A sound wave of wavelength ` lambda` travels towards the right horizontally with a velocity `V`. It strikes and reflects from a vertical plane surface , travelling at a speed `v` towards the left . The number of positive crests striking in a time interval of `3 s` on the wall is

Options

A. `3(V + v)//lambda`
B. `3(V - v)//lambda`
C. `(V + v)// 3 lambda`
D. `(V - v)// 3 lambda`

Explanation

To solve the problem, we need to determine the number of positive crests of a sound wave that strike a vertical plane surface in a time interval of 3 seconds. Here’s how we can approach the solution step by step:

### Step 1: Understand the given parameters
- Wavelength of the sound wave: \( \lambda \)
- Velocity of the sound wave: \( V \)
- Velocity of the vertical plane surface: \( v \)
- Time interval: \( t = 3 \, \text{s} \)

### Step 2: Determine the relative velocity of the sound wave with respect to the wall
When the sound wave strikes the wall, it travels towards the wall at speed \( V \), while the wall is moving towards the sound wave at speed \( v \). Therefore, the relative velocity \( V_{rel} \) of the sound wave with respect to the wall is given by:
\[ V_{rel} = V + v \]

### Step 3: Calculate the frequency of the sound wave
The frequency \( f \) of the sound wave can be calculated using the formula:
\[ f = \frac{V}{\lambda} \]
However, since the wall is moving towards the sound wave, the apparent frequency \( f' \) that the wall perceives will be:
\[ f' = \frac{V_{rel}}{\lambda} = \frac{V + v}{\lambda} \]

### Step 4: Calculate the number of crests striking the wall in 3 seconds
The number of crests striking the wall in one second is equal to the apparent frequency \( f' \). Therefore, in 3 seconds, the total number of crests \( N \) striking the wall will be:
\[ N = f' \times t = \left(\frac{V + v}{\lambda}\right) \times 3 \]

### Final Expression
Thus, the total number of positive crests striking the wall in 3 seconds is:
\[ N = \frac{3(V + v)}{\lambda} \]

### Summary
The number of positive crests striking the wall in a time interval of 3 seconds is given by:
\[ N = \frac{3(V + v)}{\lambda} \]

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