Two identical sinusoidal waves each of amplitude 10 mm with a phase difference of `90^(@)` are travelling in the same direction in a string. The amplitude of the resultant wave is

Options

A. 5 mm
B. `10sqrt2` mm
C. 15 mm
D. 20 mm

Explanation

To find the amplitude of the resultant wave formed by two identical sinusoidal waves with a phase difference of 90 degrees, we can follow these steps:

### Step 1: Understand the Problem
We have two identical sinusoidal waves, each with an amplitude \( a = 10 \, \text{mm} \), and they have a phase difference of \( 90^\circ \).

### Step 2: Use the Formula for Resultant Amplitude
When two waves of the same amplitude \( a \) interfere with a phase difference \( \phi \), the amplitude \( A \) of the resultant wave can be calculated using the formula:
\[ A = \sqrt{a^2 + a^2 + 2a^2 \cos(\phi)} \]
Since the phase difference \( \phi = 90^\circ \), we know that \( \cos(90^\circ) = 0 \).

### Step 3: Simplify the Formula
Substituting the values into the formula:
\[ A = \sqrt{a^2 + a^2 + 2a^2 \cdot 0} \]
This simplifies to:
\[ A = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \]

### Step 4: Substitute the Amplitude
Now, substituting the value of \( a = 10 \, \text{mm} \):
\[ A = 10 \sqrt{2} \, \text{mm} \]

### Step 5: Calculate the Result
Calculating \( 10 \sqrt{2} \):
\[ A \approx 10 \times 1.414 \approx 14.14 \, \text{mm} \]

### Final Answer
The amplitude of the resultant wave is approximately \( 14.14 \, \text{mm} \).

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