Another question about dispersion (and wavenumber)

[EmilyRuck]

Hello!
I still would like to thank those who participated to my previous thread about group velocity and dispersion. Now there is a (maybe) simpler question.
A sinusoidal, electro-magnetic plane wave in the vacuum propagates in a certain direction with the following wavenumber, which is supposed to equal the propagation constant:

[itex]\beta = k = \omega \sqrt{\mu_0 \epsilon_0}[/itex]

so

[itex]\omega = \displaystyle \frac{k}{\sqrt{\mu_0 \epsilon_0}}[/itex]

[itex]\displaystyle \frac{d \omega}{dk}= \displaystyle \frac{1}{\sqrt{\mu_0 \epsilon_0}} = c = v_g[/itex]

and the medium is non dispersive, because [itex]v_g[/itex] is always the same for all frequencies.
But even in this case, the wavenumber is frequency-dependent. Maybe this question is not strictly related to dispersion itself, but to the waves in general: why should the wavenumber linearly increase with frequency? I know that, being [itex]v_g = d\omega / dk[/itex], if [itex]\omega[/itex] were not dependent on [itex]k[/itex], group velocity would be always zero.

But what is the physical meaning of an increasing wavenumber with frequency? Why do waves with higher frequencies propagate with greater wavenumbers?

 

[EmilyRuck]

Maybe it is a problem of phase shift: let [itex]z[/itex] be the direction of propagation. Each wave propagates as described by the term

[itex]E(z) = E_0 e^{-j \beta z}[/itex]

After the same distance [itex]z[/itex], waves with higher frequencies have necessarily a greater phase shift than waves at lower frequencies, because they oscillate more frequently and so their phase has to change more rapidly. May this be the reason?

 

[blue_leaf77]

why should the wavenumber linearly increase with frequency?

That's the most common situation, but in some rare cases ##k## can be a decreasing function of ##\omega##. For a propagation inside a medium of refractive index ##n(\omega)##, the wavenumber is
$$
k(\omega) = \frac{\omega n(\omega)}{c}
$$
The most commonly encountered situation is when ##n(\omega)## increases with frequency, this is called normal dispersion. Within certain frequency region of a given material, however, there are instances in which ##n(\omega)## decreases with frequency, this is called anomalous dispersion. Depending on how fast it decreases, the wavenumber ##k(\omega)## may also decrease with frequency.

 

[Shinji83]

A wave is the propagation of a perturbation, in this case an electromagnetic field.
If you have a source emitting a sinusoidal wave at a certain frequency ω it means that, at the source, in a time T=2π/ω the field has completed a full oscillation.
When at the source the field has completed a full period the value that it had at t=0 (that is the perturbation that was created in the source position at time t=0) has propagated of the length v*T.
That means that looking at things in the space and so in the k domain a full oscillation occupies a length v*T.
If frequency increases since v is the same you get that in the same length v*T as before now you have more oscillations, not just a single one.
Hence k increases.

k=ω/v

k increases linearly with the frequency in non despersive mediums.
In despersive mediums k still varies with the frequency but the relationship is not linear anymore.
For example if the velocity decreases with increasing frequency (normal despersion) you'll get that k will increase faster compared to a non despersive medium linear growth.

 

[vanhees71]

Have a look on simple classical dispersion theory first. Although not really right, because in fact one has to use quantum mechanics to describe it correctly, it gives pretty similar results as the quantum treatment. You find a nice treatment in the Feynman Lectures vol. II.

 

[drvrm]

why should the wavenumber linearly increase with frequency?

Wave number is number of waves which fits in a length traversed by a wave in unit time-so its natural that in normal situation as frequency increases -more number of waves will be there-as wavelength decreases-its as simple as that.

 

[nasu]

Wave-number is either the number of waves in unit length or in 2π units of length. (there are two definitions, similar to having frequency and angular frequency for time domain).
The distance traveled by the wave in unit time is not related.
What you describe there will be
## \frac{c \cdot 1s}{\lambda}=\frac{c \cdot 1s}{c \cdot T} ##
which will actually show how many time periods fit in 1 second.

 

[drvrm]

2π units of length.

i could not get you how you can call 2. π a distance or unit of length?

 

[nasu]

I did not call 2π a unit of length. Sorry for the confusion. I was talking about two definitions of the wave-number.
The most common one in physics is
k=2π/λ, where λ is the wavelength. For λ in meters, k is in m^-1.
If the wavelength is 2π meters, k will be 1m^-1. If λ=π meters, k will be 2 m^-1 and so on.
So what I was trying to say is that the wave-number can be seen as showing how many wavelengths fit in a distance of 2π meters.

Similar to angular frequency ω, which can be seen as how many periods fit in an interval of 2π seconds.
We also have the "regular" frequency. f=ω/(2π), which shows how many cycles "fit" in 1 second.

Similar to f there is another definition of k, used in spectroscopy (and labeled with different letters, like "nu" or "sigma"). This is
ν=1/λ, with no 2π. So it would indicate how many wavelengths in 1m (or cm or whatever). They usually use cm^-1 in spectrosocy graphs.

 

[drvrm]

k=2π/λ, where λ is the wavelength. For λ in meters, k is in m-1.

agreed that your above definition is correct - perhaps i am feeling that it gives a correct dimension of k which should be L raised to -1 as k the wave vector is related to momentum P= (h/2.Pi).k ; actually the factor 2.Pi is dimensionless.
if you talk about wave number used in spectrosopy that is nu = 1/wavelength i.e. is correct so one can call the k as 2.Pi.nu; wave number can not be replaced by normal frequency of oscillation which is related with speed of the wave that is frequency f = speed/wavelength which is different than nu .
so now the definitions are clear so if k to be viewed as related to frequency of a wave one will write k=( 2.pi. f )/c then we will be dimensionally correct. now one can see the relation ship of wave vector with frequency.(might be i could not post clearly the above in my previous comments)