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: Understanding the Concept:
This question describes the interference of two waves based on the principle of superposition. The principle states that when two or more waves overlap, the resultant displacement at any point and at any instant is the vector sum of the displacements that each individual wave would produce at that point and instant. "Opposite phase" means the waves are perfectly out of sync, with a phase difference of \( \pi \) radians or 180\(^\circ\).
Step 2: Key Formula or Approach:
The amplitude \( R \) of the resultant wave from the interference of two waves with amplitudes \( A_1 \) and \( A_2 \) and a phase difference \( \phi \) is given by:
\[ R = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos\phi} \] In this problem, we are given:
- Equal amplitude: \( A_1 = A_2 = A \)
- Opposite phase: \( \phi = \pi \) radians (or 180\(^\circ\))
Step 3: Detailed Explanation:
Substitute the given values into the formula for the resultant amplitude:
\[ R = \sqrt{A^2 + A^2 + 2(A)(A)\cos(\pi)} \] We know that \( \cos(\pi) = -1 \).
\[ R = \sqrt{2A^2 + 2A^2(-1)} \] \[ R = \sqrt{2A^2 - 2A^2} \] \[ R = \sqrt{0} = 0 \] This situation is known as perfect destructive interference. The crest of one wave perfectly cancels the trough of the other, resulting in zero amplitude.
Alternatively, one could think of the superposition directly. If one wave's displacement is \( y_1 = A \sin(\omega t) \), the other wave in opposite phase will have a displacement \( y_2 = A \sin(\omega t + \pi) = -A \sin(\omega t) \). The resultant displacement is \( y = y_1 + y_2 = A \sin(\omega t) - A \sin(\omega t) = 0 \). The amplitude is therefore 0.
Step 4: Final Answer:
The amplitude of the resultant wave is 0. This corresponds to option (D).