A person with vibrating tuning fork of frequency 338 Hz is moving towards a vertical wall with speed of `2 ms^(-1)` Velocity of sound in air is `340 ms^(-1)` The number of beats heard per second is

Options

A. 2
B. 4
C. 6
D. 8

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the given values
- Frequency of the tuning fork (f) = 338 Hz
- Speed of the person (vs) = 2 m/s
- Velocity of sound in air (v) = 340 m/s

### Step 2: Calculate the apparent frequency (f') heard by the wall
When the source of sound (the person with the tuning fork) is moving towards a stationary observer (the wall), the apparent frequency can be calculated using the formula:
\[ f' = \frac{v}{v - v_s} \cdot f \]
Substituting the values:
\[ f' = \frac{340}{340 - 2} \cdot 338 \]
\[ f' = \frac{340}{338} \cdot 338 \]
\[ f' = 340 \text{ Hz} \]
So, the frequency heard by the wall is 340 Hz.

### Step 3: Calculate the apparent frequency (f'') heard by the person after reflection
Now, the wall acts as a source of sound, and the person is moving towards it. The new apparent frequency can be calculated using:
\[ f'' = \frac{v + v_o}{v - v_s} \cdot f' \]
Here, \(v_o\) (the velocity of the observer, which is the person) is also 2 m/s. Substituting the values:
\[ f'' = \frac{340 + 2}{340 - 0} \cdot 340 \]
\[ f'' = \frac{342}{340} \cdot 340 \]
\[ f'' = 342 \text{ Hz} \]

### Step 4: Calculate the beat frequency
The beat frequency is the difference between the two frequencies:
\[ \text{Beat frequency} = f'' - f' \]
Substituting the values:
\[ \text{Beat frequency} = 342 - 340 = 2 \text{ Hz} \]

### Step 5: Conclusion
The number of beats heard per second is 2 Hz.

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