Speed of a wave

A Onde is a disturbance which moves in a medium. It is possible to associate two to him speeds of wave .

In a homogeneous medium, the propagation in a given direction of a wave monochromatic, or sinusoidal, results in a simple translation of the sinusoid at a speed called speed of phase or celerity. The more complicated waves can generally be regarded as sums of sinusoids of various frequencies or pulsations (see spectral Analyze). If the speed of phase does not depend on the frequency, the resulting wave undergoes also a total translation without deformation. In the contrary case, the components disperse. One can then often identify groups of waves of which the maximum moves with a speed of group different from celerities from the components.

Speed of phase

The speed of phase of a Onde is the speed to which the phase of the wave is propagated in space. If one selects any particular point of the wave (for example the peak), it will give the impression to move in space at the speed of phase. The speed of phase is expressed according to the pulsation of the wave ω and the Nombre of wave K:

$v\_p\; =\; \backslash \; frac\; \{\backslash \; Omega\}\; \{K\}\; \backslash ,$

Indeed, $\backslash \; psi\; is\; a\; monochromatic\; wave\; =\; \backslash \; psi\_\; \{0\}\; cos\; (\backslash \; Omega\; T\; -\; kx+\; \backslash \; phi\_\; \{0\})\; \backslash ,$, place on a Surface of wave, i.e the whole of the points having the same value of $\backslash \; psi\; \backslash ,$, consequently the same value of the phase $\backslash \; phi\; \backslash ,$, it is the Plan of phase. The plan of phase $\backslash \; phi\; \backslash ,$ is in $x\; \backslash ,$ at time $t\; \backslash ,$:

$\backslash \; phi\; =\; \backslash \; Omega\; T\; -\; kx+\; \backslash \; phi\_\; \{0\}\; \backslash ,$

and in $x+dx\; \backslash ,$ at time $t+dt\; \backslash ,$:

$\backslash \; phi\; =\; \backslash \; Omega\; (t+dt)\; -\; K\; (x+dx)\; +\; \backslash \; phi\_\; \{0\}\; \backslash ,$

from where $0\; =\; \backslash \; Omega\; dt\; -\; K\; dx\; \backslash ,$

i.e. $v\_\; \{\backslash \; phi\}\; =\; \backslash \; frac\; \{dx\}\; \{dt\}\; =\; \backslash \; frac\; \{\backslash \; Omega\}\; \{K\}\; \backslash ,$

Case of the electromagnetic waves

In this case the speed of phase, equal to a constant $c$ in the vacuum, decreases in a transparent medium. This reduction is characterized by the index medium:

$n\; =\; \backslash \; frac\; \{C\}\; \{v\_\; \{\backslash \; phi\}\}\; \backslash ,$

Moreover, the medium is dispersive: this index depends on the number of wave (thus the wavelength), which results in introducing the concept speed of group.

The speed of phase of a electromagnetic Onde can be higher than the Speed of light in the Vide in certain circumstances, but that does not imply a transfer of energy or information at an high speed to that of the light. Indeed, the speed of phase is a fictitious measurement: as for a cord, one with the impression of a movement along the cord whereas all displacements are perpendicular for him. The speed of phase thus does not measure the displacement of any real physical quantity.

Speed of group

According to what precedes, the speed of phase of a monochromatic wave is equal to the report/ratio of its pulsation to its number of wave. Let us consider the simplest case of two superimposed waves of close pulsations and amplitude unit (the phases which do not intervene are ignored):

$f\; (T,\; X)\; =\; \backslash \; cos\; (\backslash \; omega\_1\; T\; -\; k\_1\; X)\; +\; \backslash \; cos\; (\backslash \; omega\_2\; T\; -\; k\_2\; X)\; \backslash ,$

According to trigonometry, a sum of cosine is transformed into a product of cosine:

$f\; (T,\; X)\; =\; 2\; \backslash \; cos\; \{1\; \backslash \; over\; 2\}\; -\; \backslash \; omega\_1)\; T\; -\; (k\_2\; -\; k\_1)\; X\; \backslash \; cos\; \{1\; \backslash \; over\; 2\}\; +\; \backslash \; omega\_1)\; T\; -\; (k\_2\; +\; k\_1)\; x$

It appears a phenomenon of beat in which a sinusoid of characteristics close to those of the two components, is modulated by a sinusoid moreover low frequency whose celerity is $\{\backslash \; omega\_2\; -\; \backslash \; omega\_1\}\; \backslash \; over\; \{k\_2\; -\; k\_1\}$. This quantity represents the speed of group. When the two starting frequencies are close, this one is roughly equal to

$v\_g\; =\; \backslash \; frac\; \{\backslash \; mathrm\; D\; \backslash \; Omega\}\; \{\backslash \; mathrm\; dk\}$

In the general case of more than two superimposed monochromatic waves, this speed of group relates to an envelope more complicated than a sinusoid.

The function which expresses the pulsation according to the number of wave defines the Relation of dispersion. When the pulsation is directly proportional to the number of wave, which means that the speed of phase is independent of the pulsation, then the speed of group is equal at this speed of common phase. In the contrary case, the envelope of the wave becomes deformed during the propagation.

Case of the electromagnetic waves

In the case of an electromagnetic wave, the speed of phase and the speed of group are bound by the relation:

$v\_g\; v\_p\; =\; \backslash \; frac\; \{c^2\}\; \{n^2\}\; \backslash ,$

with $c$ Speed of light in the vacuum.

Dispersion due at the speed of group is an important effect taken into account for the propagation of information by fiberoptics.

The speed of group is generally presented like the speed to which energy or information is transported by a wave. This description is generally valid, although it is all the same possible to carry out experiments in which the speed of impulses Laser sent in specific materials is higher than the transmission speed of the signal.

See too

Relation of dispersion

Related articles

• Wave

• Propagation the waves
• Directivity

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