Concept Of Group Velocity Research Articles

How light conquers darkness (W.R. Hamilton and the concept of group velocity)

How light conquers darkness (W.R. Hamilton and the concept of group velocity)

Diffusivity measurement method based on the concept of group velocity

Diffusivity measurement method based on the concept of group velocity

Как свет побеждает тьму (У. Р. Гамильтон и понятие групповой скорости)

Как свет побеждает тьму (У. Р. Гамильтон и понятие групповой скорости)

Energy dispersion in a barotropic atmosphere

AbstractThe propagation of energy on the sphere as described by the linear barotropic vorticity equation is studied. For simple flows, the concept of group velocity is examined. Planetary‐scale vorticity anomalies exhibit a surprising lack of dispersion in a hemispheric domain. On the sphere, there is more dispersion and a significant cross‐equatorial propagation of energy. Planetary‐scale vorticity sources tend to produce an orderly downstream train of waves which exhibit a strong tilt (NE‐SW for sources in the northern hemisphere) and cross the equator. Zonally elongated sources produce an almost north‐south train of waves. For realistic zonal flows with equatorial easterlies, large‐scale transients cross the equator, but the long‐term forced response in the opposite hemisphere is small.Comparison of the results with observations of atmospheric wave behaviour is made, and some possible implications are presented.

Energy dispersion in a barotropic atmosphere

The propagation of energy on the sphere as described by the linear barotropic vorticity equation is studied. For simple flows, the concept of group velocity is examined. Planetary-scale vorticity anomalies exhibit a surprising lack of dispersion in a hemispheric domain. On the sphere, there is more dispersion and a significant cross-equatorial propagation of energy. Planetary-scale vorticity sources tend to produce an orderly downstream train of waves which exhibit a strong tilt (NE-SW for sources in the northern hemisphere) and cross the equator. Zonally elongated sources produce an almost north-south train of waves. For realistic zonal flows with equatorial easterlies, large-scale transients cross the equator, but the long-term forced response in the opposite hemisphere is small. Comparison of the results with observations of atmospheric wave behaviour is made, and some possible implications are presented.

Simple Demonstration of the Concept of Group Velocity

Simple Demonstration of the Concept of Group Velocity

A note on waves, wavegroups, and local influence

Abstract The observed meridional motions of atmospheric systems or wavegroups are shown to be related to the shape‐preserving (normal mode, eigenfunction) waves discussed in linearized perturbation theory through the concepts of group velocity, dispersion and interference pattern. Although all phase velocities are east‐west, the group velocity vector can lie in any direction. Phase velocities with a meridional component can occur if the Coriolis parameter is replaced by a linear function of latitude only because the proper periodic boundary condition cannot then be imposed. Orographic effects make the phase velocity non‐existent in a hydrostatic atmosphere, but the concept of group velocity remains valid.

Concept of Group Velocity in Resonant Pulse Propagation

It is shown that the prediction of Garrett and McCumber that a light pulse's velocity may exceed the speed of light in a resonantly absorbing medium is due to asymmetric absorption of energy from the light pulse. More energy is absorbed from the trailing half of the pulse than from the front half, causing the center of gravity of the pulse to move at a velocity greater than the phase velocity of light. It is also shown that in certain cases a pulse's maximum will propagate at the group velocity even when the pulse as a whole is distorted by dispersion.

Energy velocity of a system

An energy velocity, representing the overall rate of energy flow in a system, is formulated in terms of moments of a waveform. For simple linear systems, it reduces to a weighted average of the group velocity, and so may be used as an alternate derivation of the group velocity concept.

Compression of transmitted pulses in plasmas

The compression of frequency-modulated RF pulses is discussed when the dispersive property of a lossless, isotropic, and homogeneous plasma is used to enhance the resolution of transmitted pulses at the receiver. Using the basic concept of group velocity, an expression is obtained for the carrier phase variation of a rectangular envelope pulse so that the pulse collapses on itself at the receiver. Unlike the chirp radar pulse, the frequency variation is not linear and the optimum detection of transmitted pulses depends on both the plasma frequency and the distance between the transmitter and the receiver. The method of convolution was used to obtain numerical results, which indicate that the formulation given results in compression for short propagation distances.

Non-linear dispersive waves

A general theory is developed for studying changes of a wave train governed by non-linear partial differential equations. The technique is to average over the local oscillations in the medium and so obtain differential equations for the variations in amplitude, wave number, etc. It corresponds to the Krylov-Bogoliubov averaging technique for the ordinary differential equations of non-linear vibrations. The equations obtained in this way are hyperbolic and can be handled by the usual theory of quasi-linear hyperbolic systems, involving the theory of characteristics and shock waves. In this case the ‘shocks’ are abrupt changes in the amplitude, wave number, etc. They do not involve dissipation, but it turns out that frequency plays the role corresponding to entropy in ordinary gas dynamic shocks. It is not clear whether these shocks will really occur in practice. However, they have a number of interesting properties and seem to be relevant to the discussion of so-called collisionless shocks in plasma dynamics. The main applications envisaged are to water waves and plasma dynamics, and the theory is developed using typical equations from these areas. If the original equations are linear, this theory predicts the usual description of dispersive waves in terms of group velocity, so it may be considered as an extension of the group velocity concept to non-linear problems. Mathematically, the theory may be considered as an extension of some of the methods and ideas for the non-linear ordinary differential equations of vibration theory to partial differential equations.

Storm Location at Sea—An Illustration of Group Velocity

When the waves from a storm at sea travel outwards, the long waves arrive first at any distant fixed point and are succeeded by shorter and shorter ones. Observation of the wavelength of the arriving train as a function of time determines the distance away of the storm center. A description of the phenomenon involves an account of the progress of an arbitrary disturbance in a dispersive medium, and it is suggested that this might prove an interesting illustration for teaching purposes of the concept of group velocity.

SOME EFFECTS OF WAVE-LENGTH VARIATIONS OF THE LONG WAVES IN THE UPPER WESTERLIES

In an attempt to use the equation developed by Rossby for the motion of waves in a single-layer barotropic atmosphere as a prognostic tool, the effect of the upstream variation of wave length is studied. With the aid of the concept of group velocity an expression is obtained for trough displacement which takes into account the change of wave length with time and the acceleration of the long waves. Tests of the results indicate that the inclusion of the upstream wave-length variation in the forecast of trough displacement gives a significant improvement in the forecast verification. The results can be expressed qualitatively as follows: If the wave length of the long waves increases upstream at the initial moment, the eastward speed of the wave under consideration decreases with time. If the wave length decreases upstream at the initial moment, the eastward speed of the wave increases with time.

LXXXV.Interference calculations and wave groups

Summary The full scope of interference calculations depends on the utilization of physically non - real wave components. The Saddle Point Method is truly a stationary phase calculation in that it emphasizes a set of damped waves of constant phase, while the Kelvin (Stationary Phase) Method is shown to really imply constancy of amplitude for the interfering wavelets. On the basis of this last remark the Kelvin Method is extended to cases where the orthodox formulation would be inaccurate. For the canonical form corresponding to the most general statement of the Steepest Descent scheme the Kelvin Method yields a Fourier Integral. The concept of group velocity for non-linear argument functions of x, t may be extended in two different ways denoted by U1 and U2. Errors are pointed out in the contribution of Green to this subject. A theorem due to Havelock is proved in an alternative manner and certain generalizations indicated, including the case of media with potential energies of type It is demonstrated that...