Phase velocity

File:Wave group.gif

Frequency dispersion in groups of gravity waves on the surface of deep water. The Template:Colorbull red square moves with the phase velocity, and the Template:Colorbull green circles propagate with the group velocity. In this deep-water case, the phase velocity is twice the group velocity. The red square overtakes two green circles when moving from the left to the right of the figure.Template:Paragraph New waves seem to emerge at the back of a wave group, grow in amplitude until they are at the center of the group, and vanish at the wave group front.Template:Paragraph For surface gravity waves, the water particle velocities are much smaller than the phase velocity, in most cases.

File:Wave packet propagation (phase faster than group, nondispersive).gif

Propagation of a wave packet demonstrating a phase velocity greater than the group velocity.

File:Wave opposite-group-phase-velocity.gif

This shows a wave with the group velocity and phase velocity going in different directions.<ref name=nemirovsky2012negative>Template:Cite journal</ref> The group velocity is positive (i.e., the envelope of the wave moves rightward), while the phase velocity is negative (i.e., the peaks and troughs move leftward).

The phase velocity of a wave is the rate at which the wave propagates in any medium. 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 Template:Mvar (lambda) and time period Template:Mvar as

<math>v_\mathrm{p} = \frac{\lambda}{T}.</math>

Equivalently, in terms of the wave's angular frequency Template:Mvar, which specifies angular change per unit of time, and wavenumber (or angular wave number) Template:Mvar, which represent the angular change per unit of space,

<math>v_\mathrm{p} = \frac{\omega}{k}.</math>

To gain some basic intuition for this equation, we consider a propagating (cosine) wave Template:Math. We want to see how fast a particular phase of the wave travels. For example, we can choose Template:Math, the phase of the first crest. This implies Template:Math, and so Template:Math.

Formally, we let the phase Template:Math and see immediately that Template:Math and Template:Math. So, it immediately follows that

<math> \frac{\partial x}{\partial t} = -\frac{ \partial \phi }{\partial t} \frac{\partial x}{\partial \phi} = \frac{\omega}{k}.</math>

As a result, we observe an inverse relation between the angular frequency and wavevector. If the wave has higher frequency oscillations, the wavelength must be shortened for the phase velocity to remain constant.<ref name="mathpages1">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Additionally, the phase velocity of electromagnetic radiation may – under certain circumstances (for example anomalous dispersion) – exceed the speed of light in vacuum, but this does not indicate any superluminal information or energy transfer.Template:Citation needed It was theoretically described by physicists such as Arnold Sommerfeld and Léon Brillouin.

The previous definition of phase velocity has been demonstrated for an isolated wave. However, such a definition can be extended to a beat of waves, or to a signal composed of multiple waves. For this it is necessary to mathematically write the beat or signal as a low frequency envelope multiplying a carrier. Thus the phase velocity of the carrier determines the phase velocity of the wave set.<ref name="electroagenda">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Group velocity

File:Wavepacket1.gif

A superposition of 1D plane waves (blue) each traveling at a different phase velocity (traced by blue dots) results in a Gaussian wave packet (red) that propagates at the group velocity (traced by the red line).

The group velocity of a collection of waves is defined as

<math> v_g = \frac{\partial \omega} {\partial k}.</math>

When multiple sinusoidal waves are propagating together, the resultant superposition of the waves can result in an "envelope" wave as well as a "carrier" wave that lies inside the envelope. This commonly appears in wireless communication when modulation (a change in amplitude and/or phase) is employed to send data. To gain some intuition for this definition, we consider a superposition of (cosine) waves Template:Mvar with their respective angular frequencies and wavevectors.

<math>\begin{align}

f(x, t) &= \cos(k_1 x - \omega_1 t) + \cos(k_2 x - \omega_2 t)\\ &= 2\cos\left(\frac{(k_2-k_1)x-(\omega_2-\omega_1)t}{2}\right)\cos\left(\frac{(k_2+k_1)x-(\omega_2+\omega_1)t}{2}\right)\\ &= 2f_1(x,t)f_2(x,t). \end{align}</math>

So, we have a product of two waves: an envelope wave formed by Template:Math and a carrier wave formed by Template:Math. We call the velocity of the envelope wave the group velocity. We see that the phase velocity of Template:Math is

<math> \frac{\omega_2 - \omega_1}{k_2-k_1}.</math>

In the continuous differential case, this becomes the definition of the group velocity.

Refractive index

In the context of electromagnetics and optics, the frequency is some function Template:Math of the wave number, so in general, the phase velocity and the group velocity depend on specific medium and frequency. The ratio between the speed of light c and the phase velocity v_p is known as the refractive index, Template:Math.

In this way, we can obtain another form for group velocity for electromagnetics. Writing Template:Math, a quick way to derive this form is to observe

<math> k = \frac{1}{c}\omega n(\omega) \implies dk = \frac{1}{c}\left(n(\omega) + \omega \frac{\partial}{\partial \omega}n(\omega)\right)d\omega.</math>

We can then rearrange the above to obtain

<math> v_g = \frac{\partial w}{\partial k} = \frac{c}{n+\omega\frac{\partial n}{\partial \omega}}.</math>

From this formula, we see that the group velocity is equal to the phase velocity only when the refractive index is independent of frequency <math display=inline>\partial n / \partial\omega = 0</math>. When this occurs, the medium is called non-dispersive, as opposed to dispersive, where various properties of the medium depend on the frequency Template:Mvar. The relation <math>\omega(k)</math> is known as the dispersion relation of the medium.

See also

• Cherenkov radiation
• Dispersion (optics)
• Group velocity
• Propagation delay
• Shear wave splitting
• Wave propagation
• Wave propagation speed
• Planck constant
• Speed of light
• Matter wave#Phase velocity

References

Bibliography

• Crawford jr., Frank S. (1968). Waves (Berkeley Physics Course, Vol. 3), McGraw-Hill, Template:ISBN Free online version
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