Question

If two waves of equal amplitude \( A \) and opposite phase interfere, the amplitude of the resultant wave is

• A. \( A \)
• B. \( 2A \)
• C. \( \dfrac{A}{2} \)
• D. \( 0 \)
• E. \( A^2 \)

Hint:

Destructive interference occurs when phase difference is \( \pi \), leading to zero resultant amplitude if amplitudes are equal.

Answer 1

The Correct Option is D

Step 1: Recall the principle of superposition.
When two waves interfere, the resultant displacement is the algebraic sum of individual displacements.

Step 2: Write the expression for resultant amplitude.
For two waves of amplitude \( A \) and phase difference \( \phi \), resultant amplitude is: \[ R = 2A \cos\left(\frac{\phi}{2}\right) \]

Step 3: Use the given phase condition.
Opposite phase means: \[ \phi = \pi \]

Step 4: Substitute into the formula.
\[ R = 2A \cos\left(\frac{\pi}{2}\right) \] \[ R = 2A \cdot 0 = 0 \]

Step 5: Interpret physically.
Waves cancel each other completely → destructive interference.

Step 6: Key observation.
Equal amplitude + phase difference \( \pi \) always gives zero amplitude.

Step 7: Final conclusion.
Hence, resultant amplitude is: \[ \boxed{0} \] Therefore, the correct option is \[ \boxed{(4)\ 0} \]