Two waves of same frequency and same amplitude reach a common point of the medium simultaneously. If the amplitude of resultant wave is zero then the path difference between the waves will be –

Options

A. `lambda`
B. `lambda//2`
C. `2pi`
D. `(3pi)/(4)`

Answer

To solve the problem, we need to find the path difference between two waves that results in a net amplitude of zero when they interfere at a common point in the medium. Here’s a step-by-step solution:

### Step 1: Understand the Condition for Zero Resultant Amplitude
When two waves of the same frequency and amplitude interfere, the resultant amplitude can be calculated using the formula:
\[ A_{\text{net}} = A_1 + A_2 + 2A_1 A_2 \cos(\phi) \]
Where \( A_1 \) and \( A_2 \) are the amplitudes of the two waves, and \( \phi \) is the phase difference between them.

### Step 2: Set the Resultant Amplitude to Zero
Since we know that the resultant amplitude \( A_{\text{net}} \) is zero, we can set up the equation:
\[ 0 = A_1 + A_2 + 2A_1 A_2 \cos(\phi) \]

### Step 3: Substitute Equal Amplitudes
Given that both waves have the same amplitude, we can let \( A_1 = A \) and \( A_2 = A \). Thus, the equation becomes:
\[ 0 = A + A + 2A^2 \cos(\phi) \]
This simplifies to:
\[ 0 = 2A + 2A^2 \cos(\phi) \]

### Step 4: Solve for Cosine
Dividing the entire equation by 2:
\[ 0 = A + A^2 \cos(\phi) \]
Rearranging gives us:
\[ A^2 \cos(\phi) = -A \]
Since \( A \neq 0 \) (the amplitude is not zero), we can divide both sides by \( A \):
\[ A \cos(\phi) = -1 \]
Thus:
\[ \cos(\phi) = -1 \]

### Step 5: Determine the Phase Difference
The cosine function equals -1 at odd multiples of \( \pi \):
\[ \phi = (2n + 1)\pi \]
where \( n \) is an integer. The simplest case is when \( n = 0 \), which gives:
\[ \phi = \pi \]

### Step 6: Relate Phase Difference to Path Difference
The phase difference \( \phi \) is related to the path difference \( \Delta x \) by the formula:
\[ \phi = \frac{2\pi}{\lambda} \Delta x \]
Substituting \( \phi = \pi \):
\[ \pi = \frac{2\pi}{\lambda} \Delta x \]

### Step 7: Solve for Path Difference
Now, we can solve for the path difference \( \Delta x \):
\[ \Delta x = \frac{\pi \lambda}{2\pi} = \frac{\lambda}{2} \]

### Final Answer
Therefore, the path difference between the two waves when the resultant amplitude is zero is:
\[ \Delta x = \frac{\lambda}{2} \]

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