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The science
The wave equation
For small disturbances of a thin liquid layer the surface height u(x, y, t) obeys utt = c² ∇²u - γ ut. The first term is the restoring tendency of the surface, the second is viscous damping. In two dimensions a point disturbance produces a ring whose crest moves outward at r = c t, followed by a trailing wake. Energy, E = ½ ∫ (ut² + c² |∇u|²) dA, decays as e-γ t when the damping is uniform and the basin is closed.
Real ripples are dispersive
On real water the speed depends on wavelength: c² = (g/k + σk/ρ) tanh(kh), with surface tension σ = 0.0728 N/m. Long waves are gravity waves, short ones are capillary ripples, and the slowest wave (about 0.23 m/s) has a wavelength near 1.7 cm. The first graph shows this curve; the simulated field uses one constant c (a non-dispersive model), so a pulse keeps its shape instead of spreading into short ripples ahead of long waves.
Honest simplifications: a single wave speed for all wavelengths; lily pads are modelled as extra damping discs, not as floating solids; the shore reflection slider is a numerical sponge, not real wave breaking; heights are small enough for the linear equation to hold (millimetres).
What to try
- Choose Single stone and watch the dashed ring: the crest of the ring follows r = c t.
- Choose Calm basin and see waves reflect from the shore and interfere into standing patterns.
- Raise damping and check that the energy line on the third graph follows the dashed slope.
- Drag a lily pad into the path of the ring and see how much of the wave it absorbs.