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By the end of this section, you will be able to:
  • Explain the mechanism behind sound-reducing headphones
  • Describe resonance in a tube closed at one end and open at the other end
  • Describe resonance in a tube open at both ends

Interference is the hallmark of waves, all of which exhibit constructive and destructive interference exactly analogous to that seen for water waves. In fact, one way to prove something “is a wave” is to observe interference effects. Since sound is a wave, we expect it to exhibit interference.

Interference of sound waves

In Waves , we discussed the interference of wave functions that differ only in a phase shift. We found that the wave function resulting from the superposition of y 1 ( x , t ) = A sin ( k x ω t + ϕ ) and y 2 ( x , t ) = A sin ( k x ω t ) is

y ( x , t ) = [ 2 A cos ( ϕ 2 ) ] sin ( k x ω t + ϕ 2 ) .

One way for two identical waves that are initially in phase to become out of phase with one another is to have the waves travel different distances; that is, they have different path lengths. Sound waves provide an excellent example of a phase shift due to a path difference. As we have discussed, sound waves can basically be modeled as longitudinal waves, where the molecules of the medium oscillate around an equilibrium position, or as pressure waves.

When the waves leave the speakers, they move out as spherical waves ( [link] ). The waves interfere; constructive inference is produced by the combination of two crests or two troughs, as shown. Destructive interference is produced by the combination of a trough and a crest.

A drawing of two speakers that act as sources of the same frequency sound waves. Points of high-intensity sound which result from two crests (compression) or two troughs (rarefaction) overlapping are shown. In addition, points of constructive interference are indicated.
When sound waves are produced by a speaker, they travel at the speed of sound and move out as spherical waves. Here, two speakers produce the same steady tone (frequency). The result is points of high-intensity sound (highlighted), which result from two crests (compression) or two troughs (rarefaction) overlapping. Destructive interference results from a crest and trough overlapping. The points where there is constructive interference in the figure occur because the two waves are in phase at those points. Points of destructive interference ( [link] ) are the result of the two waves being out of phase.
Top picture is a drawing of two speakers being driven by a single signal generator. The sound waves produced by the speakers are in phase and are of a single frequency. The constructive interference is marked by the red and blue dots, the destructive interference is marked by black dots. Figure A corresponds to the situation when difference in the path lengths is one wavelength, resulting in total constructive interference and a resulting amplitude equal to twice the original amplitude.
Two speakers being driven by a single signal generator. The sound waves produced by the speakers are in phase and are of a single frequency. The sound waves interfere with each other. When two crests or two troughs coincide, there is constructive interference, marked by the red and blue dots. When a trough and a crest coincide, destructive interference occurs, marked by black dots. The phase difference is due to the path lengths traveled by the individual waves. Two identical waves travel two different path lengths to a point P . (a) The difference in the path lengths is one wavelength, resulting in total constructive interference and a resulting amplitude equal to twice the original amplitude. (b) The difference in the path lengths is less than one wavelength but greater than one half a wavelength, resulting in an amplitude greater than zero and less than twice the original amplitude. (c) The difference in the path lengths is one half of a wavelength, resulting in total destructive interference and a resulting amplitude of zero.
Practice Key Terms 3

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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