What happens when two waves are in the same place? Let us study the principle of superposition and interference.
Basic Knowledge
Principle of superposition: The displacement at a point where two or more waves overlap is the algebraic sum of their individual displacements. After passing through each other, the waves return to their original waveforms.
Results of superposition:
・Constructive interference (constructive interference): When crest overlaps crest (or trough overlaps trough), the displacement becomes larger
・Destructive interference (destructive interference): When crest overlaps trough, the displacement becomes smaller (they cancel each other)
Standing wave (stationary wave): When two waves with the same amplitude, wavelength, and frequency travel in opposite directions and overlap, a standing wave that appears not to move is formed.
・Node (node): A point where displacement is always zero
・Antinode (antinode): A point that vibrates with maximum displacement
The distances between adjacent nodes and between adjacent antinodes are both λ/2.
Principle of superposition（Algebraic sum of displacements. Waves return to their original forms after passing through each other）
Interference (interference)（A phenomenon in which two overlapping waves reinforce or weaken each other）
Standing wave (stationary wave)（A stationary-looking wave formed by overlapping identical waves traveling in opposite directions）
Node (node)（A point where displacement is always zero in a standing wave）
Antinode (antinode)（A point where displacement is maximum in a standing wave）
Path difference（The difference between the distances from two sources to a point）
Further Study (Background and Significance)
Interference conditions for waves from two sources S₁ and S₂:
・Constructive interference: Path difference = mλ (m is an integer, when the sources are in phase)
・Destructive interference: Path difference = (m + 1/2)λ
String instruments use standing waves. Both ends of the string are fixed ends (nodes), and antinodes form between them. Frequencies satisfying string length L = nλ/2 (n = 1, 2, 3…) produce resonance. n = 1 is the fundamental vibration, while n = 2, 3, … are overtones.
Light interference can be observed in thin films (the colors of soap bubbles) and slit experiments. Young's double-slit experiment produced alternating bright and dark fringes, providing evidence of the wave nature of light.
Noise-canceling headphones are a practical example of destructive interference: they combine sound waves in opposite phases to cancel noise.
- Superposition: algebraic sum of displacements
- Constructive interference: crest + crest, trough + trough
- Destructive interference: crest + trough
- Standing wave: superposition of identical waves traveling in opposite directions
- Distance between a node and an adjacent antinode = λ/4
- Distance between adjacent nodes (or antinodes) = λ/2
- String resonance: L = nλ/2
Important Notes (Common Confusions)
① Superposition is the algebraic sum of displacements (directions must be considered), not a simple addition of amplitudes. ② A standing wave is not a “wave that does not travel”; it is the superposition of two traveling waves. ③ The distance between nodes is λ/2, while the distance between a node and an antinode is λ/4. ④ The path-difference interference conditions apply when the sources are in phase. When they are out of phase, the conditions for reinforcement and weakening are reversed.
Practice
- The distance between adjacent nodes of a standing wave is 0.4 m. Find the wavelength of the original wave.
- What happens to the displacement when a crest overlaps a trough?
- A string 0.6 m long is vibrating in its fundamental mode. Find the wavelength.