If we let subscripts denote partial derivatives, then the simplest wave equation for rightward propagating waves reads: At = -cAx . While any initial condition just moves to the right with speed c without changing form, a particularly useful form of the solution is sinusoidal: A = A0sin(kx-wt) where lambda (wave length) =2pi/k. The wavenumber, k, tells us how quickly the wave oscillates in space, while is the frequency, w, tells us how quickly the wave oscillates in time. If we know the propagation speed in the equation (say c=5 units per second) then if we choose the wavelength, and hence k, we must choose w so that (w=ck make a new equation for this). For this simplest of cases the relation is linear. In a more general physical setting (waves generated by a pebble dropped in a pond, for example) we do not expect the simple linear relation to hold, and indeed for the simple wave equation above to apply, but we do still expect wave solutions (experience tells us to) but perhaps with a more complicated relation between w and k. For a given physical situation we aim to find some function, w(k), called a dispersion relation, that determines just how w depends on k. The reason for this name will become clearer as we go on.
For the next example consider a traveling wave pulse that can be broken up into two simpler component waves, y1 and y2, where:
y1 = A sin(k1x-w1t)
y2 = A sin(k2x-w2t)
We assume that the two wave numbers are nearly, but not exactly, the same. Similarly the two frequencies are assumed to be nearly, but not exactly, the same. Recall the addition formula for sines which allows us to combine these eqations into:
y = y1 + y2
= A sin(kavx-wavt)sin(kmodx-wmodt)
Where kav and wav are the averages of wave number and frequency, respectively. wmod is a slower version called the modulation frequency and kmod is half the difference of the wave numbers of the component waves.
Graphing the wave pulse and the modulation curve we see something familiar: our wave pulse resembles a kind of beat phenomenon. The modulation curve is an “envelope” for all of the little wave crests that make up the wave pulse. This kind of wave pulse is commonly called a wave packet.
Combining the equations for period and wavelength we can calculate:
vp = w/k , which is the phase velocity and it represents the speed at which the individual component waves are traveling.
We can calculate the speed at which the wave packet travels by using the phase velocity but using kmod and wmod. Since kmod and wmod are infact differences we can take the limit as k goes to zero and derive that:
vg = d/dk w/k
Group velocity, along with the concept of phase velocity, is a very important idea in the theory of waves. All of these concepts carry over to cases where our traveling wave pulse is no longer the superposition (sum) of just two simpler component waves, but rather many or possibly infinitely many component waves. We will now use the ideas of phase and group velocity to analyze different dispersion relations.
The next simplest wave equation to consider is the Partial Differential Equation (PDE):
At = bAxxx -cAx
You can convince yourself that only odd numbered space derivatives can appear if we do not wish our waves to spontaneously decay. Assuming solutions of the form: A = A0sin(kx-wt)
Substituting and solving the differential equation gives the dispersion relation: w= bk3+ck
Taking the values of c and b to be 1 and 0.1 respectively, our dispersion relation becomes: w = k+0.1k3. Which looks like:
Calculating the phase and group velocities we get:
vp = 1+0.1k2 and vg = 1+0.3k2
Suppose that we define a set of initial conditions so that our wave pulse initially looks like this:
And obeys the differential equation, Applying Fourier analysis, we can track the individual wave components that make up the wave pulse and determine what the pulse will look like at any point in the future. A little further ahead in time, the wave pulse looks like:
And further ahead:
We can now make an important observation: Waves of shorter wavelengths travel faster than waves of longer wavelengths. Because waves of different wavelengths travel at different velocities, the pulse “spreads out” as time goes on. This phenomenon is called dispersion, and hence the reason why the function is called a dispersion relation.
So far, we have only looked at our wave pulse from a spatial perspective, looking at a “snap-shot” of the wave pulse in time. We can also consider a contour plot, which is a set of level curves taken in the x-t plane. (If you aren’t familiar with level curves, thing of a contour plot as a type of topographical map, such as those found in a world atlas. The only difference is that we are measuring wave amplitude instead of mountain height.)
The contour plot can tell us a lot about our wave pulse. The colour gradient tells us the amplitude of the individual waves at a point in space-time. The width of the bands of colour, (in the x-direction), tell us the wavelengths of the individual component waves. As we noticed before, we can see that the waves of shorter wavelengths travel faster than waves of longer wavelengths. In general, the energy of the wave pulse travels in the same direction as the group velocity.
We now consider what happens when we make one small change in our dispersion relation. This time we keep c = 1, but we now let b = −0.1. The dispersion relation now becomes: w = k - 0.1k3. A graph that looks like:
With vp = 1-0.1k2 and vg = 1 -0.3k2. Taking the same initial conditions as before we have our familiar looking wave pulse:
After some time:
We see again this time that the waves of shorter wavelengths are traveling faster than waves of longer wavelengths, but now, many of the component waves are traveling in a direction that is opposite that of the wave packet itself.
The contour plot gives us a better picture of what’s going on:
This confirms what we deduced earlier.
Therefore, we see that if we keep the same initial conditions, but change the dispersion relation ever so slightly, we end up with remarkably different results. We have also seen that there are many possible outcomes when considering traveling wave pulses. The component waves, and the wave packet and energy, may travel at different speeds and in the same or opposite directions. (In fact, for special kinds of 2D waves, called internal gravity waves, the phase and group velocities can be perpendicular to each other!)
One last thing to note, is that when the phase and group velocities are always equal, all of the component waves and the wave packet itself travel at the same speed. Hence, the waves are non-dispersive.