Problem 75 - University Physics with Modern Physics

Question

The sound source of a ship's sonar system operates at a frequency of 22.0 \(\mathrm{kHz}\) . 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 directly toward the ship at 4.95 \(\mathrm{m} / \mathrm{s}^{\prime \prime}\) The ship is at rest in the water.

Step-by-Step Solution

Answer

(a) The wavelength is approximately 0.06736 meters. (b) The frequency difference is 73.5 Hz.

Step 1: Find Wavelength of the Waves

First, let's consider the formula for the wavelength \( \lambda \) of the sound waves. The formula is given by \( \lambda = \frac{v}{f} \), where \( v \) is the speed of sound in water, and \( f \) is the frequency of the sound source. We know that \( v = 1482 \ \mathrm{m/s} \) and \( f = 22.0 \ \mathrm{kHz} = 22,000 \ \mathrm{Hz} \). Substituting these values, we find: \[ \lambda = \frac{1482}{22000} \approx 0.06736 \ \mathrm{m} \]. So, the wavelength of the emitted waves is approximately \( 0.06736 \ \mathrm{m} \).

Step 2: Determine the Frequency Change Due to Whale Movement

For part b, we'll use the Doppler effect to determine the frequency observed due to the whale's movement. The formula for the observed frequency \( f' \) is given by:\[ f' = f \left( \frac{v + v_o}{v} \right) \],where \( v \) is the speed of sound in water, \( f \) is the original frequency (22,000 Hz), and \( v_o \) is the speed of the observer (whale) moving toward the source at \( 4.95 \ \mathrm{m/s} \). Substituting the known values:\[ f' = 22000 \left( \frac{1482 + 4.95}{1482} \right) = 22000 \times \left(1.003341 \right) \approx 22073.5 \ \mathrm{Hz} \].

Step 3: Calculate Frequency Difference

Finally, to determine the difference in frequency between the directly radiated waves and the reflected waves. The reflected frequency \( f_{ref} \) will be the observed frequency from Step 2:\[ \Delta f = f_{ref} - f = 22073.5 - 22000 = 73.5 \ \mathrm{Hz} \].Thus, the frequency difference is \( 73.5 \ \mathrm{Hz} \).

Key Concepts

Wavelength Calculation

Understanding the wavelength of a sound wave is essential because it helps in describing the physical characteristics of the sound in a given medium. The wavelength (\( \lambda \)) of a wave is the distance over which the wave's shape repeats, and it can be determined using 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 source.

For sound traveling in water at a speed of 1482 \( \mathrm{m/s} \) and with a frequency of 22,000 \( \mathrm{Hz} \), the wavelength can be calculated by substituting these values into the formula:
\[\lambda = \frac{1482}{22000} \approx 0.06736 \ \mathrm{m}\]This calculation tells us that the wavelength of the sonar waves emitted by the ship is approximately 0.06736 meters.

Speed of Sound in Water

The speed of sound in water is an important factor in underwater acoustics and engineering. It influences how sound waves travel through water and affects applications such as sonar and underwater communication.Several factors influence the speed of sound in water:

• Temperature: As the temperature of water increases, the speed of sound also increases.
• Salinity: Higher salinity can increase the speed of sound.
• Pressure: As water depth increases, pressure increases, which can also increase sound speed.

For this exercise, the speed of sound in water is given as 1482 \( \mathrm{m/s} \) at a uniform temperature of 20°C. This standard value allows us to calculate related attributes, such as wavelength and frequency shift, more easily.

Frequency Shift Calculation

To understand how motion affects the frequency of sound waves, we can use the Doppler Effect. This phenomenon occurs when there is a relative motion between a source of sound and an observer, leading to a perceived change in frequency.In this problem, the key is to find the shift in frequency when a whale moves toward the sound source. The formula for calculating the observed frequency (\( f' \)) is:
\[f' = f \left( \frac{v + v_o}{v} \right)\]where:

• \( f \) is the original frequency of the sound source (22,000 \( \mathrm{Hz} \)).
• \( v \) is the speed of sound in water (1482 \( \mathrm{m/s} \)).
• \( v_o \) is the speed of the observer moving toward the source (4.95 \( \mathrm{m/s} \)).

Substituting these values, we find:
\[f' = 22000 \left( \frac{1482 + 4.95}{1482} \right) \approx 22073.5 \ \mathrm{Hz}\]The observed frequency (\( f' \)) is slightly higher than the original frequency due to the whale moving towards the ship. The frequency difference between the directly emitted and the reflected waves is:
\[\Delta f = f_{ref} - f = 22073.5 - 22000 = 73.5 \ \mathrm{Hz}\]This reveals how the motion of objects in water creates a detectable frequency shift, which is crucial for sonar applications.