Question

A stationary source emits sound of frequency \(f_{0}=492 \mathrm{~Hz} .\) The sound is reflected by a large car approaching the source with a speed of \(2 \mathrm{~ms}^{-1}\). The reflected signal is received by the source and superposed with the original. What will be the beat frequency of the resulting signal in \(\mathrm{Hz}\) ? (Given that the speed of sound in air is \(330 \mathrm{~ms}^{-1}\) and the car reflects the sound at the frequency \(i\) has received.

Short answer

The beat frequency is the absolute difference between the original frequency (492 Hz) and the reflected frequency (after applying the Doppler effect formula).

Step-by-step solution

Calculate the frequency heard by the moving car

The car is approaching the source, so the frequency heard by the car will be higher than the original frequency. Use the Doppler effect formula for a stationary source and moving observer: \[ f' = f_0 \left( \frac{v}{v - v_o} \right) \] where \( f_0 = 492 \, \text{Hz} \) is the original frequency, \( v = 330 \, \text{ms}^{-1} \) is the speed of sound, and \( v_o = 2 \, \text{ms}^{-1} \) is the speed of the car. Substituting the given values:

Substitute values into the Doppler effect formula

By plugging the given values into the Doppler effect formula, we can find the frequency heard by the car: \[ f' = 492 \left( \frac{330}{330 - 2} \right) \] \[ f' = 492 \left( \frac{330}{328} \right) \] Calculate the value of \( f' \).

Calculate the reflected frequency

The reflected frequency \( f'' \) will be the same as the frequency \( f' \) heard by the car because the car reflects the sound at the frequency it has received.

Calculate the beat frequency

The beat frequency is the absolute difference between the original frequency and the reflected frequency. Use the formula: \[ f_{\text{beat}} = |f_0 - f''| \] Since the reflected frequency \( f'' \) is the same as the frequency heard by the car \( f' \), we can use the calculated value of \( f' \) to find the beat frequency.

Final calculation and result

After calculating \( f' \), subtract \( f_0 \) from \( f' \) to get the beat frequency: \[ f_{\text{beat}} = |492 - f'| \] This will give you the beat frequency in Hz.

Key concepts

Beat Frequency

When two sound waves with slightly different frequencies intersect, this results in a phenomenon known as beat frequency. It's like having two tuning forks, each vibrating at a slightly different note, nearby each other. The sound you would hear is a wavering volume, or rather a 'beat,' which is the regular fluctuation between loud and soft sound. This happens because the sound waves alternatively reinforce and cancel each other, creating peaks (constructive interference) and troughs (destructive interference).

The beat frequency, denoted as \( f_{beat} \) , is mathematically represented as the absolute value of the difference between the two frequencies, \[ f_{beat} = |f_{1} - f_{2}| \]. When the original sound and its reflection mix, as in the textbook problem provided, the beat frequency is what a listener would perceive as a throbbing or pulsing effect.

Sound Reflection

Sound reflection is the bouncing back of sound waves when they hit an obstacle that doesn't absorb the energy of the sound wave. You're probably familiar with the echo you hear when you shout in an empty hall or mountain range. When sound reflects off an object, like a large car in our problem, the sound wave can return with the same frequency at which it was received. This is essential for calculating beat frequency when the reflected wave interferes with the original wave.

Frequency Shift

The term frequency shift refers to a change in the frequency of a wave for an observer moving relative to the source of the wave. In the context of sound, this is easily noticed when a car or train passes by you, and you perceive a change in the pitch of the sound before and after it passes. This change in pitch corresponds to a change in frequency, which is a direct consequence of the motion between the source and observer, described by the Doppler effect.

Doppler Effect Formula

The Doppler effect formula mathematically explains how the motion of a source or an observer affects the observed frequency of a wave. For a stationary source and a moving observer, the frequency observed (f') is given by: \[ f' = f_{0} \left( \frac{v}{v - v_{o}} \right) \] where \( f_{0} \) is the emitted frequency, \( v \) is the speed of sound, and \( v_{o} \) is the speed of the observer. When the observer is moving towards the source, as in our exercise, the observed frequency increases. Conversely, if the observer were moving away, the observed frequency would decrease.