Interference
Section: Superposition | Syllabus: Cambridge AS Level Physics (9702)
What is Interference?
When two or more waves overlap in the same region of space, they combine according to the principle of superposition. The resulting pattern of reinforcement and cancellation is called interference.
Interference The phenomenon that occurs when two or more waves superpose to produce a resultant wave with a displacement that is the sum of the individual displacements.Interference can result in:
- Constructive interference: Waves combine to produce a larger resultant amplitude
- Destructive interference: Waves combine to produce a smaller resultant amplitude (or zero)
Coherence
For a stable, observable interference pattern, the wave sources must be coherent.
Coherence Two wave sources are coherent if they have: (1) the same waveform, (2) the same frequency (or wavelength), and (3) a constant phase difference. For electromagnetic waves, they must also be polarised in the same plane. Why Coherence MattersNon-coherent sources have randomly varying phase differences. The interference pattern shifts rapidly and averages out-no stable pattern is observed. Coherent sources maintain fixed positions of maxima and minima.
Achieving Coherent Light Sources
- Young's double slit: Single source → two slits → two coherent secondary sources (same origin ensures same frequency and constant phase)
- Laser: Inherently coherent; all waves in phase and same polarisation
Two independent light bulbs are NOT coherent-they emit light with random phase variations. Only sources derived from a single source, or laser light, can produce stable interference patterns.
Constructive and Destructive Interference
Constructive Interference
Constructive interference occurs when two waves arrive in phase at a point. The crests of one wave align with the crests of the other.
Conditions for Constructive Interference:Path difference = nλ (where n = 0, 1, 2, 3, ...)
Phase difference = 0, 2π, 4π, ... (or 0°, 360°, 720°, ...)
Result: Maximum amplitude = sum of individual amplitudes
Destructive Interference
Destructive interference occurs when two waves arrive in antiphase at a point. The crests of one wave align with the troughs of the other.
Conditions for Destructive Interference:Path difference = (n + (1/2))λ (where n = 0, 1, 2, 3, ...)
i.e., (λ/2), (3λ/2), (5λ/2), ...
Phase difference = π, 3π, 5π, ... (or 180°, 540°, 900°, ...)
Result: Minimum amplitude Complete Cancellation
Complete destructive interference (zero resultant amplitude) only occurs when the two interfering waves have identical amplitudes. If the amplitudes are different, the resultant will be the difference of the amplitudes.
Path Difference and Phase Difference
The path difference is the difference in distance travelled by two waves from their sources to a point. The phase difference at that point depends on the path difference.
Relationship:Phase difference = (2π/λ) × path difference
or in degrees: Phase difference = (360°/λ) × path difference
Amplitude and Intensity
The intensity of a wave is proportional to the square of its amplitude.
I ∝ A^2where I = intensity and A = amplitude
For two identical waves of amplitude A:
- Constructive interference: Resultant amplitude = 2A, so intensity ∝ (2A)^2 = 4A^2 (4× single wave intensity)
- Destructive interference: Resultant amplitude = 0, so intensity = 0
Conditions for Observable Interference Fringes
To observe a clear two-source interference pattern, the following conditions must be met:
- The sources must be coherent (same frequency, constant phase difference)
- The sources should have approximately equal amplitudes (for clear dark fringes)
- For transverse waves, the waves must be polarised in the same plane (or unpolarised)
- The sources must be close together relative to the distance to the screen (for resolvable fringes with light)
Two-Source Interference Experiments
1. Water Waves in a Ripple Tank
Two dippers oscillating in phase (driven by same motor) create circular waves that overlap.
FIG 8.15: Two-Source Interference in Ripple TankShow: two point sources (dippers) producing circular wavefronts. Mark: lines of constructive interference (where crests meet crests) radiating outward as solid lines, lines of destructive interference (where crests meet troughs) as dashed lines. Label maxima and minima regions. Show the characteristic pattern of alternating maximum and minimum amplitude lines.
Observations:
- Lines of maximum disturbance (constructive) where crests meet crests
- Lines of calm water (destructive) where crests meet troughs
- Pattern is stable because sources are coherent
2. Sound Waves from Two Loudspeakers
Two loudspeakers connected with same polarity to same signal generator produce coherent sound waves.
Observations:
- Walking perpendicular to the speaker line: alternating loud and quiet regions
- Spacing depends on wavelength and speaker separation
- Pure tones produce clearest patterns
3. Microwaves
One transmitter with double slit creates two coherent secondary sources.
FIG 8.16: Microwave Double-Slit InterferenceShow: microwave transmitter → double slit in metal barrier → two coherent sources → receiver/detector that can be moved perpendicular to the wave direction. Show regions of maxima and minima. Include connection to amplifier/loudspeaker for detection.
Observations:
- Detector moved across pattern: alternating maxima and minima
- Can calculate wavelength from spacing and geometry
- Confirms microwaves travel at c = 3.0 × 10⁸ m s⁻¹
4. Light: Young's Double-Slit Experiment
Crucial evidence for wave nature of light (Thomas Young, 1803).
FIG 8.17: Young's Double-Slit SetupShow: monochromatic light source → single slit S (to create coherence) → double slits S₁ and S₂ (separation a) → screen at distance D. Show wavefronts from S₁ and S₂ overlapping. On screen, show alternating bright and dark fringes with fringe separation x marked. Include path difference illustration showing paths from S₁ and S₂ to a point on screen.
FIG 8.18: Double-Slit Fringe PatternShow intensity distribution: equally-spaced bright fringes with dark fringes between. Below, show intensity vs position graph with regular maxima and minima. Mark central maximum (path difference = 0), and label fringe separation x.
Observations:
- Alternating bright and dark fringes on screen
- Central fringe is bright (path difference = 0)
- Fringes equally spaced
- Bright fringes: path difference = nλ
- Dark fringes: path difference = (n + ½)λ
Young's Double-Slit Equation
The fringe spacing relates to wavelength, slit separation, and screen distance.
FIG 8.19: Derivation of Young's EquationShow geometry: two slits S₁ and S₂ separated by distance a, screen at distance D. Two paths from slits to point P on screen (at distance y from centre). Show similar triangles: path difference z relates to a and D, fringe spacing x relates to D. Derivation uses: z/a ≈ y/D and for adjacent bright fringes, z = λ, y = x, giving λ = ax/D.
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