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The math of Huygens' principle (2013)

Page history last edited by Matt Harrington 9 years, 12 months ago

Working Content  > The wave model  > Huygens' principle and the wave model

 

Prerequisites:

 

The basic idea of Huygens' wave model is that some small (approximately a point) source starts a wave (like an oscillating electric charge for light or an oscillating surface for sound). Each point on the wave then acts as an out going circular (in 2D) or spherical wave (in 3D). 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 produce the obvious new wavefront.     

 

      

 

We really only have to worry about the wavelets 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

From our discussion of waves on an elastic string, we know 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 - v0t) we will get a pulse moving to larger values of x with a velocity v0.

 

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, it will expand into a circle or a sphere with velocity v0 if we write the function f (r - v0t). Since we are writing the magnitude of r, not a vector, this f will have its peak whenever its argument, r - v0t = 0. So it will have a peak at all points that have r = v0t; an expanding circle or sphere.

 

The energy factor

This isn't quite right. In 1D all the energy of the wave 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 get 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 circumference of a circle grows proportional to r, and the area of the surface of a sphere grows proportional to r2, we have to thin the energy density by these factors as the wave expands.

 

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:

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

 

For a sinusoidally oscillating small source, we need a factor that has units of 1/distance to be able to put r - v0t 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-v0t) = kr - ωt  (where ω = kv0).

 

This results in an outgoing wave that looks like this:

Now that we've been very careful about conserving energy by putting in the r factors, 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, so the changes in amplitude from the numerator is huge compared to the percentage changes from the denominator -- which are at most λ/r, usually a very 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:


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:

These are just two sinusoidal oscillation shifted from each other. We know that if the shifts differ by 180o (π radians) the will be opposite and cancel. If they differ by 360o (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:

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.

 

Follow-ons:

 

Joe Redish 4/25/12

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