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Chapter 12: Problem 65

The sound source of a ship's sonar system operates at a frequency of \(22.0 \mathrm{kHz}\). The speed of sound in water (assumed to be at a uniform \(20^{\circ} \mathrm{C}\) ) is \(1482 \mathrm{~m} / \mathrm{s}\). (a) What is the wavelength of the waves emitted by the source? (b) What is the difference in frequency between the directly radiated waves and the waves reflected from a whale traveling straight toward the ship at \(4.95 \mathrm{~m} / \mathrm{s} ?\) The ship is at rest in the water.

Short Answer

Expert verified
a) Wavelength is 0.06736 m. b) Frequency difference is 76.31 Hz.

Step by step solution

01

Calculating the Wavelength

To find the wavelength \( \lambda \) of the sound waves emitted by the source, we use the formula \( \lambda = \frac{v}{f} \), where \( v \) is the speed of sound in water and \( f \) is the frequency. Given, \( v = 1482 \text{ m/s} \) and \( f = 22.0 \text{ kHz} = 22000 \text{ Hz} \). Plugging in these values, we get: \[ \lambda = \frac{1482 \text{ m/s}}{22000 \text{ Hz}} \approx 0.06736 \text{ m} \]
02

Finding the Frequency of Reflected Waves

When sound waves are reflected off a moving object (the whale), the frequency changes due to the Doppler effect. The formula for the observed frequency \( f' \) when the observer is moving towards a stationary source is: \[ f' = f \left( \frac{v + v_o}{v} \right) \] where \( v_o \) is the speed of the observer (whale). Given \( f = 22000 \text{ Hz} \), \( v = 1482 \text{ m/s} \), and \( v_o = 4.95 \text{ m/s} \), the frequency of the reflected waves is: \[ f'' = f' = 22000 \left( \frac{1482 + 4.95}{1482} \right) \approx 22076.31 \text{ Hz} \]
03

Calculating the Frequency Difference

The frequency difference \( \Delta f \) is the difference between the reflected frequency and the original frequency of the waves: \[ \Delta f = f'' - f = 22076.31 \text{ Hz} - 22000 \text{ Hz} \approx 76.31 \text{ Hz} \]
04

Conclusion

The solutions to the problems are: (a) The wavelength of the waves emitted by the source is approximately \( 0.06736 \text{ m} \). (b) The difference in frequency between the directly radiated waves and those reflected from the whale is about \( 76.31 \text{ Hz} \).

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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 a type of mechanical wave that travels through a medium such as air, water, or another solid, liquid, or gas. These waves are created by vibrations, which result in areas of compression and rarefaction in the medium. One of the key characteristics of sound waves is their ability to travel at different speeds depending on the medium. For instance, in water, sound waves travel at approximately 1482 meters per second at 20掳C. This speed is significantly faster than the speed of sound in air, which is around 343 meters per second under similar conditions.

Sound waves have two main properties: frequency and wavelength. Frequency, measured in Hertz (Hz), describes how many wave cycles pass a point per second, while wavelength is the distance between consecutive points of the same phase in the cycle, such as compression to compression. These properties are crucial for understanding phenomena such as the Doppler Effect, which is the change in frequency observed when a sound source moves relative to an observer.
Frequency Calculation
Frequency calculation is vital in understanding how sound waves behave, especially in applications like sonar systems. The Doppler Effect plays a crucial role here. When a sound source or observer is moving, the frequency of the sound changes. This change explains how animals like whales are detected by sonar on ships.

Frequency is given by the number of oscillations per second and is calculated using the formula:
  • Without any movement, the frequency (\( f \)) remains as the source frequency.
  • With an observer in motion, the observed frequency (\( f' \)) changes according to the equation:\[ f' = f \left( \frac{v + v_o}{v} \right) \]
Here, \( v \) is the speed of sound in the medium and \( v_o \) is the speed of the observer. For a whale moving towards a stationary ship, the formula helps determine how much higher the frequency appears due to the approaching whale. This increase in frequency allows for accurate measurement and detection through sonar.
Wavelength Determination
Wavelength determination is an essential aspect when dealing with sound waves, particularly for applications such as sonar. Wavelength is inversely proportional to frequency, meaning as the frequency increases, the wavelength decreases. It is 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 a sonar system, knowing the wavelength helps in understanding how sound waves interact with their environment.

In the given exercise, the wavelength of the sound waves is determined to be approximately 0.06736 meters. This is found by dividing the speed of sound in water (1482 m/s) by the transmitter frequency (22000 Hz). Understanding this relationship is key to interpreting how sound waves travel and are used in technology such as sonar systems. This knowledge can also help in designing equipment to cater to specific scenarios based on their wavelength capacities.

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Most popular questions from this chapter

While sitting in your car by the side of a country road, you see your friend, who happens to have an identical car with an identical horn, approaching you. You blow your horn, which has a frequency of \(260 \mathrm{~Hz}\); your friend begins to blow his horn as well, and you hear a beat frequency of \(6.0 \mathrm{~Hz}\). How fast is your friend approaching you?

The equation describing a transverse wave on a string is $$y(x, t)=(1.50 \mathrm{~mm}) \sin \left[\left(157 \mathrm{~s}^{-1}\right) t-\left(41.9 \mathrm{~m}^{-1}\right) x\right]$$ Find (a) the wavelength, frequency, and amplitude of this wave, (b) the speed and direction of motion of the wave, and (c) the transverse displacement of a point on the string when \(t=0.100 \mathrm{~s}\) and at a position \(x=0.135 \mathrm{~m}\)

If an earthquake wave having a wavelength of \(13 \mathrm{~km}\) causes the ground to vibrate 10.0 times each minute, what is the speed of the wave?

The end of a stopped pipe is to be cut off so that the pipe will be open. If the stopped pipe has a total length \(L\). what fraction of \(L\) should be cut off so that the fundamental mode of the resulting open pipe has the same frequency as the fifth harmonic \((n=5)\) of the original stopped pipe?

A police siren of frequency \(f_{\text {siren }}\) is attached to a vibrating platform. The platform and siren oscillate up and down in simple harmonic motion with amplitude \(A_{\mathrm{p}}\) and frequency \(f_{\mathrm{p}}\). (a) Find the maximum and minimum sound frequencies that you would hear at a position directly above the siren. (b) At what point in the motion of the platform is the maximum frequency heard? The minimum frequency? Explain.
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