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Chapter 16: Problem 86

The siren on an ambulance is emitting a sound whose frequency is 2450 Hz. The speed of sound is 343 m/s. (a) If the ambulance is stationary and you (the 鈥渙bserver鈥�) are sitting in a parked car, what are the wavelength and the frequency of the sound you hear? (b) Suppose that the ambulance is moving toward you at a speed of 26.8 m/s. Determine the wavelength and the frequency of the sound you hear. (c) If the ambulance is moving toward you at a speed of 26.8 m/s and you are moving toward it at a speed of 14.0 m/s, find the wavelength and frequency of the sound you hear.

Short Answer

Expert verified
(a) Frequency: 2450 Hz, Wavelength: 0.14 m; (b) Frequency: 2661 Hz, Wavelength: 0.13 m; (c) Frequency: 2782 Hz, Wavelength: 0.123 m.

Step by step solution

01

Understand the stationary scenario

When both the source and observer are stationary, the frequency of the sound heard by the observer is the same as the frequency emitted by the source. Thus, the frequency is 2450 Hz.
02

Calculate wavelength for stationary case

Use the formula for wavelength: \( \lambda = \frac{v}{f} \), where \( v \) is the speed of sound (343 m/s) and \( f \) is the frequency (2450 Hz). Substitute the given values: \( \lambda = \frac{343}{2450} \approx 0.14 \) meters.
03

Describe the scenario with moving ambulance

For a moving source, use the formula to calculate the observed frequency: \( f' = f \times \frac{v + v_o}{v - v_s} \). Since the observer is stationary, \( v_o = 0 \). Therefore, \( f' = 2450 \times \frac{343}{343 - 26.8} \).
04

Calculate observed frequency with moving ambulance

Compute \( f' \) where \( f = 2450 \text{ Hz} \), \( v = 343 \text{ m/s} \), and \( v_s = 26.8 \text{ m/s} \):\[ f' = 2450 \times \frac{343}{316.2} \approx 2661 \text{ Hz} \].
05

Calculate wavelength with moving ambulance

Use the formula \( \lambda = \frac{v}{f'} \) for the observed frequency \( f' = 2661 \text{ Hz} \) calculated previously: \( \lambda = \frac{343}{2661} \approx 0.13 \text{ meters} \).
06

Describe the scenario where both move towards each other

Now both the ambulance \( v_s = 26.8 \text{ m/s} \) and the observer \( v_o = 14.0 \text{ m/s} \) are moving towards each other, impacting both frequency and wavelength.
07

Calculate frequency for both moving

Adjust the Doppler effect formula: \( f' = f \times \frac{v + v_o}{v - v_s} \). Substitute \( f = 2450 \text{ Hz} \), \( v = 343 \text{ m/s} \), \( v_o = 14.0 \text{ m/s} \), and \( v_s = 26.8 \text{ m/s} \):\[ f' = 2450 \times \frac{343 + 14}{343 - 26.8} \approx 2782 \text{ Hz} \].
08

Calculate wavelength for both moving

Apply the observed frequency to find the wavelength: \( \lambda = \frac{v}{f'} \) with \( f' = 2782 \text{ Hz} \): \( \lambda = \frac{343}{2782} \approx 0.123 \text{ meters} \).

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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 pass a point in one second. It is measured in hertz (Hz). In our problem, the ambulance's siren emits sound at a frequency of 2450 Hz. When the source and observer are stationary, the frequency detected by the observer remains unchanged at 2450 Hz. This is because sound waves travel at a constant speed in a given medium, and there are no relative movements to alter the waves' perceivable properties.
Wavelength Calculation
The wavelength of a sound wave is the distance between consecutive points of similar phase in the wave, such as crest to crest. It can be calculated using the formula: \[ \lambda = \frac{v}{f} \] where \( \lambda \) is the wavelength, \( v \) is the speed of sound, and \( f \) is the frequency. For the stationary observer and source scenario, with the speed of sound as 343 m/s and a frequency of 2450 Hz, the wavelength is approximately 0.14 meters. Changes in circumstances, such as movement, will affect this calculation as they modify the perceived or actual frequency.
Ambulance Scenario
The scenario with an ambulance often illustrates the Doppler Effect 鈥� the change in frequency or wavelength of a wave concerning an observer moving relative to its source. When an ambulance approaches, the sound frequency appears higher due to the compressed sound waves, and conversely, it appears lower as it moves away. In our exercise, when the ambulance moves toward you at 26.8 m/s, the perceived frequency increases to 2661 Hz. This results from the ambulance closing the gap faster than the speed of sound, effectively reducing the sound wave length reaching your ears.
Moving Observer
When an observer moves toward or away from a sound source, the frequency they perceive also changes due to the Doppler Effect. Unlike the stationary scenario, here the observer's movement impacts the compressing or stretching of sound waves. For our exercise, when you, as the observer, move towards the ambulance at 14 m/s, alongside the ambulance moving towards you, the perceived frequency increases to 2782 Hz. This happens because your movement adds to the compression of sound waves, accentuating the already increased frequency due to the ambulance's motion.
Stationary Source and Observer
In events where both the source of sound and the observer are stationary, the frequency perceived by the observer matches exactly with that of the source. There are no shifts or compressions in the sound wave, because neither party is moving relative to the other. In our problem's first part, with the ambulance parked and the observer sitting still, the sound frequency observed remains at 2450 Hz, and the wavelength is calculated to be 0.14 meters. This condition allows a clear, unchanged reception of sound, ideal for understanding baseline sound properties.

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Most popular questions from this chapter

A dish of lasagna is being heated in a microwave oven. The effective area of the lasagna that is exposed to the microwaves is 2.2 \(\times 10^{-2} \mathrm{m}^{2}\) . The mass of the lasagna is 0.35 kg, and its specific heat capacity is 3200 \(\mathrm{J} / \mathrm{kg} \cdot \mathrm{C}^{\circ}\) ). The temperature rises by 72 \(\mathrm{C}^{\circ}\) in 8.0 minutes. What is the intensity of the microwaves in the oven?

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Multiple-Concept Example 11 provides a model for solving this type of problem. A wireless transmitting microphone is mounted on a small platform that can roll down an incline, directly away from a loudspeaker that is mounted at the top of the incline. The loudspeaker broadcasts a tone that has a fixed frequency of \(1.000 \times 10^{4} \mathrm{Hz}\) , and the speed of sound is 343 \(\mathrm{m} / \mathrm{s}\) . At a time of 1.5 s following the release of the platform, the microphone detects a frequency of 9939 \(\mathrm{Hz}\) . At a time of 3.5 s following the release of the platform, the microphone detects a frequency of 9857 Hz. What is the acceleration (assumed constant) of the platform?

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