Question

A siren emitting a sound of frequency \(1000 \mathrm{~Hz}\) moves away from you toward a cliff at a speed of \(10.0 \mathrm{~m} / \mathrm{s} .(a)\) What is the frequency of the sound you hear coming directly from the siren? ( \(b\) ) What is the frequency of the sound you hear reflected off the cliff? ( \(c\) ) Find the beat frequency. Could you hear the beats? Take the speed of sound in air as \(330 \mathrm{~m} / \mathrm{s}\)

Short answer

The frequency of the sound heard directly from the siren is around 970 Hz while the frequency of the sound reflected off the cliff is approximately 1007 Hz. The beat frequency is roughly 37 Hz, which is within the audible range for humans, so yes, you should be able to hear the beats.

Step-by-step solution

Calculate the frequency heard directly from the siren

Use the formula for the Doppler effect when the source is moving away: \(f' = f \frac{v}{v + v_s}\), where \(f' = 1000 Hz\) is the observed frequency, \(f = 1000 Hz\) is the source frequency, \(v = 330 m/s\) is the speed of sound, and \(v_s = 10.0 m/s\) is the speed of the source.

Calculate the frequency of the sound reflected off the cliff

Since the frequency heard directly from the siren has been calculated, this becomes the 'source' frequency for the sound reflected off the cliff. Using the same formula as in step 1, but this time considering the sound wave moving towards you, the equation changes to \(f' = f \frac{v}{v - v_s}\).

Calculate the beat frequency

The beat frequency is simply the absolute difference between two frequencies. This can be calculated by taking the absolute difference of the two frequencies obtained in the prior steps. The audible range for humans is typically 20 Hz to 20,000 Hz. Therefore, if the beat frequency falls within this range, then yes, you should be able to hear the beats.

Key concepts

Frequency Calculation

Understanding frequency calculation with the Doppler Effect is crucial. When an observer perceives a sound, the frequency can change if either the sound source or observer is moving. This phenomenon is explained by the Doppler Effect.

For the scenario in the exercise, where the siren moves away, the frequency calculation becomes important to understand the change in frequency heard. The formula used is: \[ f' = f \frac{v}{v + v_s} \]

• \(f'\) is the observed frequency.
• \(f\) is the emitted frequency, 1000 Hz in this case.
• \(v\) is the velocity of sound in air, given as 330 m/s.
• \(v_s\) is the speed of the source, given as 10 m/s.

This formula demonstrates how the observed frequency decreases when the source moves away from the observer. Plugging in the values into the formula gives an adjusted frequency due to the source moving away, which is lower than the emitted frequency due to the source's movement.

Beat Frequency

Beat frequency is a fascinating concept that occurs when two sound waves with slightly different frequencies interfere with each other. This results in a fluctuating sound - called beats - that oscillates at a frequency equal to the difference between the two frequencies.

Once you have calculated the direct frequency from the siren and the frequency reflected off the cliff, you find the beat frequency using:\[ f_{beat} = |f'_1 - f'_2|\]Here:

• \(f'_1\) is the frequency heard directly from the siren.
• \(f'_2\) is the frequency of the reflected sound.

This absolute value represents how often the peaks and troughs of these sound waves align over time. If this beat frequency is within human audible range (20 Hz - 20,000 Hz), the beats can be heard. The calculated beat frequency in this problem will inform you if it is detectable by human ears.

Sound Wave Reflection

Sound wave reflection is when sound bounces off surfaces, similar to how we see light reflecting off a mirror. When a sound wave, like that from a siren, hits a barrier such as a cliff, it reflects back, carrying information about the original sound frequency.

In the given exercise, once the siren's emitted sound reaches the cliff, it acts as a new source. To find the frequency of the reflected sound, you recalculate using the Doppler Effect to account for the direction of the reflected wave moving towards the observer:\[ f' = f \frac{v}{v - v_s} \]
Using this adjusted formula reflects the situation appropriately. It considers now the frequency change as the wave travels back to you, making the frequency closer to the original because the source is effectively moving towards you. By understanding this reflection, you understand why the observed frequency changes, highlighting the intricate effects of motion on sound.