PHYS·04 Echo — the real wave equation on a grid: tap to drop small pulses that travel outward, bounce off the pool walls, cross back over the water, and bounce again; flick on the bench to throw a wave packet in the direction you aim

Physics Lab · Experiment 04 · wave equation

Waves that come back.

03 painted a standing pattern; this pool is the real thing. Tap and a small pulse sets off across the water, hits a wall, reflects, crosses back, and bounces again — fainter each pass, the way a shout dies in a courtyard. Down at the bench you can flick a packet in exactly the direction you aim.

utt = c²∇²u
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[00]The pool

the actual PDE this time — waves that travel, not a pattern that shimmers

Equation
utt = c²∇²u · finite differences on a grid, two substeps per frame — every ripple you see is the solver, not an effect
Walls
Neumann boundaries (∂u/∂n = 0) — a crest arrives as a crest and leaves as a crest, like water against the pool edge
Momentum
a flick writes the pulse into two moments of time — now and just-before, offset along your aim — and the equation does the throwing
Stability
Courant kept at c² = 0.28 ≤ ½ — cross it and the pool explodes into static
Damping
every step multiplies by slightly-less-than-one — the slider decides how many wall trips an echo survives
Status
🟢 echoing · pauses offscreen · reduced-motion turns taps into a photograph of the echo one second later
[01]Six returns

everything a bounded pool knows how to do

01 · the equation

Acceleration equals curvature

Each cell accelerates toward the average of its neighbours. That single local rule — no rays, no reflection code — produces travel, bouncing, diffraction, everything on this page.

utt = c²∇²u
02 · the bounce

Walls for free

Reflection isn't programmed. Mirror the edge cells and the maths turns waves around by itself.

03 · the flick

Aim is two timestamps

To throw a packet, write the same bump into "now" and, slightly upstream, into "a moment ago". The solver reads that as velocity and carries it off in your direction.

04 · crossings

Echoes pass through echoes

Two returning wavefronts meet, brighten where crests agree, cancel where they don't, and continue unharmed — 03's superposition, now in motion.

05 · stability

The speed limit

Information may cross at most one cell per step. Break Courant and the sim doesn't drift — it detonates.

c²·Δt²/Δx² ≤ ½
06 · going quiet

Why courtyards stop shouting

Each pass costs a little energy. Set damping low and one tap keeps the pool ringing for a dozen wall trips; set it high and echoes die at the first return. The slider is the room's furniture.

[02]The bench

tap = drop a pulse · flick = throw a packet where you aim · damping slider quiets the hero pool too

damping 18
Field note: 03 and 04 look like siblings but are different animals. Interference summed sin(kd − ωt) per source — a closed-form answer with no walls and no travel time. Echo integrates the wave equation itself, so pulses have a position, a speed, and a history: they take real time to cross the pool, and the walls give them back. Same LUT, same warm water — but this one remembers where you touched it.