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- Quantum Mechanics:Wave packet, phase velocity and group
- Optics Express
- Phase Velocity and Group Velocity Civil Engineering (CE) Notes | EduRev

*According to quantum physics, particles sometimes behave like waves. In some cases, waves behave like particles.*

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## Quantum Mechanics:Wave packet, phase velocity and group

The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the modulation or envelope of the wave—propagates through space. For example, if a stone is thrown into the middle of a very still pond, a circular pattern of waves with a quiescent center appears in the water, also known as a capillary wave. The expanding ring of waves is the wave group , within which one can discern individual wavelets of differing wavelengths traveling at different speeds.

The shorter waves travel faster than the group as a whole, [ citation needed ] [ dubious — discuss ] but their amplitudes diminish as they approach the leading edge of the group.

The longer waves travel more slowly, and their amplitudes diminish as they emerge from the trailing boundary of the group. The group velocity v g is defined by the equation: [2] [3] [4] [5]. One derivation of the formula for group velocity is as follows.

By the superposition principle , the wavepacket at any time t is. There are two factors in this expression. Part of the previous derivation is the Taylor series approximation that:. As a result, the envelope of the wave packet not only moves, but also distorts, in a manner that can be described by the material's group velocity dispersion. Loosely speaking, different frequency-components of the wavepacket travel at different speeds, with the faster components moving towards the front of the wavepacket and the slower moving towards the back.

Eventually, the wave packet gets stretched out. This is an important effect in the propagation of signals through optical fibers and in the design of high-power, short-pulse lasers. The idea of a group velocity distinct from a wave's phase velocity was first proposed by W. Hamilton in , and the first full treatment was by Rayleigh in his "Theory of Sound" in For waves traveling through three dimensions, such as light waves, sound waves, and matter waves, the formulas for phase and group velocity are generalized in a straightforward way: [10].

If the waves are propagating through an anisotropic i. 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 or gainful medium, this does not always hold. In these cases the group velocity may not be a well-defined quantity, or may not be a meaningful quantity. An example concerning the transmission of electromagnetic waves through an atomic gas is given by Loudon.

Despite this ambiguity, a common way to extend the concept of group velocity to complex media is to consider spatially damped plane wave solutions inside the medium, which are characterized by a complex-valued wavevector. Then, the imaginary part of the wavevector is arbitrarily discarded and the usual formula for group velocity is applied to the real part of wavevector, i.

It can be shown that this generalization of group velocity continues to be related to the apparent speed of the peak of a wavepacket. The above generalization of group velocity for complex media can behave strangely, and the example of anomalous dispersion serves as a good illustration. Since the s, various experiments have verified that it is possible for the group velocity as defined above of laser light pulses sent through lossy materials, or gainful materials, to significantly exceed the speed of light in vacuum c.

The peaks of wavepackets were also seen to move faster than c. In all these cases, however, there is no possibility that signals could be carried faster than the speed of light in vacuum , since the high value of v g does not help to speed up the true motion of the sharp wavefront that would occur at the start of any real signal.

Essentially the seemingly superluminal transmission is an artifact of the narrow band approximation used above to define group velocity and happens because of resonance phenomena in the intervening medium.

In a wide band analysis it is seen that the apparently paradoxical speed of propagation of the signal envelope is actually the result of local interference of a wider band of frequencies over many cycles, all of which propagate perfectly causally and at phase velocity. The result is akin to the fact that shadows can travel faster than light, even if the light causing them always propagates at light speed; since the phenomenon being measured is only loosely connected with causality, it does not necessarily respect the rules of causal propagation, even if it under normal circumstances does so and leads to a common intuition.

From Wikipedia, the free encyclopedia. Physical quantity. Frequency dispersion in groups of gravity waves on the surface of deep water. In this deep-water case, the phase velocity is twice the group velocity. The red square overtakes two green circles when moving from the left to the right of the figure. A wave packet. The envelope of the wave packet. The envelope moves at the group velocity. See also: Plane wave. Optics Express. Bibcode : OExpr..

Archived from the original PDF on 16 October Retrieved 10 October Whitham Introduction to Quantum Mechanics. Prentice Hall. Ferry CRC Press. Bibcode : qmid. Wave Propagation in Periodic Structures. New York: McGraw Hill. The Quantum Theory of Light. Solar Physics. Bibcode : SoPh.. Bibcode : Sci Retrieved Physics of Fluids B: Plasma Physics. Bibcode : PhFlB Physical Review E.

Bibcode : PhRvE.. Crawford jr. Waves Berkeley Physics Course, Vol. Freeman and Company, p. Biot, M. Velocities of waves. Phase Group Front Signal. Categories : Radio frequency propagation Optics Wave mechanics Physical quantities Mathematical physics. Hidden categories: Articles with short description Short description matches Wikidata All articles with unsourced statements Articles with unsourced statements from February All accuracy disputes Articles with disputed statements from February Namespaces Article Talk.

Views Read Edit View history. Help Learn to edit Community portal Recent changes Upload file. Download as PDF Printable version. Wikimedia Commons. Wave propagation Dispersion water waves Dispersion optics Wave propagation speed Group delay Group velocity dispersion Group delay dispersion.

## Optics Express

The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the modulation or envelope of the wave—propagates through space. For example, if a stone is thrown into the middle of a very still pond, a circular pattern of waves with a quiescent center appears in the water, also known as a capillary wave. The expanding ring of waves is the wave group , within which one can discern individual wavelets of differing wavelengths traveling at different speeds. The shorter waves travel faster than the group as a whole, [ citation needed ] [ dubious — discuss ] but their amplitudes diminish as they approach the leading edge of the group. The longer waves travel more slowly, and their amplitudes diminish as they emerge from the trailing boundary of the group. The group velocity v g is defined by the equation: [2] [3] [4] [5]. One derivation of the formula for group velocity is as follows.

Meanings of wave velocity. Group Velocity. Phase Velocity. Fourier Analysis. Spectral density. Power Spectrum. Spectral measurements. 2.

## Phase Velocity and Group Velocity Civil Engineering (CE) Notes | EduRev

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*In Alford and Zhao , henceforth Part I , time series of depth-integrated energy E and flux F were computed at 80 historical moorings for the first two baroclinic modes. They describe a globally propagating field of near-inertial and semidiurnal internal waves. Knowledge of the group velocity proves very powerful, because it yields internal-wave intrinsic frequency via the dispersion relation—enabling, for example, estimation of horizontal wavenumber from single-point measurements.*

Wave packet, phase velocity and group velocity. The velocities of the individual waves which superpose to produce the wave packet representing the particle.

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