when a source of sound of frequency f crosses stationary observer with a speed ` v_(s) (lt lt ` speed of sound v) , the apparent change in frequency ` Δf` is given by

Options

A. `(2fv_(s))/v`
B. `2fs v_(s)`
C. `(2fv_(s))/v_(s)`
D. `(fv_(s))/v`

Answer

To solve the problem of finding the apparent change in frequency \( \Delta f \) when a source of sound of frequency \( f \) crosses a stationary observer with speed \( v_s \) (where \( v_s \) is much less than the speed of sound \( v \)), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Frequencies**:
- The frequency of the source is \( f \).
- When the source approaches the observer, the apparent frequency \( f_1 \) is given by the formula:
\[ f_1 = f \frac{v}{v - v_s} \]
- When the source moves away from the observer, the apparent frequency \( f_2 \) is given by:
\[ f_2 = f \frac{v}{v + v_s} \]

2. **Calculate the Change in Frequency**:
- The change in frequency \( \Delta f \) is defined as:
\[ \Delta f = f_1 - f_2 \]
- Substituting the expressions for \( f_1 \) and \( f_2 \):
\[ \Delta f = f \frac{v}{v - v_s} - f \frac{v}{v + v_s} \]

3. **Factor Out \( f \)**:
- We can factor \( f \) out of the equation:
\[ \Delta f = f \left( \frac{v}{v - v_s} - \frac{v}{v + v_s} \right) \]

4. **Combine the Fractions**:
- To combine the fractions, we need a common denominator:
\[ \Delta f = f \left( \frac{v(v + v_s) - v(v - v_s)}{(v - v_s)(v + v_s)} \right) \]
- Simplifying the numerator:
\[ v(v + v_s) - v(v - v_s) = v^2 + vv_s - v^2 + vv_s = 2vv_s \]
- Thus, we have:
\[ \Delta f = f \left( \frac{2vv_s}{(v - v_s)(v + v_s)} \right) \]

5. **Approximate for Small \( v_s \)**:
- Since \( v_s \) is much smaller than \( v \) (i.e., \( v_s \ll v \)), we can approximate:
\[ (v - v_s)(v + v_s) \approx v^2 \]
- Therefore, the equation simplifies to:
\[ \Delta f \approx f \left( \frac{2vv_s}{v^2} \right) = \frac{2f v_s}{v} \]

6. **Final Result**:
- The apparent change in frequency \( \Delta f \) is given by:
\[ \Delta f = \frac{2f v_s}{v} \]

### Summary:
The apparent change in frequency when a source of sound crosses a stationary observer is given by:
\[ \Delta f = \frac{2f v_s}{v} \]