Two waves of slightly different frequencies `f_(1) and f_(2) (f_(1) gt f_(2))` with zero phase difference , same amplitudes ,travelling in the same direction superimpose .

Options

A. Phenomenon of beats is always observed by human ear.
B. Intensity of resultant wave is a constant.
C. Intensity of resultant wave varies periodically with time with maximum intensity `4 a^(2)` and minimum intensity zero.
D. A maxima appears at a time `1//[2 (f_(1) - f_(2))]` later ( or earlier ) than a minima appears .

Explanation

To solve the problem of two waves of slightly different frequencies \( f_1 \) and \( f_2 \) (where \( f_1 > f_2 \)), with zero phase difference and the same amplitudes, we will analyze the situation step by step.

### Step-by-Step Solution:

1. **Understanding Wave Properties**:
- We have two waves described by the following equations:
\[ y_1 = A \sin(2\pi f_1 t) \]
\[ y_2 = A \sin(2\pi f_2 t) \]
where \( A \) is the amplitude of the waves.

2. **Superposition of Waves**:
- When these two waves superimpose, the resultant wave \( y \) can be expressed as:
\[ y = y_1 + y_2 = A \sin(2\pi f_1 t) + A \sin(2\pi f_2 t) \]

3. **Using the Trigonometric Identity**:
- We can use the trigonometric identity for the sum of sines:
\[ \sin a + \sin b = 2 \sin\left(\frac{a+b}{2}\right) \cos\left(\frac{a-b}{2}\right) \]
- Applying this identity, we get:
\[ y = 2A \sin\left(\pi (f_1 + f_2) t\right) \cos\left(\pi (f_1 - f_2) t\right) \]

4. **Resultant Amplitude and Intensity**:
- The amplitude of the resultant wave varies with time and is given by:
\[ A_{resultant} = 2A \cos\left(\pi (f_1 - f_2) t\right) \]
- The intensity \( I \) of the wave is proportional to the square of the amplitude:
\[ I \propto A_{resultant}^2 = (2A \cos(\pi (f_1 - f_2) t))^2 = 4A^2 \cos^2(\pi (f_1 - f_2) t) \]
- This indicates that the intensity varies periodically with time, reaching a maximum of \( 4A^2 \) and a minimum of \( 0 \).

5. **Beat Frequency**:
- The beat frequency \( f_{beat} \) is given by the difference in frequencies:
\[ f_{beat} = f_1 - f_2 \]
- The time period of the beats \( T_{beat} \) is:
\[ T_{beat} = \frac{1}{f_{beat}} = \frac{1}{f_1 - f_2} \]

6. **Time Difference Between Maxima and Minima**:
- Maxima occur at times \( t = \frac{n}{f_{beat}} \) (where \( n \) is an integer).
- Minima occur at times \( t = \frac{(2n + 1)}{2f_{beat}} \).
- The time difference between a maximum and the nearest minimum is:
\[ \Delta t = \frac{1}{2f_{beat}} = \frac{1}{2(f_1 - f_2)} \]

### Conclusion:
- The correct options based on the analysis are:
- The intensity of the resultant wave varies periodically with time (correct).
- A maxima appears at a time \( \frac{1}{2(f_1 - f_2)} \) later or earlier than a minima appears (correct).