Beats are heard due to interference of two sound waves. The intensity of sound changes from maximum to minimum in 100 milli second. The difference in frequencies of two waves is :

Question

Accepted Answer

Beats are a phenomenon that occur when two sound waves of slightly different frequencies interfere with each other. The beat frequency, which is the rate at which the intensity of the sound changes from maximum to minimum, is equal to the absolute difference between the frequencies of the two waves.

1. **Explanation for Option 3 (5.0 Hz):**
   - The problem states that the intensity of sound changes from maximum to minimum in 100 milliseconds. This means that one complete cycle of the beat (from maximum to minimum and back to maximum) takes 200 milliseconds (since it takes 100 milliseconds to go from maximum to minimum and another 100 milliseconds to return to maximum).
   - The beat frequency is the reciprocal of the time period of one complete beat cycle. Therefore, the beat frequency is $ \frac{1}{0.2 \text{ seconds}} = 5.0 \text{ Hz} $.
   - This calculation shows that the difference in frequencies of the two waves is 5.0 Hz, which corresponds to Option 3.

2. **Explanation for why other options are incorrect:**
   - **Option 1 (No difference):** If there were no difference in frequencies, there would be no beats heard. Beats occur due to the interference of two waves with slightly different frequencies. Therefore, this option is incorrect.
   - **Option 2 (2.5 Hz):** If the frequency difference were 2.5 Hz, the time period for one complete beat cycle would be $ \frac{1}{2.5} \text{ seconds} = 0.4 \text{ seconds} $, or 400 milliseconds. This does not match the given condition of 200 milliseconds for a complete beat cycle. Hence, this option is incorrect.
   - **Option 4 (10.0 Hz):** If the frequency difference were 10.0 Hz, the time period for one complete beat cycle would be $ \frac{1}{10} \text{ seconds} = 0.1 \text{ seconds} $, or 100 milliseconds. This would mean the intensity changes from maximum to minimum and back to maximum in 100 milliseconds, which contradicts the given condition. Therefore, this option is incorrect.

In conclusion, the correct answer is Option 3 (5.0 Hz), as it accurately reflects the difference in frequencies based on the given time period for the beat cycle.