# Huygens' principle

#### Prerequisites

Recall that when we looked at waves moving on a 1D elastic string, it was quite a bit more complicated than the oscillation of a mass on a spring (the simple harmonic oscillator). Instead of having one oscillator, we had an infinite number of oscillators (that we modeled as beads) labeled by a continuous variable that we called $x$, which basically specified which bead we were looking at.

Now that we are talking not about elastic strings but about water waves, light waves, and sound waves, we have to move from 1D into 2D and 3D. This is even more complicated. We'll take it slowly, a step at a time, moving first into 2D.

We are going to see that "what is waving" in light are in fact electromagnetic fields. But these are difficult to get a picture of, both since they propagate in three dimensions and because they're abstract; we don't have any concrete personal images of them.

We began our discussion of waves with a one-point-particle example: the mass on a spring. We then extended to a one-dimensional example (corresponding to infinitely many one-point particles): waves on an elastic string. Rather than leap right away into electromagnetic waves in 3D, lets try to see if we can make sense of a toy model two-dimensional example where we can draw on concrete experience: ripples on the surface of deep water.

## Starting simple: The water wave analogy and pulses

The mechanism is not quite the same as our 1D wave on an elastic string. The water wave is indeed transverse, mostly, but the water particles really move in little vertical circles, not just perpendicular to the surface of the water. We will ignore this complication here and treat the waves on the surface of water as if the water is just oscillating up and down. (Remember: our point is not to describe the motion of waves on water, but to construct a conceptual model that helps in thinking about the complex superpositions that happen with waves in 2 and 3D.)

If you drop a small stone in a pond or drip a drop of water from a faucet into a basin, it will produce an outgoing circular ripple — a propagating pulse in 2D. If we do it multiple times we will get propagating circular rings — a string of pulses. This is shown using the PhET simulation, wave interference shown in the figure at the right. The rings are caused by the faucet dripping at a regular interval and they move outward at a constant speed.

Since the water can go either up (positive) or down (negative) adding two signals can make either a stronger one or a weaker one, solving the problem with the particle model observed by Thomas Young. In the second image at the right, we see the result of adding together two sources of waves. The pattern is quite complex. To see how this solves the problem and produces an interference pattern, we need to create a model of how the waves propagate and how they superpose.

## Waves driven by a sinusoidal oscillation

Since colors and tones correspond to particular frequencies of sine waves, we need to think not just about successive pulses, but about sinusoidal oscillations: where the driver of the wave oscillates like a harmonic oscillator, $\sin{\omega t}$. That oscillation is propagated outward through the medium, just like a traveling sine wave on a string. But now, instead of a wave moving along a 1D string, we have to think of it moving out in 2D (or 3D) like the rings of water waves in the images above.

Whatever it is that's waving — water, sound, or light — Huygens' key idea is the wavefront, like the outgoing raised ring of water that spreads outward when you drop a rock in a pond. Just as each bead in our model of the elastic string moves up and down to drive the next bead along in the same way as the initiating driver moved the first bead, each bit of water moving up and down serves as a source for outgoing circular waves on the surface of the water. This idea is the key to Huygens' wave model.

If we put a rod into the water and let it be driven up and down by a rotating wheel, we will get sinusoidal circular waves moving outward. The up-down displacement of the water will look something like $A\sin{(kr - \omega t)} = A\sin{[k(r-v_0 t)]}$ where $v_0$ is the speed with which the pulses travel in the medium and where $r$ is the distance from the source. (See Reading the content in a sinusoidal wave.)

For one pulse, it makes sense to think about the wavefront as the growing circle (in 2D — or a sphere in 3D) of equal amplitude that propagates outward from the source. For sinusoidal waves, it is more useful to think of the wavefront as lines (in 2D) or surfaces (in 3D) of equal phase — the amplitude is staying the same because the argument of the sine wave stays the same if $r$ grows like $v_0t$.

## Huygens' model: the foothold ideas

Here are the foothold ideas of Huygens' model. We'll see how these works in detail once we add the math in the follow-on page, Huygens' principle -- the math.

• Just as each point on a string acts like a demonstrators hand moving the string up and down, each oscillating point on a wavefront acts like a source for a new spherical wavefront going out. These secondary waves are referred to as wavelets.

We need wavelets here whereas we didn't on the elastic string since we are in 2D or 3D.

• The sum of the wavelets from the wavefronts at any one instant in time creates new wavefronts for later times.

In 2D and 3D, there can be many sources of waves moving on, not just one. This is what makes the Huygens' theory complex and hard to calculate. We have to add the wavelets together using the idea of superposition — that when we have multiple waves (or wavelets) arriving at the same point at the same time, we just add them up. The problem is, there could be a lot of waves to be added up. (We'll never explicitly consider more than two.) Calculating a result typically involves doing a complicated integral in 3D. This is what Fresnel figured out to win the French Academy Prize in 1817.

• The speed of the wavelets depends on the medium, not on the amplitude.

We'll stick with this assumption for now, though for some kinds of waves and some kinds of media, the speed of the waves can depend on the wavelength. Such systems are called dispersive. An example of this is light in glass. This is why prisms work to separate colors.

## The Huygens' model for the case of light

In the case of light, there are some interesting results.

• In the Huygens' model, rays are interpreted as the lines that are perpendicular to the wavefronts of the model.
• If we assume that light propagates more slowly in dense media (in contrast to Newton's model), the Huygens' model can derive the basic principles of the ray model, including both the mirror rule and Snell's law using just geometry. (See the page Huygens and the basic results of the ray model (technical).)
• Multiple waves from different sources pass through each other and just add their values to get a result. Superposition holds, just as it did for pulses on an elastic string and as it does for electric fields (because light is and electric field).

These results mean that Huygens' model can not only reproduce the results of the ray model, but they also allow for interference. Having more sources can result in cancellations at particular points, not just enhancements.

In the follow-ons, we work through the examples of how the Huygens' model correctly predicts the interference pattern of light at two slits, and even the diffraction pattern (spreading out) of light from a single narrow slit.

Joe Redish 4/23/12 and 7/3/19

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