Problem 44 - Calculus Early Transcendentals

Question

If a sound with frequency \(f_{s}\) is produced by a source traveling along a line with speed \(v_{s}\) and an observer is traveling with speed \(v_{o}\) along the same line from the opposite direction toward the source, then the frequency of the sound heard by the observer is $$f_{o}=\left(\frac{c+v_{o}}{c-v_{s}}\right) f_{s}$$ where \(c\) is the speed of sound, about 332 \(\mathrm{m} / \mathrm{s}\) . (This is the Doppler effect.) Suppose that, at a particular moment, you are in a train traveling at 34 \(\mathrm{m} / \mathrm{s}\) and accelerating at 1.2 \(\mathrm{m} / \mathrm{s}^{2}\) A train is approaching you from the opposite direction on the other track at 40 \(\mathrm{m} / \mathrm{s}\) , accelerating at 1.4 \(\mathrm{m} / \mathrm{s}^{2}\) , and sounds its whistle, which has a frequency of 460 \(\mathrm{Hz}\) . At that instant, what is the perceived frequency that you hear and how fast is it changing?

Step-by-Step Solution

Answer

The perceived frequency is approximately 578 Hz, and it increases at 5.2 Hz/s.

Step 1: Identify Given Values

We are given the following values: - Speed of sound, \( c = 332 \, \text{m/s} \)- Observer's speed, \( v_{o} = 34 \, \text{m/s} \)- Source's speed, \( v_{s} = 40 \, \text{m/s} \)- Frequency of source, \( f_{s} = 460 \, \text{Hz} \)- Observer's acceleration, \( a_{o} = 1.2 \, \text{m/s}^2 \)- Source's acceleration, \( a_{s} = 1.4 \, \text{m/s}^2 \).

Step 2: Calculate the Perceived Frequency

Use the Doppler effect formula: \[ f_{o} = \left(\frac{c+v_{o}}{c-v_{s}}\right) f_{s} \]Substitute the given values:\[ f_{o} = \left(\frac{332 + 34}{332 - 40}\right) \times 460 \]Calculate the result:\[ f_{o} = \left(\frac{366}{292}\right) \times 460 \approx 577.8 \, \text{Hz} \].

Step 3: Calculate the Rate of Change of Frequency

To find out how fast the frequency is changing, we need the time rate of change of \( f_{o} \). Differentiating the Doppler effect formula with respect to time involves both velocities and accelerations:\[ \frac{df_{o}}{dt} = \frac{d}{dt} \left(\frac{c + v_{o}}{c - v_{s}}\right) f_{s} \]The derivative involves:\[ \frac{d}{dt} \left(\frac{c+v_{o}}{c-v_{s}}\right) = \left(\frac{a_{o} (c-v_{s}) + a_{s} (c+v_{o})}{(c-v_{s})^2}\right) \]Substitute \( a_{o} \) and \( a_{s} \) to get:\[ \frac{d}{dt} \left(\frac{366}{292}\right) \approx \frac{1.2 \times 292 + 1.4 \times 366}{292^2} \approx 0.0113 \, \text{s}^{-1} \]Thus,\[ \frac{df_{o}}{dt} = 460 \times 0.0113 \approx 5.2 \, \text{Hz/s} \].

Key Concepts

Frequency

Frequency, in the context of sound, refers to the number of vibrations or cycles a sound wave undergoes per second. It is measured in hertz (Hz), where one hertz is equal to one cycle per second.
When considering the Doppler Effect, the perceived frequency can change depending on the relative motion between the sound source and the observer.
This is why an approaching sound, like a train whistle, might sound higher in pitch as it nears you, and lower as it moves away.

• When both the source and the observer are moving, as in the exercise, the change in frequency is influenced by both velocities.
• The Doppler Effect formula shows how the source frequency is modified based on observer and source speeds, leading to a perceived frequency shift.
• Changes in frequency also depend on the acceleration of both the source and the observer, which impacts their speeds over time.

Grasping this concept helps us understand phenomena like ambulance sirens changing pitch as they move past or how astronomical observations can tell us about distant galaxies.

Sound Waves

Sound waves are mechanical waves that travel through a medium such as air, water, or solids. They are produced by vibrating objects and propagate by causing particles in the medium to oscillate.
These waves are characterized by their wavelength, frequency, and amplitude.
Sound travels at a speed of approximately 332 m/s in air, but this can vary depending on temperature and pressure.

• The speed of sound forms a crucial part of the Doppler Effect equation. It serves as a baseline for measuring how the relative speeds of the source and observer alter the perceived frequency.
• Because sound waves need a medium, they cannot travel through a vacuum like space.
• The understanding of sound as waves also explains phenomena such as echoes and sound refraction, which occur when sound waves bounce back from surfaces or bend around obstacles.

Understanding these properties helps explain why sound changes when sources move relative to us, such as the changing pitch of a passing siren.

Velocity

Velocity is the speed of an object in a specified direction. In the context of the Doppler Effect, both the velocity of the sound source and the observer affect the frequency of the sound as heard by the observer.
The velocity of the observer or source can be positive or negative depending on whether the movement is towards or away from each other.

• In the Doppler Effect formula, both observer velocity (\(v_o\)) and source velocity (\(v_s\)) influence the adjustment of the original frequency (\(f_s\)).
• To account for changing velocities, a calculation involving the accelerations of the source and observer is required, essentially influencing how the perceived frequency shifts over time.
• This encompasses both the fixed and variable components of motion to offer a comprehensive picture of sound perception changes.

By understanding velocity, we're better equipped to predict and calculate how sounds will change in frequency as sources and observers move relative to each other, resulting in experiences described by the Doppler Effect.