Phase difference, path difference, and coherence

Two waves leaving one oscillator reach a point at different times, and phase is what a delay costs. A sinusoid advances ω radians every second, so an arrival-time difference Δt appears as δ = ω Δt = 2π f Δt. In a uniform medium of speed v the extra travel time for extra distance Δr is Δr/v, and since v = fλ the frequency cancels: δ = 2π f Δr/v = 2π Δr/λ. That is the familiar conversion, and it carries a warning inside it — λ must be the wavelength in the medium the wave actually crossed, not in air or vacuum. A 1.00 kHz tone in air at 343 m s⁻¹ has λ = 0.343 m, so an extra 0.100 m of path is Δt = 2.92 × 10⁻⁴ s and δ = 2π(1000 Hz)(2.92 × 10⁻⁴ s) = 1.83 rad, which is 0.58π: noticeably out of step, but well short of the π that would cancel.

Write δ = (2π/λ)Δr + Δφ₀ and fix what every symbol means before substituting anything. Both entries have to count the same thing — how far wave 2 sits behind wave 1 — or adding them is meaningless. Choose an order for the sources and keep it: let Δr = r₂ − r₁, so a positive Δr means wave 2 travelled further and therefore lags. Then Δφ₀ must be the lag wave 2 already carried when it left, Δφ₀ = φ₀₁ − φ₀₂, and not the other subtraction: write φ₀₂ − φ₀₁ into this formula and every source offset enters with the wrong sign. Two speakers fed from one amplifier have Δφ₀ = 0; swap the leads on the second and Δφ₀ = π, permanently and everywhere in the room. A wave reflected at a boundary with a denser medium comes away π behind, so that π enters the ledger with a plus sign when it is wave 2 that reflected and a minus sign when it is wave 1. The convention is not decoration. Relabel the sources half way through a problem and δ changes sign, which cos δ forgives but which ruins any statement about which arrival is ahead.

The ledger assumes Δφ₀ is a constant. No real source obliges indefinitely. An atom radiates for a finite time and restarts with a fresh phase; a laser's phase diffuses; even a quartz oscillator drifts. Call τ_c the coherence time, the interval over which the phase stays predictable to within about a radian. It is not a separate property to be measured on its own: a wave train of duration τ_c cannot have a sharp frequency, and Fourier analysis ties the two by Δν τ_c ≈ 1. Narrow the spectrum and the phase holds longer, in strict inverse proportion to the bandwidth. A quartz-referenced 1.00 kHz tone stable to 0.01 Hz has τ_c ≈ 100 s. A green line from a discharge lamp, Doppler-broadened to Δν ≈ 1.5 GHz, has τ_c ≈ 0.67 ns. Sunlight through a 10 nm filter at 550 nm has Δν ≈ 9.9 × 10¹² Hz, so τ_c ≈ 0.10 ps.

Multiply the coherence time by the wave speed and the result is a length, L_c = c τ_c. Substituting Δν = c Δλ/λ² turns it into the working form L_c ≈ λ²/Δλ, and the ratio λ/Δλ counts the fringes available before the contrast dies. The rule of use is blunt: fringes need |Δr| ≪ L_c, they weaken as |Δr| approaches L_c, and beyond it nothing stable survives. White light, λ ≈ 550 nm with Δλ ≈ 300 nm, has L_c ≈ 1.0 µm — about two fringes, which is exactly why a white-light pattern is a few coloured bands and no more. The same lamp behind a 10 nm filter gives L_c = 30 µm and roughly 55 fringes. An isolated Doppler-broadened line at Δν = 1.5 GHz gives L_c = 0.20 m, and a frequency-stabilised laser hundreds of metres.

Now the headline result. Let two sources at the same nominal frequency run from separate oscillators. Their offset Δφ₀(t) is not fixed: it wanders on the scale of τ_c and, over any longer observation, takes every value in [0, 2π) with equal probability. The interference term carries cos δ, so a slow detector records the average (1/T)∫₀ᵀ cos δ(t) dt, and a uniformly distributed phase makes that integral zero. The intensities simply add, everywhere, and no fringes form. Model the wander instead as Gaussian about a mean δ with standard deviation σ, and the average of the cosine is the cosine of that mean reduced by the factor e^(−σ²/2) — the full term at σ = 0, about half of it at σ = 1.2 rad, and 1% of it by σ = 3 rad. None of this is about mismatched frequencies: two generators agreeing to 0.002 Hz at 440 Hz still sweep δ through 2π every 500 s, so a one-second exposure sees a pattern and a ten-minute one sees none. Detect fast enough and even two separate lasers interfere.

Temporal coherence asks how far apart in time two samples of one beam may be taken; spatial coherence asks how far apart across it. An extended source is a crowd of independent emitters, and each one presents its own Δφ₀ to a pair of slits. Two points of the source separated by a, viewed from distance R, reach slits separated by d with path differences that differ by roughly ad/R. Once that difference reaches λ the patterns from the two edges of the source sit a whole fringe apart, the emitters between them fill that interval evenly, and the sum of all of them is flat — so fringes survive only while d ≲ λR/a. That length, ℓₜ ≈ λR/a, is the transverse coherence width, equivalently λ/θₛ with θₛ the angular width of the source. The Sun subtends 9.3 × 10⁻³ rad, so at 550 nm its coherence width at the ground is about 59 µm: pinholes closer together than that make fringes in raw sunlight. Read backwards, the same argument turns an interferometer into a ruler for θₛ, which is how stellar diameters are measured.