Destructive Interference
Destructive interference occurs when two waves overlap out of phase — with the crest of one aligning with the trough of the other — causing their displacements to cancel partially or completely, reducing the resultant amplitude. When two waves of equal amplitude are exactly 180° out of phase, the resultant amplitude is zero, meaning complete cancellation. This principle underlies active noise cancellation in headphones, anti-reflective optical coatings, and acoustic dead spots in concert halls.
Key Formula
A_result = |A1 - A2|
LaTeX: A_{result} = |A_1 - A_2|
| Symbol | Meaning | Unit |
|---|---|---|
| A_{result} | Amplitude of the resultant wave | m |
| A_1 | Amplitude of wave 1 | m |
| A_2 | Amplitude of wave 2 | m |
Worked Example
Problem
Two sound waves arrive at a microphone: wave 1 has amplitude 0.6 m, wave 2 has amplitude 0.6 m, and they are exactly 180° out of phase. What is the resultant amplitude?
Solution
Step 1: Identify amplitudes. A1 = 0.6 m, A2 = 0.6 m. Step 2: Since the phase difference is 180°, apply destructive interference: A_result = |A1 − A2|. Step 3: A_result = |0.6 − 0.6| = 0 m. Conclusion: Complete cancellation occurs.
Answer
Resultant amplitude = 0 m (complete cancellation)
Conditions for Destructive Interference
| Path Difference | Phase Difference | Resultant (equal amplitudes) | Example |
|---|---|---|---|
| λ/2 | 180° | 0 (complete cancellation) | Noise-cancelling headphones |
| 3λ/2 | 540° | 0 (complete cancellation) | Anti-reflective coatings |
| 5λ/2 | 900° | 0 (complete cancellation) | Acoustic dead zones |
| Non-integer × λ/2 | Partial | Reduced amplitude | Partial sound attenuation |
Interactive Tools
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Related Terms
Constructive Interference
Constructive interference occurs when two or more waves overlap in such a way that their displacements add together, producing a resultant wave with greater amplitude than either individual wave. This phenomenon arises when the waves are in phase — that is, their crests and troughs align — leading to a net increase in energy at that point. It is fundamental to technologies such as noise-cancelling headphones (in reverse), optical coatings, and phased-array antennas.
Acoustic Resonance
Acoustic resonance occurs when an object or air column vibrates at its natural frequency in response to an external sound source at that same frequency, resulting in a dramatic amplification of the sound. The phenomenon arises when standing waves are set up within the resonating object, with nodes and antinodes at fixed positions. Acoustic resonance is exploited in all musical instruments — strings, pipes, and percussion — as well as in architectural acoustics, industrial machinery fault detection, and medical imaging.
Sound Beats
Sound beats are periodic variations in amplitude — heard as a rhythmic pulsing or "wah-wah" sound — that occur when two sound waves of slightly different frequencies interfere. The beat frequency equals the absolute difference between the two source frequencies, and the sound alternately gets louder (constructive interference) and quieter (destructive interference). Musicians use beats to tune instruments: when no beats are heard, the two sources are in tune; as beats slow to zero, the frequencies converge.
From Latin "destructivus" (tending to destroy) and "interferere" (to strike between). The term gained precise scientific meaning through Thomas Young's wave optics work (1801) and was further formalized by Augustin-Jean Fresnel.