Two sound sources emitting sound each of wavelength `lambda` are fixed at a given distance apart. A listener moves with a velocity `u` along the line joining the two suorces. The number of beats heard by him per second is

Options

A. `2u//lambda`
B. `u//lambda`
C. `(u)/(3lambda)`
D. `(2lambda)/(u)`

Explanation

To solve the problem of how many beats the listener hears per second when moving between two sound sources emitting sound of wavelength \( \lambda \), we can follow these steps:

### Step 1: Understand the situation
We have two sound sources, \( S_1 \) and \( S_2 \), emitting sound waves of wavelength \( \lambda \). The listener is moving with a velocity \( u \) along the line joining the two sources.

### Step 2: Calculate the frequency of the sound
The frequency \( f \) of a sound wave is related to its wavelength \( \lambda \) and the speed of sound \( v \) in the medium by the formula:
\[ f = \frac{v}{\lambda} \]
Let’s denote the frequency of both sources as \( f_0 = \frac{v}{\lambda} \).

### Step 3: Determine the apparent frequency heard by the listener
As the listener moves towards one source and away from the other, the apparent frequency changes due to the Doppler effect.

1. **For Source \( S_1 \)** (moving towards):
The apparent frequency \( f_1 \) heard by the listener from source \( S_1 \) is given by:
\[ f_1 = f_0 + \frac{u}{\lambda} \quad \text{(since the listener is moving towards the source)} \]

2. **For Source \( S_2 \)** (moving away):
The apparent frequency \( f_2 \) heard by the listener from source \( S_2 \) is given by:
\[ f_2 = f_0 - \frac{u}{\lambda} \quad \text{(since the listener is moving away from the source)} \]

### Step 4: Calculate the beat frequency
The beat frequency \( f_b \) is the difference between the two frequencies heard by the listener:
\[ f_b = |f_1 - f_2| \]
Substituting the expressions for \( f_1 \) and \( f_2 \):
\[ f_b = \left( f_0 + \frac{u}{\lambda} \right) - \left( f_0 - \frac{u}{\lambda} \right) \]
This simplifies to:
\[ f_b = \frac{u}{\lambda} + \frac{u}{\lambda} = \frac{2u}{\lambda} \]

### Final Answer
The number of beats heard by the listener per second is:
\[ \boxed{\frac{2u}{\lambda}} \]