Waves in a Bathtub: The Wave Equation, Reflection and Superposition

A ripple is a disturbance that travels, not water that travels

When you press the surface of bathwater, you don't actually push water outward across the tub — you push it down, and gravity and surface tension pull it back up, overshooting past level and pulling the neighbouring water down in turn. That handoff — down, up, down, up — repeats outward from where you pressed, and it is the pattern of motion, not the water itself, that races across the tub. A rubber duck floating nearby bobs up and down as the ripple passes but ends up almost exactly where it started; it is the disturbance that travels, while the water mostly stays in place.

The wave equation: one rule, everywhere at once

The whole surface obeys a single, simple rule everywhere at once: the wave equation. It says that how fast the height of a point on the water is accelerating depends on how curved the surface is around it — a point sitting in a dip surrounded by higher water gets pulled up, and a point sitting on a bump surrounded by lower water gets pulled down. Apply that one local rule at every point on the tub, over and over, and the large-scale behaviour you see — expanding circular ripples, bouncing off walls, crossing through each other — emerges automatically, with no global instruction telling the wave where to go.

∂²h/∂t²  =  c² · ∇²h

h  = water surface height above rest level
c  = wave speed  ≈  √(g · depth)   for shallow water
∇²h = curvature of the surface (how "dented" it is around a point)

Superposition: why two ripples pass right through each other

Press the water in two different spots and you get two expanding rings that meet in the middle — and where they overlap, they don't collide or bounce apart, they simply add together. This is the principle of superposition, and it holds because the wave equation is linear: the total displacement is just the sum of what each wave would have produced on its own. Where two crests meet the water rises extra high (constructive interference); where a crest meets a trough they can momentarily cancel to nearly flat water (destructive interference). Then each ripple continues on exactly as if the other had never been there, completely unaffected by the encounter.

Bouncing off the walls

A bathtub wall cannot move, so when a ripple reaches it the wave's energy has nowhere to go but back into the water — the wall reflects the wave, sending it back at an angle equal to the one it arrived at, exactly the way a mirror reflects a ray of light. Reflected ripples then superpose with the ripples still spreading outward from the original splash, producing the complicated crisscross patterns you see once a real bathtub has been disturbed for more than a second or two — every criss-cross line is just two or more waves adding together at that instant.

Frequently asked questions

Try it live

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