Superposition and Interference

Saul Beceiro-Novo

Oscillatory Motion and Waves and Physics of Hearing.

125 Superposition and Interference

Learning Objectives

Explain how superposition leads to standing waves and beats.
Describe the mathematical basis for overtones and beat frequency.

Water surface of a river is shown, with mountains in the background. There are small ripples over the water surface. — Figure 125.1: These waves result from the superposition of several waves from different sources, producing a complex pattern. (credit: waterborough, Wikimedia Commons)

Waves in nature often appear much more complex than simple sinusoidal shapes. As shown in Figure 125.1, most waveforms result from the combination—or superposition—of many individual waves. Despite their complexity, these waves follow straightforward rules. When two or more waves meet at a point in space, their displacements add algebraically.

For waves that overlap exactly in phase, their amplitudes combine to produce constructive interference. An example is shown in Figure 125.2, where two identical waves align crest-to-crest and trough-to-trough, resulting in a wave with twice the amplitude.

The graph shows two identical waves that arrive at the same point exactly in phase. The crests of the two waves are precisely aligned, as are the troughs. The amplitude of each wave being X . It produces pure constructive wave. The disturbances add resulting in a new wave with twice the amplitude of the individual waves that is two X but of same wavelength. — Figure 125.2: Constructive interference occurs when two identical waves superimpose in phase, doubling the amplitude.

In contrast, Figure 125.3 shows destructive interference, where the crest of one wave aligns with the trough of another. Their displacements cancel out, producing zero net amplitude.

The graph shows two identical waves that arrive exactly out of phase. The crests of one wave are aligned with the trough of another wave. Each wave has amplitude equal to X. As the disturbances are in the opposite directions, they cancel out each other, resulting in zero amplitude which is shown as the third figure showing a green straight line, that is, the waves cancel each other producing pure destructive interference. — Figure 125.3: Destructive interference results when two identical waves overlap out of phase, canceling each other completely.

Most wave interactions involve partial alignment, producing a mix of constructive and destructive interference across different locations or times. For example, in a room with multiple audio speakers, certain regions may sound louder due to constructive interference, while others sound muted due to destructive interference.

When waves of different shapes and frequencies combine, the resulting waveform becomes more irregular, as shown in Figure 125.4.

The graph shows two non-identical waves with different frequencies and wavelengths. In the first graph only one crest and one trough of the wave are seen. In the second figure five crests are seen in the same length. When they superimpose, the disturbance add and subtract, producing a more complicated looking wave with highly irregular amplitude and wavelength due to combined effect of constructive and destructive interference. — Figure 125.4: Superposition of non-identical waves leads to complex patterns involving both constructive and destructive interference.

Standing Waves

Sometimes waves appear stationary even though they result from moving disturbances. These are called standing waves, created by the superposition of two identical waves traveling in opposite directions. A familiar example is the vibration of fluid surfaces in a container, such as milk in a refrigerator. Standing waves also occur in strings, like those on musical instruments.

Standing wave combinations of two waves is shown. At the time t is equal to zero. The waves are in the same phase so the amplitude of the superimposed wave is double that of wave one and two. In the second figure at time t is equal to one fourth of time period T , the waves are in opposite phase so their super imposed figure is a straight line. Again at the time t is equal to half the time period the waves are in the same phase and the process is repeated at t is equal to three fourth of time period and at the end of the time period T. — Figure 125.5: Standing wave formed from two waves of equal amplitude traveling in opposite directions. The pattern oscillates in place due to alternating constructive and destructive interference.

In standing waves, fixed points called nodes remain stationary, while the points of maximum motion are known as antinodes. Strings fixed at both ends form standing waves with quantized frequencies. The fundamental, or lowest frequency, corresponds to the longest wavelength:

λ_{1} = 2 L

f_{1} = \frac{v_{w}}{λ_{1}} = \frac{v_{w}}{2 L}

Higher-frequency modes are called harmonics or overtones. The second harmonic (first overtone) has half the wavelength of the fundamental:

λ_{2} = L \Rightarrow f_{2} = \frac{v_{w}}{L} = 2 f_{1}

And the third harmonic:

f_{3} = 3 f_{1}

The graph shows a wave with wavelength lambda one equal to L, which has two loops. There three nodes and two antinodes in the figure. The length of one loop is L. — Figure 125.6:A string oscillating in its fundamental mode with two nodes and one antinode.

first overtone is shown as the wave length if lambda two is L and there are three nodes and two antinodes in the figure. For first overtone the frequency f two is equal to two times f one. — Figure 125.7: Standing waves showing the first and second overtones (harmonics), with increasing numbers of nodes and antinodes.

Beats

When two sound waves of similar frequency are combined, the result is a wave whose amplitude varies periodically. This phenomenon is known as a beat. Listeners perceive the volume fluctuating—an effect often heard when tuning musical instruments or listening to two closely pitched jet engines.

The general form for a wave of frequency $f$ is:

x = X \cos (2 π f t)

If we add two such waves with frequencies $f_{1}$ and $f_{2}$ , the result is:

x = X \cos (2 π f_{1} t) + X \cos (2 π f_{2} t)

Using a trigonometric identity:

x = 2 X \cos (π f_{B} t) \cos (2 π f_{ave} t)

where:

f_{B} = | f_{1} - f_{2} | and f_{ave} = \frac{f_{1} + f_{2}}{2}

The result is a wave with an average frequency and a slowly varying amplitude at the beat frequency $f_{B}$ . This phenomenon is useful in a range of applications, including sound tuning, radar, and ultrasound imaging.

Making Career Connections

Piano tuners use beats routinely in their work. When comparing a note with a tuning fork, they listen for beats and adjust the string until the beats go away (to zero frequency). For example, if the tuning fork has a $256 Hz$ frequency and two beats per second are heard, then the other frequency is either $254$ or $258 Hz$ . Most keys hit multiple strings, and these strings are actually adjusted until they have nearly the same frequency and give a slow beat for richness. Twelve-string guitars and mandolins are also tuned using beats.

While beats may sometimes be annoying in audible sounds, we will find that beats have many applications. Observing beats is a very useful way to compare similar frequencies. There are applications of beats as apparently disparate as in ultrasonic imaging and radar speed traps.

Check Your Understanding

Imagine you are holding one end of a jump rope, and your friend holds the other. If your friend holds her end still, you can move your end up and down, creating a transverse wave. If your friend then begins to move her end up and down, generating a wave in the opposite direction, what resultant wave forms would you expect to see in the jump rope?

The rope would alternate between having waves with amplitudes two times the original amplitude and reaching equilibrium with no amplitude at all. The wavelengths will result in both constructive and destructive interference

Check Your Understanding

Define nodes and antinodes.

Nodes are areas of wave interference where there is no motion. Antinodes are areas of wave interference where the motion is at its maximum point.

Check Your Understanding

You hook up a stereo system. When you test the system, you notice that in one corner of the room, the sounds seem dull. In another area, the sounds seem excessively loud. Describe how the sound moving about the room could result in these effects.

With multiple speakers putting out sounds into the room, and these sounds bouncing off walls, there is bound to be some wave interference. In the dull areas, the interference is probably mostly destructive. In the louder areas, the interference is probably mostly constructive.

PhET Explorations: Wave Interference

Make waves with a dripping faucet, audio speaker, or laser! Add a second source or a pair of slits to create an interference pattern.

Section Summary

Superposition is the combination of two waves at the same location.
Constructive interference occurs when two identical waves are superimposed in phase.
Destructive interference occurs when two identical waves are superimposed exactly out of phase.
A standing wave is one in which two waves superimpose to produce a wave that varies in amplitude but does not propagate.
Nodes are points of no motion in standing waves.
An antinode is the location of maximum amplitude of a standing wave.
Waves on a string are resonant standing waves with a fundamental frequency and can occur at higher multiples of the fundamental, called overtones or harmonics.
Beats occur when waves of similar frequencies $f_{1}$ and $f_{2}$ are superimposed. The resulting amplitude oscillates with a beat frequency given by
$f_{B} =∣ f_{1} - f_{2} ∣ .$

Conceptual Questions

Speakers in stereo systems have two color-coded terminals to indicate how to hook up the wires. If the wires are reversed, the speaker moves in a direction opposite that of a properly connected speaker. Explain why it is important to have both speakers connected the same way.

Problems & Exercises

A car has two horns, one emitting a frequency of 199 Hz and the other emitting a frequency of 203 Hz. What beat frequency do they produce?
The middle-C hammer of a piano hits two strings, producing beats of 1.50 Hz. One of the strings is tuned to 260.00 Hz. What frequencies could the other string have?
Two tuning forks having frequencies of 460 and 464 Hz are struck simultaneously. What average frequency will you hear, and what will the beat frequency be?
Twin jet engines on an airplane are producing an average sound frequency of 4100 Hz with a beat frequency of 0.500 Hz. What are their individual frequencies?
A wave traveling on a Slinky® that is stretched to 4 m takes 2.4 s to travel the length of the Slinky and back again. (a) What is the speed of the wave? (b) Using the same Slinky stretched to the same length, a standing wave is created which consists of three antinodes and four nodes. At what frequency must the Slinky be oscillating?
Three adjacent keys on a piano (F, F-sharp, and G) are struck simultaneously, producing frequencies of 349, 370, and 392 Hz. What beat frequencies are produced by this discordant combination?

Glossary

antinode: the location of maximum amplitude in standing waves

beat frequency: the frequency of the amplitude fluctuations of a wave

constructive interference: when two waves arrive at the same point exactly in phase; that is, the crests of the two waves are precisely aligned, as are the troughs

destructive interference: when two identical waves arrive at the same point exactly out of phase; that is, precisely aligned crest to trough

fundamental frequency: the lowest frequency of a periodic waveform

nodes: the points where the string does not move; more generally, nodes are where the wave disturbance is zero in a standing wave

overtones: multiples of the fundamental frequency of a sound

superposition: the phenomenon that occurs when two or more waves arrive at the same point

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Standing Waves

Beats

Making Career Connections

Check Your Understanding

PhET Explorations: Wave Interference

Section Summary

Conceptual Questions

Problems & Exercises

Glossary

License

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