Gaussian wave packet (red) that propagates at the group velocity (traced by the red line). The group velocity of a collection of waves is defined as =.
1D Gaussian wave packet, shown in the complex plane, for =, =, =, =. The overall group velocity is positive, and the wave packet moves as it disperses. The inverse Fourier transform is still a Gaussian, but now the parameter a has become complex, and there is an overall normalization factor.
A superposition of 1D plane waves (blue) each traveling at a different phase velocity (traced by blue dots) results in a Gaussian wave packet (red) that propagates at the group velocity (traced by the red line). The group velocity of a collection of waves is defined as =.
Animation: phase and group velocity of electrons This animation portrays the de Broglie phase and group velocities (in slow motion) of three free electrons traveling over a field 0.4 ångströms in width. The momentum per unit mass (proper velocity) of the middle electron is lightspeed, so that its group velocity is 0.707 c. The top electron ...
Dispersion of gravity waves on a fluid surface. Phase and group velocity divided by shallow-water phase velocity √ gh as a function of relative depth h / λ. Blue lines (A): phase velocity; Red lines (B): group velocity; Black dashed line (C): phase and group velocity √ gh valid in shallow water.
Then the group velocity of the plane wave is defined as = = =, which agrees with the formula for the classical velocity of the particle. The group velocity is the (approximate) speed at which the whole wave packet propagates, while the phase velocity is the speed at which the individual peaks in the wave packet move. [ 5 ]
If the group velocity (see below) is wavelength-independent, this equation can be simplified as: [14] (,) = (+), showing that the envelope moves with the group velocity and retains its shape. Otherwise, in cases where the group velocity varies with wavelength, the pulse shape changes in a manner often described using an envelope equation .
Inherent in these equations is a relationship between the angular frequency ω and the wave number k. Numerical methods are used to find the phase velocity c p = fλ = ω/k, and the group velocity c g = dω/dk, as functions of d/λ or fd. c l and c t are the longitudinal wave and shear wave velocities respectively.