Two sources of sound `S_(1)` and `S_(2)` produce sound waves of same frequency `660 Hz`. A listener is moving from source `S_(1)` towards `S_(2)` with a constant speed `u m//s` and he hears `10` beats/s. The velocity of sound is `330 m//s`. Then, `u` equals :

Options

A. `5.5 m//s`
B. `15.0 m//s`
C. `2.5 m//s`
D. `10.0 m//s`

Answer

To solve the problem, we will use the concept of apparent frequency and the beat frequency produced when two sound waves of slightly different frequencies interfere with each other.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Frequency of both sources, \( f = 660 \, \text{Hz} \)
- Velocity of sound, \( v = 330 \, \text{m/s} \)
- Beat frequency, \( f_b = 10 \, \text{beats/s} \)
- Speed of the listener, \( u \, \text{m/s} \) (to be determined)

2. **Apparent Frequency from Source \( S_1 \):**
The listener is moving away from source \( S_1 \). The formula for the apparent frequency \( f_1 \) when the observer is moving away from the source is:
\[ f_1 = f \left( \frac{v - u}{v} \right) \]

3. **Apparent Frequency from Source \( S_2 \):**
The listener is moving towards source \( S_2 \). The formula for the apparent frequency \( f_2 \) when the observer is moving towards the source is:
\[ f_2 = f \left( \frac{v + u}{v} \right) \]

4. **Beat Frequency Relationship:**
The beat frequency is given by the difference in the apparent frequencies:
\[ f_b = |f_2 - f_1| = 10 \]
Substituting the expressions for \( f_1 \) and \( f_2 \):
\[ |f \left( \frac{v + u}{v} \right) - f \left( \frac{v - u}{v} \right)| = 10 \]

5. **Simplifying the Equation:**
Factoring out \( f \):
\[ f \left( \frac{(v + u) - (v - u)}{v} \right) = 10 \]
This simplifies to:
\[ f \left( \frac{2u}{v} \right) = 10 \]

6. **Substituting Known Values:**
Substitute \( f = 660 \, \text{Hz} \) and \( v = 330 \, \text{m/s} \):
\[ 660 \left( \frac{2u}{330} \right) = 10 \]

7. **Solving for \( u \):**
Rearranging gives:
\[ \frac{1320u}{330} = 10 \]
Simplifying:
\[ 4u = 10 \implies u = \frac{10}{4} = 2.5 \, \text{m/s} \]

### Final Answer:
The speed of the listener \( u \) is \( 2.5 \, \text{m/s} \).

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