Problem 21
Question
A sound wave is traveling in seawater, where the adiabatic bulk modulus and density are \(2.31 \mathrm{X} 10^{9} \mathrm{~Pa}\) and \(1025 \mathrm{~kg} / \mathrm{m}^{3}\), respectively. The wavelength of the sound is \(3.35 \mathrm{~m}\). A tuning fork is struck underwater and vibrates at \(440.0 \mathrm{~Hz} .\) What would be the beat frequency heard by an underwater swimmer?
Step-by-Step Solution
Verified Answer
The beat frequency is approximately 8.28 Hz.
1Step 1: Determine the speed of sound in seawater
The speed of sound in a medium can be calculated using the formula \( v = \sqrt{\frac{K}{\rho}} \), where \( K \) is the adiabatic bulk modulus and \( \rho \) is the density of the medium. Plugging in the values:\[ v = \sqrt{\frac{2.31 \times 10^9 \text{ Pa}}{1025 \text{ kg/m}^3}} \]Calculate the result to find the speed of sound in seawater.
2Step 2: Calculate speed of sound
Computing the expression from Step 1:\[ v = \sqrt{\frac{2.31 \times 10^9}{1025}} = \sqrt{2253658.537} = 1501.22 \text{ m/s} \]Therefore, the speed of sound in seawater is approximately 1501 m/s.
3Step 3: Determine wavelength of tuning fork
Given the frequency of the tuning fork is \( f = 440.0 \text{ Hz} \), you can use the speed of sound to calculate the wavelength using the formula \( \lambda = \frac{v}{f} \):\[ \lambda = \frac{1501.22 \text{ m/s}}{440.0 \text{ Hz}} \]Calculate the value to find the wavelength of the tuning fork.
4Step 4: Calculate tuning fork wavelength
Substitute the values from Step 3:\[ \lambda = \frac{1501.22}{440.0} = 3.41 \text{ m} \]The wavelength of the tuning fork is approximately 3.41 m.
5Step 5: Calculate beat frequency
The beat frequency is the absolute difference between two frequencies. Since the speed of sound and wavelength are known, the frequency of the sound wave can be calculated using \( f = \frac{v}{\lambda_\text{wave}} \), where \( \lambda_\text{wave} = 3.35 \text{ m} \):\[ f = \frac{1501.22}{3.35} \approx 448.28 \text{ Hz} \]The beat frequency is the absolute difference between the two frequencies:\[ f_\text{beat} = |448.28 \text{ Hz} - 440.0 \text{ Hz}| = 8.28 \text{ Hz} \]
6Step 6: Final Step: Conclusion
The calculated beat frequency heard by an underwater swimmer is approximately 8.28 Hz.
Key Concepts
Beat FrequencySpeed of SoundWavelengthTuning Fork VibrationUnderwater Acoustics
Beat Frequency
When two sound waves of slightly different frequencies travel through the same medium, their interference produces a phenomenon known as beat frequency. This interference results in fluctuations in the loudness or amplitude of the sound that we hear.
- **Beat Frequency Formula**: The beat frequency (\( f_{\text{beat}} \)) is the absolute difference between the two frequencies, given by the formula:\[ f_{\text{beat}} = |f_1 - f_2| \]- **Example in Exercise**: In the context of our exercise, we have two frequencies: one produced by the tuning fork (440 Hz) and one calculated for the sound wave with wavelength 3.35 m (approximately 448.28 Hz). Thus, the beat frequency is:\[ f_{\text{beat}} = |448.28 \text{ Hz} - 440 \text{ Hz}| = 8.28 \text{ Hz} \]- **Perception**: This means an underwater swimmer would hear the frequency changes as beats at roughly 8.28 Hz.
- **Beat Frequency Formula**: The beat frequency (\( f_{\text{beat}} \)) is the absolute difference between the two frequencies, given by the formula:\[ f_{\text{beat}} = |f_1 - f_2| \]- **Example in Exercise**: In the context of our exercise, we have two frequencies: one produced by the tuning fork (440 Hz) and one calculated for the sound wave with wavelength 3.35 m (approximately 448.28 Hz). Thus, the beat frequency is:\[ f_{\text{beat}} = |448.28 \text{ Hz} - 440 \text{ Hz}| = 8.28 \text{ Hz} \]- **Perception**: This means an underwater swimmer would hear the frequency changes as beats at roughly 8.28 Hz.
Speed of Sound
The speed of sound varies depending on the medium through which the sound wave travels. We often find that sound travels faster in liquids and solids than in gases due to the particles being closer together.
- **Key Factors**: The speed of sound in a medium is determined by its adiabatic bulk modulus (a measure of the medium's resistance to compression) and its density.- **Formula**: Speed of sound (\( v \)) can be calculated as:\[ v = \sqrt{\frac{K}{\rho}} \]where \( K \) is the bulk modulus and \( \rho \) is the density.- **Exercise Context**: For seawater, given \( K = 2.31 \times 10^9 \text{ Pa} \) and \( \rho = 1025 \text{ kg/m}^3 \), the speed of sound is:\[ v = \sqrt{\frac{2.31 \times 10^9}{1025}} \approx 1501 \text{ m/s} \]This value reflects how efficiently sound can travel in water, and this efficiency is a key factor in underwater acoustics.
- **Key Factors**: The speed of sound in a medium is determined by its adiabatic bulk modulus (a measure of the medium's resistance to compression) and its density.- **Formula**: Speed of sound (\( v \)) can be calculated as:\[ v = \sqrt{\frac{K}{\rho}} \]where \( K \) is the bulk modulus and \( \rho \) is the density.- **Exercise Context**: For seawater, given \( K = 2.31 \times 10^9 \text{ Pa} \) and \( \rho = 1025 \text{ kg/m}^3 \), the speed of sound is:\[ v = \sqrt{\frac{2.31 \times 10^9}{1025}} \approx 1501 \text{ m/s} \]This value reflects how efficiently sound can travel in water, and this efficiency is a key factor in underwater acoustics.
Wavelength
The wavelength of a sound wave is the distance over which the wave's shape repeats. It is directly related to the speed of sound and the frequency of the wave.
- **Relation**: Wavelength (\( \lambda \)) can be calculated using the formula:\[ \lambda = \frac{v}{f} \]where \( v \) is the speed of sound and \( f \) is the frequency.- **Exercise Calculation**: In the problem, the tuning fork vibrates with a frequency of \( 440 \text{ Hz} \). Using the speed of sound calculated as \( 1501 \text{ m/s} \), the wavelength is:\[ \lambda = \frac{1501.22}{440.0} \approx 3.41 \text{ m} \]- **Significance**: Understanding wavelength is crucial because it defines how sound propagates in different environments. This understanding helps in designing audio equipment and in fields like underwater acoustics.
- **Relation**: Wavelength (\( \lambda \)) can be calculated using the formula:\[ \lambda = \frac{v}{f} \]where \( v \) is the speed of sound and \( f \) is the frequency.- **Exercise Calculation**: In the problem, the tuning fork vibrates with a frequency of \( 440 \text{ Hz} \). Using the speed of sound calculated as \( 1501 \text{ m/s} \), the wavelength is:\[ \lambda = \frac{1501.22}{440.0} \approx 3.41 \text{ m} \]- **Significance**: Understanding wavelength is crucial because it defines how sound propagates in different environments. This understanding helps in designing audio equipment and in fields like underwater acoustics.
Tuning Fork Vibration
A tuning fork is a practical tool used to produce a sound of a specific pitch. When struck, it vibrates producing sound waves in the surrounding medium.
- **Sound Production**: The two prongs or "tines" of the fork oscillate back and forth. This movement creates periodic compressions and rarefactions in the air or water, generating sound waves.- **Calculating Pitch**: The frequency of the tuning fork determines the pitch of the sound it produces. In our case, the tuning fork vibrates at \( 440 \text{ Hz} \), corresponding to the musical note A above middle C.- **Role in Exercise**: In the underwater exercise, this specific frequency interacts with waves of other frequencies, leading to the observed beat frequency heard underwater.
- **Sound Production**: The two prongs or "tines" of the fork oscillate back and forth. This movement creates periodic compressions and rarefactions in the air or water, generating sound waves.- **Calculating Pitch**: The frequency of the tuning fork determines the pitch of the sound it produces. In our case, the tuning fork vibrates at \( 440 \text{ Hz} \), corresponding to the musical note A above middle C.- **Role in Exercise**: In the underwater exercise, this specific frequency interacts with waves of other frequencies, leading to the observed beat frequency heard underwater.
Underwater Acoustics
Underwater acoustics is the study of how sound behaves in water. This field is crucial for applications ranging from submarine communication to marine biology.
- **Sound Behavior**: Sound travels differently in water compared to air. Water is denser, allowing sound to travel faster and over longer distances. However, factors like temperature, salinity, and pressure can affect this speed. - **Exercise Relevance**: The example provided uses a tuning fork underwater, allowing us to explore how frequencies interact and create beat frequencies under these conditions. - **Applications**: Understanding underwater acoustics aids in a variety of technological advancements, including sonar development and environmental monitoring. Such knowledge is essential for efficient communication and navigation in underwater environments.
- **Sound Behavior**: Sound travels differently in water compared to air. Water is denser, allowing sound to travel faster and over longer distances. However, factors like temperature, salinity, and pressure can affect this speed. - **Exercise Relevance**: The example provided uses a tuning fork underwater, allowing us to explore how frequencies interact and create beat frequencies under these conditions. - **Applications**: Understanding underwater acoustics aids in a variety of technological advancements, including sonar development and environmental monitoring. Such knowledge is essential for efficient communication and navigation in underwater environments.
Other exercises in this chapter
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