Two sound sources of frequency `9Hz` and `11Hz` are sounded together then which plot is correct after superposition of sound waves.

Options

A.
B.
C.
D.

Answer

To solve the problem of determining the correct plot after the superposition of sound waves from two sources with frequencies of 9 Hz and 11 Hz, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Frequencies**:
- The two sound sources have frequencies \( f_1 = 9 \, \text{Hz} \) and \( f_2 = 11 \, \text{Hz} \).

2. **Calculate the Beat Frequency**:
- The beat frequency \( f_b \) is given by the absolute difference of the two frequencies:
\[ f_b = |f_2 - f_1| = |11 \, \text{Hz} - 9 \, \text{Hz}| = 2 \, \text{Hz} \]

3. **Determine the Time Period of the Beats**:
- The time period \( T_b \) of the beats is the reciprocal of the beat frequency:
\[ T_b = \frac{1}{f_b} = \frac{1}{2 \, \text{Hz}} = 0.5 \, \text{seconds} \]

4. **Analyze the Waveforms**:
- Each wave oscillates according to its frequency. The wave with frequency 9 Hz completes 9 cycles in 1 second, while the wave with frequency 11 Hz completes 11 cycles in 1 second.
- Over a time period of 2 seconds, the 9 Hz wave completes \( 9 \times 2 = 18 \) cycles, and the 11 Hz wave completes \( 11 \times 2 = 22 \) cycles.

5. **Determine the Number of Beats**:
- The number of beats produced in 2 seconds can be calculated using the beat frequency:
\[ \text{Number of beats in 2 seconds} = f_b \times 2 = 2 \, \text{Hz} \times 2 \, \text{seconds} = 4 \, \text{beats} \]
- This means that in 2 seconds, there will be 4 instances where the sound waves constructively and destructively interfere.

6. **Select the Correct Plot**:
- The correct plot will show the oscillation pattern reflecting 4 beats in 2 seconds, which corresponds to the superposition of the two sound waves.

### Conclusion:
Based on the calculations, the correct plot after the superposition of the sound waves from the two sources of frequencies 9 Hz and 11 Hz is the one that shows 4 beats over a duration of 2 seconds.