Problem 37 - Pearson Physics

Question

An emergency vehicle blowing its siren is moving away from you with a speed of \(23 \mathrm{~m} / \mathrm{s}\). The sound you hear has a frequency of \(590 \mathrm{~Hz}\). What is the frequency produced by the siren?

Step-by-Step Solution

Answer

The frequency produced by the siren is approximately 629 Hz.

Step 1: Understand the Doppler Effect Formula

The problem involves the Doppler effect, which describes the change in frequency or wavelength of a wave in relation to an observer who is moving relative to the wave source. The formula to calculate the observed frequency is \( f' = \frac{f \cdot (v + v_0)}{v + v_s} \), where \( f' \) is the observed frequency (590 Hz), \( f \) is the source frequency, \( v \) is the speed of sound in air (approximately 343 m/s at room temperature), \( v_0 \) is the speed of the observer (0 m/s, since you are stationary), and \( v_s \) is the speed of the source (23 m/s, moving away from the observer).

Step 2: Rearrange the Formula to Solve for the Source Frequency

Rearrange the Doppler effect formula to solve for the source frequency \( f \):\[ f = \frac{f' \cdot (v + v_s)}{v} \]

Step 3: Substitute the Known Values

Substitute the given values into the formula: \( f' = 590 \text{ Hz} \), \( v = 343 \text{ m/s} \), \( v_s = 23 \text{ m/s} \):\[ f = \frac{590 \times (343 + 23)}{343} \]

Step 4: Calculate the Source Frequency

Perform the calculation: \[ f = \frac{590 \times 366}{343} \] \( f \approx 629 \text{ Hz} \). Therefore, the frequency produced by the siren is approximately 629 Hz.

Key Concepts

Frequency Calculation and the Doppler Effect

The Doppler Effect occurs when there is a change in frequency or wavelength of sound waves as the source moves relative to an observer. This is why the frequency you hear from a siren differs if the vehicle is moving toward or away from you. For frequency calculation, we need to adjust the observed frequency to find out the actual frequency emitted by the source.

The general formula used in these calculations is \( f' = \frac{f \cdot (v + v_0)}{v + v_s} \), where:

• \( f' \) is the observed frequency
• \( f \) is the actual frequency
• \( v \) is the speed of sound in air
• \( v_0 \) is the speed of the observer
• \( v_s \) is the speed of the source

In the exercise, the observer is stationary, so \( v_0 = 0 \), and the source is moving away, which affects the frequency you hear. To find the source frequency \( f \), we can rearrange the formula: \[ f = \frac{f' \cdot (v + v_s)}{v} \]This calculation helps us understand the true frequency being emitted, despite the relative motion creating a change in what we perceive.

Understanding Sound Waves

Sound waves are vibrations that travel through a medium, such as air, and reach our ears to be interpreted as sound. These waves consist of alternating areas of high pressure (compressions) and low pressure (rarefactions).

Some key properties of sound waves include:

• Frequency: Determines the pitch of the sound. Higher frequencies are higher-pitched.
• Wavelength: The distance between two consecutive compressions or rarefactions. Shorter wavelengths mean higher frequencies.
• Speed: The speed of sound can vary based on the medium it's traveling through, but it is approximately 343 m/s in air at room temperature.

These properties are crucial in understanding phenomena like the Doppler Effect, where the apparent change in frequency when the source of the sound moves relative to an observer is explained by the alterations in these wave characteristics.

The Role of the Wave Source in the Doppler Effect

The wave source is the origin of the sound waves, such as a siren on an emergency vehicle. It's crucial in the context of the Doppler Effect since the motion of the source directly influences the frequency received by the observer.

When the source moves:

• Towards the observer: The sound waves are compressed, leading to a higher observed frequency.
• Away from the observer: The sound waves are spread out, resulting in a lower observed frequency.

The speed and direction of the wave source must be taken into account when calculating the observed frequency. In the given example, the emergency vehicle (wave source) is moving away at 23 m/s which causes the frequency heard (590 Hz) to be less than the actual frequency produced by the siren (approximately 629 Hz). This understanding is essential for interpreting how the Doppler Effect works in real-world scenarios.