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Propagation of a wave packet demonstrating a phase velocity greater than the group velocity. This shows a wave with the group velocity and phase velocity going in different directions. [1] The group velocity is positive (i.e., the envelope of the wave moves rightward), while the phase velocity is negative (i.e., the peaks and troughs move ...
Propagation of a wave packet demonstrating a phase velocity greater than the group velocity. This shows a wave with the group velocity and phase velocity going in different directions. The group velocity is positive, while the phase velocity is negative. [1] The phase velocity of a wave is the rate at which the wave propagates in any medium.
The group velocity is depicted by the red lines (marked B) in the two figures above. In shallow water, the group velocity is equal to the shallow-water phase velocity. This is because shallow water waves are not dispersive. In deep water, the group velocity is equal to half the phase velocity: {{math|c g = 1 / 2 c p. [7]
Download as PDF; Printable version; In other projects ... The group velocity is one half the phase velocity. A wave in which the group and phase velocities differ is ...
Ideas related to wave packets – modulation, carrier waves, phase velocity, and group velocity – date from the mid-1800s. The idea of a group velocity distinct from a wave's phase velocity was first proposed by W.R. Hamilton in 1839, and the first full treatment was by Rayleigh in his "Theory of Sound" in 1877.
It is possible to calculate the group velocity from the refractive-index curve n(ω) or more directly from the wavenumber k = ωn/c, where ω is the radian frequency ω = 2πf. Whereas one expression for the phase velocity is v p = ω/k, the group velocity can be expressed using the derivative: v g = dω/dk. Or in terms of the phase velocity v p,
The deep-water wavelength is L 0 = g/(2π)T 2 and the deep-water phase speed is c 0 = L 0 /T. The grey line corresponds with the shallow-water limit c p =c g = √(gh). The phase velocity c p is related to the wavelength L through c p = L/T. Consequently, for fixed period T, L/L 0 varies identical to c p /c 0 with depth changes.
In optics, group-velocity dispersion (GVD) is a characteristic of a dispersive medium, used most often to determine how the medium affects the duration of an optical pulse traveling through it. Formally, GVD is defined as the derivative of the inverse of group velocity of light in a material with respect to angular frequency, [1] [2]