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 of finding the number of positive crests striking a vertical plane surface in a time interval of 3 seconds, we can follow these steps:

### Step 1: Understand the Problem
We have a sound wave traveling towards a wall and reflecting off it. We need to find out how many positive crests (or waves) hit the wall in a given time period.

### Step 2: Identify Given Data
- Wavelength of the sound wave: \( \lambda \)
- Speed of the sound wave towards the wall: \( V \)
- Speed of the sound wave after reflection: \( v \)
- Time interval: \( t = 3 \, \text{s} \)

### Step 3: Calculate the Relative Velocity
When the wave strikes the wall, it travels towards the wall with speed \( V \) and reflects back with speed \( v \). Therefore, the effective speed at which the wave approaches the wall is the sum of these two speeds:
\[ \text{Relative velocity} = V + v \]

### Step 4: Calculate the Distance Traveled in 3 Seconds
The total distance traveled by the wave in 3 seconds can be calculated using:
\[ \text{Distance} = \text{Relative velocity} \times \text{Time} \]
Substituting the values, we get:
\[ \text{Distance} = (V + v) \times 3 \]

### Step 5: Calculate the Number of Waves
The number of waves (or positive crests) striking the wall can be calculated by dividing the total distance traveled by the wavelength \( \lambda \):
\[ \text{Number of waves} = \frac{\text{Distance}}{\lambda} = \frac{(V + v) \times 3}{\lambda} \]

### Final Answer
Thus, the number of positive crests striking the wall in a time interval of 3 seconds is:
\[ \text{Number of positive crests} = \frac{3(V + v)}{\lambda} \]

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