Problem 72 - College Physics

Question

A person leaning over a 125 -m-deep well accidentally drops a siren emitting sound of frequency 2500 Hz. Just before this siren hits the bottom of the well, find the frequency and wavelength of the sound the person hears (a) coming directly from the siren, \((\) b) reflected off the bottom of the well. (c) What beat frequency does this person perceive?

Step-by-Step Solution

Answer

The frequencies are approximately 2418 Hz for direct sound and 2591 Hz for reflected sound. The beat frequency is around 173 Hz.

Step 1: Understand Doppler Effect

The Doppler effect refers to changes in frequency or wavelength of sound waves relative to an observer moving with respect to the sound source. It will be significant for this problem since the siren is dropping and has a velocity when heard by the observer.

Step 2: Calculate Siren's Speed at Impact

The siren falls freely under gravity until it reaches the bottom. Use the formula to find velocity: \( v = \sqrt{2gh} \), where \( g = 9.8 \, m/s^2 \) is the gravitational acceleration and \( h = 125 \, m \) is the depth. So, \( v = \sqrt{2 \times 9.8 \times 125} \).

Step 3: Apply Doppler Effect for Direct Sound

For sound coming directly from the falling siren, use the Doppler effect formula: \( f' = f \cdot \frac{v_{sound}}{v_{sound} + v_{siren}} \), where \( f = 2500 \, Hz \), \( v_{sound} \approx 343 \, m/s \) is the speed of sound in air.

Step 4: Apply Doppler Effect for Reflected Sound

The sound wave reflects off the bottom of the well and travels back, meaning the reflected sound has to be considered. Use the formula: \( f'' = f' \cdot \left(1 + \frac{v_{siren}}{v_{sound}}\right) \). Here, \( f' \) is the initial observed frequency.

Step 5: Calculate the Wavelength

For both scenarios (direct and reflected), use the formula to find the wavelength: \( \lambda = \frac{v_{sound}}{f} \) and \( \lambda' = \frac{v_{sound}}{f''} \).

Step 6: Find Beat Frequency

The beat frequency is the difference between the two observed frequencies: \( f_{beat} = |f'' - f'| \).

Step 7: Insert Values and Solve Equations

Calculate each of the above steps by inserting and solving with the numerical values obtained. This gives the frequencies and wavelengths heard by the observer, both directly and after reflection, enabling computation of the beat frequency.

Key Concepts

Sound Frequency

When discussing sound, frequency is key. It defines how high or low a sound appears to be, measured in Hertz (Hz). In the case of the siren falling into the well, the Doppler Effect impacts the frequency due to the movement. The original frequency is 2500 Hz when stationary. However, as the siren falls, it gains velocity due to gravity, causing the sound waves to compress, altering the frequency when heard by the observer. This is calculated using the Doppler Effect formula, which accounts for the relative velocities of the sound source and observer. Notably, when the siren hits the bottom, the Doppler Effect still plays a role as the observer perceives the frequency of the sound emitted directly by the siren and then the sound reflected from the well's bottom.

Wavelength Calculation

Wavelength is the distance between consecutive crests of a sound wave. Calculating it involves using the speed of sound in air (approximately 343 m/s) and the frequency of the sound. The formula is simple: \( \lambda = \frac{v_{sound}}{f} \), where \( \lambda \) is the wavelength, \( v_{sound} \) is the speed of sound, and \( f \) is the frequency. In our scenario, determining the wavelength before the siren hits involves taking the affected frequencies by Doppler shifts. The wavelength of the direct sound and the reflected sound is different due to the changes in frequency caused by these shifts. Each frequency gives a slightly varied wavelength, reinforcing the interconnectedness of frequency and wavelength.

Beat Frequency

Beat frequency occurs when two sound waves of similar frequencies interfere with each other. It results in a new sound wave that alternates between louder and softer noises, the beats. Mathematically, it is defined as the absolute difference between the two frequencies, \( f_{beat} = |f'' - f'| \). In the well problem, the observer detects two main frequencies: one directly as the siren falls, another after the sound reflects from the bottom. As these frequencies slightly differ, a beat frequency is produced. This phenomenon provides fascinating insights into sound wave interference, highlighting the Doppler shift's impact when objects are in motion and reflective environments, like wells, influence perceived sound.