Wave Interference in Light, Sound, and Water: A Comparative Guide

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.