The frequency of sound reaching a stationery listener behind a moving source is

Options

A. lower than source frequency
B. higher than source frequency
C. zero
D. same as the frequency of the source

Answer

To solve the problem of determining the frequency of sound reaching a stationary listener behind a moving source, we can use the Doppler effect formula. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Doppler Effect
The Doppler effect describes the change in frequency (or wavelength) of a wave in relation to an observer moving relative to the source of the wave. In this case, we have a stationary listener and a moving source of sound.

### Step 2: Identify the Variables
- Let \( f_0 \) be the frequency of the source.
- Let \( v \) be the speed of sound in air.
- Let \( v_s \) be the speed of the source.
- Since the listener is stationary, \( v_{observer} = 0 \).

### Step 3: Use the Doppler Effect Formula
The formula for the apparent frequency \( f' \) heard by a stationary observer when the source is moving away is given by:

\[ f' = f_0 \frac{v + v_{observer}}{v + v_s} \]

Since the observer is stationary, this simplifies to:

\[ f' = f_0 \frac{v}{v + v_s} \]

### Step 4: Analyze the Situation
In this scenario, since the source is moving away from the listener, \( v_s \) is positive. Therefore, the formula becomes:

\[ f' = f_0 \frac{v}{v + v_s} \]

### Step 5: Determine the Relationship Between \( f' \) and \( f_0 \)
As \( v_s \) (the speed of the source) increases, the denominator \( (v + v_s) \) increases, which means that the fraction \( \frac{v}{v + v_s} \) decreases. Consequently, the apparent frequency \( f' \) will be less than the original frequency \( f_0 \).

### Conclusion
Thus, the frequency of sound reaching a stationary listener behind a moving source is lower than the frequency of the source.

### Final Answer
The frequency of sound reaching a stationary listener behind a moving source is lower than the source frequency.

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