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Chapter 16 | Oscillatory Motion and Waves 715
Figure 16.42 Beats are produced by the superposition of two waves of slightly different frequencies but identical amplitudes. The waves alternate in time between constructive interference and destructive interference, giving the resulting wave a time-varying amplitude.
The wave resulting from the superposition of two similar-frequency waves has a frequency that is the average of the two. This wave fluctuates in amplitude, or beats, with a frequency called the beat frequency. We can determine the beat frequency by adding two waves together mathematically. Note that a wave can be represented at one point in space as
where is the frequency of the wave. Adding two waves that have different frequencies but identical amplitudes produces a resultant
(16.69)
(16.70) (16.71)
(16.72)
More specifically,
Using a trigonometric identity, it can be shown that
where
(16.73) amplitude and the average frequency of the two superimposed waves, but it also fluctuates in overall amplitude at the beat
is the beat frequency, and is the average of and . These results mean that the resultant wave has twice the
frequency . The first cosine term in the expression effectively causes the amplitude to go up and down. The second cosine
term is the wave with frequency . This result is valid for all types of waves. However, if it is a sound wave, providing the two frequencies are similar, then what we hear is an average frequency that gets louder and softer (or warbles) at the beat frequency.
Real World Connections: Tuning Forks
The MIT physics demo (http://openstaxcollege.org/l/31tuningforks/) entitled “Tuning Forks: Resonance and Beat Frequency” provides a qualitative picture of how wave interference produces beats.
Description: Two identical forks and sounding boxes are placed next to each other. Striking one tuning fork will cause the other to resonate at the same frequency. When a weight is attached to one tuning fork, they are no longer identical. Thus, one will not cause the other to resonate. When two different forks are struck at the same time, the interference of their pitches produces beats.
Real World Connections: Jump Rop
This is a fun activity with which to learn about interference and superposition. Take a jump rope and hold it at the two ends with one of your friends. While each of you is holding the rope, snap your hands to produce a wave from each side. Record your observations and see if they match with the following:
a. One wave starts from the right end and travels to the left end of the rope.
b. Another wave starts at the left end and travels to the right end of the rope.
c. The waves travel at the same speed.
d. The shape of the waves depends on the way the person snaps his or her hands.
e. There is a region of overlap.
f. The shapes of the waves are identical to their original shapes after they overlap.
Now, snap the rope up and down and ask your friend to snap his or her end of the rope sideways. The resultant that one sees here is the vector sum of two individual displacements.
This activity illustrates superposition and interference. When two or more waves interact with each other at a point, the
