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Waves in Two and Three Dimensions
6/2/08 Michael Fowler
Introduction
So far, we’ve looked at waves in one dimension, traveling along a string or sound waves going down a narrow tube. But waves in higher dimensions than one are very familiar—water waves on the surface of a pond, or sound waves moving out from a source in three dimensions.
It is pleasant to find that these waves in higher dimensions satisfy wave equations which are a very natural extension of the one we found for a string, and—very important—they also satisfy the Principle of Superposition, in other words, if waves meet, you just add the contribution from each wave. In the next two paragraphs, we go into more detail, but this Principle of Superposition is the crucial lesson.
The Wave Equation and Superposition in One Dimension
For waves on a string, we found
and it turned out that sound
waves in a tube satisfied the same equation. Before going to higher dimensions, I just
want to focus on one crucial feature of this wave equation: it’s linear, which just means that if you
find two different solutions 
and 
then 
is also a solution, as we proved earlier.
This important property is easy to interpret visually: if you can draw two wave
solutions, then at each point on the string simply add the displacement of one wave to the
other—you just add the waves together—this also is a solution. So, for example, as two traveling waves moving
along the string in opposite directions meet each other, the displacement of the
string at any point at any instant is just the sum of the displacements it
would have had from the two waves singly. This simple addition of the displacements is
termed “interference”, doubtless because if the waves meeting have displacement
in opposite directions, the string will be displaced less than by a single
wave. It’s also called the Principle
of Superposition.
The Wave Equation and Superposition in More Dimensions
What happens in higher dimensions? Let’s consider two dimensions, for example waves
in an elastic sheet like a drumhead. If
the rest position for the elastic sheet is the (x, y) plane, so when it’s
vibrating it’s moving up and down in the z-direction,
its configuration at any instant of time is a function 
.
In fact, we could do the same thing we did for the string, looking
at the total forces on a little bit and applying 
term measured that
curvature, the rate of change of slope. In two dimensions, thinking of a small
square of the elastic sheet, things are more complicated. Visualize the bit of sheet to be momentarily
like a tiny patch on a balloon, you’ll see it curves in two directions, and
tension forces must be tugging all around the edges. The total force on the little square comes
about because the tension forces on opposite sides are out of line if the
surface is curving around, now we have to add two sets of almost-opposite forces from the two pairs of sides. I’m not going to go through all the math
here, but I hope it’s at least plausible that the equation is:
.
The physics of this equation is that the acceleration of a tiny bit of the sheet comes from out-of-balance tensions caused by the sheet curving around in both the x- and y-directions, this is why there are the two terms on the left hand side.
Remarkably, this same equation comes out for water waves (at least for small amplitudes), sound waves, and even the electromagnetic waves we now know as radio, microwaves, light, X-rays: so it’s called the Wave Equation.
And, going to three dimensions is easy: add one more term to give
.
This sum of partial differentiations is so common in physics that there’s a shorthand:
Just as we found in one dimension traveling harmonic waves 
, with 
, you can verify that the three-dimensional
equation has harmonic solutions
and now
, where 
In fact, 
is a vector in the
direction the wave is moving. The
electric and magnetic fields in a radio wave or light wave have just this form
(or, closer to the source, a very similar equivalent expression for outgoing
spheres of waves, rather than plane waves).
It’s important to realize that this more complicated equation is still a linear equation—the principle of superposition still holds. If two waves on an elastic sheet, or the surface of a pond, meet each other, the result at any point is given by simply adding the displacements from the individual waves. (Assuming as always small waves, so the water waves don’t fall apart into foam.)
We’ll begin by thinking about waves propagating freely in two and three dimensions, than later consider waves in restricted areas, such as a drum head.
How Does a Wave Propagate in Two and Three Dimensions?
A one-dimensional wave doesn’t have a choice: it just moves along the line (well, it could get partly reflected by some change in the line and part of it go backwards). But when we go to higher dimensions, how a wave disturbance starting in some localized region spreads out is far from obvious. But we can begin by recalling some simple cases: dropping a pebble into still water causes an outward moving circle of ripples. If we grant that light is a wave, we notice a beam of light changes direction on going from air into glass. Of course, it’s not immediately evident that light is a wave: we’ll talk a lot more about that later.
Huygen’s Picture of Wave Propagation
If a point source of light is switched on, the wavefront is an expanding sphere centered at the source. Huygens suggested that this could be understood if at any instant in time each point on the wavefront was regarded as a source of secondary wavelets, and the new wavefront a moment later was to be regarded as built up from the sum of these wavelets. For a light shining continuously, this process just keeps repeating.
What use is this idea? For one thing, it explains refraction—the change in direction of a wavefront on entering a different medium, such as a ray of light going from air into glass.
If the light moves more slowly in the glass, velocity v instead of c, with v < c, then Huygen’s picture explains Snell’s Law, that the ratio of the sines of the angles to the normal of incident and transmitted beams is constant, and in fact is the ratio c/v.
This is evident from the diagram below: in the time the wavelet centered at A has propagated to C, that from B has reached D, the ratio of lengths AC/BD being c/v. But the angles in Snell’s Law are in fact the angles ABC, BCD, and those right-angled triangles have a common hypotenuse BC, from which the Law follows.
Huygens’ picture also provides a ready explanation of what happens when a plane wave front encounters a barrier with one narrow opening: and by narrow, we mean small compared with the wavelength of the wave. It’s easy to arrange this for water waves: it’s found that on the other side of the barrier, the waves spread out in circular fashion form the small hole.
Two-Slit Interference: How Young measured the Wavelength of Light
If the slit is wider than a wavelength or so, the pattern gets more complicated, as we would expect from Huygens’ ideas, because now the waves on the far side arise from a line of sources, not what amounts to one point. To investigate this further, consider the simplest possible next case: a barrier with two small holes in it, so on the far side we’re looking at waves radiating outwards from, effectively, two point sources.
For two synchronized sources generating harmonic waves, at any point in the tank equally distant from the two sources (the central line in the picture above), the waves will add, the water will be maximally disturbed. For light waves, there will be a maximum in brightness at the center of a screen as shown in the diagram:
For light waves passing through two narrow slits and shining on a screen (on the right) there will be another bright spot at a point P away from the center C2 of the screen, provided the distances of P from the two slits differ by a whole number of wavelengths: