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Chapter 16: Problem 3

(I) \((a)\) Calculate the wavelengths in air at \(20^{\circ} \mathrm{C}\) for sounds in the maximum range of human hearing, 20 \(\mathrm{Hz}\) to \(20,000 \mathrm{Hz}\) . (b) What is the wavelength of a 15 -MHz ultrasonic wave?

Short Answer

Expert verified
The wavelengths are 17.15 m for 20 Hz, 0.01715 m for 20,000 Hz, and 2.287 x 10^-5 m for 15 MHz.

Step by step solution

01

Understand the speed of sound at 20掳C

At 20掳C, the speed of sound in air is approximately 343 m/s. We'll use this information to calculate the wavelengths of sounds.
02

Write the formula for wavelength

The formula to calculate wavelength \( \lambda \) is given by: \[ \lambda = \frac{v}{f} \] where \( v \) is the speed of sound and \( f \) is the frequency.
03

Calculate the wavelength for 20 Hz

Using the formula, substitute \( v = 343 \) m/s and \( f = 20 \) Hz. \[ \lambda = \frac{343}{20} = 17.15 \text{ m} \] The wavelength of a 20 Hz sound is 17.15 m.
04

Calculate the wavelength for 20,000 Hz

Using the same formula, substitute \( f = 20,000 \) Hz. \[ \lambda = \frac{343}{20000} = 0.01715 \text{ m} \] The wavelength of a 20,000 Hz sound is 0.01715 m.
05

Calculate the wavelength for 15 MHz ultrasonic wave

First, convert 15 MHz to Hz: \( 15 \times 10^6 \) Hz. Then use the formula with \( v = 343 \) m/s and \( f = 15 \times 10^6 \) Hz. \[ \lambda = \frac{343}{15 \times 10^6} = 2.287 \times 10^{-5} \text{ m} \] The wavelength of a 15 MHz wave is \( 2.287 \times 10^{-5} \) m.

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Key Concepts

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

Speed of Sound
The speed of sound is how fast sound waves travel through a medium, such as air. At 20掳C, the speed of sound in air is about 343 meters per second (m/s). This speed can change depending on the medium and the temperature of the environment. For instance, sound can travel faster in waters than in air because water molecules are closer together.

To calculate speed, scientists consider factors like:
  • The air temperature: Warmer air can speed up sound.
  • The medium: Sound moves quicker in liquids and even faster in solids.
  • Humidity: More moisture in air can increase sound speed.
Understanding the speed of sound is vital when calculating wavelengths of various frequencies of sound waves, as it influences how long or short a sound wave is.
Frequency
Frequency refers to the number of times a wave oscillates or repeats per second. It is measured in Hertz (Hz). One Hertz equals one cycle per second. The range of human hearing spans from 20 Hz to 20,000 Hz, allowing us to hear everything from low rumbles to high-pitched squeaks.

Whenever you're dealing with frequency, remember:
  • High frequency waves have shorter wavelengths.
  • Low frequency waves have longer wavelengths.
  • The formula for calculating wavelength is crucial: \[\lambda = \frac{v}{f}\]
Where \( \lambda \) is the wavelength, \( v \) the speed of sound, and \( f \) the frequency. Comprehending this formula allows you to determine how far apart the peaks of waves are, influencing how we perceive sound.
Ultrasonic Waves
Ultrasonic waves are sound waves with frequencies above the range of human hearing, typically above 20,000 Hz. These waves are used in various technologies, such as medical imaging and industrial cleaning. For instance, in ultrasound technology, waves in the megahertz range (1 MHz = 1 million Hz) are used, enabling detailed internal images in medical diagnostics.

Ultrasonic waves are harnessed for:
  • Ultrasound imaging: Creating images of body organs.
  • Cleaning delicate items: Breaks dirt and contaminants from surfaces.
  • Measuring distances: As in sonar, for underwater exploration.
Calculations involving ultrasonic waves, such as finding the wavelength of a 15 MHz wave, follow the same principles as regular sound waves. By using the equation \( \lambda = \frac{v}{f} \) with the speed of sound in air and the ultrasonic frequency, we can determine how compact the wave is.

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Most popular questions from this chapter

For large concerts, loudspeakers are sometimes used to amplify a singer's sound. The human brain interprets sounds that arrive within \(50 \mathrm{~ms}\) of the original sound as if they came from the same source. Thus if the sound from a loudspeaker reaches a listener first, it would sound as if the loudspeaker is the source of the sound. Conversely, if the singer is heard first and the loudspeaker adds to the sound within \(50 \mathrm{~ms}\), the sound would seem to come from the singer, who would now seem to be singing louder. The second situation is desired. Because the signal to the loudspeaker travels at the speed of light \(\left(3 \times 10^{8} \mathrm{~m} / \mathrm{s}\right),\) which is much faster than the speed of sound, a delay is added to the signal sent to the loudspeaker. How much delay must be added if the loudspeaker is \(3.0 \mathrm{~m}\) behind the singer and we want its sound to arrive 30 ms after the singer's?

A wave on the surface of the ocean with wavelength \(44 \mathrm{~m}\) is traveling east at a speed of \(18 \mathrm{~m} / \mathrm{s}\) relative to the ocean floor. If, on this stretch of ocean surface, a powerboat is moving at \(15 \mathrm{~m} / \mathrm{s}\) (relative to the ocean floor), how often does the boat encounter a wave crest, if the boat is traveling (a) west, and (b) east?

(I) The predominant frequency of a certain fire truck's siren is 1350 Hz when at rest. What frequency do you detect if you move with a speed of 30.0 \(\mathrm{m} / \mathrm{s}(a)\) toward the fire truck, and \((b)\) away from it?

(II) If the amplitude of a sound wave is made 2.5 times greater, (a) by what factor will the intensity increase? (b) By how many dB will the sound level increase?

What is the sound level of a sound whose intensity is \(2.0 \times 10^{-6} \mathrm{~W} / \mathrm{m}^{2} ?\)
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