On this page:
Introduction The Principle of Superposition Interference: When Waves Combine Young's Double-Slit Experiment Coherence Path Difference and Phase Difference Single-Slit Diffraction Diffraction Gratings Refraction: Bending at Boundaries Total Internal Reflection Dispersion: Separating Colors Polarization Real-World ApplicationsImagine you're in a bathtub. You make a wave by pushing water at one end. Your friend makes another wave at the other end. The two waves travel toward each other across the water. What happens when they meet in the middle?
Do they bounce off each other like billiard balls colliding? Do they destroy each other? Or something else entirely?
The answer is beautiful: they pass right through each other. While overlapping, their effects add together. After they've passed through, each wave continues as if nothing happened.
This simple behavior, called superposition, leads to some of the most stunning phenomena in physics. Interference patterns in light prove it's a wave. Diffraction allows sound to bend around corners. Anti-reflective coatings on eyeglasses. Holography. Even the colors you see in soap bubbles.
Let's explore what happens when waves interact with each other and with boundaries.
y_total = y₁ + y₂ + y₃ + ...
When two or more waves overlap at a point, the total displacement at that point equals the sum of the displacements from each wave.
If both waves push upward at a point, the result is a larger upward displacement. If one pushes up while the other pushes down, they partially cancel. If they push with equal strength in opposite directions, they cancel completely—resulting in zero displacement at that instant.
This adding up of displacements is how noise-cancelling headphones work. They detect incoming sound waves and generate opposite waves that cancel them through destructive interference.
Waves meet in phase (crest with crest). Amplitudes add together → larger amplitude.
Path difference = nλ (n = 0, 1, 2, ...)
Sound: louder. Light: brighter.
Waves meet out of phase (crest meets trough). Amplitudes subtract → smaller or zero amplitude.
Path difference = (n + ½)λ (n = 0, 1, 2, ...)
Sound: quieter or silent. Light: dimmer or dark.
In 1801, Thomas Young proved that light is a wave.
Light passing through two slits produced an interference pattern — alternating bright and dark bands (fringes) on the screen. Only waves create interference patterns.
Fringe separation: Δy = λD/d
Bright fringes: y = nλD/d
Dark fringes: y = (n + ½)λD/d
λ = wavelength, D = distance to screen, d = slit separation
Example: Red light (650 nm) through slits separated by 0.30 mm, screen 2.0 m away. Find fringe spacing.
Δy = λD/d = (650 × 10⁻⁹)(2.0)/(0.30 × 10⁻³) = 4.3 mm
Coherent sources maintain a constant phase relationship. They must have:
Lasers are highly coherent. Two independent light bulbs aren't coherent — any interference pattern would shift trillions of times per second, appearing as a blur.
In Young's experiment, using one light source to illuminate both slits automatically creates coherent sources.
Phase difference (radians) = (2π/λ) × path difference
Example: Two speakers emit 500 Hz sound. You stand 3.0 m from one and 3.4 m from the other. Is the interference constructive or destructive? (v_sound = 340 m/s)
λ = v/f = 340/500 = 0.68 m
Path difference = 0.4 m → 0.4/0.68 ≈ 0.59 ≈ λ/2 → nearly destructive interference
Diffraction is the spreading of waves when they pass through an opening or around an obstacle.
Single-slit diffraction pattern: Wide central maximum with progressively dimmer fringes on either side.
Dark fringes: a sin θ = nλ (n = 1, 2, 3, ...)
a = slit width, θ = angle from center
Example: Light (600 nm) through a 0.02 mm slit. Find angle to first dark fringe.
a sin θ = nλ → (0.02 × 10⁻³) sin θ = (1)(600 × 10⁻⁹)
sin θ = 0.03 → θ = 1.72°
Key observations: Narrower slit → more spreading. Longer wavelength → more spreading.
Grating equation: d sin θ = nλ
d = slit spacing, n = order (0, 1, 2, ...)
Example: A grating with 500 lines/mm is illuminated by 550 nm light. Find angle to first-order maximum.
d = 1/(500 × 10³) = 2.0 × 10⁻⁶ m
sin θ = (1)(550 × 10⁻⁹)/(2.0 × 10⁻⁶) = 0.275 → θ = 15.9°
Application: Spectrometers use diffraction gratings to separate light into component wavelengths. Each element emits unique wavelengths — a "fingerprint" — allowing astronomers to identify elements in distant stars.
Snell's Law: n₁ sin θ₁ = n₂ sin θ₂
Refractive index: n = c/v
Common refractive indices:
Example: Light travels from air (n = 1.0) into glass (n = 1.5) at 40° to the normal. Find refraction angle.
(1.0) sin 40° = (1.5) sin θ₂ → sin θ₂ = 0.643/1.5 = 0.429 → θ₂ = 25.4°
Important: Frequency stays constant. Wavelength changes because speed changes: λ₂/λ₁ = v₂/v₁ = n₁/n₂
Critical angle: sin θ_c = n₂/n₁ (where n₁ > n₂)
Example: Find critical angle for glass (n = 1.5) to air (n = 1.0).
sin θ_c = 1.0/1.5 = 0.667 → θ_c = 41.8°
Light hitting the boundary at > 41.8° reflects completely back into the glass.
Applications:
Dispersion: The separation of white light into component colors because different wavelengths have different refractive indices.
Rainbow: Water droplets act as tiny prisms. Sunlight enters, refracts, reflects off the back surface, and refracts again on exit, creating the rainbow arc.
Malus's Law: I = I₀ cos²θ
Polarization applies only to transverse waves. Light oscillates perpendicular to its direction of travel.
Unpolarized light: Oscillations in all perpendicular directions randomly.
Polarized light: Oscillations restricted to one specific plane.
Methods: Polarizing filters, reflection, scattering.
Example: Polarized light (intensity I₀) passes through a polarizer at 30° to its polarization direction. Find transmitted intensity.
I = I₀ cos²(30°) = I₀(0.866)² = 0.75I₀