Two sound waves travel in the same direction in a medium. The amplitude of each wave is A and the phase difference between the two waves is `120^(@)`. The resultant amplitude will be

Options

A. `sqrt(2) A`
B. `2 A`
C. 3A
D. A

Explanation

To find the resultant amplitude of two sound waves traveling in the same direction with a given phase difference, we can use the principle of superposition. Here's a step-by-step solution to the problem:

### Step 1: Identify the given values
- Amplitude of each wave, \( A_1 = A \) and \( A_2 = A \)
- Phase difference, \( \Delta \phi = 120^\circ \)

### Step 2: Use the formula for resultant amplitude
The formula for the resultant amplitude \( R \) of two waves with amplitudes \( A_1 \) and \( A_2 \) and a phase difference \( \Delta \phi \) is given by:
\[ R = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos(\Delta \phi)} \]

### Step 3: Substitute the values into the formula
Since both amplitudes are equal, we can substitute \( A_1 = A \) and \( A_2 = A \):
\[ R = \sqrt{A^2 + A^2 + 2 A A \cos(120^\circ)} \]

### Step 4: Simplify the equation
This simplifies to:
\[ R = \sqrt{2A^2 + 2A^2 \cos(120^\circ)} \]
Now, we need to find \( \cos(120^\circ) \):
\[ \cos(120^\circ) = -\frac{1}{2} \]
Substituting this value into the equation:
\[ R = \sqrt{2A^2 + 2A^2 \left(-\frac{1}{2}\right)} \]
\[ R = \sqrt{2A^2 - A^2} = \sqrt{A^2} = A \]

### Step 5: Conclusion
The resultant amplitude \( R \) is:
\[ R = A \]

### Final Answer
The resultant amplitude of the two sound waves is \( A \).

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