Wave Interference: How Superposition Builds Young's Fringes

Superposition: waves just add

Every wave phenomenon in this simulation reduces to one rule: where two or more waves overlap, the total displacement is the sum of the individual displacements. That is the superposition principle, and it holds for any linear medium -- water, sound, light, the electromagnetic field. It costs nothing to state and it is the entire explanation for interference: crests adding to crests make a bigger crest (constructive interference), a crest overlapping a trough cancels toward flat water (destructive interference), and everything in between blends continuously.

For two point sources emitting in phase, whether a given point sees constructive or destructive interference depends only on the path-length difference, Δr -- how much farther that point is from one source than the other, measured in wavelengths λ:

constructive (bright fringe):  delta r = m * lambda           m = 0, +-1, +-2, ...
destructive (dark fringe):    delta r = (m + 1/2) * lambda   m = 0, +-1, +-2, ...

Along the perpendicular bisector between the sources, delta r = 0 for every point, so that whole line is a bright fringe (m = 0) -- the central maximum. Moving off-axis, delta r grows, and the point sweeps through bright and dark fringes in order, tracing out the classic hyperbolic interference pattern radiating from a two-source setup.

Young's double-slit experiment

Thomas Young ran the light version of this in 1801, and it is the experiment that settled (for over a century) that light is a wave. Two narrow, closely spaced slits illuminated by a single coherent source act as two synchronised point emitters -- exactly the two-source picture above -- and the resulting fringes on a distant screen are spaced by:

y = m * lambda * L / d

y   fringe position on the screen, measured from the centre
L   distance from the slits to the screen
d   slit separation
m   fringe order (0, 1, 2, ...)

The pattern only forms cleanly if the two sources are coherent -- same frequency, fixed phase relationship -- which is why Young split a single beam into two rather than using two separate lamps; two independent ordinary sources drift in relative phase too fast for any fringe to hold still. Lasers make coherent double-slit setups trivial today, but in 1801 splitting one wavefront in two was the only practical trick available.

Standing waves and beats: interference in one dimension

Interference is not limited to two separated sources. A wave reflecting off a fixed boundary interferes with its own reflection travelling the other way; when the two counter-propagating waves have equal amplitude, the sum is a standing wave -- fixed points (nodes) that never move and points between them (antinodes) that oscillate with maximum amplitude, but no visible left-or-right travel. This is exactly how a guitar string or an organ pipe picks its resonant frequencies: only wavelengths that fit the boundary conditions with a node at each fixed end survive.

A related effect, beats, appears when two sources have slightly different frequencies f1 and f2 instead of the same one. Their sum drifts in and out of phase at the difference frequency, producing a slow throb in amplitude at |f1 - f2| -- audible as the classic wavering "wah-wah-wah" when tuning two guitar strings toward the same pitch, and exploited by piano tuners for centuries before electronic tuners existed.

Simulating the field

The canvas on this page evaluates the superposition directly at every pixel, every frame: for each source i at distance r_i from a point, add a travelling-wave contribution A * sin(k*r_i - omega*t) / sqrt(r_i) (the 1/sqrt(r) factor is the correct 2-D geometric spreading for a circular wave, keeping total energy roughly conserved as the wavefront expands). Summing contributions from every source at every pixel is O(sources x pixels) per frame, cheap enough for real time with a handful of sources, and it reproduces the exact hyperbolic fringe geometry without ever explicitly computing path differences -- the interference falls out of the arithmetic for free.

Frequently asked questions

Try it live

Everything above runs in your browser — open Wave Interference and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.