Dispersion Relations: Most Famous DR & Contexts Explained

[DaTario]

Hi All,

The equation:
$v = \lambda f$
is presented as a dispersion relation (DR) for it is a formula that specifies the velocity of a wave of certain frequency.
This equation seems to be the most famous DR in physics.
My questions are the following:
What is the second most famous DR? Which context this second DR fits in?

Best wishes,
DaTario

 

[ZapperZ]

Hi All,

The equation:
$v = \lambda f$
is presented as a dispersion relation (DR) for it is a formula that specifies the velocity of a wave of certain frequency.
This equation seems to be the most famous DR in physics.
My questions are the following:
What is the second most famous DR? Which context this second DR fits in?

Best wishes,
DaTario

Where did you discover that this is "the most famous DR in physics"?

For many of us who study Solid State/Condensed matter physics, the E vs k dispersion relation is more common (it is almost the "figure of merit" in this field since this is how we describe the band structure of materials). But one could argue that E vs k dispersion relation is itself a distant cousin of v vs. λ.

Zz.

 

[Pythagorean]

I can see how, in a breadth sense, that equation is so popular. It's an easy equation that explains a lot when you consider geometry. I got my undergrad in physics before moving on to neuroscience and that's the equation that sticks out to me for DR. So it's popularity could be a statement of its easy digestibility more so than it being more fundamental or meaningful that other relations.

 

[DaTario]

Where did you discover that this is "the most famous DR in physics"?
Zz.

I said "seems to be", not "I have discovered".

Once I was in a congress of laser applications in Santa Catarina (Brazil) and one of the works presented there was about a study of the atmosphere´s level of polution. They developed a set of laser beams, each one with a different color, all aimed to the sky, along the same line. The dispersion relation was known and taken into account in a manner that the slowest colors were emitted before and the faster ones, after. The time intervals were precisely adjusted so that at a given altitude, say 5 km above the ground, all the pulses would be at the same volume, producing a sort of explosion. From the collected light of this explosion they were able to infer the level of polution in the air.

I had no contact with the dispersion relation they used.

 

[Delta2]

Hi All,

The equation:
$v = \lambda f$
is presented as a dispersion relation (DR) for it is a formula that specifies the velocity of a wave of certain frequency.
This equation seems to be the most famous DR in physics.
My questions are the following:
What is the second most famous DR? Which context this second DR fits in?

Best wishes,
DaTario

I haven't studied dispersion so much but the equation you mention is mostly known as "the fundamental equation of wave physics".

If the velocity v is constant (depends only on the medium that the wave propagates) then there is no dispersion at all. When the frequency changes then the wavelength changes according to that equation as to keep the velocity constant.

If the velocity v depends on the frequency or wavelength like it is for example in optics $v=\frac{c}{n(\lambda)}$ where $n(\lambda)$ the refractive index of the medium that depends also on the wavelength $\lambda$ then we have dispersion.

 

[blue_leaf77]

This equation seems to be the most famous DR in physics.

Classifying DR by its popularity lacks physical content. "The simplest" is probably the more fitting term. $v=\lambda f$ with $v=\textrm{constant}$ describes the relation between frequency and wavelength of EM wave in free space.

 

[DaTario]

Classifying DR by its popularity lacks physical content. "The simplest" is probably the more fitting term. $v=\lambda f$ with $v=\textrm{constant}$ describes the relation between frequency and wavelength of EM wave in free space.

And it also applies to the homogeneous string with fixed ends.

 

[Andy Resnick]

Where did you discover that this is "the most famous DR in physics"?

For many of us who study Solid State/Condensed matter physics, the E vs k dispersion relation is more common (it is almost the "figure of merit" in this field since this is how we describe the band structure of materials). But one could argue that E vs k dispersion relation is itself a distant cousin of v vs. λ.

I agree, a dispersion relation is a functional relationship between energy and momentum, and this can be expressed for a variety of particles and systems- electrons, holes, phonons, photons, transverse waves, longitudinal waves, surface waves, etc.

 

[DaTario]

I agree, a dispersion relation is a functional relationship between energy and momentum, and this can be expressed for a variety of particles and systems- electrons, holes, phonons, photons, transverse waves, longitudinal waves, surface waves, etc.

I have seen some dispersion relations expressed as an $$ \omega(k) $$ function, which seems to be what you are referring to.
If we have $$\omega(k) = Constant \,\times k$$ it qualifies the medium as a non dispersive one, for $$ \omega / k $$, the phase velocity, will be constant.
A more complete dispersion equation for a string, for instance, is pointed out in wikipedia as $$\omega^2 = \frac{T}{\mu} k^2 + \alpha k^4. $$

 

[ZapperZ]

I have seen some dispersion relations expressed as an $$ \omega(k) $$ function, which seems to be what you are referring to.
If we have $$\omega(k) = Constant \,\times k$$ it qualifies the medium as a non dispersive one, for $$ \omega / k $$, the phase velocity, will be constant.

Since

$E=\hbar \omega$

This is no different then E vs k dispersion relation that I mentioned earlier. In solid state physics, this is the band structure of the material. Look for example, the band structure for some typical metals:

http://what-when-how.com/electronic...stals-fundamentals-of-electron-theory-part-3/

No "constant phase velocity" or group velocity anywhere.

Zz.

 

[DaTario]

Since

$E=\hbar \omega$

This is no different then E vs k dispersion relation that I mentioned earlier. In solid state physics, this is the band structure of the material. Look for example, the band structure for some typical metals:

http://what-when-how.com/electronic...stals-fundamentals-of-electron-theory-part-3/

No "constant phase velocity" or group velocity anywhere.

Zz.

Yes, I see. This extension of the concept of dispersion relation seems to be due to the work of De Broglie, who assigned wave atributes to the microscopic particle's phenomenology.

The index of refraction $n(f)$ seems also to be another door to enter this subject.

 

[ZapperZ]

Yes, I see. This extension of the concept of dispersion relation seems to be due to the work of De Broglie, who assigned wave atributes to the microscopic particle's phenomenology.

Excuse me? "Microscopic particle's phenomenology"?

It is the solution to the many-body Hamiltonian, or to the Density Functional Theory. I wouldn't call those "phenomenology" anymore than the Schrodinger equation.

Zz.

 

[DaTario]

The intelectual act of De Broglie was to suppose that the microscopic particles would be correctly described by using wave formalism and atributes. The band structure arises from collective effects of particles, which create periodic or quasi periodic potentials, on the energy levels.

 

[DaTario]

Classifying DR by its popularity lacks physical content. "The simplest" is probably the more fitting term. $v=\lambda f$ with $v=\textrm{constant}$ describes the relation between frequency and wavelength of EM wave in free space.

The point in the OP was not to give importance to popularity. Sorry if this was the impression left. The point was to ask for contributions in the field of dispersion relations. I notice that in introductory books of physics, even in A.P. French book, examples of dispersion relations appear very rarely.