what is group velocity and phase velocity in waves?

Answer 1

The group velocity of a wave is the velocity with which the variations in the shape of the wave's amplitude (known as the modulation or envelope of the wave) propagate through space. The group velocity is defined by the equation:

The group velocity is often thought of as the velocity at which energy or information is conveyed along a wave. In most cases this is accurate, and the group velocity can be thought of as the signal velocity of the waveform. However, if the wave is travelling through an absorptive medium, this does not always hold. For example, it is possible to design experiments where the group velocity of laser light pulses sent through specially prepared materials significantly exceeds the speed of light in vacuum. However, superluminal communication is not possible, since the signal velocity remains less than the speed of light. It is also possible to reduce the group velocity to zero, stopping the pulse, or have negative group velocity, making the pulse appear to propagate backwards.

The function ω(k), which gives ω as a function of k, is known as the dispersion relation. If ω is directly proportional to k, then the group velocity is exactly equal to the phase velocity. Otherwise, the envelope of the wave will become distorted as it propagates. This "group velocity dispersion" is an important effect in the propagation of signals through optical fibers and in the design of short pulse lasers.

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.


The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the velocity at which the phase of any one frequency component of the wave will propagate. You could pick one particular phase of the wave (for example the crest) and it would appear to travel at the phase velocity. The phase velocity is given in terms of the wave's angular frequency ω and wave vector k


Note that the phase velocity is not necessarily the same as the group velocity of the wave, which is the rate that changes in amplitude (known as the envelope of the wave) will propagate.

The phase velocity of electromagnetic radiation may under certain circumstances (e.g. in the case of anomalous dispersion) exceed the speed of light in a vacuum, but this does not indicate any superluminal information or energy transfer. It was theoretically described by physicists such as Arnold Sommerfeld and Leon Brillouin. See dispersion for a full discussion of wave velocities.

Answer 2

Phase Velocity Vs Group Velocity

Answer 3

Wave Velocity Definition

Answer 4

The group and phase velocities are prosperities of any wave-packet. By wave-packet, I mean a wave that is somewhat localized in a given region in space. For example, the ripples on pond when you toss a rock in it. The ripples do not spread across the whole pond, but are localized as a set of rings propagating away from the impact sight. A packet can only be formed by a superposition of many plane waves, called Fourier harmonics. The phase velocity is the velocity at which each harmonic propagates. The group velocity is the velocity at which the "center of mass" of packet as a whole propagates. By "center of mass," I mean the expectation value of the coordinate. If the wave is only one-dimensional, the expectation value is \[ = \int dx x |\psi|^2, \] where \psi is the wave-function of the packet. If the wave is "non-dispersive," the phase velocity and group velocities are equal, and do not depend on the wave-length of the wave. Light in a vacuum and sound (at small amplitudes) are two examples of non-dispersive waves. A wave packet, in a non-dispersive medium, does not change its shape as it propagates. "Dispersive" waves, on the other hand, are waves where the velocity does depend on the wavelength. The phase and group velocities are not equal when the wave is dispersive. Examples of these are de Broglie waves, and surface waves on water. When a wave is dispersive, a wave-packet will change shape as it propagates. The group velocity can be very different from the phase velocity, only when the phase velocity has a very large derivative. In physical systems, the derivative of the phase velocity is usually only large when the medium is strongly absorbing. As a result, the wave will be damped before the effect can be measured. In the experiments where light packets move faster than light, a second laser is used to "burn a hole" in the absorbing material so that light can propagate where it normally would be absorbed.

Answer 5

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what is group velocity and phase velocity in waves?