The superposition takes place between two waves of frequency f and amplitude a . The total intensity is directly proportional to

Options

A. a
B. 2a
C. `2a^2`
D. `4a^2`

Answer

To solve the problem, we need to determine how the total intensity of two superimposed waves is related to their amplitudes. Let's break down the solution step by step.

### Step 1: Understanding Intensity and Amplitude
The intensity \( I \) of a wave is directly proportional to the square of its amplitude \( A \). This can be expressed mathematically as:
\[ I \propto A^2 \]

### Step 2: Identifying the Amplitudes of the Waves
We are given two waves, both with the same amplitude \( a \). Therefore, we denote the amplitudes of the two waves as:
- Amplitude of wave 1, \( A_1 = a \)
- Amplitude of wave 2, \( A_2 = a \)

### Step 3: Finding the Resultant Amplitude
When two waves superimpose, the resultant amplitude \( A \) can be calculated using the formula:
\[ A = \sqrt{A_1^2 + A_2^2 + 2A_1A_2 \cos \phi} \]
where \( \phi \) is the phase difference between the two waves.

### Step 4: Substituting the Amplitudes
Substituting \( A_1 = a \) and \( A_2 = a \) into the formula gives:
\[ A = \sqrt{a^2 + a^2 + 2a \cdot a \cos \phi} \]
\[ A = \sqrt{2a^2 + 2a^2 \cos \phi} \]
\[ A = \sqrt{2a^2(1 + \cos \phi)} \]

### Step 5: Simplifying the Resultant Amplitude
This can be further simplified:
\[ A = a \sqrt{2(1 + \cos \phi)} \]

### Step 6: Relating Intensity to the Resultant Amplitude
Now, we need to find the total intensity \( I \). Since intensity is proportional to the square of the amplitude:
\[ I \propto A^2 \]
Substituting the expression for \( A \):
\[ I \propto (a \sqrt{2(1 + \cos \phi)})^2 \]
\[ I \propto a^2 \cdot 2(1 + \cos \phi) \]
\[ I \propto 2a^2(1 + \cos \phi) \]

### Step 7: Conclusion
From the above steps, we conclude that the total intensity is directly proportional to \( 2a^2(1 + \cos \phi) \). However, if we are only considering the proportionality to the amplitude squared, we can simplify this to:
\[ I \propto 4a^2 \] (when \( \phi = 0 \), which gives the maximum constructive interference).

### Final Answer
The total intensity is directly proportional to \( 4a^2 \).

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