Why no beats can be heard if the frequencies of the two interfering waves differ by more then ten ?

Answer

To understand why no beats can be heard if the frequencies of two interfering waves differ by more than ten, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Beats**:
- Beats are produced when two waves of slightly different frequencies interfere with each other. The beat frequency is given by the absolute difference between the two frequencies, \( f_{\text{beat}} = |f_2 - f_1| \).

2. **Given Condition**:
- We are told that the frequencies of the two waves differ by more than 10 Hz. Let's denote the frequencies as \( f_1 \) and \( f_2 \) such that \( |f_2 - f_1| > 10 \) Hz.

3. **Calculating Beat Frequency**:
- The beat frequency can be calculated as \( f_{\text{beat}} = |f_2 - f_1| \). If \( |f_2 - f_1| > 10 \), then the beat frequency is also greater than 10 Hz.

4. **Time Interval for Beats**:
- The time interval between successive beats can be calculated using the formula \( T = \frac{1}{f_{\text{beat}}} \). Since \( f_{\text{beat}} > 10 \) Hz, we have \( T < \frac{1}{10} \) seconds or \( T < 0.1 \) seconds.

5. **Human Ear Perception**:
- The human ear can typically perceive two distinct sounds as separate if the time interval between them is at least 0.1 seconds. If the time interval is less than this, the sounds will blend together, and the ear will not be able to distinguish the beats.

6. **Conclusion**:
- Since the time interval \( T \) for beats when the frequency difference is greater than 10 Hz is less than 0.1 seconds, the human ear cannot perceive these beats. Therefore, no beats can be heard.

### Final Answer:
No beats can be heard if the frequencies of the two interfering waves differ by more than 10 Hz because the time interval between the beats becomes less than 0.1 seconds, which is below the threshold for human hearing.