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Interference

When two particles come together, like two balls or two cars, they generally collide in some manner. When two waves come together, they do not collide but jumble together in a process called interference. It is not as complicated as it sounds. For example, consider a wave on a lake (amplitude 7 cm, wavelength 4 m) arriving from the north, such that at point A at time 12:30 (exactly) its height would be +5 cm if it were the only wave around (Figure 11-13). Now another wave (amplitude 10 cm, wavelength 6 m) arrives from the east, such that at time 12:30 its height would be –2 cm if it were the only wave around. The resulting height of the water at 12:30 at point A is (+5 + –2) cm = +3 cm. (These heights are measured relative to the equilibrium height of the water.) Three seconds later, let's say, wave 1 would have height +7 cm and wave 2, –1 cm. Then the new height, with both waves, would be (+2 + –1) cm = +1 cm.
 

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Figure 11-13
 

Principle of Superposition: When two waves come together, the resulting displacement of the medium is the sum of the individual displacements.

Figure 11-14 shows another example of this, in which two wave pulses come together, one from the right and one from the left.
 

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Figure 11-14
 

Figure 11-15 shows a third example of interference on water. In this example a wave train from the left encounters a wave train from the right with the same wavelength. Water skaters are sitting on the water at points A, B, and C.
 

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Figure 11-15
 

The skater at point A experiences the up–down–up–down–up from the wave on the left and the up–down–up–down–up from the wave on the right. Superposition tells us that the resulting motion is added, so A experiences increased amplitude UP–DOWN–UP–DOWN–UP. This is called constructive interference, when two wave forms add in such a way as to create maximal displacement of the medium. The two waves are said to be in phase, and point A is called an antinode. Their relative phase is said to be 0. The resulting amplitude is the sum of the individual amplitudes.
 
For the water skater at C, the left wave is up–down–up–down–up, and the right wave is down–up–down–up–down. By superposition we see that the resulting motion is no displacement at all. This is called destructive interference, when the wave forms tend to cancel and give minimal displacement. The two waves are said to be out of phase, and point C is called a node. Their relative phase is said to be 180˚. The resulting amplitude is the difference of the individual amplitudes.

For the water skater at B, the relative phase is between 0˚ and 180˚, and the interference is neither in phase nor out of phase but somewhere in between. The resulting amplitude is somewhere in between as well.

Figure 11-16 shows a fourth example of interference, this time with sound waves. Stereo speakers are at points 1 and 2, both producing a pure tone (sine wave) of wavelength λ in phase. Alice and Bob are listening to the speakers.
 

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Figure 11-16

 

Alice is an equal distance from both speakers. Figure 11-17 shows the waves traveling from speakers to Alice. Since the waves begin in phase and travel equal distances, they arrive in phase. If only speaker 1 were making sound, the wave arriving at her ear would look like that shown in Figure 11-18. If only speaker 2 were making a sound, the wave arriving at her ear would look like that shown in the same figure. Note that these figures show displacement of air particles versus time. Figure 11-19 shows what Alice hears with both speakers. She experiences constructive interference, so she hears a sound of greater intensity than that from one speaker.
 

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Figure 11-17

 

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Figure 11-18

 

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Figure 11-19

 

Bob, however, is further away from speaker 1 than from speaker 2 by half a wavelength. What difference does half a wavelength make? The wave arriving from speaker 1 has further to go, so when a peak is coming from speaker 2, a trough is just arriving from speaker 1. (See Figure 11-20.) Figure 11-21 shows the sound waves from the two speakers. Figure 11-22 shows what Bob hears, nothing.
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Figure 11-20
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Figure 11-21 Figure 11-22

Note: In real life it is difficult to hear this phenomenon fully, since the amplitudes of the two speakers are rarely exactly the same. However, architects of orchestra halls often must guard against the possibility of “dead spots”, where certain notes made by the concert master’s violin, for example, cannot be heard well.


Calculations of this type get complicated fairly quickly, beyond the scope of the MCAT. But you should recognize constructive and destructive interference when you see it (or hear it). In any wave phenomenon, when there are bands of strong and weak amplitude, you should suspect that constructive and destructive interference is the cause. And you should recognize this:
 
Constructive interference occurs when two waves differ by no wavelengths, or one. or two. or cetera. Destructive interference occurs when two waves differ by half a wavelength, or 3/2. or 5/2. or cetera.





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