Group speed

Dispersion of water waves (red dots move with phase velocity, green dots move with group velocity). In this case, the phase velocity is twice the group velocity.

Group velocity is a quantity that characterizes the propagation velocity of a “group of waves” - that is, a more or less well-localized quasimonochromatic wave (waves with a fairly narrow spectrum). It is usually interpreted as the velocity of the maximum amplitude envelope of the quasimonochromatic wave packet (or train of waves). In the case of considering the propagation of waves in space with a dimension greater than unity, it is usually assumed that a wave packet is close in shape to a plane wave ^[1] .

The group velocity in many important cases determines the rate of energy and information transfer by a quasi-sinusoidal wave (although this statement in the general case requires serious refinements and reservations).

Group velocity is determined by the dynamics of the physical system in which the wave propagates (of a specific medium, of a specific field, etc.). In most cases, the linearity of this system is implied (exactly or approximately).

For one-dimensional waves, the group velocity is calculated from the dispersion law :

v_{gr}=d\omega /dk

{\ displaystyle v_ {gr} = d \ omega / dk}

{\ displaystyle v_ {gr} = d \ omega / dk}

,

Where $\omega$ ${\ displaystyle \ omega}$ $\ omega$ - angular frequency $k$ ${\ displaystyle k}$ $k$ Is the wave number .

The group velocity of waves in space (for example, three-dimensional or two-dimensional) is determined by the frequency gradient along the wave vector ${\vec {k}}$ ${\ displaystyle {\ vec {k}}}$ ${\ vec k}$ :

{\vec {v}}_{gr}=\nabla _{\vec {k}}\omega

{\ displaystyle {\ vec {v}} _ {gr} = \ nabla _ {\ vec {k}} \ omega}

{\ vec v} _ {{gr}} = \ nabla _ {{\ vec k}} \ omega

or (for three-dimensional space):

(v_{gr})_{x}=\partial \omega /\partial k_{x},

{\ displaystyle (v_ {gr}) _ {x} = \ partial \ omega / \ partial k_ {x},}

(v _ {{gr}}) _ {x} = \ partial \ omega / \ partial k_ {x},

(v_{gr})_{y}=\partial \omega /\partial k_{y},

{\ displaystyle (v_ {gr}) _ {y} = \ partial \ omega / \ partial k_ {y},}

(v _ {{gr}}) _ {y} = \ partial \ omega / \ partial k_ {y},

(v_{gr})_{z}=\partial \omega /\partial k_{z}.

{\ displaystyle (v_ {gr}) _ {z} = \ partial \ omega / \ partial k_ {z}.}

(v _ {{gr}}) _ {z} = \ partial \ omega / \ partial k_ {z}.

Note: the group velocity, generally speaking, depends on the wave vector (in the one-dimensional case, on the wave number), that is, generally speaking, it is different for different sizes and for different directions of the wave vector.

Content

Special cases

In one-dimensional media without dispersion, the group velocity formally coincides with the phase velocity only in the case of one-dimensional waves.

In dissipative (absorbing) media, the group velocity decreases with increasing frequency in the case of normal phase velocity dispersion and, conversely, increases in media with anomalous dispersion . In this case, the group velocity can overcome the speed of light, as well as negative anomalous dispersion, when the group velocity is opposite to the phase velocity. In dissipative structures (for example, plasmonic), the group velocity can have any value: less than the speed of light, more than the speed of light, be negative with respect to phase velocity, go through infinity. Such a group velocity is a kinematic quantity (as well as a phase velocity) and determines the speed of beat transfer of two monochromatic waves that are infinitely close in frequency (as Stokes considered it). For Hamiltonian systems (closed systems without dissipation) in the general case, S.M. Rytov (JETP, 7, 930, 1947) proved a theorem stating that the group velocity coincides with the transfer rate of electromagnetic energy by a monochromatic wave (Leontovich-Lighthill-Rytov theorem). Negative (with respect to phase velocity) group velocity in such non-dissipative media and structures corresponds to backward waves. In dissipative media and structures, the direction of energy motion is determined by the Poynting vector or the direction of wave attenuation.

If the dispersion properties of the medium are such that the wave packet propagates in it without significant changes in the shape of its envelope, the group velocity can usually be interpreted as the speed of transfer of the “energy” of the wave and the speed with which signals carrying information can be transmitted using the wave packet ( that is, the “speed of causality”).

In the classical limit of quantum-mechanical equations, the velocity of a classical particle is the group velocity of the corresponding quantum-mechanical wave function. One of a pair of canonical Hamilton equations :

{\dot {q}}_{i}=\partial H/\partial p_{i}

{\ displaystyle {\ dot {q}} _ {i} = \ partial H / \ partial p_ {i}}

- there is, therefore, the classical limit of the above expression for group velocity; this is especially clear in Cartesian coordinates, given ${\vec {p}}=\hbar {\vec {k}},\ H(p,q)=\hbar \omega (k,q).$ ${\ displaystyle {\ vec {p}} = \ hbar {\ vec {k}}, \ H (p, q) = \ hbar \ omega (k, q).}$

History

The idea of a group velocity different from the phase velocity of a wave was first proposed by Hamilton in 1839. The first fairly complete consideration was made by Rayleigh in his Theory of Sound in 1877 ^[2] .

Notes

↑ Miller M.A., Suvorov E.V. Group speed // Physical Encyclopedia / Ch. ed. A.M. Prokhorov . - M .: Soviet Encyclopedia , 1988.- T. 1. - S. 544-545. - 704 s. - 100,000 copies.
↑ Brillouin, Léon (1960), Wave Propagation and Group Velocity , New York: Academic Press Inc., OCLC 537250

Literature

Huizem J.B. Linear and nonlinear waves. - M .: World. - 1977.
Ablowitz MJ & Segur H. Solitons and the Inverse Scattering Transform. - SIAM Philadelphia. - 1981.
Rabinovich MI, Trubetskov DI Oscillations and waves in linear and nonlinear systems. - Kewver-Academic Publ., Amsterdam. - 1989.
Ostrovsky LA and Potapov AI Modulated Waves. Theory and Applications. - Jonh Hopkins Uni Press, Baltimore - London. - 1999.

[ФЭ1-1] Miller M.A., Suvorov E.V. Group speed // Physical Encyclopedia / Ch. ed. A.M. Prokhorov . - M .: Soviet Encyclopedia , 1988.- T. 1. - S. 544-545. - 704 s. - 100,000 copies.

[autogenerated1-2] Brillouin, Léon (1960), Wave Propagation and Group Velocity , New York: Academic Press Inc., OCLC 537250