*6/2/08 Michael Fowler*

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.

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**.

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 *square*
of the elastic sheet, with tension pulling all around the edge. Remember that the net force on the bit of
string came about because the string was curving around, so the tensions at the
opposite ends tugged in slightly different directions, and didn’t cancel. The _{} 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—

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.

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.

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

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.

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 *C*_{2} of the screen, *provided the distances of P from the two
slits differ by a whole number of wavelengths*:

On the other hand, at a point approximately half way from
the center of the screen to *P* the
waves from the two sources will arrive at the screen exactly *out* of phase: the crest of one will
arrive with the trough of the other, they will cancel, and there will be no
light. Evidently, then, we will see on
the screen *a series of bright areas and
dark areas*, the brightest spots being at the points where the waves from
the two slits arrive exactly in phase.

There is a Flash animation of this
pattern formation here.

This pattern, generated by what is called ** interference**
between the waves, and also referred to as a

Young used the pattern to *find the wavelengths* of red and violet light. His method can be understood from the diagram
above. We did the experiment in class
with a slit separation of about 0.2 mm., giving bright spots on the screen
about 3 cm apart, with a screen 10 m from the slits.

That is to say, in the diagram above we had _{} and we found _{} (within a percent or
two). Looking at the diagram, it’s clear
that the angle to *P* from the slits is
very small, in fact it’s _{} So the diagram as
drawn is very exaggerated!

Now, the line *S*_{1}*Q* is perpendicular to the light rays
setting off for *P* (they are *extremely *close to parallel). The angle between *S*_{1}*Q* and *S*_{1}*S*_{2} is the same as
that between *C*_{1}*P* and *C*_{1}*C*_{2},
that is, _{} This means that the
lengths *S*_{1}*Q* and *S*_{1}*S*_{2}
are effectively equal, and therefore that

_{}

This is very accurate for such a small angle, and for the
data as given here the wavelength of the light _{}

About ten years after Young’s result a French civil engineer, Augustin Fresnel, independently developed a wave theory of light, and gave a more complete mathematical analysis. This was disputed by the famous French mathematician Simeon Poisson, who pointed out that if the wave theory were true, one could prove mathematically that in the sharp shadow of a small round object, there would be a bright spot in the center, because the waves coming around the circumference all around would add there. This seemed ridiculous—but French physicist Francois Arago actually did the experiment, and found the spot! The wave theory of light had arrived.