I think the time dispersion is the collective name of a number of physical processes in the medium. This is primarily due to different times of relaxation processes. The relaxation mechanisms at different frequencies are also different.

The physical meaning of relaxation processes can usually be described through the dynamics of continuous media. We must also not forget about spatial dispersion at the microscale.

Dispersion means that different frequencies travel at different speeds. The reason for this is partly as described by Kirill Zeyde but it can also be due to structure, such as in periodic materials, or in waveguide.

When there is dispersion signal shapes do not remain constant as a signal travels, unless the signal is a pure infinite sine wave, in which case it doesn't qualify as a signal (it contains no or almost no information). All useful signals contain several frequencies, and if these travel at different speeds the signal shape changes with distance.

If the ratio of w/k is constant for all frequencies there is no dispersion - the graph is a straight line through the origin and the slope is the velocity of any signal.

If w/k is not constant the line is not straight and/or does not go through the origin. The slope is the speed that a signal can be sent at close to that frequency, and the ratio w/k is the speed that the wave pattern travels. For instance, in waveguide, a puse several wavelengths long might travel up the waveguide at half the speed of light, while a wave pattern at twice the speed of light travels through the pulse, starting at the back of the pulse and fading when it reaches the front. [Ripples travelling out from a stone dropped in a pond do the same https://www.youtube.com/watch?v=dsrUxhaaWks ]. Over distance the pulse shape (envelope) will change because of dispersion. This can be seen using excel for instance by doing a Fourier transform of a pulse to get the frequency components, then adding a quadratic phase change between them (any non-linear relation will do, a linear phase change represents non-dispersive travel) and adding them back together (doing the inverse Fourier transform).

A medium will be disperse waves with different frequencies when it changes the speed of wave with the frequency of the wave.

This means that the speed of the wave in the medium v= c/ sqr epsilonr, where c is the speed of light in the space, epsilonr is the dielectric consatnt of the materials. An example is the white light dispersion when it passes through glass prism. Due to dispersion the light is dispersed in different directions according to its frequency. So, dispersion of the different frequency wave in the medium is due to the dependence refractive index on the frequency of the wave.

Best wishes

Basically, the dispersion (in the time scale) is determined by Delt t dependent probability of photon illumination by the atom (molecule) of the medium, captured a photon before time Delt t.

In the frequency scale, the intensity (the dielectric parameter "eps", or refraction index "n") of this process occurs to be frequency-dependent. Therefore, "Dispersion" is defined often as the frequency-dependent speed of light (EM waves) in such media.

Thank you for your answers

Dear Amel Benahmed

In addition to all interesting answers in this thread, I would like to say that the dispertion relation for elastic waves also accounts for the anisotropic properties of the solids & allows to study not only microscopic crystals, but also macroscopic bodies such as the earth.

The dispersion law for elastic waves in solid is called the "Christoffel equation"

Please check for example the following link:

Article Solving the Christoffel equation: Phase and group velocities