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Difficulty: MediumWaves, Sound, and Light

An acoustic engineer tests a noise-reduction device that emits a secondary sound wave exactly half a wavelength (180∘180^\circ) out of phase with an incoming noise wave of identical frequency and amplitude. What occurs when these two sound waves overlap in the air?

  1. A
    The waves undergo constructive interference, resulting in a sound wave with twice the original amplitude.
  2. The waves undergo destructive interference, resulting in the cancellation of sound with zero net amplitude.Answer
  3. C
    The waves undergo refraction, resulting in a permanent doubling of the wave's frequency.
  4. D
    The waves undergo dispersion, resulting in the sound separating into different component frequencies.

Answer

The waves undergo destructive interference, resulting in the cancellation of sound with zero net amplitude.
When two waves with equal frequency and amplitude meet half a wavelength (180∘180^\circ) out of phase, the crests of the first wave precisely coincide with the troughs of the second wave. According to the principle of superposition, their individual displacements subtract from each other, resulting in complete destructive interference and zero resultant amplitude.

Step-by-Step Solution

1
Analyze the phase relationship of the two sound waves.
The two waves have identical frequency and amplitude but are half a wavelength (180∘180^\circ) out of phase.
Determining phase alignment indicates whether wave crests and troughs align constructively or destructively.
2
Apply the principle of wave superposition to calculate the net displacement.
Every positive peak (crest) of the primary wave perfectly aligns with an equal negative peak (trough) of the secondary wave, summing to zero net displacement (A−A=0A - A = 0).
The principle of superposition states that overlapping waves combine algebraically at every point in space.
3
Identify the corresponding wave phenomenon.
Destructive interference.
Complete destructive interference occurs when opposing wave displacements of equal magnitude cancel each other out.

Key Concept

Destructive Wave Interference and Superposition
Estimated Time:1m 0s
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Topics

Waves, Sound, and Light

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