When beats are produced by two progressive waves of nearly the same frequency, which one of the following if correct?

Options

A. The particle vibrates simple harmonically , with the frequency equal to the difference in the component frequencies
B. The amplitude of vibration at any point changes simple harmonically with a frequency equal to the difference in the frequencies of the two waves
C. The frequency of beats depends upon the position , where the observer is
D. The frequency of beats changes at the time progresses

Explanation

To solve the question regarding the production of beats by two progressive waves of nearly the same frequency, we can follow these steps:

### Step-by-step Solution:

1. **Understanding Beats**:
- Beats occur when two waves of slightly different frequencies interfere with each other. The result is a new wave pattern where the amplitude varies periodically.

2. **Wave Equations**:
- Let’s denote the two waves as:
- Wave 1: \( y_1 = A \sin(\omega_1 t) \)
- Wave 2: \( y_2 = A \sin(\omega_2 t) \)
- Here, \( \omega_1 \) and \( \omega_2 \) are the angular frequencies of the two waves.

3. **Superposition Principle**:
- According to the superposition principle, the resultant wave \( y \) at a point is given by:
\[ y = y_1 + y_2 = A \sin(\omega_1 t) + A \sin(\omega_2 t) \]

4. **Using Trigonometric Identities**:
- We can use the trigonometric identity:
\[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
- Applying this to our waves:
\[ y = 2A \sin\left(\frac{\omega_1 + \omega_2}{2} t\right) \cos\left(\frac{\omega_1 - \omega_2}{2} t\right) \]

5. **Identifying Beat Frequency**:
- The term \( \cos\left(\frac{\omega_1 - \omega_2}{2} t\right) \) represents the modulation of amplitude, and the frequency of this modulation is:
\[ f_{\text{beat}} = \frac{\omega_1 - \omega_2}{2\pi} \]
- The beat frequency is thus the absolute difference between the two frequencies:
\[ f_{\text{beat}} = |f_1 - f_2| \]

6. **Conclusion**:
- The amplitude of the resultant wave changes harmonically with a frequency equal to the difference in frequencies of the two waves. Therefore, the correct statement is that the frequency of the beats produced is equal to the difference in the frequencies of the two waves.

### Final Answer:
The correct option is that the frequency of the beats produced is equal to the difference in the frequencies of the two waves.

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