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Chapter 14: Problem 86

You are heading toward an island in your speedboat when you see a friend standing on shore. You sound the boat's horn to get your friend's attention. Is the wavelength of the sound produced by the horn greater than, less than, or equal to the wavelength of the sound heard by your friend? Explain.

Short Answer

Expert verified
The wavelength heard by your friend is less than the wavelength produced.

Step by step solution

01

Understand the Doppler Effect

The Doppler Effect describes how the frequency (and consequently the wavelength) of waves changes when the source of the wave is moving relative to the observer. When the source approaches the observer, the waves are compressed, leading to a higher frequency and a shorter wavelength. Conversely, when the source moves away, the waves are stretched, resulting in a lower frequency and a longer wavelength.
02

Identify the Situation

In this case, you are on a boat moving toward the island where your friend is standing. You sound the horn (the source of the sound) as you move closer to your friend (the observer).
03

Apply the Doppler Effect

Since you are moving toward your friend, the sound waves of your horn will be compressed as they travel to your friend. This will increase the frequency of the sound your friend hears compared to the frequency at which it was produced.
04

Compare the Wavelengths

Using the relation between frequency and wavelength, \[ v = f \lambda \] where \( v \) is the speed of sound, \( f \) is the frequency, and \( \lambda \) is the wavelength. As the frequency perceived by your friend increases, the wavelength of the sound they hear decreases.

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Key Concepts

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

Understanding Wavelength
The wavelength of a wave is the distance between successive crests, troughs, or identical points of the wave. It is often denoted by the Greek letter \( \lambda \). Wavelengths are crucial in understanding the properties of waves, including sound waves. In the context of the Doppler Effect, when the source of a sound wave is moving relative to an observer:
  • The wavelength becomes shorter if the source approaches the observer.
  • The wavelength becomes longer if the source moves away.
For the exercise scenario, as you drive your speedboat towards your friend, the sound waves of the horn are compressed. This means the wavelength of the sound reaching your friend is shorter than what was originally emitted.
The Role of Frequency
Frequency refers to how often waves pass a point in a given time period and is measured in hertz (Hz). It is a key factor in how we perceive sound, influencing the pitch of the sound. Relating this to the Doppler Effect:
  • The frequency increases when the sound source approaches the observer.
  • The frequency decreases when the source moves away from the observer.
Using the relationship \( v = f \lambda \), we can see that for a constant speed of sound \( v \), if the frequency \( f \) increases, the wavelength \( \lambda \) must decrease. That's exactly what happens in the exercise: as the source of the sound (the boat) moves closer to your friend, the frequency they perceive is higher, and the wavelength is consequently shorter.
Understanding Sound Waves in Motion
Sound waves are a type of mechanical wave that require a medium to travel through, such as air, water, or solid materials. These waves are created by the vibration of objects and propagate as longitudinal waves. In the context of this exercise, to truly grasp how the Doppler Effect and sound waves interact:
  • Remember that sound waves from the boat's horn travel through the air to your friend onshore.
  • As the boat moves toward the shore, the speed of sound remains constant, but the frequency and wavelength change due to relative motion.
The sound waves are compressed because of the boat's motion towards your friend, leading to a higher pitch sound than originally produced. This is a direct demonstration of the Doppler Effect in real-life situations, providing a clear understanding of how motion affects wave properties.

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Most popular questions from this chapter

The fundamental frequency of an organ pipe that is closed at one end and open at the other end is \(261.6 \mathrm{~Hz}\) (middle \(C\) ). The second harmonic of an organ pipe that is open at both ends has the same frequency. What are the lengths of these two pipes?

Ten violins playing simultaneously with the same intensity combine to give an intensity level of \(70 \mathrm{~dB}\). (a) What is the intensity level of each violin? (b) If the number of violins is increased to 100 , will the intensity level be more than, less than, or equal to \(80 \mathrm{~dB}\) ? Explain.

A person with perfect pitch sits on a park bench listening to the 450 -Hz horn of a moving car. (a) If the person detects a frequency of \(470 \mathrm{~Hz}\), is the car approaching or moving away? Explain. (b) How fast is the car moving?

The fundamental frequency of a pipe that is open at both ends is \(200 \mathrm{~Hz}\). If you cut the pipe in half, will the fundamental frequency of each half be greater than, less than, or equal to \(200 \mathrm{~Hz}\) ? Explain.

Dolphins of the open ocean are classified as Type II Odontocetes (toothed whales). These animals use ultrasonic "clicks" with a frequency of \(55 \mathrm{kHz}\) to navigate and find prey. (a) Suppose a dolphin sends out a series of clicks that are reflected back from the bottom of the ocean \(75 \mathrm{~m}\) below. How much time elapses before the dolphin hears the echoes of the clicks? (The speed of sound in seawater is approximately \(1530 \mathrm{~m} / \mathrm{s}\) ) (b) What is the wavelength of a \(55-\mathrm{kHz}\) sound in the ocean?
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