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If you play the lowest two notes of a piano you may hear the notes separately, but you may also hear a beating pattern, like aaaaaaaah–oooooooo–aaaaaaaah–oooooooo–aaaaaaaah about twice a second. Try this if a piano is available.

If you have a guitar, you can hear this effect by playing an A on the fifth fret of the sixth string and an A on the fifth string open. If the two strings are slightly out of tune, you will hear a single note that gets louder and quieter, louder and quieter. This is called beats.

What is happening? The first two graphs of Figure 12-13 show the two notes which have similar frequency. At t = 0, they are in phase, and the amplitude of the combination is large. This is shown in third graph, where their sum is shown. A little while later, however, the two waves are out of phase, and the amplitude of the sum is a minimum. So this is the origin of the loud-soft-loud sound of the two notes. The beat period is shown in the third graph, and beat frequency is given by

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Figure 12-13




Jessica is tuning a guitar by comparing notes to a piano that she knows is in tune. She plays an A on the piano (220 Hz) and loudly plucks the A-string. She hears a loud-soft ringing whose maxima are separated by 3 seconds.

  1. What is the guitar string’s current fundamental frequency?
  2. Jessica tightens the string (increases the tension) of the guitar slightly, and the beat gets faster. Should she continue to tighten the string?
  1. The beat period is 3 s, so the beat frequency is 0.33 Hz. The string may be producing 220.33 Hz or 119.67 Hz, that is, too sharp or too flat.
  2. By tightening the string, Jessica increases its frequency. If the resulting frequency were closer to 220 Hz, the beat period would get longer. She should reduce the tension in the string.

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