Group velocity
The group velocity is the speed at which a wave packet travels as a whole.
A wave packet is a wave whose amplitude is equal to zero only in a limited region of space. The amplitude curve is called the envelope of the wave packet. One can imagine a wave packet as a superposition of individual waves with different frequencies via a Fourier series. They spread each with a certain phase velocity, which can be frequency-dependent. However, the envelope moves with the group velocity. The shape of the envelope may change in the presence of the dispersion during the advance of the wave packet.
The group velocity is derived from the angular frequency of the wave and the circular wave number to
The group velocity is to be distinguished from the phase velocity, which indicates the speed at which points of constant phase move. The phase velocity is
Of wavelength and frequency.
Substituting in the definition of the group velocity is given by applying the product rule the Rayleigh relationship
You can also write to the wavelength as
Often arises the group velocity than the velocity before which the wave packet transported energy or information through space. This is true in most cases, namely, whenever losses can be neglected, so that the group velocity can be understood as the signal speed of the wave packet. However, with light pulses in very lossy media, the phase speed be significantly higher than the group velocity, and even larger than the speed of light in vacuum. However, information transfer faster than light is not possible since the signal velocity is always at most equal to ( in the case of lossy media is the signal speed is not the same as the group velocity ). Correct would be to speak in this connection of the front velocity. This is the speed at which the wavefronts move (i.e., areas of the same amplitude ), and discontinuities of the shaft. It is defined as a limit value of the phase velocity of the wave number k infinitely large, however, the group velocity is the speed of the movement of the envelope of the wave package. This subtle difference into perspective the idea of the transfer faster than light at a negative group velocity. Crucial for the transmission of information is the front velocity, it can never reach superluminal. See the link " experiment for signal transmission with" speed of light " ."
The function that describes how by dependent, is called the dispersion equation. Is proportional to the group velocity is identical to the phase speed. In the other case widens the envelope of the wave packet as it propagates. This happens for example with signals in optical fibers.