However, the question remains: What causes dispersion?

[Studiot]

The simple wave equation is

∂²η/∂x² = (1/c²)(∂²η/∂t²

The solution of this has velocity c independent of frequency. This is called the phase velocity.

However more complicated wave equations exist where the solution does not include a constant c but have velocity dependent upon frequency.

eg

∂⁴η/∂x⁴ = -(ρS/YI²)(∂²η/∂t²

These commonly arise from non linearities in the restoring force.

Since different frequency waves then travel at different velocities, a wave shape distorting effect called dispersion occurs.
The power in a wave depends upon its shape so if the shape changes the power changes, hence the term dispersion.

Sorry I don't have acces to my latex editor at the moment.

Please note SophieCentaur's comment that polarisation only occurs in trnasverse waves, not longitudinal ones and also that power loss only occurs at the instant of polarisation and not thereafter.

With waves with dispersion as every frequency has a different phase velocity we sometimes group velocity which refers to a (small) renge of frequencies that have a small range of velocities and therefore a signal containing only this range of frequencies will pass a substantial distance undistorted. They will, of course disperse eventually.

 

[snorkack]

I find this interesting but conflicting with the understanding I had so far.

The speed of a mechanical wave in a medium is:

(a) Directly proportional to restoring (elastic) force of the material, i.e. how much the medium resists strain or deformation and hence with which force it recover the equilibrium position.
(b) Inversely proportional to inertia of the material.

For example:

- In a string, (a) is tension and (b) is the longitudinal density.
- In sound, (a) is compressibility and (b) is volume density.

In the case of EM waves, I had thought that permittivity (which is related to how easily the material polarizes) plays the role (b) of inertia, not (a). In fact, permittivity is placed in the denominator of the formula, like density, thus clearly denoting that the speed is inversely proportional to this factor. Of course, one could say that speed is proportional to how low permittivity is, but that is a convoluted approach.

While it is true that dipoles align against the E field and hence weaken it, it does not seem that such is the reason for an EM wave’s retardation, since the latter is an oscillation, so the dipoles align in one direction and then in the opposite direction, thus acting like an antenna that, instead of attenuating the E field, simply re-radiates the wave, thus leaving the E field unaffected.

Is this understanding correct or should I change it?

Compare an oscillator circuit including a capacitor.

A dielectric with dielectric permittivity will decrease the voltage of a capacitor, while leaving the charges unchanged. This also means that the increased capacity of the capacitor decreases the frequency of the oscillator circuit. That is the restoring force side - not the inertia side of the equation.

Likewise consider that the Hertz antenna is a limiting case of oscillator circuit - the plates are converted to ends of the antenna in surrounding medium. The frequency would respond to the permittivity of the surrounding medium in the same way.

 

[Studiot]

@snorkack

I'm sorry, I don't follow where any of your examples are dealing with dispersion?

 

[Saw]

Compare an oscillator circuit including a capacitor.

A dielectric with dielectric permittivity will decrease the voltage of a capacitor, while leaving the charges unchanged. This also means that the increased capacity of the capacitor decreases the frequency of the oscillator circuit. That is the restoring force side - not the inertia side of the equation.

You bring in a good comparison, the role of the same dielectric when faced not with an electromagnetic wave but with a similar phenomenon, an oscillating current.

I fully agree that in this second example capacitance (which is a function of the permittivity of the dielectric material and of the geometry of the capacitor) plays against capacitative reactance, which is usually defined as a kind of inertia that opposes change in tension. To sum up, permittivity plays against inertia. Somehow we could say that here higher permittivity means better “conductivity” of the oscillation that the AC is. And the same applies to frequency: the higher the frequency, the lower the reactance and the higher the “conduction” of the oscillation.

However, the fact is that in the situation that is the object of our discussion (the dielectric is faced with an EM wave) permittivity and frequency play the opposite roles. The higher the permittivity and the higher the frequency, the lower the velocity, which looks more like inertia (since it plays against velocity) than like restoring force (which would play in favor of velocity). Someone could argue: “but you are comparing apples with pears: in one case you talk about the preservation of changes in tension and frequency; in the other, of the preservation of velocity". And I would agree. That is why I tended to believe that the two situations admit different solutions: in one case (AC - tension), permittivity acts against inertia, in the other (EM wave – velocity) it would be on inertia side.