Dispersion is the phenomenon by which, in a given medium, the propagation speed of a wave depends on its wavelength. The particular law that, case by case, describes how the speed depends on the wavelength is called the dispersion law (or dispersion relation).

The dispersion law can be represented in many ways, because waves are described by many different quantities, not all independent of each other (see fundamental quantities of waves). In applications it is usual to give the dependence of the frequency on the wavelength.

How dispersion shows itself

To understand the essence of this property of the medium, let us look at the animations below:

• in both animations the black line is the sum of the two coloured lines. Each coloured line is a simple sinusoid, but the two have slightly different wavelengths (in a ratio of 11:10).
• Both animations start from the same initial state, that is, the coloured waves are superposed so as to form exactly the same pattern at the initial instant.

We observe that:

1. in the non-dispersive medium the two coloured components advance at the same speed. Consequently their relative phase stays the same and, as a result, the black line (the sum of the components) also advances at the same speed without changing shape during propagation;
2. in the dispersive medium, on the contrary, the blue sinusoid advances faster than the red one, so their relative phase changes over time. As a result, as time passes, the sum signal (black line) does not simply translate in space, but changes shape completely.

To highlight the propagation speed of the signal, in both animations the maximum of the resulting wave is marked with a black dot (in an acoustic example this would coincide with the point where the beat between the two components has maximum intensity).

superposition of two harmonic waves of different frequency
non-dispersive medium	dispersive medium

Consequences and applications

So far the only apparent consequence of dispersion seems to be that the resulting signal propagates at two different speeds in the two cases. In fact, the two speeds have different names in physics:

• the speed at which each harmonic component propagates is called the phase velocity, while
• the speed at which the beat between the two (or, more generally, the envelope of the resulting wave) propagates is called the group velocity.

Note that, in the case above, the group velocity is greater than both phase velocities, but with a different dispersion (that is, by changing the dependence of the phase velocity on the wavelength) it is possible to produce superpositions in which the group velocity is smaller than or equal to the phase velocity, or even a stationary beat with zero group velocity. In a non-dispersive medium, on the other hand, phase and group velocities are always equal.

Far from being a mere curiosity, this phenomenon becomes extremely important in telecommunications (both those based on electromagnetic waves and acoustic ones, such as speech or music). This becomes clearer when more than two frequencies are superposed. The following example sketches the propagation of white light in a dispersive medium. The colours of the components correspond to the wavelengths. As for real visible light, red corresponds to the longest wavelength and violet to the shortest.

Non-dispersive medium	Dispersive medium
Waves of all colours propagate at the same speed	Waves of different colours propagate at different speeds

It is now easy to see why this phenomenon matters in applications: suppose the white curve is a signal we want to transmit from one place to another (for example a radio transmission, or a spoken word, if we are thinking of acoustic waves). Clearly, if the transmission line is non-dispersive, the signal propagates without changing shape, and can therefore be transmitted at a distance faithfully to the original, without distortion. Indeed, the envelope of the white curve moves to the right in the animation. When the medium is dispersive, instead, the original signal cannot be received, because it has changed shape along the way.

Both animations start from the same initial state, in which the superposition of the monochromatic waves produces a signal with a well-defined envelope (similar to that of the first animations on the page); but while in the non-dispersive medium the maximum is preserved, moving as a whole to the right, in the second animation the signal soon spreads out, losing its initial character.

Further reading and links

Diagram of dispersion in Newton's prism

In a very famous experiment, Newton showed that sunlight is actually a mixture of light of various colours. Using a prism and exploiting the phenomenon of refraction, he managed to split a beam of sunlight into its component colours.

The phenomenon can be explained simply by assuming that the refractive index depends on the "colour of the light" striking the boundary surface between the two media. According to the wave theory, which associates each "colour" with a well-defined wavelength within the electromagnetic spectrum, this dependence must be attributed to a dependence of the propagation speed in the denser medium on the wavelength of the wave itself.

effect of optical dispersion in water drops

• A natural phenomenon in which dispersion produces a spectacular effect is the rainbow, to which a separate page is devoted, illustrating some of its details.
• Waves propagating at the boundary surface between two media are also generally dispersive, even if the two media taken separately are not. The best-known example is waves on the surface of water. It is well known that longer waves propagate faster than shorter ones. To find out the particular dispersion relation in this case, see the page wave equation in water.
• Another type of surface wave (which therefore propagates with dispersion) is the so-called Rayleigh wave, observed at the surface of the Earth's crust during earthquakes. On this subject see speed of mechanical waves and transverse and longitudinal waves.
  
  
• Dispersion in glass and transparent materials is particularly relevant for all optical instruments (camera lenses, binoculars and telescopes). Indeed, if dispersion is not adequately compensated, these instruments can show image artefacts and distortions known as "chromatic aberration". In photographs taken with cheap cameras it is evident at the boundary between a dark object and a bright one (typically the sky), along which small coloured lines can be seen.
• Even when an infinite medium has no dispersion, it is possible to build confined configurations that show dispersion. This is the case of waveguides. They are "tracks" able to propagate a certain type of wave along a fixed path. For electromagnetic waves, optical fibres and certain types of antenna are waveguides; for acoustic waves, the bores of wind instruments are waveguides. As illustrated in detail on the page waveguides, a common feature of these devices is that each propagation mode has a cutoff frequency below which it cannot propagate, except in some cases for the fundamental mode (such as the plane wave in a pipe, or the fundamental mode of an optical fibre). In the language of dispersion, we can say that this happens because, as the cutoff frequency is approached, the group velocity becomes zero. Note that, in these cases, dispersion depends not only on the physical properties of the medium carrying the wave, but also on the particular geometric configuration of the walls bounding it (in particular on their shape and size).
• As for waveguides, we also observe dispersion when a wave propagates in a discrete rather than a continuous medium. More precisely, the phenomenon becomes relevant for waves whose wavelength is comparable to the distance between two successive discrete elements of the medium. Otherwise the medium can be treated as a continuum. Think of a solid as an atomic lattice or, more simply, of a chain of harmonic oscillators. As described in more detail on the page wave equation in the chain of oscillators, in this case too waves of different wavelengths propagate at different speeds, and there are "forbidden bands", corresponding to frequency ranges in which waves cannot propagate in the medium at all.