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Sandbox Physics

Waves · superposition

Wave Interference

Two point sources emit in sync, with surface height showing the instantaneous field. They reinforce at integer-wavelength path differences and cancel at half wavelengths. The upright screen shows time-averaged intensity.

Interactive modelWave Interference
Path difference0.0 px
In wavelengths / λ\lambda0.00
Probe field+0.00
Fringe spacing0 px

Physics tutorial

Wave interference: how phase becomes fringes

BackgroundLinear waves add amplitudes when they meet. A detector records the time average of the squared result, turning path difference into intensity contrast.

Why it mattersInterference measures tiny changes in distance, refractive index, and phase, and prepares the intuition needed for quantum amplitudes.

Start with the essentials

Focus question
If both sources emit waves, why do some locations remain dark?
One-sentence intuition
Energies do not add first; signed field amplitudes add first, and the result is then squared.

Core mathematical model

Field and intensity

u=u1+u2,I∝⟨u2⟩tu=u_1+u_2,\qquad I\propto\langle u^2\rangle_t

In-phase amplitudes reinforce and opposite amplitudes cancel, while intensity remains nonnegative.

Far-field fringe spacing

Δy≈λLd\Delta y\approx\frac{\lambda L}{d}

Larger wavelength λ\lambda spreads fringes; larger source separation dd compresses them.

Common difficulties

Cancellation does not destroy energy

Typical misconceptionThe two waves erase energy at a dark fringe.

Better mental modelInterference redistributes energy; what is missing from dark regions appears in neighboring bright regions.

Stable phase is required

Typical misconceptionAny two equal-frequency sources make a stable pattern.

Better mental modelRelative phase must remain stable during observation or time averaging washes the pattern out.

Run the experiment

  1. 01

    Find zero path difference

    Keep the sources in phase and place the probe on their bisector.

    What to observe: Path difference is zero and the central region reinforces.
  2. 02

    Lengthen the wavelength

    Increase λ\lambda gradually.

    What to observe: Wavefront and bright-fringe spacing both grow.
  3. 03

    Separate the sources

    Hold λ\lambda fixed and increase dd.

    What to observe: The screen fringes become denser.