Velocity| Velocity |
In Classical Mechanics [> s.a. classical
mechanics.]
* Idea: The time derivative
of position, va:=
dqa/dt; @ Kiang
CAA(04)ap/03 [classical
mechanics, special relativity, general relativity].
@ References: Lévy-Leblond AJP(80)
[notions, in special relativity, and rapidity]; Braginsky
et
al PRD(00)gq/99 [speed
meter]; Smith SHPMP(03)
[v can be considered instantaneous]; Bolós CMP(07)
[relative velocity
in curved spacetime]; Casey AJP(07)
[areal velocity, for non-planar problems]; > s.a. tachyons.
In Quantum Mechanics
* Weak velocity: The
expression 
a|v|b
/ a|b,
where v is
the velocity operator and |b and
|a the
states of a particle before and after a velocity measurement.
@ References: Rohrlich & Aharonov PRA(02)qp/01 [superluminal
weak v].
For Waves > s.a. dispersion;
wave phenomena; constants [speed
of light]; schwarzschild-de sitter [vp < 0].
* Energy transfer velocity:
@ in Panofsky & Phillips 62.
* Front velocity: For
waves that have a well-defined front, such as shock waves; vf :=
limk to infty 
(k)/k;
This is the concept that
is relevant for causality considerations.
* Group velocity: For a wave packet centered at k = k0,
vg:= d/dk |
k_0 ;
First mentioned by Hamilton (1839), reintroduced by Stokes (1876) and Rayleigh
(1877).
* Group velocity, generalized: vphi:=
l/(d
/d),
with l a
length, and the
phase change over l.
* Phase velocity: For a wave of wave number k and frequency ,
= A exp{i k(x–vpt)},
or
vp:= /
k = Re [1/n()]
.
* Signal velocity: The
speed of information vs;
Satisfies vg 
vs vf,
but no general definition is known.
* Remark: Some claim that it is vs that should be less than c,
others that it is vf.
* Relationships: For linear waves in a homogeneous medium, vg = ve.
@ General references: in Shore CP(03)gq.
@ Phase velocity: Lakhtakia et al PLA(05)
[negative, for electromagnetic waves in curved spacetime]; Rousseaux et al NJP(08)-a0711 [in
a water tank].
@ Group velocity: in Stratton 41; McDonald AJP(98)
[and energy]; Dolgov & Khriplovich
PLA(98)ht/97 [and
front velocity]; Bers AJP(00)
[= ve];
McDonald AJP(01)
[negative]; Amelino-Camelia et al JCAP(03)ht/02 [in
non-commutative spacetime].
@ Related topics: Drozdov & Stahlhofen
a0704 [local concept];
Mayo & Kerstein PLA(07) [front speed speedup in random media].
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5 jul 2008