Mathematical Treatment of Wave Interference: Phases, Path Difference, and Patterns
1. Basics and assumptions
- Consider two monochromatic, coherent waves of the same angular frequency ω and wavenumber k:
- y1 = A1 cos(kx1 − ωt + φ1)
- y2 = A2 cos(kx2 − ωt + φ2)
- Assume linear superposition applies (medium is linear, no significant damping between sources and observation point).
2. Phase and path difference
- Instantaneous phase of each wave at the observation point:
- θ1 = kx1 − ωt + φ1
- θ2 = kx2 − ωt + φ2
- Phase difference Δθ = θ2 − θ1 = k(x2 − x1) + (φ2 − φ1) = kΔx + Δφ.
- Path difference Δx relates to Δθ by Δθ = (2π/λ)Δx, since k = 2π/λ.
3. Resultant amplitude (equal amplitudes for simplicity)
- For A1 = A2 = A, resultant displacement:
- y = y1 + y2 = 2A cos(Δθ/2) cos(kx − ωt + average phase)
- Resultant amplitude Ar = 2A |cos(Δθ/2)|.
- Intensity I ∝ Ar^2 ∝ 4A^2 cos^2(Δθ/2).
4. Constructive and destructive interference
- Constructive: Δθ = 2mπ → Δx = mλ → Ar = 2A (max intensity).
- Destructive: Δθ = (2m+1)π → Δx = (m + ⁄2)λ → Ar = 0 (min intensity).
5. Unequal amplitudes
- For A1 ≠ A2, use phasors or trigonometric identity:
- Ar = sqrt(A1^2 + A2^2 + 2A1A2 cos Δθ)
- I ∝ Ar^2 = A1^2 + A2^2 + 2A1A2 cos Δθ
6. Two-source interference pattern on a screen (double-slit geometry)
- Geometry: two slits separated by distance d, screen distance D >> d.
- Path difference to a point at angle θ: Δx ≈ d sin θ.
- Phase difference Δθ = (2π/λ) d sin θ.
- Bright fringes: d sin θ = mλ. Dark fringes: d sin θ = (m + ⁄2)λ.
- Fringe spacing (small-angle approximation): y_m ≈ m(λD/d), fringe spacing Δy ≈ λD/d.
7. Thin-film interference (phase shift on reflection)
- When reflecting from a medium with higher refractive index, a π phase shift occurs.
- Effective phase difference includes path through film (2π·2nt/λ) and any π shifts.
- Condition for constructive/destructive depends on whether one or two π shifts occur; use Δθ_total = (4πnt/λ) + phase_shifts.
8. Coherence and visibility
- Visibility V = (Imax − Imin)/(Imax + Imin) = (2A1A2)/(A1^2 + A2^2) for two waves.
- Temporal coherence requires stable phase relation over observation time; spatial coherence relates to source size.
9. Summary formulas
- Δθ = (2π/λ)Δx + Δφ
- Ar = sqrt(A1^2 + A2^2 + 2A1A2 cos Δθ)
- I ∝ Ar^2
- Double-slit: d sin θ = mλ (bright), Δy ≈ λD/d
If you want, I can derive any of the steps symbolically (phasor method, trig identities) or work a numeric example.