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Chapter 17: Problem 65

A \(2000 \mathrm{~Hz}\) siren and a civil defense official are both at rest with respect to the ground. What frequency does the official hear if the wind is blowing at \(12 \mathrm{~m} / \mathrm{s}\) (a) from source to official and (b) from official to source?

Short Answer

Expert verified
(a) 2070 Hz, (b) 1930 Hz.

Step by step solution

01

Understanding the Problem

We need to calculate the frequency heard by the official when a wind is blowing between the sound source (siren) and the listener (official). We use the Doppler effect in an environment where wind affects the sound's speed.
02

Identifying the Needed Formula

The apparent frequency \(f'\) is given by the formula: \[ f' = f \times \frac{v + v_o}{v - v_s} \]where \(f\) is the source frequency, \(v\) is the speed of sound in air, \(v_o\) is the speed of the observer, and \(v_s\) is the speed of the source. In this case, both the observer and source are at rest, so \(v_o = v_s = 0\), but we must adjust \(v\) for wind speed.
03

Calculating Sound Speed with Wind

Given that the normal speed of sound in air is approximately \(343 \, \text{m/s}\), we adjust this by the wind speed. When the wind blows from source to official, the sound speed is \(v = 343 + 12 = 355 \, \text{m/s}\). When it blows from official to source, it is \(v = 343 - 12 = 331 \, \text{m/s}\).
04

Applying Formula - Wind from Source to Official

Plug the values into the formula for the first case (wind from source to official):\[ f' = 2000 \, \text{Hz} \times \frac{355}{343} \approx 2070 \, \text{Hz} \]The official hears a frequency of approximately \(2070 \, \text{Hz}\).
05

Applying Formula - Wind from Official to Source

For the second case (wind from official to source), plug the values into the formula:\[ f' = 2000 \, \text{Hz} \times \frac{331}{343} \approx 1930 \, \text{Hz} \]The official hears a frequency of approximately \(1930 \, \text{Hz}\).

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Key Concepts

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

Sound Frequency
Sound frequency refers to the number of sound wave cycles that occur in one second. It is measured in hertz (Hz). The higher the frequency, the higher the pitch of the sound. In the context of the Doppler Effect, frequency is what changes due to the relative motion between the source of the sound and the observer. Even if both the source and observer are at rest, other factors such as wind can alter the perceived frequency.

For instance, if a siren emits sound at a frequency of 2000 Hz, the observer will hear this exact frequency when there鈥檚 no wind. However, if wind is involved, the frequency can appear higher or lower depending on the wind's direction relative to the listener and the sound's source. This occurs due to changes in the effective speed of sound, which modifies how frequently the sound waves reach the observer.
Speed of Sound
The speed of sound is the rate at which sound waves travel through a medium, typically air. At room temperature, this speed is approximately 343 meters per second (m/s). However, factors like wind can adjust this speed, affecting how sound reaches a listener.
  • When there is no wind, sound travels at its typical speed in the medium.
  • If the wind is moving in the same direction as the sound, it increases the speed of sound relative to an observer downstream, making the sound waves reach the observer more quickly.
  • Conversely, if the wind is moving against the sound direction, it decreases the speed of sound relative to an observer upstream, meaning sound waves take longer to reach them.
These adjustments in speed directly influence the frequency the observer perceives due to the altered number of wave cycles that hit them in a second.
Wind Effect on Sound
Wind can significantly influence how sound travels through the air. It essentially adds to or subtracts from the natural speed of sound, depending on its direction.

When wind blows from the sound source towards the observer, it effectively boosts the speed of the sound waves. This results in a higher frequency than when there is no wind, as shown in the example where the official hears a frequency of 2070 Hz instead of the original 2000 Hz.

If the wind blows from the observer towards the source, it causes a decrease in the speed of sound relative to the observer, meaning the frequency of the sound appears lower. In the exercise's situation, this results in the frequency dropping to around 1930 Hz.

It's important to consider this effect because wind can cause noticeable variations in the perceived sound, even when neither the source nor the observer is actually moving relative to the ground.

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Most popular questions from this chapter

Ultrasound, which consists ultrasour of sound waves with frequencies above the human audible range, can Artery be used to produce an image of the interior of a human body. Moreover, ultrasound can be used to measure Figure 17-47 Problem \(83 .\) the speed of the blood in the body; it does so by comparing the frequency of the ultrasound sent into the body with the frequency of the ultrasound reflected back to the body's surface by the blood. As the blood pulses, this detected frequency varies. Suppose that an ultrasound image of the arm of a patient shows an artery that is angled at \(\theta=20^{\circ}\) to the ultrasound's line of travel (Fig. 17-47). Suppose also that the frequency of the ultrasound reflected by the blood in the artery is increased by a maximum of \(5495 \mathrm{~Hz}\) from the original ultrasound frequency of \(5.000000 \mathrm{MHz} .\) (a) In Fig. \(17-47\), is the direction of the blood flow rightward of leftward? (b) The speed of sound in the human arm is \(1540 \mathrm{~m} / \mathrm{s}\). What is the maximum speed of the blood? (Hint: The Doppler effect is caused by the component of the blood's velocity along the ultrasound's direction of travel.) (c) If angle \(\theta\) were greater, would the reflected frequency be greater or less?

A tube \(1.20 \mathrm{~m}\) long is closed at one end. A stretched wire is placed near the open end. The wire is \(0.330 \mathrm{~m}\) long and has a mass of \(9.60 \mathrm{~g}\). It is fixed at both ends and oscillates in its fundamental mode. By resonance, it sets the air column in the tube into oscillation at that column's fundamental frequency. Find (a) that frequency and (b) the tension in the wire.

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?

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?

Earthquakes generate sound waves inside Earth. Unlike a gas, Earth can experience both transverse (S) and longitudinal (P) sound waves. Typically, the speed of S waves is about \(4.5 \mathrm{~km} / \mathrm{s}\), and that of \(\mathrm{P}\) waves \(8.0 \mathrm{~km} / \mathrm{s}\). A seismograph records \(\mathrm{P}\) and \(\mathrm{S}\) waves from an earthquake. The first \(\mathrm{P}\) waves arrive \(3.0 \mathrm{~min}\) before the first S waves. If the waves travel in a straight line, how far away did the earthquake occur?
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