When waves from two sources overlap, interference occurs: the waves reinforce or weaken one another depending on the location. The relationship between path difference and wavelength is the key.
Basic Knowledge
Interference: When two waves with the same frequency (coherent sources) overlap, the displacement at point P is determined by the path difference Δ.
When two sources S1, S2 vibrate in phase:
・Condition for constructive interference（maximum amplitude）: Δ = |S₁P − S₂P| = mλ（m = 0, ±1, ±2, …）
・Condition for destructive interference（amplitude 0）: Δ = (m + 1/2)λ（m = 0, ±1, ±2, …）
When the sources vibrate out of phase, the conditions for reinforcement and weakening are reversed.
Relationship between phase difference and path difference: δφ = 2π·Δ/λ. Constructive interference: δφ = 2πm; destructive interference: δφ = (2m+1)π.
Interference（A phenomenon in which the amplitude becomes stronger or weaker through the superposition of two waves）
Path difference Δ（The difference between the distances from two wave sources to the observation point）
Coherent（Waves with a constant phase difference. Necessary for forming interference fringes）
Constructive / Destructive interference（When the path difference is an integer multiple / half-integer multiple of the wavelength）
Interference fringe（A pattern of alternating bright and dark regions produced by constructive and destructive interference）
Further Study (Background・Significance)
Interference occurs in all phenomena with wave properties, including water waves, sound waves, light waves, radio waves, and electron waves (quantum mechanics). Interference is evidence of wave behavior, and the observation of interference fringes even with electron beams (the double-slit experiment) is direct evidence of wave-particle duality in quantum mechanics.
Noise-canceling headphones use interference of sound waves: they output sound in the opposite phase to the noise picked up by a microphone, causing the sounds to weaken one another.
Because the condition for constructive interference “Δ = mλ” appears for every integer multiple of the wavelength, the wavelength can be measured precisely from the spacing of interference fringes (with nm-order precision in the case of light).
- Constructive interference (in-phase sources): Δ = mλ
- Destructive interference (in-phase sources): Δ = (m+1/2)λ
- For out-of-phase sources, the conditions for reinforcement and weakening are reversed
- Phase difference δφ = 2πΔ/λ
- Interference is evidence of wave behavior (it applies to light, sound, and electrons)
Points to Note (Easily Confused)
① Do not confuse the conditions for in-phase sources and out-of-phase sources. Check whether the problem states that they “vibrate in phase.” ② “Δ = mλ produces constructive interference” applies only to in-phase sources. ③ Path difference is the “absolute value of the difference in lengths.” Calculate it as |S₁P − S₂P| without worrying about the sign. ④ Constructive and destructive interference concern amplitude; energy is redistributed depending on the location (it does not disappear).
Practice
- Waves with a wavelength of 4.0 cm from two in-phase sources reach point P with a path difference of 6.0 cm. Do they interfere constructively or destructively at P?
- Express the condition for destructive interference at a point whose distances from two in-phase sources of wavelength λ are d1 and d2.
- Explain why “coherent” waves are necessary for interference to occur.