When two progressive waves of intensity `I_(1)` and `I_(2)` but slightly different frequencies superpose, the resultant intensity flutuates between :-

Options

A. `(sqrt(I_(1))+sqrt(I_(2)))^(2)" and "(sqrt(I_(1))-sqrt(I_(2)))^(2)`
B. `(sqrt(I_(1))+sqrt(I_(2)))" and "(sqrt(I_(1))-sqrt(I_(2)))`
C. `(I_(1)+I_(2))" and "(sqrt(I_(1))-sqrt(I_(2)))`
D. `(I_(1))/(I_(2))"and"(I_(2))/(I_(1))`

Answer

To solve the problem of determining the resultant intensity when two progressive waves of intensities \(I_1\) and \(I_2\) with slightly different frequencies superpose, we can follow these steps:

### Step 1: Understand the Superposition of Waves
When two waves superpose, the resultant wave can be expressed in terms of their amplitudes and phase difference. The two waves can be represented as:
- Wave 1: \(A_1 \sin(\omega_1 t - kx)\)
- Wave 2: \(A_2 \sin(\omega_2 t - kx + \phi)\)

### Step 2: Resultant Amplitude Calculation
The resultant amplitude \(A_R\) of the two waves can be calculated using the formula:
\[ A_R = \sqrt{A_1^2 + A_2^2 + 2A_1A_2 \cos(\phi)} \]
where \(\phi\) is the phase difference between the two waves.

### Step 3: Relate Amplitude to Intensity
Since intensity \(I\) is proportional to the square of the amplitude, we can express the intensities in terms of amplitudes:
\[ I_1 \propto A_1^2 \quad \text{and} \quad I_2 \propto A_2^2 \]
Thus, we can write:
\[ A_1 = \sqrt{I_1} \quad \text{and} \quad A_2 = \sqrt{I_2} \]

### Step 4: Substitute Amplitudes into Resultant Intensity
Substituting \(A_1\) and \(A_2\) into the resultant amplitude formula gives:
\[ A_R = \sqrt{I_1 + I_2 + 2\sqrt{I_1 I_2} \cos(\phi)} \]

### Step 5: Calculate Resultant Intensity
The resultant intensity \(I_R\) can then be expressed as:
\[ I_R = A_R^2 = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos(\phi) \]

### Step 6: Determine Maximum and Minimum Intensity
The value of \(\cos(\phi)\) varies between -1 and 1. Therefore, the resultant intensity will fluctuate between:
- Minimum intensity when \(\cos(\phi) = -1\):
\[ I_{\text{min}} = I_1 + I_2 - 2\sqrt{I_1 I_2} \]
- Maximum intensity when \(\cos(\phi) = 1\):
\[ I_{\text{max}} = I_1 + I_2 + 2\sqrt{I_1 I_2} \]

### Conclusion
Thus, the resultant intensity fluctuates between:
\[ I_1 + I_2 - 2\sqrt{I_1 I_2} \quad \text{and} \quad I_1 + I_2 + 2\sqrt{I_1 I_2} \]

### Final Answer
The resultant intensity fluctuates between \(I_1 + I_2 - 2\sqrt{I_1 I_2}\) and \(I_1 + I_2 + 2\sqrt{I_1 I_2}\).

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