Book cover for Physics

Physics

John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler

ISBN #9781118486894

10th Edition

2,562 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section explores the principle of linear superposition, which states that overlapping waves add their displacements together. This principle underlies many phenomena in acoustics, including interference (constructive and destructive), diffraction, and beat frequencies. Further, the formation of standing waves—both transverse (as in strings) and longitudinal (as in air columns)—is central to understanding harmonics and the production of complex sound waves, which in turn explain why musical instruments have distinct timbres.

Learning Objectives

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Key Concepts

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Example Problems

Example 1

In Figure $17.7,$ suppose that the separation between speakers $A$ and $B$ is $5.00 \mathrm{m}$ and the speakers are vibrating in phase. They are playing identical $125-\mathrm{Hz}$ tones, and the speed of sound is $343 \mathrm{m} / \mathrm{s} .$ What is the largest possible distance between speaker $\mathrm{B}$ and the observer at $\mathrm{C},$ such that he observes destructive interference?

Example 2

Two speakers, one directly behind the other, are each generating a 245-Hz sound wave. What is the smallest separation distance between the speakers that will produce destructive interference at a listener standing in front of them? The speed of sound is $343 \mathrm{m} / \mathrm{s}$.

Example 3

The drawing graphs a string on which two rectangular pulses are traveling at a constant speed of $1 \mathrm{cm} / \mathrm{s}$ at time $t=0 \mathrm{s}$. Using the principle of linear superposition, draw the shape of the string at $t=1 \mathrm{s}, 2 \mathrm{s}, 3 \mathrm{s},$ and $4 \mathrm{s}$.

Example 4

Loudspeakers A and B are vibrating in phase and are playing the same tone, which has a frequency of $250 \mathrm{Hz}$. They are set up as in Figure 17.7, and point C is located as shown there. However, the distance between the speakers and the distance between speaker B and point C have the same value d. The speed of sound is $343 \mathrm{m} / \mathrm{s} .$ What is the smallest value of d, such that constructive interference occurs at point C?

Example 5

Two waves are traveling in opposite directions on the same string. The displacements caused by the individual waves are given by $y_{1}=(24.0 \mathrm{mm}) \sin (9.00 \pi t-1.25 \pi x)$ and $y_{2}=(35.0 \mathrm{mm}) \sin (2.88 \pi t+$ $0.400 \pi x$ ). Note that the phase angles $(9.00 \pi t-1.25 \pi x)$ and $(2.88 \pi t+$ $0.400 \pi x$ ) are in radians, $t$ is in seconds, and $x$ is in meters. At $t=4.00 \mathrm{s}$ what is the net displacement (in $\mathrm{mm}$ ) of the string at (a) $x=2.16 \mathrm{m}$ and (b) $x=2.56 \mathrm{m} ?$ Be sure to include the algebraic sign $(+$ or $-)$ with your answers.
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