Presentation: interference

This is your neighbor. You know, the guy who plays his stereo system way too loud. (Video link: "Maxell Tape: Blown Away (1979).")

Butler: "The usual sir?"
Blown Away Guy: "Please."
(Tape player starts blaring Richard Wagner's Walkürenritt ("Ride of the Valkyries").)
Narrator: "Even after 500 plays, our high-fidelity tape still delivers...high fidelity."

Maybe we can do something about that, next time we just happen to be in his apartment (invited or not), with some minor adjustments to his stereo system wiring.

Last semester we discussed the behavior of sound waves, and so far this semester have been extending those concepts to model the behavior of electromagnetic radiation. Here we specifically look at the superposition of two waves in general, first sound, then later extending these concepts to visible light in a subsequent presentation.

First, defining a few terms.

Here we have two speakers, which are our sources of two sound waves. Since they are plugged into the same frequency source, they will generate sound waves of the same frequency f (which is depends only on the source), same speed v (which depends only on the medium), and thus the same wavelength λ (which depends on both f and v). If the speakers are wired the same way--red and black wires to red and black plugs--then they will oscillate in phase, with both speaker cones moving forward and backwards in unison.

However, if the speakers are wired with opposite polarities--here, the speaker on the left is wired with black and red wires to red and black plugs--then they will oscillate out of phase, with one speaker cone moving backwards while the other is moving forwards, and then forwards while the other is moving backwards.

When we have two in phase sound sources with speaker cones that move in unison with each other, then the waves they generate will have crests and troughs that line up with each other. The superposition of these two waves will result in constructive interference, which will be a single louder wave.

If instead we have two out of phase sources with speaker cones that move contrary to each other, then the waves they generate will have crests and troughs that line up with the other speaker's troughs and crests. The superposition of these two waves will result in destructive interference--which would ideally be silence--but more realistically would be a single wave that is much quieter. (This is what would result if you switched the speaker wire polarities for one side of your neighbor's stereo system.)

Now let's consider two in phase sound speakers, but for an observer located at a position where the distance from each speaker--the path length--is different.

Here waves from the left speaker travel approximately 0.81 m, while the path length for the waves from the right speaker is about 0.63 m. The path difference ∆l is the (absolute value) of how much farther one wave travels than the other, so in this case ∆l = 0.81 m - 0.63 m = 0.18 m.

Even with in phase speakers we can get either constructive or destructive interference, if the waves from each speaker travel different path lengths, resulting in certain path differences ∆l. For two in phase speakers where one wave travels a half-wavelength longer than the other, the path difference is (1/2)λ, and as a result crests and troughs line up with the other speaker's troughs and crests: destructive interference.

For two in phase speakers where one wave travels a whole wavelength longer than the other, the path difference is λ, and as a result crests and troughs line up with the other speaker's crests and troughs: constructive interference.

Second, mixing up the source phases and path difference conditions for constructive and destructive interference.

Here are two cases where both source phases and path differences matter. The top example is where two sources with a half-wavelength path difference results in constructive interference. The bottom example is where two sources with a whole wavelength path difference results in destructive interference. So how can we account for cases like these?

Whether constructive or destructive interference occurs depends on both the sources (how the waves start out, whether in phase or out of phase) and the path difference ∆l (how far each wave travels farther than the other, whether a whole wavelength or a half-wavelength longer than the other). There are four different cases:

• For two in phase sources, if each wave travels a whole wavelength longer than the other, then constructive interference occurs (this is the solid black line.)
• For two in phase sources, if each wave travels a half-wavelength longer than the other, then destructive interference occurs (this is the dashed black line.)
• For two out of phase sources, if each wave travels a whole wavelength longer than the other, then destructive interference occurs (this is the solid red line.)
• For two out of phase sources, if each wave travels a half-wavelength longer than the other, then constructive interference occurs (this is the dashed red line.)

Right now these different conditions look rather intimidating, so we'll make sure to be able to practice applying these conditions to various scenarios of in phase sources and out of phase sources with different shifted positions. Remember, there are only four unique cases of different phases and path differences.