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Lesson 22: Wave Interference

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The Principle of Superposition

When waves interfere, they pass through each other completely unaffected. However, during the time when they are actually interfering, what we observe can appear quite different. The waves can appear much larger or much smaller than they actually are. The Principle of Superposition helps us to understand this phenomenon.

The Principle of Superposition says that when two waves interfere with each other, the overall amplitude of the wave is the sum of the individual amplitudes.

When two crests or two troughs interfere, the result is constructive interference. The overall amplitude is greater than that of either of the two crests or the two troughs. When a crest and a trough interfere, the result is destructive interference. The overall amplitude is smaller than that of either of the crest or the trough.

Constructive Interference

The diagrams below show rectangular pulses rather than entire curved waves. This makes it much easier for us to see what is happening. The more complicated, full waves will be addressed afterwards.

The green crest has an amplitude of +2 and is moving to the right. This green crest caused the medium (the black line) to be raised two units.The pink crest has an amplitude of +3 and is moving to the left. This pink crest caused the medium (the black line) to be raised three units.At this point the two crests appear to be getting closer and closer to each other.

For a very brief moment, the two crests are superimposed on each other. The green crest still causes the medium to be raised two units and the pink crest still causes the medium to be raised three units. This has been shown with a dotted line because you would not actually be able to see the separate pulses at this point.

The overall displacement of the two crests is one large crest with an amplitude of five units. This is shown with the black line. When two crests interfere, we have a supercrest. When two troughs interfere, we have a supertrough.

Once the pulses have interfered with each other, they carry on in their original direction with their original amplitude. They have passed through each other unaffected.

A green wave and a pink wave are undergoing constructive interference as they pass through each other. At all points, only the overall amplitude of the wave (black line) can be seen. The amplitude can be determined using the principle of superposition.

Destructive Interference

The green crest has an amplitude of +2 and is moving to the right. This green crest caused the medium (the black line) to be raised two units.

The pink trough has an amplitude of -3 and is moving to the left. This pink crest caused the medium (the black line) to be lowered three units.

At this point the two pulses appear to be getting closer and closer to each other.

For a very brief moment, the two pulses are superimposed on each other. The green crest still causes the medium to be raised two units and the pink crest still causes the medium to be lowered three units. This has been shown with a dotted line because you would not actually be able to see the separate pulses at this point.

The overall displacement of the two pulses is one large crest with an amplitude of -1 units. This is shown with the black line.

Once the pulses have interfered with each other, they carry on in their original direction with their original amplitude. They have passed through each other unaffected.

A green wave and a pink wave are undergoing destructive interference as they pass through each other. At all points, only the overall amplitude of the wave (black line) can be seen. The amplitude can be determined using the principle of superposition. If both the green wave and the pink wave had the same initial amplitude, they would cancel each other out and the overall amplitude would be zero.

This video of constructive and destructive interference provides an excellent example of pulses similar to the ones above interfering. The video clearly illustrates that the amplitude of the resulting wave is the sum of the individual waves.

Practice Questions

Decide whether constructive interference or destructive interference is occurring at each of the points indicated.

A.
B.
C.
D.
E.

A)Destructive interference
B)Destructive interference
C)Constructive interference
D)Constructive interference
E)Destructive interference

tanding Waves

Standing waves are waves that appear to be standing still. They occur when two waves with the same wavelength and the same amplitude pass through each other in opposite directions. Generally, this occurs when a wave reflects back upon itself after coming in contact with a fixed point or surface.

This picture shows two identical waves passing in opposite directions (the red and the green lines). Constructive interference results and a series of supercrests and supertroughs are produced (black line). They are producing the first part of a standing wave.

As the green wave moves right and the red wave moves left, we get the image shown to the left. Destructive interference results and the waves cancel each other out.

A standing wave is produced when the two situations continuously alternate between constructive interference and destructive interference.

An understanding of standing waves is necessary in order to understand how musical instruments produce sound.

Watch this slow motion video of how standing waves are produced. In musical instruments, the waves will be moving much more quickly and the standing wave (if we could see it) would appear blurred.

This link shows the production of a standing wave. Though the animation is slow in order to convey how a standing wave is produced, try to imagine what the standing wave would look like if the waves were passing through each other at a much faster speed.

Complete the multimedia element entitled The Interference of Waves. When you have finished working through this MME, return here to continue the activity.

Beats

The interference of two sounds waves that have similar frequencies results in beats. Beats are alternating periods of high and low amplitude. In the case of sound, the beats would be alternating periods of loud and quiet. How quickly these beats occur depends upon how close the frequencies of the two sources are.

To understand how beats work, consider the diagram below:

We can see that where two troughs come together, we will get constructive interference and a supertrough is formed. This represents a region of high amplitude. Where a crest and a trough come together (the orange regions), destructive interference occurs. This represents a region of low amplitude. These high and low amplitude regions will always alternate. The frequency of the beats gets lower as the frequencies of the waves become closer together.

Beat Frequency =

Where f1 and f2 are the frequencies of the two waves (measured in Hertz),

N is the number of beats heard,

and Īt is the time interval (measured in seconds).

Use the student exploration guide to investigate sound beats.

Practice Questions

1. Two trumpet players are trying to tune their instruments. When they are in tune, they will both be playing the same note and no beats will be heard. The more experienced player plays a note with a frequency of 520 Hz. When the second musician begins to play, the note sounds lower and they can hear 20 beats in 4.0 s. With what frequency is the second trumpeter playing?

Given:

F1 = 520 Hz
N = 20
Īt = 4.0 s

Required:

F2 = ?

Solution:
Beat Frequency =

Because of the absolute value signs, there will be two possible solutions to the equation above. F2 could equal 515 Hz or 525 Hz. However, in the question we were told that āthe note sounds lowerā; consequently, it is at a lower frequency. Therefore, the second player is playing a note with a frequency of 515 Hz.