17.4

Beats

CONCEPTS AT A GLANCE In situations where waves with the same frequency overlap, we have seen how the principle of linear superposition leads to constructive and destructive interference and how it explains diffraction. We will see in this section that two overlapping waves with slightly different frequencies give rise to the phenomenon of beats. However, as the Concepts-at-a-Glance chart in Figure 17-14 illustrates, the principle of linear superposition again provides an explanation of what happens when the waves overlap. This chart is an expansion of that in Figure 17-9.

Figure 17-14 CONCEPTS AT A GLANCE This figure, which is an extension of the concept chart in Figure 17-9, shows that the principle of linear superposition can also be used to explain the phenomenon of beats. Here, concert pianist Mordecai Shehori tunes his piano with the aid of software and beats. (Andrea Mohin/New York Times Pictures)

Need more practice?

Interactive LearningWare17.1
A sound wave is traveling in neon, assumed to be an ideal gas and for which the ratio of the specific heats is . The mass of a neon atom is 3.35 × 10−26 kg, the temperature is 373 K, and the wavelength of the sound is 1.14 m. A tuning fork is struck and vibrates at 440.0 Hz in neon. What would be the beat frequency produced by the two sounds?

A tuning fork has the property of producing a single-frequency sound wave when struck with a sharp blow. Figure 17-15 shows sound waves coming from two tuning forks placed side by side. The tuning forks in the drawing are identical, and each is designed to produce a 440-Hz tone. However, a small piece of putty has been attached to one fork, whose frequency is lowered to 438 Hz because of the added mass. When the forks are sounded simultaneously, the loudness of the resulting sound rises and falls periodically—faint, then loud, then faint, then loud, and so on. The periodic variations in loudness are called beats and result from the interference between two sound waves with slightly different frequencies.

Figure 17-15 Two tuning forks have slightly different frequencies of 440 and 438 Hz. The phenomenon of beats occurs when the forks are sounded simultaneously. The sound waves are not drawn to scale.

For clarity, Figure 17-15 shows the condensations and rarefactions of the sound waves separately. In reality, however, the waves spread out and overlap. In accord with the principle of linear superposition, the ear detects the combined total of the two. Notice that there are places where the waves interfere constructively and places where they interfere destructively. When a region of constructive interference reaches the ear, a loud sound is heard. When a region of destructive interference arrives, the sound intensity drops to zero (assuming each of the waves has the same amplitude). The number of times per second that the loudness rises and falls is the beat frequency and is the difference between the two sound frequencies. Thus, in the situation illustrated in Figure 17-15, an observer hears the sound loudness rise and fall at the rate of 2 times per second (440 Hz − 438 Hz).
Figure 17-16 helps to explain why the beat frequency is the difference between the two frequencies. The drawing displays graphical representations of the pressure patterns of a 10-Hz wave and a 12-Hz wave, along with the pressure pattern that results when the two overlap. These frequencies have been chosen for convenience, even though they lie below the audio range and are inaudible. Audible sound waves behave in exactly the same way. The top two drawings, in blue, show the pressure variations in a one-second interval of each wave. The third drawing, in red, shows the result of adding together the blue patterns according to the principle of linear superposition. Notice that the amplitude in the red drawing is not constant, as it is in the individual waves. Instead, the amplitude changes from a minimum to a maximum, back to a minimum, and so on. When such pressure variations reach the ear and occur in the audible frequency range, they produce a loud sound when the amplitude is a maximum and a faint sound when the amplitude is a minimum. Two loud–faint cycles, or beats, occur in the one-second interval shown in the drawing, corresponding to a beat frequency of 2 Hz. Thus, the beat frequency is the difference between the frequencies of the individual waves, or 12 Hz − 10 Hz = 2 Hz.

Figure 17-16 A 10-Hz and a 12-Hz sound wave, when added together, produce a wave with a beat frequency of 2 Hz. The drawings show the pressure patterns (in blue) of the individual waves and the pressure pattern (in red) that results when the two overlap. The time interval shown is one second.

The physics of tuning a musical instrument. Musicians often tune their instruments by listening to a beat frequency. For instance, a guitar player plucks an out-of-tune string along with a tone that has the correct frequency. He then adjusts the string tension until the beats vanish, ensuring that the string is vibrating at the correct frequency.

Check Your Understanding 2

Tuning fork A (frequency unknown) and tuning fork B (frequency = 384 Hz) produce 6 beats in 2 seconds. When a small piece of putty is attached to tuning fork A, as in Figure 17-15, the beat frequency decreases. What is the frequency of tuning fork A? (The answer is given at the end of the book.)
Background: This question deals with the concept of beat frequency, its relationship to the frequencies of the sounds producing the beats, and how it changes when one of them changes.
For similar questions (including conceptual counterparts), consult Self-Assessment Test 17.1, which is described at the left.

Test your understanding of the material in Sections 17.1, 17.2, 17.3 and 17.4:

· | The Principle of Linear Superposition
· | Interference of Sound Waves
· | Diffraction
· | Beats

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