Two-Source Interference Lab
H2 Physics · Superposition — two sources, one pattern, every fringe from Δ
Top-down field · S₁ and S₂ radiating · screen at x = D
Along the screen — intensity and the measured fringe spacing
ⓘ more
Solid curve uses the exact path lengths S₁P = √(D² + (y − a/2)²) and S₂P = √(D² + (y + a/2)²). Dashed curve uses the syllabus small-angle form, which replaces Δ with ay/D. They sit on top of each other while D ≫ a; push a up or D down and the dashed one starts to lag, which is exactly the limit of λ = ax/D.
Experiment
two dippers on one bar, water waves
Controls
Probe P on the screen
- position y
- 0.0 mm
- S₁P
- 600.52061 mm
- S₂P
- 600.52061 mm
- Δ = |S₂P − S₁P|
- 0.0 mm
- Δ / λ
- 0.000
Δ is the thick amber piece of the longer path. Integer Δ/λ → crests meet crests.
Fringe spacing — measured vs formula
- measured off the pattern
- 60.354 mm
- λD / a
- 60.000 mm
- difference
- +0.59 %
- λ back out: λ = ax/D
- 5.0 mm
- λ actually set
- 5.0 mm
Measured spacing matches λD/a to 0.6% — the formula is the small-angle case.
Conditions for fringes to be seen
(I_max − I_min) / (I_max + I_min). 1.000 = perfect fringes, 0.000 = no pattern recorded at all.
ⓘ the other two conditions
Same wave type and same plane of polarisation — two waves that cannot cancel each other never make a dark fringe. And a must be small enough that λD/a is wide enough to see; drag a to its top end and the fringes crowd together until the pattern greys out on its own.
What this confirms
- ›LO (a) — superposition only — every pixel is A₁² + A₂² + 2A₁A₂cos(2πΔ/λ), i.e. the two displacements added and squared. Bright is A₁+A₂, dark is A₁−A₂. Nothing new is created; energy is only redistributed.
- ›LO (b) — path difference and phase difference — the probe reads S₁P, S₂P and Δ from true geometry. Δ = nλ gives BRIGHT, Δ = (n+½)λ gives DARK, every time, at any preset.
- ›LO (h) — conditions for observable fringes — break same-frequency or constant-phase and the contrast readout falls to 0.000: a pattern still exists at each instant but nothing steady is recorded. Break equal amplitude and the fringes survive with contrast 0.471 — poor, not absent.
- ›LO (i) — λ = ax/D — the fringe spacing is found by locating adjacent maxima on the computed curve, not by the formula. It matches λD/a, and feeding it back through λ = ax/D returns the λ you set — the actual double-slit measurement.
- ›LO (g) — same physics for every wave — water, microwaves, sound and light differ only in λ, a and D. The pattern depends on the ratios a/λ and D/a, which is why light needs slits a fraction of a millimetre apart to give fringes millimetres wide.
- ›Antinodal and nodal lines are hyperbolae — each drawn locus is the exact set of points with constant Δ, so it has S₁ and S₂ as foci. They fan out from the sources and cut the screen at the bright and dark fringes.
- ›Idealisations kept honest — both sources are modelled as point sources of constant amplitude (no 1/√r fall-off and no single-slit envelope), so the fringes stay equally bright across the field. Where fringes get finer than one screen pixel the shading fades to grey rather than faking detail — that is genuinely what an instrument of that resolution records.