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In dispersive media the phase velocity is not necessarily the same as the group velocity. The phase velocity varies with frequency. The phase velocity is the rate at which the phase of the wave propagates in space. The group velocity is the rate at which the wave envelope, i.e. the changes in amplitude, propagates. The wave envelope is the ...
It is also occasionally referred to as temporal frequency for clarity and to distinguish it from spatial frequency. Ordinary frequency is related to angular frequency (symbol ω, with SI unit radian per second) by a factor of 2 π. The period (symbol T) is the interval of time between events, so the period is the reciprocal of the frequency: T ...
In time and frequency, the purpose of a phase comparison is generally to determine the frequency offset (difference between signal cycles) with respect to a reference. [3] A phase comparison can be made by connecting two signals to a two-channel oscilloscope. The oscilloscope will display two sine signals, as shown in the graphic to the right.
In the context of electromagnetics and optics, the frequency is some function ω(k) of the wave number, so in general, the phase velocity and the group velocity depend on specific medium and frequency. The ratio between the speed of light c and the phase velocity v p is known as the refractive index, n = c / v p = ck / ω.
The basic definitions here translate as the common physics quantities location (phase), speed (angular velocity), acceleration (chirpyness). If a waveform is defined as: = (())
Phase velocity is the rate at which the phase of the wave propagates in space: any given phase of the wave (for example, the crest) will appear to travel at the phase velocity. The phase velocity is given in terms of the wavelength λ (lambda) and period T as v p = λ T . {\displaystyle v_{\mathrm {p} }={\frac {\lambda }{T}}.}
Frequency dispersion of surface gravity waves on deep water. The red square moves with the phase velocity, and the green dots propagate with the group velocity. In this deep-water case, the phase velocity is twice the group velocity. The red square traverses the figure in the time it takes the green dot to traverse half.
A so-called eigenmode is a solution that oscillates in time with a well-defined constant angular frequency ω, so that the temporal part of the wave function takes the form e −iωt = cos(ωt) − i sin(ωt), and the amplitude is a function f(x) of the spatial variable x, giving a separation of variables for the wave function: (,) = ().