# Waves that meet out of phase

### Constructive and Destructive Interference When the two individual waves are exactly in phase the result is large amplitude. When the two gray waves become exactly out of phase the. So, the pressure fluctuations where the two waves meet have twice the This is called "destructive interference" where two waves are "exactly out of phase". Out of phase waves are two waves whose functions reach 0 value for different x values. The amount of phase shift is denoted in degrees or radians. If two waves . We will explore how to hear this difference in detail in Lab 7. The most important requirement for interference is to have at least two waves. One wave alone behaves just as we have been discussing.

We shall see that there are many ways to create a pair of waves to demonstrate interference. The simplest way to create two sound waves is to use two speakers. If we place them side-by-side, point them in the same direction and play the same frequency, we have just the situation described above to produce constructive interference: If we stand in front of the two speakers, we will hear a tone louder than the individual speakers would produce.

The two waves are in phase. Now imagine that we start moving on of the speakers back: At some point, the two waves will be out of phase — that is, the peaks of one line up with the valleys of the other creating the conditions for destructive interference.

## Interference of Waves

If we stand in front of the speakers right now, we will not hear anything! This must be experienced to really appreciate. Equally as strange, if you now block one speaker, the destructive interference goes away and you hear the unblocked speaker.

In other words, the sound gets louder as you block one speaker! How far back must we move the speaker to go from constructive to destructive interference?

We know that the distance between peaks in a wave is equal to the wavelength. If we look back at the first two figures in this section, we see that the waves are shifted by half of a wavelength. So, in the example with the speakers, we must move the speaker back by one half of a wavelength. What happens if we keep moving the speaker back? At some point the peaks of the two waves will again line up: At this position, we will again have constructive interference! As the speaker is moved back the waves alternate between constructive and destructive interference.

### Superposition of Waves

What the example of the speakers shows is that it is the separation of the two speakers that determines whether there will be constructive or destructive interference. If the speakers are at the same position, there will be constructive interference at all points directly in front of the speaker. If the speakers are separated by half a wavelength, then there is destructive interference, regardless of how far or close you are to the speakers.

However, it already has become apparent that this is not the whole story, because if you keep moving the speaker you again can achieve constructive interference. This can be fairly easily incorporated into our picture by saying that if the separation of the speakers in a multiple of a wavelength then there will be constructive interference. Note that zero separation can always be considered a multiple of a wavelength. What about destructive interference? We know that if the speakers are separated by half a wavelength there is destructive interference. However, if we move an additional full wavelength, we will still have destructive interference. So, this case is a bit hard to state, but if the separation is equal to half a wavelength plus a multiple of a wavelength, there will be destructive interference. Thus, we have described the conditions under which we will have constructive and destructive interference for two waves with the same frequency traveling in the same direction.

Unfortunately, the conditions have been expressed in a cumbersome way that is not easily applied to more complex situations. So, before going on to other examples, we need a more mathematically concise way of stating the conditions for constructive and destructive interference. The proper way to define the conditions for having constructive or destructive interference requires knowing the distance from the observation point to the source of each of the two waves.

Since there must be two waves for interference to occur, there are also two distances involved, R1 and R2. For two waves traveling in the same direction, these two distances are as follows: When we discussed interference above, it became apparent that it was the separation between the two speakers that determined whether the interference was constructive or destructive. From this diagram, we see that the separation is given by R1 — R2. So, really, it is the difference in path length from each source to the observer that determines whether the interference is constructive or destructive.

I emphasize this point, because it is true in all situations involving interference. The only difficulty lies in properly applying this concept. With this more rigorous statement about interference, we can now right down mathematically the conditions for interference: We saw that when the two speakers are right next to each other, we have constructive interference.

But, we also saw that if we move one speaker by a whole wavelength, we still have constructive interference. Remember that we use the Greek letter l for wavelength. With this, our condition for constructive interference can be written: Here, the variable n is used to specify an integer and can take on any value, as long as it is an integer.

This ensures that we only add whole numbers of wavelengths.

## Introduction to Computer Music: Volume One

Once we have the condition for constructive interference, destructive interference is a straightforward extension. The basic requirement for destructive interference is that the two waves are shifted by half a wavelength. This means that the path difference for the two waves must be: But, since we can always shift a wave by one full wavelength, the full condition for destructive interference becomes: Now that we have mathematical statements for the requirements for constructive and destructive interference, we can apply them to a new situation and see what happens.

Waves with the same frequency traveling in opposite directions. To create two waves traveling in opposite directions, we can take our two speakers and point them at each other, as shown in the figure above. We again want to find the conditions for constructive and destructive interference.

Let's set up a situation: We'll say that the wavelength of the sound is 1m. Finally, and most importantly, the speakers' diaphragms are vibrating synchronously moving outward and inward together.

Since the distance from the speakers to you is the same, the condensations of the wave coming from one speaker are always meeting the condensations from the other at the same time. As a result, the rarefactions are also always meeting rarefactions. One principle of sound is linear superposition, which states that the combined pattern of the waves is the sum of the individual wave patterns. So, the pressure fluctuations where the two waves meet have twice the amplitude of the individual waves.

An increase in amplitude results in a louder sound. When this situation occurs it is said to be "exactly in phase" and to exhibit "constructive interference".

### Wave interference - Wikipedia

But, if we slightly change one of the variables, the resulting sound is nearly the opposite of what it was. Let's say we move one of the speakers. We'll assume that the volume on this speaker is turned up so that the amplitude remains constant.

This movement causes the condensations from one speaker to meet the rarefactions from the other sound wave and vice versa. Again referring to the principle of linear superposition, the result is a cancellation of the two waves. The rarefactions from one wave are offset by the condensation from the other wave producing constant air pressure.