A sound source is moving towards stationary listener with `1/20 ` th of the speed of sound. The ratio of apparent to real frequency is

Options

A. `(9/10)^2`
B. `20/19`
C. `11/10`
D. `(11/10)^2`

Explanation

To find the ratio of the apparent frequency to the real frequency when a sound source is moving towards a stationary listener, we can use the Doppler effect formula. Here’s the step-by-step solution:

### Step 1: Identify the Variables
- Let \( f_s \) be the real frequency of the sound source.
- The speed of sound \( c \) is taken as a constant.
- The speed of the source \( v_s \) is given as \( \frac{c}{20} \).
- The speed of the observer \( v_o \) is \( 0 \) since the listener is stationary.

### Step 2: Write the Doppler Effect Formula
The formula for the observed frequency \( f' \) when the source is moving towards a stationary observer is given by:
\[ f' = f_s \frac{c + v_o}{c - v_s} \]
Since \( v_o = 0 \), the formula simplifies to:
\[ f' = f_s \frac{c}{c - v_s} \]

### Step 3: Substitute the Values
Substituting \( v_s = \frac{c}{20} \) into the formula:
\[ f' = f_s \frac{c}{c - \frac{c}{20}} \]

### Step 4: Simplify the Denominator
Now, simplify the denominator:
\[ c - \frac{c}{20} = c \left(1 - \frac{1}{20}\right) = c \left(\frac{20 - 1}{20}\right) = c \left(\frac{19}{20}\right) \]

### Step 5: Substitute Back into the Formula
Now substituting back into the formula for \( f' \):
\[ f' = f_s \frac{c}{\frac{19c}{20}} = f_s \cdot \frac{20}{19} \]

### Step 6: Find the Ratio of Apparent to Real Frequency
The ratio of the apparent frequency \( f' \) to the real frequency \( f_s \) is:
\[ \frac{f'}{f_s} = \frac{20}{19} \]

### Conclusion
Thus, the ratio of the apparent frequency to the real frequency is:
\[ \frac{f'}{f_s} = \frac{20}{19} \]