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Chapter 17: Problem 16

Two sound waves, from two different sources with the same frequency, \(540 \mathrm{~Hz}\), travel in the same direction at \(330 \mathrm{~m} / \mathrm{s}\). The sources are in phase. What is the phase difference of the waves at a point that is \(4.40 \mathrm{~m}\) from one source and \(4.00 \mathrm{~m}\) from the other?

Short Answer

Expert verified
The phase difference is approximately 4.11 radians.

Step by step solution

01

Determine the Wavelength of the Sound Waves

First, calculate the wavelength \( \lambda \) of the sound waves using the formula \( \lambda = \frac{v}{f} \), where \( v \) is the speed of sound \( 330 \, \text{m/s} \) and \( f \) is the frequency \( 540 \, \text{Hz} \). Thus, \( \lambda = \frac{330}{540} \approx 0.6111 \, \text{m} \).
02

Calculate the Path Difference

The path difference between the two waves is the difference in the distances they travel from their respective sources to the point of interest. This is \( 4.40 \, \text{m} - 4.00 \, \text{m} = 0.40 \, \text{m} \).
03

Relate Path Difference to Phase Difference

The phase difference \( \Delta \phi \) is related to the path difference \( \Delta x \) and the wavelength \( \lambda \) by \( \Delta \phi = \frac{2\pi \Delta x}{\lambda} \). Here, substitute \( \Delta x = 0.40 \, \text{m} \) and \( \lambda = 0.6111 \, \text{m} \) to find \( \Delta \phi = \frac{2\pi \times 0.40}{0.6111} \approx 4.11 \, \text{radians} \).
04

Simplify the Phase Difference

Phase differences are simplified to the range \( 0 \) to \( 2\pi \) by taking the remainder when divided by \( 2\pi \). Here, \( 4.11 \, \text{radians} \) is already within this range, so it is the final phase difference between the waves.

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Key Concepts

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

Sound Waves
Sound waves are vibrations that travel through the air or any medium as a result of disturbances, like someone speaking or music playing. They are longitudinal waves, which means the particle displacement is parallel to the direction the wave travels.
These waves require a medium like air, water, or solids to travel, unlike light waves, which can travel through a vacuum. Each sound wave is characterized by its frequency, speed, and wavelength.
  • Frequency: Measured in Hertz (Hz), it tells us how many wave cycles pass a point per second.
  • Speed: The rate at which the sound wave travels through a medium.
  • Wavelength: The physical length of one cycle of the wave.
Understanding sound waves is essential when analyzing their interactions, such as interference and phase differences.
Wave Interference
When two or more sound waves meet while traveling through the same medium, interference occurs. This interaction can be constructive or destructive, depending on the phase relationship of the waves.
Interference determines how the waves combine, affecting the amplitude and intensity of the resulting sound.
  • Constructive Interference: Occurs when waves are in phase, meaning their peaks and troughs align perfectly, resulting in a higher amplitude.
  • Destructive Interference: Occurs when waves are out of phase, meaning the peak of one wave aligns with the trough of another, canceling each other out and reducing amplitude.
Observing interference is crucial to understanding how phase differences occur and are measured.
Wavelength
Wavelength, represented by the Greek letter \( \lambda \), is defined as the distance between two consecutive points in phase on a wave, such as peak to peak or trough to trough. It is a fundamental property that influences how waves interact over distance.
The formula \( \lambda = \frac{v}{f} \) helps calculate the wavelength, where \( v \) is the speed of the wave and \( f \) is the frequency.
  • Shorter Wavelengths: Lead to higher frequency and potentially more rapid interference patterns.
  • Longer Wavelengths: Result in lower frequency, influencing how sound travels and behaves over different distances.
Accurately determining the wavelength is crucial in problems like the one in the original exercise, where wavelength is needed to calculate phase difference.
Path Difference
Path difference refers to the difference in the distance traveled by two waves from their respective sources to a common point. This difference plays a significant role in determining the phase difference between waves.
In the context of the exercise, this is calculated as the difference in distance from each source to the point of interest.
  • Positive Path Difference: Generally indicates the presence of constructive interference if aligned properly.
  • Negative Path Difference: Could result in potential destructive interference, depending on further calculations.
The knowledge of path difference \, \( \Delta x \), \, is crucial for assessing how these waves interfere and the resulting phase difference, thus determining the nature of the combined sounds at the point of interest.

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Most popular questions from this chapter

\(A F\) fet plane passes over you at a height of \(5000 \mathrm{~m}\) and a speed of Mach \(1.5 .\) (a) Find the Mach cone angle (the sound speed is \(331 \mathrm{~m} / \mathrm{s}\) ). (b) How long after the jet passes directly overhead does the shock wave reach you?

A violin string \(30.0 \mathrm{~cm}\) long with linear density \(0.650 \mathrm{~g} / \mathrm{m}\) is placed near a loudspeaker that is fed by an audio oscillator of variable frequency. It is found that the string is set into oscillation only at the frequencies 880 and \(1320 \mathrm{~Hz}\) as the frequency of the oscillator is varied over the range \(500-1500 \mathrm{~Hz}\). What is the tension in the string?

A continuous sinusoidal longitudinal wave is sent along a very long coiled spring from an attached oscillating source. The wave travels in the negative direction of an \(x\) axis; the source frequency is \(25 \mathrm{~Hz}\); at any instant the distance between successive points of maximum expansion in the spring is \(24 \mathrm{~cm}\); the maximum longitudinal displacement of a spring particle is \(0.30 \mathrm{~cm} ;\) and the particle at \(x=0\) has zero displacement at time \(t=0\). If the wave is written in the form \(s(x, t)=s_{m} \cos (k x \pm \omega t)\), what are (a) \(s_{m}\), (b) \(k\), (c) \(\omega\), (d) the wave speed, and (e) the correct choice of sign in front of \(\omega ?\) Incident

A girl is sitting near the open window of a train that is moving at a velocity of \(10.00 \mathrm{~m} / \mathrm{s}\) to the east. The girl's uncle stands near the tracks and watches the train move away. The locomotive whistle emits sound at frequency \(500.0 \mathrm{~Hz}\). The air is still. (a) What frequency does the uncle hear? (b) What frequency does the girl hear? A wind begins to blow from the east at \(10.00\) \(\mathrm{m} / \mathrm{s}\). (c) What frequency does the uncle now hear? (d) What frequency does the girl now hear?

A person on a railroad car blows a trumpet note at \(440 \mathrm{~Hz}\). The car is moving toward a wall at \(20.0 \mathrm{~m} / \mathrm{s}\). Find the sound frequency (a) at the wall and (b) reflected back to the trumpeter.
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