Phase speed


The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, the crest) will appear to travel at the phase velocity. The phase velocity is given in terms of the wavelength λ (lambda) and period T as

v_\mathrm{p} = \frac{\lambda}{T}.

Or, equivalently, in terms of the wave's angular frequency ω, which specifies the number of oscillations per unit of time, and wavenumber k, which specifies the number of oscillations per unit of space, by

v_\mathrm{p} = \frac{\omega}{k}.

To understand where this equation comes from, imagine a basic sine wave, A cos (kxωt). Given time t, the source produces ωt oscillations. At the same time, the initial wave front propagates away from the source through the space to the distance x to fit the same amount of oscillations, kx = ωt. So that the propagation velocity v is v = x/t = ω/k. The wave propagates faster when higher frequency oscillations are distributed less densely in space.[2] Formally, Φ = kxωt is the phase. Since ω = −dΦ/dt and k = +dΦ/dx, the wave velocity is v = dx/dt = ω/k.

Relation to group velocity, refractive index and transmission speed

Since a pure sine wave cannot convey any information, some change in amplitude or frequency, known as modulation, is required. By combining two sines with slightly different frequencies and wavelengths,

\cos[(k-\Delta k)x-(\omega-\Delta\omega)t]\; +\; \cos[(k+\Delta k)x-(\omega+\Delta\omega)t] = 2\; \cos(\Delta kx-\Delta\omega t)\; \cos(kx-\omega t),

the amplitude becomes a sinusoid with phase speed of vg = Δωk. It is this modulation that represents the signal content. Since each amplitude envelope contains a group of internal waves, this speed is usually called the group velocity.[2] In reality, the vp = ω/k and vg = dω/dk ratios are determined by the media. The relation between phase speed, vp, and speed of light, c, is known as refractive index, n = c/vp = ck/ω. Taking the derivative of ω = ck/n, we get the group speed,

\frac{\text{d}\omega}{\text{d}k} = \frac{c}{n} - \frac{ck}{n^2}\cdot\frac{\text{d}n}{\text{d}k}.

Noting that c/n = vp, this shows that group speed is equal to phase speed only when the refractive index is a constant: dn/dk = 0.[2] Otherwise, when the phase velocity varies with frequency, velocities differ and the medium is called dispersive and the function, \omega(k), from which the group velocity is derived is known as a dispersion relation. The phase velocity of electromagnetic radiation may – under certain circumstances (for example anomalous dispersion) – exceed the speed of light in a vacuum, but this does not indicate any superluminal information or energy transfer. It was theoretically described by physicists such as Arnold Sommerfeld and Léon Brillouin. See dispersion for a full discussion of wave velocities.

See also

References

Footnotes

Other

External links

  • Subluminal – a Java applet
  • Simulation – a Java applet by Paul Falstad
  • Group and Phase Velocity – Java applet showing the difference between group and phase velocity.


Velocities of waves
Phase velocityGroup velocityFront velocitySignal velocity
fr:Vitesse d'une onde#Vitesse de phase

nl:Voortplantingssnelheid#Fase- en groepssnelheid

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