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A sound wave of frequency f propagating ...

A sound wave of frequency `f` propagating through air with a velocity `c`, is reflected from a surface which is moving away from the source with a constant speed `v`. Find the frequency of the reflected wave, measured by the observer at the position of the source.

A

`(f(c-v))/(c+v)`

B

`(f(c-v))/(c-v)`

C

`(f(c+2v))/(c+v)`

D

`(f(c-v))/(c-2v)`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to find the frequency of the reflected sound wave as measured by an observer at the position of the source. We will use the Doppler effect principles for sound waves. ### Step-by-Step Solution: 1. **Identify the Given Parameters:** - Frequency of the emitted sound wave: \( f \) - Velocity of sound in air: \( c \) - Velocity of the moving surface (reflector): \( v \) 2. **Determine the Frequency Received by the Moving Reflector:** - The reflector is moving away from the source. According to the Doppler effect, the frequency \( f' \) received by the moving reflector can be calculated using the formula: \[ f' = \left( \frac{c - v}{c} \right) f \] - Here, \( c - v \) is used because the reflector is moving away from the source. 3. **Determine the Frequency of the Reflected Wave:** - Now, the reflector acts as a new source of sound, emitting the frequency \( f' \) that it received. The observer (the original source) is stationary, and the new source (the reflector) is moving away with speed \( v \). - The frequency \( f'' \) received by the observer (the original source) can be calculated using the formula: \[ f'' = \left( \frac{c}{c + v} \right) f' \] - Here, \( c \) is the velocity of sound, and \( c + v \) is used because the new source (reflector) is moving away from the observer. 4. **Substituting \( f' \) into the Equation for \( f'' \):** - Substitute the expression for \( f' \) into the equation for \( f'' \): \[ f'' = \left( \frac{c}{c + v} \right) \left( \frac{c - v}{c} \right) f \] - Simplifying this gives: \[ f'' = \left( \frac{c - v}{c + v} \right) f \] 5. **Final Result:** - The frequency of the reflected wave measured by the observer at the position of the source is: \[ f'' = \frac{c - v}{c + v} f \]
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