How do we define how fast a wave is going? The question at first glance seems obvious. When we discussed harmonic waves in a previous post, we observed that the velocity of the wave could be measured by measuring how far one of the peaks of the wave travels in a certain amount of time. There are a number of subtle points that arise when talking about wave velocity, however, including the possibility of light traveling at faster than the 'speed of light'! In this post we'll try and define the velocity of a wave, and explain why the question is not so easy to answer.

The velocity *v* of a harmonic wave is directly related to the angular frequency ω of the wave and its wavenumber *k*, in the form

\(v =omega/k\).

We will call this definition of wave velocity the *phase velocity* of the wave, for reasons which will hopefully become clear soon. It is customary to write this velocity as a fraction of the speed of light *c*, and the fractional coefficient is simply the refractive index *n* of the material the light is passing through, i.e.

\(v = c/n\).

The refractive index, and therefore the velocity of the light wave, is a property of the material medium. Ordinary glass, for instance, has a refractive index of about* n* = 1.5 (light travels at 2/3rds the vacuum speed); the vacuum of interstellar space has a refractive index of unity, by definition. The frequency of the wave is unchanged on entering a new material, which means that the wavelength of the wave depends on the medium.

Harmonic waves are an ideal waveform not seen in practice. How can we measure the velocity of an arbitrary wave 'pulse', as illustrated below?

As long as the shape of the pulse does not change, we can measure its velocity in a manner similar to the harmonic wave: follow one of the peaks of the pulse, and measure how far it goes in a given period of time.

This works just fine for waves on a string, or light fields traveling in vacuum: the shape of the pulse does not change as it travels*.

When one considers a light pulse traveling in matter, however, a problem arises: the shape of the wave changes as it travels through the medium! A simple simulated example of this is shown in the animation below:

We can see that while the bulk of the wave is traveling to the right, the 'ups and downs' of the wave are moving *within* the bulk. Now there's no fixed feature of the wave that we can use to measure the velocity!

The origin of this shape change is what is known as *dispersion*. The atoms in a material respond differently to light of different frequencies, and the wavelength and hence the phase velocity of light is different at each frequency. As we have mentioned in a previous basics post, we can mathematically represent any pulse of light as a collection of harmonic waves of different amplitudes. This means that different parts of the pulse are traveling at different velocities, resulting in a change of pulse shape.

A very famous and well-known example of dispersion is illustrated on the cover of a classic Pink Floyd album:

Because the glass of the prism is dispersive, different frequencies of the incoming white light are bent at different angles on entering and leaving the prism, resulting in a separation of the colors of the light.

Dispersion is characterized mathematically by what is called a *dispersion relation*, a functional relationship between the frequency of a wave and its wavenumber in the medium, i.e. ω=ω(k).

What can we say about wave velocity, then? Looking at the animation above again, we note that even though individual features of the wave are changing, the entire 'group' of features are moving to the right with what seems to be a well-defined velocity. Can we quantify this velocity?

It turns out we can! Suppose our wave consists of only of harmonic waves that lie within a range of frequencies Δω, and a corresponding range of wavenumbers Δk. We can define the group velocity of the wave as:

\(v_g=Delta omega/Delta k\).

It is to be noted that this definition is different from the phase velocity of the wave. The phase velocity dealt with the ratio of the *center* frequency of the wave to the *center* wavenumber of the wave, while the group velocity deals with the ratio of the *range* of frequencies in the wave to the *range* of wavenumbers in the wave. This group velocity is typically a reasonable measure of how fast the bulk of the wave is traveling through a material.

Ordinarily, the group velocity is smaller than the vacuum speed of light. Unfortunately, though, this is not always the case! When a pulse has a center frequency which is strongly absorbed by the material (a situation known as *anomalous dispersion*), the group velocity can be greater than the speed of light or even a negative quantity! In these circumstances it is, like the phase velocity, a useless measure of wave velocity.

This issue with the group velocity has been known for a century now. It has been typically dismissed by noting that, in circumstances of anomalous dispersion, the wave is so strongly absorbed and distorted that one cannot define a meaningful velocity of the group. For instance, what would we call the speed of the wave based on the picture below?

The input wave had a single, well-defined peak, but the output wave has multiple peaks. Which one do we use to measure the velocity? To make a very odd analogy (I came up with it late at night), imagine sending five identical dogs single file into a forest, and timing how long they take to come out the other side. If they come out say, ten minutes later, still single file and with roughly the same separation between them, we can confidently describe the average speed of the dogs. But suppose one comes out in a minute, two more come out at five minutes, one more at ten minutes, and the last comes out an hour later! Not only is there no single time to use to calculate the speed, but we can't even be certain that the dogs came out in the same order they entered! In such a case, a single well-defined measure of 'velocity' is meaningless.

This would seem to be the end of the discussion. We can use 'group velocity' to describe the velocity of a light pulse, except in cases of anomalous dispersion when there is no well-defined velocity. However, in 2000 a group of researchers *experimentally* demonstrated a medium where the group velocity is greater than the vacuum speed of light, and the pulse shape is unchanged on passing through the medium! One of the odd consequences of this is the ability to construct situations where the pulse seems to leave the medium before it fully enters it!

We'll leave a detailed discussion of this 'superluminal' case for another post. We can ask right now, though: does this observation conflict with Einstein's relativity? The answer is a (qualified) no! Once the dispersion relation of a material is known, one can demonstrate that the leading edge of any pulse cannot travel faster than the vacuum speed of light. What is the qualification to this statement? In order to demonstrate that causality is exactly satisfied, we would need to know the complete dispersion relation, from k=0 to k = infinity!** Since we can't generate (or measure) fields with arbitrarily high frequency and/or wavenumber, there's always the chance that there exists some ultra-high-frequency violation of causality that we are unaware of...

This post is intended to give some inkling of the challenges and difficulties of defining the velocity of a wave. There are even more challenges in such definitions, including the previously mentioned superluminal light, but also including the definition of 'tunneling times' for evanescent waves. We'll take a look at these issues in later posts.

* We're considering only one-dimensional waves here. For 3-dimensional fields, there is also natural spreading (diffraction) of the wave, but this does not change the essential observations and arguments presented above.

** This point was brought to my attention by the book *Fast Light, Slow Light and Left-Handed Light*, by P.W. Milonni.