I Dispersion: Why are Sine-like Functions Fundamental?

[greypilgrim]

TL;DR Summary

Why do sine-like waves travel at constant speed in a dispersive medium, and not for example triangle waves?

Hi.
Light travelling in dispersive media is normally treated by being broken up into its harmonic constituents by Fourier analysis and those then travel at frequency-dependent, but constant speed.

However, from a mathematical point of view, there should be infinitely many other bases of the vector space of periodic functions that are not sine-like, and there is no mathematical reason why the sines should be preferred. For example, the Walsh basis consists of square waves (but there surely are also bases with smooth periodic functions).
So why is it that exactly the sine-like waves travel at constant speed?

My guess is that it comes down to the single oscillators of the medium and that they are treated as harmonic in first-order approximation, which then distinguishes sines from other bases. Does that work?

 

[Andy Resnick]

TL;DR Summary: Why do sine-like waves travel at constant speed in a dispersive medium, and not for example triangle waves?

If I understand your question correctly, the main benefit to using sinusoidal-type basis functions is that it's easy to express conservation of energy and momentum. Also, it comes down to the way a electromagnetic phenomena are modeled, for example in the case of E&M, the permittivity/permeability/refractive index are all easily specified as functions ε(ω), μ(ω) , and n(ω), which require/imply sinusoidal oscillatory-type functions for the fields.

To be sure, there are indeed other basis functions available (wavelets are a good counterexample as are Gaussian modes).

You are basically correct, as soon as harmonic oscillation is used to model any wave-like phenomenon, you are choosing a set of basis functions sin(kx-ωt) or cos(kx-ωt) or e^i(kx-ωt).

 

[greypilgrim]

If I understand your question correctly, the main benefit to using sinusoidal-type basis functions is that it's easy to express conservation of energy and momentum.

They surely are in most cases the most convenient to deal with. My question was more about why they are also the ones that travel at constant velocity, i.e. individually show no dispersion (waveform does not change while travelling through the medium).

Consequently, each other basis should not behave that nicely, i.e. the waveform of the basis function changes while passing the medium.

My question was why nature seems to favour the sine-like basis in that respect, even though the choice of basis should be mathematically equivalent. But as you say, it is probably not nature but us doing this by modelling the material as collection of harmonic oscillators that distinguishes this basis from all others.

Which begs the question: Are there materials for which the harmonic oscillator approximation is that bad such that even sinusoidal waves show dispersion?

Or maybe just theoretically, would sinusoidal waves show dispersion in a crystal where the restoring force of the individual atoms behaves as $F\left(x\right)=-k\cdot x^4$ ?

 

[Paul Colby]

Doesn't this have to do with time translation invariance of the medium? This, of course, may be traced back to energy conservation. Triangular waves aren't multiplied by a constant when the time derivative is taken.

 

[FinBurger]

The propagation in a medium is governed by a differential equation. Good basis functions turn out to be solutions to that differential equation. In most cases, the differential equations are second order due to the laws of motion*. Hence, the popularity of forms similar to exp(kr-iwt).

* You can ask the question: https://physics.stackexchange.com/q...-equations-for-fields-in-physics-of-order-two
That rabbit hole goes pretty deep. I don't pretend to understand, but it has something to do with the fact that systems with higher order derivatives tend to be non-causal and non-invariant. Imagine all the molecules in your body exploding at the speed of light. :-)

 

[greypilgrim]

The propagation in a medium is governed by a differential equation. Good basis functions turn out to be solutions to that differential equation.

Yes. But the usual derivation of the wave equation explicitly assumes the restoring force of the oscillators to be proportional to displacement, so it again comes down to the harmonic oscillator.

 

[Andy Resnick]

They surely are in most cases the most convenient to deal with. My question was more about why they are also the ones that travel at constant velocity, i.e. individually show no dispersion (waveform does not change while travelling through the medium).

That's a consequence of Maxwell's equations being linear.

Which begs the question: Are there materials for which the harmonic oscillator approximation is that bad such that even sinusoidal waves show dispersion?

That's the relevant question! Alternatively, you are asking about "strongly nonlinear media" such as liquid crystals. AFAIK, what happens isn't 'dispersion' (a frequency change requires energy non-conservation) but rather, a variety of propagation effects can be manifested. A 'common' case is when the refractive index varies with intensity: light no longer travels in straight lines.

Airy modes received some attention recently:

https://www.nature.com/articles/s41598-018-22510-7

https://iopscience.iop.org/article/10.1088/0305-4470/25/6/002

There are also new kinds of modes (basis states) called "breathers":

https://www.sciencedirect.com/science/article/pii/S0375960110009527
https://opg.optica.org/ol/fulltext.cfm?uri=ol-32-21-3206&id=144557

And, the same question in a totally different application:

https://royalsocietypublishing.org/doi/10.1098/rsta.2017.0130