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Chapter 16: Problem 48

Moving Source vs. Moving Listener. (a) A sound source producing \(1.00-\mathrm{kHz}\) waves moves toward a stationary listener at one-half the speed of sound. What frequency will the listener hear? (b) Suppose instead that the source is stationary and the listener moves toward the source at one- half the speed of sound. What frequency does the listener hear'? How does your answer compare to that in part (a)? Explain on physical grounds why the two answers differ.

Short Answer

Expert verified
(a) 2000 Hz, (b) 1500 Hz. Different outcomes due to how motion affects wave crests.

Step by step solution

01

Understand the Doppler Effect Formula for Moving Source

For a moving source, the Doppler Effect formula is given by:\[ f' = \frac{f}{1 - \frac{v_s}{v}} \]where \( f' \) is the observed frequency, \( f \) is the source frequency, \( v_s \) is the speed of the source, and \( v \) is the speed of sound. Given \( f = 1.00 \text{kHz} \) and \( v_s = \frac{v}{2} \), plug in these values to solve for \( f' \).
02

Calculate Observed Frequency for Moving Source

Substituting the given values into the formula:\[ f' = \frac{1000}{1 - \frac{1}{2}} = \frac{1000}{0.5} = 2000 \text{ Hz} \]Therefore, the listener hears a frequency of \( 2000 \text{ Hz} \) when the source moves toward the stationary listener at half the speed of sound.
03

Utilize Doppler Effect Formula for Moving Listener

For a moving listener, the Doppler Effect formula is:\[ f' = f \left( 1 + \frac{v_l}{v} \right) \]where \( v_l \) is the speed of the listener. With \( v_l = \frac{v}{2} \) and \( f = 1000 \text{ Hz} \), substitute these into the formula to find \( f' \).
04

Calculate Observed Frequency for Moving Listener

Plug in the values into the formula:\[ f' = 1000 \left( 1 + \frac{1}{2} \right) = 1000 \times 1.5 = 1500 \text{ Hz} \]Thus, the listener hears a frequency of \( 1500 \text{ Hz} \) when moving toward the stationary source at half the speed of sound.
05

Compare and Explain Differences

The frequency heard by the listener when the source is moving is \( 2000 \text{ Hz} \), while it's \( 1500 \text{ Hz} \) when the listener is moving. This difference occurs because the relative motion affects wave compression differently in each scenario: a moving source increases wave crests physically reaching the listener, whereas a moving listener simply increases the frequency of encountering existing wave crests.

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Key Concepts

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

Moving Source
When a sound source moves towards a listener, the waves it emits compress in the direction of the motion. This causes an increase in the frequency of the sound waves reaching the listener. The phenomenon is known as the Doppler Effect. In the formula for a moving source, \( f' = \frac{f}{1 - \frac{v_s}{v}} \), the observed frequency \( f' \) depends on the speed of the source \( v_s \). As the source approaches, the denominator decreases, making \( f' \) larger. For instance, when the source's speed is half the speed of sound, the frequency heard doubles. Thus, in a moving source scenario, more crests of the sound waves hit the listener per second.

Imagine a train with a whistle moving towards you; the sound pitch goes higher than when it was static. The motion squeezes the sound waves moving toward you, making sound waves catch up to each other faster.

This is why the frequency heard by the stationary listener from the moving source is higher, exemplified by a frequency shift to 2000 Hz."},{

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Most popular questions from this chapter

CP The sound from a trumpet radiates uniformly in all directions in \(20^{\circ} \mathrm{C}\) air. At a distance of 5.00 \(\mathrm{m}\) from the trumpet the sound intensity level is 52.0 \(\mathrm{dB}\) . The frequency is 587 \(\mathrm{Hz}\) . (a) What is the pressure amplitude at this distance? (b) What is the displacement amplitude? (c) At what distance is the sound intensity level 30.0 \(\mathrm{dB}\) ?

How fast (as a percentage of light speed) would a star have to be moving so that the frequency of the light we receive from it is 10.0\(\%\) higher than the frequency of the light it is emitting? Would it be moving away from us or toward us? (Assume it is moving either directly away from us or directly toward us.)

A loud factory machine produces sound having a displace ment amplitude of \(1.00 \mu \mathrm{m},\) but the frequency of this sound can be adjusted. In order to prevent ear damage to the workers, the maximum pressure amplitude of the sound waves is limited to 10.0 \(\mathrm{Pa.}\) Under the conditions of this factory, the bulk modulus of air is \(1.42 \times 10^{5}\) Pa. What is the highest-frequency sound to which this machine can be adjusted without exceeding the prescribed limit? Is this frequency audible to the workers?

Two loudspeakers, \(A\) and \(B,\) are driven by the same amplifier and emit sinusoidal waves in phase. The frequency of the waves emitted by each speaker is 172 Hz. You are 8.00 m from A. What is the closest you can be to \(B\) and be at a point of destructive interference?

A car alarm is emitting sound waves of frequency 520 \(\mathrm{Hz}\) You are on a motorcycle, traveling directly away from the car. How fast must you be traveling if you detect a frequency of 490 \(\mathrm{Hz}\) ?
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