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Chapter 12: Problem 38

A source of sound waves of frequency \(1.0 \mathrm{kHz}\) is traveling through the air at 0.50 times the speed of sound. (a) Find the frequency of the sound received by a stationary observer if the source moves toward her. (b) Repeat if the source moves away from her instead.

Short Answer

Expert verified
Answer: When the source of sound is moving towards the observer, the frequency received is \(\frac{2000}{3} \ \mathrm{Hz}\). When the source of sound is moving away from the observer, the frequency received is \(2000 \ \mathrm{Hz}\).

Step by step solution

01

Case (a): Source moving toward the observer

Since the source is moving towards the observer, the velocity of the source \(v_s\) will have the opposite sign to the velocity of sound in the air. Therefore, the source's speed \(v_s = -0.50v\). We can plug this into the Doppler effect formula: \(f_r = f_s \frac{v + v_0}{v - (-0.50v)}\), with \(f_s = 1000 \ \mathrm{Hz}\), \(v_0 = 0\), and \(v_s = -0.50v\) \(f_r = 1000 \frac{v}{v + 0.50v} = 1000 \frac{v}{1.50v}\) Now we can simplify and solve for \(f_r\): \(f_r = 1000 \cdot \frac{1}{1.5} = 1000 \cdot \frac{2}{3} = \frac{2000}{3} \ \mathrm{Hz}\) So the frequency received by the observer when the source is moving towards her is \(\frac{2000}{3} \ \mathrm{Hz}\).
02

Case (b): Source moving away from the observer

In this case, the source is moving away from the observer, so the velocity of the source \(v_s = 0.50v\). We can plug this into the Doppler effect formula: \(f_r = f_s \frac{v + v_0}{v - 0.50v}\), with \(f_s = 1000 \ \mathrm{Hz}\), \(v_0 = 0\), and \(v_s = 0.50v\) \(f_r = 1000 \frac{v}{v - 0.50v} = 1000 \frac{v}{0.50v}\) Now we can simplify and solve for \(f_r\): \(f_r = 1000 \cdot \frac{1}{0.5} = 1000 \cdot 2 = 2000 \ \mathrm{Hz}\) So the frequency received by the observer when the source is moving away from her is \(2000 \ \mathrm{Hz}\).

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Most popular questions from this chapter

Bats emit ultrasonic waves with a frequency as high as $1.0 \times 10^{5} \mathrm{Hz} .$ What is the wavelength of such a wave in air of temperature \(15^{\circ} \mathrm{C} ?\)
At what frequency \(f\) does a sound wave in air have a wavelength of $15 \mathrm{cm},$ about half the diameter of the human head? Some methods of localization work well only for frequencies below \(f\), while others work well only above \(f\). (See Conceptual Questions 4 and 5 .)
In an experiment to measure the speed of sound in air, standing waves are set up in a narrow pipe open at both ends using a speaker driven at \(702 Hz\). The length of the pipe is \(2.0 \mathrm{m} .\) What is the air temperature inside the pipe (assumed reasonably near room temperature, \(20^{\circ} \mathrm{C}\) to \(\left.35^{\circ} \mathrm{C}\right) ?\) [Hint: The standing wave is not necessarily the fundamental.]
A musician plays a string on a guitar that has a fundamental frequency of \(330.0 \mathrm{Hz}\). The string is \(65.5 \mathrm{cm}\) long and has a mass of \(0.300 \mathrm{g} .\) (a) What is the tension in the string? (b) At what speed do the waves travel on the string? (c) While the guitar string is still being plucked, another musician plays a slide whistle that is closed at one end and open at the other. He starts at a very high frequency and slowly lowers the frequency until beats, with a frequency of \(5 \mathrm{Hz}\), are heard with the guitar. What is the fundamental frequency of the slide whistle with the slide in this position? (d) How long is the open tube in the slide whistle for this frequency?
A bat emits chirping sounds of frequency \(82.0 \mathrm{kHz}\) while hunting for moths to eat. If the bat is flying toward the moth at a speed of $4.40 \mathrm{m} / \mathrm{s}\( and the moth is flying away from the bat at \)1.20 \mathrm{m} / \mathrm{s},$ what is the frequency of the sound wave reflected from the moth as observed by the bat? Assume it is a cool night with a temperature of \(10.0^{\circ} \mathrm{C} .\) [Hint: There are two Doppler shifts. Think of the moth as a receiver, which then becomes a source as it "retransmits" the reflected wave. \(]\)
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