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Chapter 17: Problem 89

Two sound waves with an amplitude of \(12 \mathrm{~nm}\) and a wavelength of \(35 \mathrm{~cm}\) travel in the same direction through a long tube, with a phase difference of \(\pi / 3\) rad. What are the (a) amplitude and (b) wavelength of the net sound wave produced by their interference? If, instead, the sound waves travel through the tube in opposite directions, what are the (c) amplitude and (d) wavelength of the net wave?

Short Answer

Expert verified
(a) Amplitude: 20.78 nm; (b) Wavelength: 35 cm; (c) Amplitude: 24 nm; (d) Wavelength: 35 cm.

Step by step solution

01

Identify the Parameters

We have two sound waves of the same amplitude (\(A = 12 \ \mathrm{nm}\)) and the same wavelength (\(\lambda = 35 \ \mathrm{cm}\)), with a phase difference of \(\phi = \frac{\pi}{3} \ \mathrm{rad}\).
02

Find Amplitude of Net Wave for Traveling in Same Direction

For two waves traveling in the same direction, the resultant amplitude \(A_r\) is given by:\[A_r = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos(\phi)}\]Plugging in the values:\[ A_r = \sqrt{12^2 + 12^2 + 2 \cdot 12 \cdot 12 \cdot \cos(\frac{\pi}{3}) }\]Since \(\cos(\frac{\pi}{3}) = 0.5\), we have:\[ A_r = \sqrt{144 + 144 + 144}\cdot 0.5 = \sqrt{432} \approx 20.78 \ \mathrm{nm}\]
03

Determine Wavelength of Net Wave When Traveling in Same Direction

The wavelength \(\lambda_r\) remains the same as the individual waves when they travel in the same direction. Thus, \(\lambda_r = 35 \ \mathrm{cm}\).
04

Find Amplitude of Net Wave for Opposite Directions

For two waves traveling in opposite directions, they form a standing wave. The resultant amplitude of this standing wave is given by twice the amplitude of one wave:\[A_s = 2A\]So, \[A_s = 2 \cdot 12 \ \mathrm{nm} = 24 \ \mathrm{nm}\].
05

Determine Wavelength of Net Wave for Opposite Directions

In a standing wave, the wavelength is still the same as that of the original waves. So, the resultant wavelength \(\lambda_s = 35 \ \mathrm{cm}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Amplitude of Sound Waves
The amplitude of a sound wave is a measure of how much energy it carries and is often associated with the loudness of the sound. In this exercise, the individual sound waves each have an amplitude of 12 nm. When combining two sound waves, the resultant amplitude can be calculated depending on their direction of travel and their phase difference.
  • If the waves travel in the same direction and have a phase difference \(\phi\) of \(\frac{\pi}{3}\) rad, the formula to determine the resultant amplitude \(A_r\) is:\[A_r = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos(\phi)}\]
  • For two waves traveling in opposite directions, a standing wave is formed, and the new amplitude \(A_s\) is simply twice the amplitude of a single wave:\[A_s = 2A\]
These formulas help us determine how two interacting waves affect the loudness, depending on their directions and phase difference.
Wavelength of Sound Waves
The wavelength of a sound wave is the distance over which the wave's shape repeats. It is typically linked to the pitch of the sound, with longer wavelengths corresponding to lower pitches. In the given exercise, each individual sound wave has a wavelength of 35 cm.
  • When sound waves travel in the same direction, interference does not change their wavelength. The resultant wavelength \(\lambda_r\) is the same as the original wavelength:\[\lambda_r = \lambda = 35 \ cm\]
  • If the waves travel in opposite directions, creating a standing wave, the net wavelength \(\lambda_s\) also remains unchanged. Standing waves efficiently maintain the original wavelengths of the traveling waves:\[\lambda_s = 35 \ cm\]
Thus, the wavelength stays constant, no matter the direction, preserving the sound's pitch during interference.
Phase Difference in Waves
The phase difference between waves determines how they interact when combined. This difference, measured in radians, affects whether waves reinforce or cancel each other. In our problem, the two sound waves have a phase difference \(\phi\) of \(\frac{\pi}{3}\) rad.
  • When waves are in phase (i.e., phase difference is a multiple of \(2\pi\)) or partially in phase, they constructively interfere, producing a larger resultant amplitude.
  • The formula \(A_r = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos(\phi)}\) accounts for this, with the term \(\cos(\phi)\) determining the level of constructive interference. Here, \(\cos(\frac{\pi}{3}) = 0.5\), indicating that the waves partially reinforce each other, resulting in a net amplitude of approximately 20.78 nm.
  • If the waves are completely out of phase (i.e., the phase difference leads to \(\cos(\phi) = -1\)), they completely cancel each other out. However, our phase difference results only in partial cancellation or reinforcement.
Phase difference is critical in determining how strongly waves interact, affecting their combined amplitude but not their wavelength.

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Most popular questions from this chapter

Organ pipe \(A\), with both ends open, has a fundamental frequency of \(300 \mathrm{~Hz}\). The third harmonic of organ pipe \(B\), with one end open, has the same frequency as the second harmonic of pipe A. How long are (a) pipe \(A\) and (b) pipe \(B\) ?

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