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Chapter 17: Problem 66

Two trains are traveling toward each other at \(30.5 \mathrm{~m} / \mathrm{s}\) relative to the ground. One train is blowing a whistle at \(500 \mathrm{~Hz}\). (a) What frequency is heard on the other train in still air? (b) What frequency is heard on the other train if the wind is blowing at \(30.5 \mathrm{~m} / \mathrm{s}\) toward the whistle and away from the listener? (c) What frequency is heard if the wind direction is reversed?

Short Answer

Expert verified
(a) Frequency with no wind: Calculate using Doppler formula. (b) Frequency with wind toward whistle: Use modified speed of sound. (c) Frequency with reversed wind: Similar calculation but with adjusted wind direction.

Step by step solution

01

Understand the Doppler Effect

In these problems, the Doppler Effect plays a crucial role. It occurs when there is a relative motion between the source of sound and the observer, affecting the frequency heard by the observer. The observed frequency changes depending on whether the source or observer is moving toward or away from each other.
02

Calculate frequency with no wind

To find the frequency heard by the listener on the other train in still air, use the formula for the Doppler Effect:\[\text{f'} = f \frac{v + v_o}{v - v_s}\]where:- \(f = 500 \) Hz is the source frequency.- \(v = 343 \text{ m/s}\) is the speed of sound in air.- \(v_o = 30.5 \text{ m/s}\) is the speed of the observer towards the source.- \(v_s = 30.5 \text{ m/s}\) is the speed of the source towards the observer.Substitute the values:\[\text{f'} = 500 \frac{343 + 30.5}{343 - 30.5}\]Calculate \( ext{f'}\) to find the frequency heard.
03

Calculate frequency with wind blowing toward whistle

If the wind is blowing toward the whistle (source) and away from the observer at \(30.5 \text{ m/s}\), it effectively reduces the relative speed of the sound towards the observer. The speed of sound relative to the still air increases by the wind speed:\[v = 343 + 30.5 = 373.5 \text{ m/s}\]Using the modified Doppler Effect formula:\[\text{f'} = 500 \frac{373.5 + 30.5}{373.5 - 30.5}\]

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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 other mediums, allowing us to hear sounds. These waves are characterized by their frequency, which determines the pitch of the sound we perceive. A high frequency corresponds to a high-pitched sound, like a whistle, whereas a low frequency results in a deeper sound, like a drum.

The speed of sound in air typically averages at 343 m/s, although this can vary depending on atmospheric conditions such as temperature and wind speed. Understanding sound waves in motion is crucial when dealing with scenarios like the Doppler Effect, where both the source and the observer might be moving.
  • Frequency: Measured in Hertz (Hz), it refers to the number of sound wave cycles per second.
  • Amplitude: The height of the sound wave, affecting the loudness.
  • Wavelength: The distance between two peaks of the wave, influencing the frequency.
Relative Motion
Relative motion occurs when an observer and a source of sound move towards or away from each other. This concept is a cornerstone of the Doppler Effect. Understanding how motion affects perceived sound frequency can be key in problem-solving. In scenarios involving relative motion between a sound source and an observer, it鈥檚 essential to consider their velocities.

When both the source and the observer move towards each other, the frequency appears higher because the sound waves are compressed. Conversely, if they are moving apart, the frequency appears lower due to the waves stretching out. This principle is what drives the changes in frequency in a Doppler Effect problem.
  • Moving toward each other increases the perceived frequency.
  • Moving away from each other decreases the perceived frequency.
  • Wind can also affect relative motion by changing the effective speed of sound.
Frequency Calculation
Frequency calculation using the Doppler Effect allows us to determine how sound frequency will be perceived under different conditions of motion. The fundamental formula for the Doppler Effect is:\[\text{f'} = f \frac{v + v_o}{v - v_s}\]Here, \
  • \(\text{f'}\): The observed frequency.
  • \(f\): The actual frequency of the sound source, which in this exercise is 500 Hz.
  • \(v\): Speed of sound in still air, generally 343 m/s.
  • \(v_o\): Speed of the observer moving towards the source.
  • \(v_s\): Speed of the source moving towards the observer.
To complete these calculations, substitute the known values into the formula and solve for \(\text{f'}\). This process must be adapted if environmental factors like wind are considered since they alter the effective speed of sound.

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Most popular questions from this chapter

A sound wave of frequency \(300 \mathrm{~Hz}\) has an intensity of \(1.00 \mu \mathrm{W} / \mathrm{m}^{2} .\) What is the amplitude of the air oscillations caused by this wave?

Hot chocolate effect. Tap a metal spoon inside a mug of water and note the frequency \(f_{i}\) you hear. Then add a spoonful of powder (say, chocolate mix or instant coffee) and tap again as you stir the powder. The frequency you hear has a lower value \(f_{s}\) because the tiny air bubbles released by the powder change the water's bulk modulus. As the bubbles reach the water surface and disappear, the frequency gradually shifts back to its initial value. During the effect, the bubbles don't appreciably change the water's density or volume or the sound's wavelength. Rather, they change the value of \(d V / d p-\) that is, the differential change in volume due to the differential change in the pressure caused by the sound wave in the water. If \(f_{s} / f_{i}=0.333,\) what is the ratio \((d V / d p)_{s} /(d V / d p)_{i} ?\)

Diagnostic ultrasound of frequency \(4.50 \mathrm{MHz}\) is used to examine tumors in soft tissue. (a) What is the wavelength in air of such a sound wave? (b) If the speed of sound in tissue is \(1500 \mathrm{~m} / \mathrm{s}\), what is the wavelength of this wave in tissue?

You have five tuning forks that oscillate at close but different resonant frequencies. What are the (a) maximum and (b) minimum number of different beat frequencies you can produce by sounding the forks two at a time, depending on how the resonant frequencies differ?

An ambulance with a siren emitting a whine at \(1600 \mathrm{~Hz}\) over takes and passes a cyclist pedaling a bike at \(2.44 \mathrm{~m} / \mathrm{s}\). After being passed, the cyclist hears a frequency of \(1590 \mathrm{~Hz}\). How fast is the ambulance moving?
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