Huygens' principle -- the math

We know if we have a single oscillating point source of waves, it produces outgoing circular (or spherical in 3D) waves, like the image of the water drops shown in the figure at the right.

The basic idea of Huygens' wave model is that on a broader oscillating wave, every oscillating point starts an outgoing wave.

If we have a broad wave front — a line or surface of whatever it is that's waving like the straight-line row of wavefronts shown coming onto the beach in the figure at the right — each point on the wave acts as a source of outgoing circular (in 2D) or spherical wave (in 3D). (When the water gets shallow the wavefronts build up and turn over.)

Huygens showed that this worked to explain the moving circular waves at in the first figure or the moving straight line waves as in the second. Here's how it works.

The result one gets at a time $t + dt$ is then the sum of all these secondary waves, or wavelets, created at the same time t. One doesn't actually want to do the integral of all these wavelets if one doesn't have to (which you might if your were doing a research project or designing an instrument), but in many cases you can just see that the result is the envelope of the wavelets. In two cases, the wavelets just keep the obvious wave going so we don't have to worry about them at all. These cases are

• the circular/spherical wave
• the plane wave.

What we mean by "the envelope of the wavelets" is shown for these two cases in the figure below as dashed lines. In these cases each wavefront creates wavelets that add up to produce the obvious new wavefront.  

So the wavelet idea is consistent with what we see for the motion of circular and straight waves.

But the place where wavelets become really interesting an useful is when we have multiple sources added together, such as trying to figure out what happens in the case of interference from multiple slits or the diffraction from a single slit. Especially in the first case, it helps us to understand what's going on if we actually see the math of what's going on.

The math of traveling waves in 2D and 3D

Let's begin with the simple toy model we built for how pulses move on an elastic string. There, we found that if a source creates a pulse of a shape $f(x)$, that to make the pulse move, we have to shift the argument of $f$. If we write $f(x - v_0t)$ we will get a pulse moving to larger values of $x$ with a velocity $v_0$.

In 2D and 3D, instead of just moving in one direction, our point source will generate a pulse that creates a circle (in 2D) or a sphere (in 3D). The pulse moves outward in increasing radius. So if we start with a shape $f(r)$ with its peak when the argument is 0, let's try writing the function $f(r - v_0t)$. Since we are writing the magnitude of $r$, not a vector, this function will have its peak whenever its argument, $r - v_0t = 0$. So it will have a peak at all points that have $r = v_0t$; an expanding circle or sphere. So it describes a pulse that expands into a circle or a sphere with velocity $v_0$. 

The energy factor

This isn't quite right. In 1D all the energy that the wave carries travels along with the pulse. The pulse doesn't change size or shape, so the energy just moves with the pulse without change. In 2D and 3D, however, the circle or sphere gets larger in size, spreading the energy out over a larger region. As a result, we have to put in a factor to decrease the energy density as the size of the pulse grows.

For example, in 3D

(Energy/unit area) x (area of sphere) = Total energy (constant)

So the energy density (energy/unit area) must look like a constant divided by the area of the sphere. Since the area of the surface of a sphere grows proportional to $r^2$, we have to thin the energy density by that factors as the wave expands.

In 2D, the energy is spread out over a line — the circumference of a circle — so 

(Energy/unit length) x (circumference of circle) = Total energy (constant)

We have used

$C = 2\pi r$     (circumference of a circle)
$A = 4\pi r^2$    (area of surface of a sphere).

In the case of both sound and light, the energy density in the sound is proportional to the square of the variable that is "waving"— pressure for sound and electric field for light. So if we are writing an equation to describe the waving variable, we only have to thin it by the square root of $r$ (in 2D) or by $r$ (in 3D). The squaring will then produce the correct "thinning" of the energy density to get the total correct. This gives us the correct result for how a wave propagates outward from a small source in 2D and 3D.  For a pulse it looks like this:

$$y(r,t) = \frac{1}{\sqrt{r}} f(r-v_0t)\quad\mathrm{(2d)}$$

$$y(r,t) = \frac{1}{r} f(r-v_0t)\quad\;\;\mathrm{(3d)}$$

We have written "$y$" for our waving variable, whatever it is.

Sinusoidal waves

For a sinusoidally oscillating small source, we need a factor that has units of 1/distance to be able to put $r - v_0t$ into the argument on sine. Recall that we like to write that factor as the wave number, $k = 2π/λ$. We then write the combination

$k(r-v_0t) = kr - ωt$  (where $ω = kv_0$).

This results in an oscillating outgoing wave that looks like this:

$$y(r,t) = \frac{1}{\sqrt{r}} A\sin{(kr-\omega t)}\quad\mathrm{(2d)}$$

$$y(r,t) = \frac{1}{r} A\sin{(kr-\omega t)}\quad\;\;\mathrm{(3d)}$$

Although we've been very careful about conserving energy by putting in the $r$ factors (because we want you to see how energy is represented in the math), in most cases we are going to ignore them. Typically, we will be combining waves from two or more sources and the numerator changes the value of y by 200% (from a max to a min) when $r$ changes by 1/2 a wavelength. In most cases of both light and sound, the wavelengths are very small compared to our distances from the sources (a few hundred nanometers for visible light, ~1 m for sound), so the changes in amplitude from the numerator is huge compared to the percentage changes from the denominator — which are of the order $λ/r$, usually a small factor.

How this makes an interference

To see how these equations tell us what to do, let's consider a plane wave on the surface of water approaching a pair of slits. As the plane wave hits the slits, it will drive the water up and down in each slit. Since the plane wave is hitting the barrier square on, the water in each slit will oscillate up and down together (in phase). Each of these will produce an outgoing circular sinusoidal wave that looks like $A sin(kr - ωt)$. (We are surpressing the fall off due to the distance factor.)

Now these two slits are each producing outgoing circles of waves that look just the same and have the same time dependence. BUT! Suppose we go to a particular point: say the small red dot indicated on the figure. This dot is different distances from each of the sources. As a result, the waves from the two sources will not necessarily reach our point at the same point on their oscillation at the same time. To see this in math, the result is:

$$y = A\sin{(k_1r - \omega t)} + A\sin{(k_2r - \omega t)}$$

Since we are at a fixed position and just considering the time dependence, we can treat the $kr$ terms as constants.  Let's call them phi.  This makes our equation look like this:

$$y = A\sin{(\phi_1 - \omega t)} + A\sin{(\phi_2 - \omega t)}$$

These are just two sinusoidal oscillation shifted from each other. We know that if the shifts differ by 180^o ($π$ radians) the oscillations will be opposite to each other and cancel. If they differ by 360^o ($2π$ radians) they will oscillate together and add. To see which, we need to look at the difference between the phases of the two waves that are adding:

$$\Delta \phi = \phi_1 - \phi_2 = kr_1 - kr_2 = \frac{2\pi}{\lambda}(r_1-r_2)$$

or

$$\Delta \phi =2\pi \frac{\Delta r}{\lambda}$$

This shows us that depending on our distances from the two sources, even though the sources are oscillating together, what we see may add or cancel depending on where we are. If, at a particular point in space the difference in the distances to the sources is a whole number of wavelengths the two waves will add together — constructive interference. If the difference in the distances is a whole number plus 1/2 wavelengths, the two waves will be out of phase and cancel at that point in space — destructive interference. This is the key idea in understanding all of interference.

We note that if our wavelength is very small, we can use the shift from constructive to destructive interference to measure very small distances. This is how x-rays (which have wavelengths of a fraction of a nanometer) can probe the structure of atoms and molecules.

We'll go over the details of how the interference works and what the interference pattern tells us in the follow-ons.

Joe Redish 4/25/12