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In a dispersive medium, the phase speed itself depends upon the frequency of the wave, making the relationship between wavelength and frequency nonlinear. In the case of electromagnetic radiation —such as light—in free space , the phase speed is the speed of light , about 3 × 10 8 m/s .
A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the dispersion relation, one can calculate the frequency-dependent phase velocity and group velocity of each sinusoidal component of a wave in the medium, as a function of frequency.
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.
is the speed of light (i.e. phase velocity) in a medium with permeability μ, and permittivity ε, and ∇ 2 is the Laplace operator. In a vacuum, v ph = c 0 = 299 792 458 m/s , a fundamental physical constant . [ 1 ]
The wavelength can be calculated as the relation between a wave's speed and ordinary frequency. λ = c f . {\displaystyle \lambda ={\frac {c}{\ f\ }}~.} For sound waves, the amplitude of the wave is the difference between the pressure of the undisturbed air and the maximum pressure caused by the wave.
The "direction of wave propagation" is the direction of a wave's energy flow, and the direction that a small wave packet will move, i.e. the direction of the group velocity. For light waves in vacuum, this is also the direction of the Poynting vector. On the other hand, the wave vector points in the direction of phase velocity.
Since the velocity of propagation is very high – about 300,000 kilometers per second – the wave of an alternating or oscillating current, even of high frequency, is of considerable length. At 60 cycles per second, the wavelength is 5,000 kilometers, and even at 100,000 hertz, the wavelength is 3 kilometers.
For example, a wavenumber in inverse centimeters can be converted to a frequency expressed in the unit gigahertz by multiplying by 29.979 2458 cm/ns (the speed of light, in centimeters per nanosecond); [5] conversely, an electromagnetic wave at 29.9792458 GHz has a wavelength of 1 cm in free space.