Interference of Light (Superposition)
Interactive demonstration of wave interference and superposition. Explore how phase difference, amplitude ratio, and wavelength affect the combined wave pattern—creating constructive and destructive interference.
- Phase Difference (φ): Controls the phase offset between the two waves (0° to 360°)
- Amplitude Ratio: Sets the amplitude of Wave 2 relative to Wave 1 (0 to 2)
- Wavelength Ratio: Adjusts Wave 2's wavelength relative to Wave 1 (creates beat patterns)
- Constructive Interference: Set φ = 0° or 360° for maximum amplitude
- Destructive Interference: Set φ = 180° with equal amplitudes for cancellation
- Phasor Diagram: Toggle to view rotating phasor arrows
- Animation: Click Start to see waves propagate over time
📚 Physics Background
🌊 Wave Superposition Principle
The superposition principle states that when two or more waves meet at a point, the resultant displacement is the algebraic sum of the individual wave displacements. This fundamental principle applies to all linear wave phenomena, including light, sound, and water waves.
For electromagnetic waves, this principle governs how light waves combine, leading to interference patterns that depend on the phase relationship between waves.
📐 Mathematical Formulation
For two sinusoidal waves with the same frequency, the electric fields are:
E₁(x, t) = A₁ cos(kx - ωt)
E₂(x, t) = A₂ cos(kx - ωt + φ)
The resultant wave is:
E_total = E₁ + E₂
Where:
- A₁, A₂ are the amplitudes of the individual waves
- k = 2π/λ is the wave number
- ω = 2πf is the angular frequency
- φ is the phase difference between waves
🎯 Interference Types
✅ Constructive Interference
Condition: φ = 0°, 360°, 720°... (path difference = nλ)
Physical meaning: When waves are in phase, their crests align with crests and troughs with troughs. The resultant amplitude is the sum of individual amplitudes, creating maximum intensity.
Result: A_resultant = A₁ + A₂ (bright fringes in optical interference)
❌ Destructive Interference
Condition: φ = 180°, 540°... (path difference = (n+½)λ)
Physical meaning: When waves are out of phase by half a wavelength, crests align with troughs. The waves cancel each other out.
Result: A_resultant = |A₁ - A₂| (dark fringes when A₁ = A₂)
〰️ Partial Interference
Condition: Any other phase difference
Physical meaning: The resultant amplitude lies between the maximum (constructive) and minimum (destructive) values, creating intermediate intensity.
🔄 Phasor Representation
Waves can be represented as rotating vectors (phasors) in the complex plane. Each phasor has:
- Length proportional to the wave amplitude
- Angle representing the phase at time t
- Rotation at angular frequency ω
The resultant phasor is the vector sum of individual phasors. This geometric approach makes it easy to visualize how phase differences affect the combined wave amplitude.
🔬 Applications of Interference
- Young's Double Slit: Classic demonstration producing bright and dark fringes
- Thin Film Interference: Colors in soap bubbles and oil slicks
- Anti-reflective Coatings: Destructive interference reduces unwanted reflections
- Interferometry: Precise measurements using interference patterns
- Holography: 3D imaging using interference of coherent light
- Noise Cancellation: Audio waves combined destructively to reduce sound
🌟 Beat Phenomenon
When two waves with slightly different frequencies interfere, they produce beats—periodic variations in amplitude. The beat frequency equals the difference between the two wave frequencies:
f_beat = |f₁ - f₂|
This effect is commonly used in music for tuning instruments and in physics for precise frequency measurements.