Interference is the phenomenon that occurs when two or more waves from coherent sources overlap in space, producing a resultant wave whose amplitude at each point is determined by the Principle of Superposition.
Principle of Superposition: When two waves meet at a point, the resultant displacement is the algebraic sum of the individual displacements:
y=y1+y2
Interference is a direct consequence of the wave nature of the medium (water, sound, light, microwaves, etc.).
Occurs when two waves arrive in phase (phase difference = 0,2π,4π,…).
- Crests meet crests; troughs meet troughs.
- Resultant amplitude = y1+y2 (maximum).
- Path difference condition: Δx=nλ where n=0,1,2,3,…
Occurs when two waves arrive completely out of phase (phase difference = π,3π,5π,…).
- Crests meet troughs.
- Resultant amplitude = ∣y1−y2∣ (minimum; zero if amplitudes are equal).
- Path difference condition: Δx=(n+21)λ where n=0,1,2,…
ϕ=λ2π×Δx
For example, a path difference of λ/2 gives a phase difference of π radians (destructive interference).
Energy is not destroyed in destructive interference. It is redistributed: energy is transferred from regions of destructive interference (minima) to regions of constructive interference (maxima). The total energy across the pattern is conserved.
For two sources to produce a stable, observable interference pattern, they must be:
- Coherent — same frequency and a constant phase difference over time.
- Monochromatic — single wavelength (especially important for light).
- Similar amplitudes — for maximum fringe visibility/contrast.
Two independent light bulbs or two independent loudspeakers cannot produce a stable pattern because their phase difference changes randomly and rapidly (incoherent sources).
- Two vibrating dippers (coherent sources) create circular ripples.
- Overlapping ripples form a pattern of nodal lines (calm water — destructive) and antinodal lines (maximum disturbance — constructive).
- Path difference at each point determines the type of interference.
- Two speakers connected to the same signal generator act as coherent sources.
- Walking along a line parallel to the speakers, one alternately hears loud (constructive) and quiet (destructive) regions.
- Condition: path difference from the two speakers determines loud/quiet points.
- A single light source illuminates two narrow slits, which act as coherent secondary sources.
- On a screen, alternating bright fringes (constructive) and dark fringes (destructive) are observed.
- This experiment provided key evidence for the wave nature of light.
- A single microwave transmitter illuminates two slits in a metal plate.
- A receiver moved along a line detects alternating strong and weak signals.
- Microwaves (wavelength ∼3 cm) make fringe spacing large enough to measure easily in a lab.
Since intensity is proportional to the square of amplitude (I∝A2):
- Constructive: resultant amplitude =2A → intensity ∝(2A)2=4A2 (four times one source).
- Destructive: resultant amplitude =0 → intensity =0.