17.2

Constructive and Destructive Interference of Sound Waves

Suppose that the sounds from two speakers overlap in the middle of a listening area, as in Figure 17-3, and that each speaker produces a sound wave of the same amplitude and frequency. For convenience, the wavelength of the sound is chosen to be . In addition, assume the diaphragms of the speakers vibrate in phase; that is, they move outward together and inward together. If the distance of each speaker from the overlap point is the same (3 m in the drawing), the condensations (C) of one wave always meet the condensations of the other when the waves come together; similarly, rarefactions (R) always meet rarefactions. According to the principle of linear superposition, the combined pattern is the sum of the individual patterns. As a result, the pressure fluctuations at the overlap point have twice the amplitude A that the individual waves have, and a listener at this spot hears a louder sound than that coming from either speaker alone. When two waves always meet condensation-to-condensation and rarefaction-to-rarefaction (or crest-to-crest and trough-to-trough), they are said to be exactly in phase and to exhibit constructive interference.

Figure 17-3 As a result of constructive interference between the two sound waves (amplitude = A), a loud sound (amplitude = 2A) is heard at an overlap point located equally distant from two in-phase speakers (C, condensation; R, rarefaction).

Now consider what happens if one of the speakers is moved. The result is surprising. In Figure 17-4, the left speaker is moved away* from the overlap point by a distance equal to one-half of the wavelength, or 0.5 m. Therefore, at the overlap point, a condensation arriving from the left meets a rarefaction arriving from the right. Likewise, a rarefaction arriving from the left meets a condensation arriving from the right. According to the principle of linear superposition, the net effect is a mutual cancellation of the two waves. The condensations from one wave offset the rarefactions from the other, leaving only a constant air pressure. A constant air pressure, devoid of condensations and rarefactions, means that a listener detects no sound. When two waves always meet condensation-to-rarefaction (or crest-to-trough), they are said to be exactly out of phase and to exhibit destructive interference.

Figure 17-4 The speakers in this drawing vibrate in phase. However, the left speaker is one-half of a wavelength farther from the overlap point than the right speaker. Because of destructive interference, no sound is heard at the overlap point (C, condensation; R, rarefaction).

When two waves meet, they interfere constructively if they always meet exactly in phase and destructively if they always meet exactly out of phase. In either case, this means that the wave patterns do not shift relative to one another as time passes. Sources that produce waves in this fashion are called coherent sources.
Destructive interference is the basis of a useful technique for reducing the loudness of undesirable sounds. For instance, Figure 17-5 shows a pair of noise-canceling headphones. Small microphones are mounted inside the headphones and detect noise such as the engine noise that an airplane pilot would hear. The headphones also contain circuitry to process the electronic signals from the microphones and reproduce the noise in a form that is exactly out of phase compared to the original. This out-of-phase version is played back through the headphone speakers and, because of destructive interference, combines with the original noise to produce a quieter background.

Figure 17-5 Noise-canceling headphones utilize destructive interference.

The physics of noise-canceling headphones.

It should be apparent that if the left speaker in Figure 17-4 were moved away from the overlap point by another one-half wavelength the , would again be in phase, and constructive interference would occur. The listener would hear a loud sound because the left wave travels one whole wavelength () farther than the right wave and, at the overlap point, condensation meets condensation and rarefaction meets rarefaction. In general, the important issue is the difference in the path lengths traveled by each wave in reaching the overlap point:

> For two wave sources vibrating in phase, a difference in path lengths that is zero or an integer number (1, 2, 3, …) of wavelengths leads to constructive interference; a difference in path lengths that is a half-integer number (, , , …) of wavelengths leads to destructive interference.

Problem solving insight

Interference effects can also be detected if the two speakers are fixed in position and the listener moves about the room. Consider Figure 17-6, where the sound waves spread outward from each speaker, as indicated by the concentric circular arcs. Each solid arc represents the middle of a condensation, and each dashed arc represents the middle of a rarefaction. Where the two waves overlap, there are places of constructive interference and places of destructive interference. Constructive interference occurs at any spot where two condensations or two rarefactions intersect, and the drawing shows four such places as solid dots. A listener stationed at any one of these locations hears a loud sound. On the other hand, destructive interference occurs at any place where a condensation and a rarefaction intersect, such as the two open dots in the picture. A listener situated at a point of destructive interference hears no sound. At locations where neither constructive nor destructive interference occurs, the two waves partially reinforce or partially cancel, depending on the position relative to the speakers. Thus, it is possible for a listener to walk about the overlap region and hear marked variations in loudness.

Figure 17-6 Two sound waves overlap in the shaded region. The solid lines denote the middle of the condensations (C), and the dashed lines denote the middle of the rarefactions (R). Constructive interference occurs at each solid dot (•) and destructive interference at each open dot (○).

The individual sound waves from the speakers in Figure 17-6 carry energy, and the energy delivered to the overlap region is the sum of the energies of the individual waves. This fact is consistent with the principle of conservation of energy, which we first encountered in Section 6.8. This principle states that energy can neither be created nor destroyed, but can only be converted from one form to another. One of the interesting consequences of interference is that the energy is redistributed, so there are places within the overlap region where the sound is loud and other places where there is no sound at all. Interference, so to speak, “robs Peter to pay Paul,” but energy is always conserved in the process. Example 1 illustrates how to decide what a listener hears.

| Example 1 | | | What Does a Listener Hear?

In Figure 17-7 two in-phase loudspeakers, A and B, are separated by 3.20 m. A listener is stationed at point C, which is 2.40 m in front of speaker B. The triangle ABC is a right triangle. Both speakers are playing identical 214-Hz tones, and the speed of sound is 343 m/s. Does the listener hear a loud sound or no sound?

Figure 17-7 Example 1 discusses whether this setup leads to constructive or destructive interference at point C for 214-Hz sound waves.

Problem solving insight To decide whether two sources of sound produce constructive or destructive interference at a point, determine the difference in path lengths between each source and that point and compare it to the wavelength of the sound.

Reasoning The listener will hear either a loud sound or no sound, depending on whether the interference occurring at point C is constructive or destructive. To determine which it is, we need to find the difference in the distances traveled by the two sound waves that reach point C and see whether the difference is an integer or half-integer number of wavelengths. In either event, the wavelength can be found from the relation (Equation 16.1).
Solution Since the triangle ABC is a right triangle, the distance AC is given by the Pythagorean theorem as . The distance BC is given as 2.40 m. Thus, the difference in the travel distances for the waves is 4.00 m − 2.40 m = 1.60 m. The wavelength of the sound is

| (16.1)

Since the difference in the distances is one wavelength, constructive interference occurs at point C, and the listener hears a loud sound.

Up to this point, we have been assuming that the speaker diaphragms vibrate synchronously, or in phase; that is, they move outward together and inward together. This may not be the case, however, and Conceptual Example 2 considers what happens then.

| Conceptual Example 2 | | | Out-of-Phase Speakers

The physics of wiring stereo speakers. To make a speaker operate, two wires must be connected between the speaker and the receiver (amplifier), as in Figure 17-8. To ensure that the diaphragms of two speakers vibrate in phase, it is necessary to make these connections in exactly the same way. If the wires for one speaker are not connected just as they are for the other speaker, the two diaphragms will vibrate out of phase. Whenever one diaphragm moves outward, the other will move inward, and vice versa. Suppose that in Figures 17-3 and 17-4 the connections are made so that the speaker diaphragms vibrate out of phase, everything else remaining the same. In each case, what kind of interference would result at the overlap point?

Figure 17-8 A loudspeaker is connected to a receiver (amplifier) by two wires. (Andy Washnik)

Reasoning and Solution Since the diaphragms are vibrating out of phase, one of them is now moving exactly opposite to the way it was moving originally. Let us assume that it is the one on the left in the drawings. The effect of this change is that every condensation originating from the left speaker becomes a rarefaction and every rarefaction becomes a condensation. As a result, in Figure 17-3 a rarefaction from the left now meets a condensation from the right at the overlap point, and destructive interference results. In Figure 17-4, a condensation from the left now meets a condensation from the right at the overlap point, leading to constructive interference. These results are opposite to those shown in the figures.
Instructions for connecting stereo systems specifically warn users to avoid out-of-phase vibration of the speaker diaphragms. One way to check your system is to play some music that is rich in low-frequency bass tones. Use the monaural mode on your receiver so the same sound comes from each speaker, and slide the speakers toward each other. If the diaphragms are moving in phase, the bass sound will either remain the same or slightly improve as the speakers come together. If the diaphragms are moving out of phase, the bass sound will decrease notice ably because of destructive interference when the speakers are right next to each other. In this event, simply interchange the wires to the terminals on one (but not both) of the speakers.

The phenomena of constructive and destructive interference are exhibited by all types of waves, not just sound waves. We will encounter interference effects again in Chapter file://../cutnell3158c27/cutnell3158c27.xml, in connection with light waves.

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.