Problem 75 - University Physics with Modern Physics

Question

The sound source of a ship's sonar system operates at a frequency of 22.0 \(\mathrm{kH}_{2}\) . 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 cmitted by the source? (b) What is the difference in frequency between the directly radiated waves and the waves reflected from a whale traveling directly toward the slip at 4.95 \(\mathrm{m} / \mathrm{s} ?\) The ship is at rest in the water.

Step-by-Step Solution

Answer

Wavelength: 6.736 cm, Frequency difference: 147.79 Hz.

Step 1: Understanding the Problem

We need to find the wavelength of sound waves and the difference in frequency of waves reflected from a whale. The sound frequency of the sonar is 22.0 kHz, and the speed of sound in water is 1482 m/s. The whale's speed towards the ship is 4.95 m/s.

Step 2: Calculating the Wavelength

The wavelength \( \lambda \) of a wave is found using the equation \( \lambda = \frac{v}{f} \), where \( v \) is the speed of sound in water and \( f \) is the frequency.Substituting values, \( \lambda = \frac{1482 \text{ m/s}}{22000 \text{ Hz}} = 0.06736 \text{ m} \). Convert to centimeters: \( 0.06736 \text{ m} \times 100 = 6.736 \text{ cm} \).

Step 3: Understanding the Doppler Effect

The Doppler effect explains how the observed frequency changes when a source and observer are moving relative to each other. The formula for observed frequency \( f' \) when the source is stationary and the observer is moving toward the source is \( f' = f \left( \frac{v + v_o}{v} \right) \), where \( v \) is the speed of sound, and \( v_o \) is the velocity of the observer.

Step 4: Calculating the Frequency Observed by the Whale

Using the equation from the Doppler effect, the frequency observed by the whale is:\[ f' = 22000 \left( \frac{1482 + 4.95}{1482} \right) \approx 22073.56 \text{ Hz} \].

Step 5: Calculating the Frequency of Reflected Waves

Now, consider the reflected waves being observed by the ship. These are the waves the whale initially heard, returning back to the ship. Using the Doppler effect formula again, for the ship at rest and the whale's speed towards the ship:\[ f'' = 22073.56 \left( \frac{1482 + 4.95}{1482} \right) = 22073.56 \left( \frac{1486.95}{1482} \right) \approx 22147.79 \text{ Hz} \].

Step 6: Calculating Frequency Difference

The difference in frequency is given by \( f'' - f = 22147.79 - 22000 = 147.79 \text{ Hz} \).

Step 7: Conclusion

The wavelength of the sound waves is 6.736 cm, and the frequency difference for the waves reflected from the whale is approximately 147.79 Hz.

Key Concepts

Wavelength Calculation

When dealing with sound waves, one fundamental concept is calculating the wavelength. The wavelength \( \lambda \) describes the physical length of one cycle of the wave. To find it, we utilize the formula \( \lambda = \frac{v}{f} \) where:

• \( v \) is the speed of sound in the medium (in this case, water).
• \( f \) is the frequency of the sound waves.

For example, in our exercise, the speed of sound is given as 1482 m/s, and the frequency is 22.0 kHz or 22000 Hz.
Substituting these values into the formula yields \( \lambda = \frac{1482}{22000} = 0.06736 \text{ meters} \).
Converting meters to centimeters, we get \( 0.06736 \times 100 = 6.736 \text{ cm} \).

Understanding this equation allows you to determine the physical size of the wave. This is crucial in many sound applications, including sonar technology.
Knowing the wavelength helps sonar operators understand how sound waves propagate through water and interact with objects they encounter.

Doppler Effect

The Doppler Effect is a phenomenon where the observed frequency of a wave changes if the source and observer are moving relative to each other.
Essential concepts include:

• If the observer is moving towards the source, the observed frequency increases.
• If the observer is moving away from the source, the observed frequency decreases.

In the sonar exercise, the ship is stationary, but the whale is moving towards it at 4.95 m/s.
To determine the frequency observed by the whale, we use the Doppler Effect formula:\[ f' = f \left( \frac{v + v_o}{v} \right) \]where:

• \( v \) is the speed of sound in water.
• \( v_o \) is the whale's speed.

Plugging in the numbers:\[ f' = 22000 \left( \frac{1482 + 4.95}{1482} \right) \approx 22073.56 \text{ Hz} \]The frequency shifts because the whale is moving towards the source, experiencing waves more frequently than if it were stationary.
This principle not only applies in sonar systems but also in everyday experiences, such as hearing the pitch change of a passing ambulance siren.

Frequency Difference

In scenarios involving the Doppler Effect, like in our sonar problem, calculating the frequency difference is crucial.
Once the whale reflects the sonar waves back towards the stationary ship, it acts as a moving source for the reflected waves.
Using the Doppler Effect formula again, we calculate the frequency of these reflected waves:\[ f'' = f' \left( \frac{v + v_o}{v} \right) \]where:

• \( f' \) is the modified frequency perceived by the moving whale.
• \( f \) and \( v \) remain constants as before.

The calculation results in a reflected frequency of approximately 22147.79 Hz.
Hence, the frequency difference observed by the ship is:\[ f'' - f = 22147.79 - 22000 = 147.79 \text{ Hz} \]Understanding the frequency difference is fundamental in analyzing how motion affects sound perception.
It is especially important in applications involving motion-based frequency shifts, such as radar and sonar systems.
It helps in determining the rate of movement and direction of the object reflecting the waves.