Water Ripple Tank

Wave equation on the water surface

For long waves in a shallow layer of depth h the surface height obeys eta_tt = div(c^2 grad eta) - gamma eta_t + q sin(wt) delta(x - x_s) with phase speed c = sqrt(g h). A dipper at frequency f therefore makes waves of wavelength lambda = c / f. The tank is a 192 x 192 grid (0.60 m wide, 3.1 mm cells); the update is the semi-implicit (symplectic) Euler scheme with time step dt = 0.5 dx / (c sqrt 2), so it is stable and conserves energy without drift.

Sources and amplitude

The dippers are soft sources, so waves from one pass through the other. In two dimensions a point source of strength q radiates eta ~ (q / 4 c^2) sqrt(2 / (pi k r)), falling like 1/sqrt r because the same energy spreads over a longer circle. The amplitude slider is the height 2 cm from a dipper; q is computed from it. The plane-wave bar uses the one-dimensional result eta = q / (2 w c).

Interference and diffraction

Two in-phase sources a distance d apart give maxima where the path difference is a whole number of wavelengths, d sin(theta) = m lambda, and nodal lines where it is a half-integer. Between the sources the two waves form a standing pattern with nodes at (2m + 1) lambda / 4 from the centre, which the second check measures. A single slit of width a sends a beam with first nulls at sin(theta) = lambda / a; a double slit multiplies that by the two-slit fringes. The dashed curve in the intensity graph adds Huygens wavelets (amplitude 1/sqrt r, phase k r) for the sources or slit openings.

Refraction at a shelf

Where the water gets shallower the wave slows (c ~ sqrt h), the frequency stays fixed, so the wavelength shrinks and the wave fronts bend towards the normal: sin(theta1) / sin(theta2) = c1 / c2 (Snell's law for water waves). Tilt the shelf edge to see it.

Lamp projection

In a real ripple tank a lamp above the water projects the surface onto a white sheet. Crests act as converging lenses and troughs as diverging lenses, so the picture brightness follows the surface curvature, I ~ 1 + K laplacian(eta). The “Lamp projection” view shows exactly that quantity from the simulated surface.

Honest simplifications

Real ripple-tank waves at a few hertz are capillary-gravity waves and are dispersive (w^2 = (g k + sigma k^3 / rho) tanh(k h)); this model keeps only the non-dispersive long-wave speed sqrt(g h), which holds when k h is small, and amplitudes are small enough for the linear equation. Viscous damping is one constant gamma. The beach is a smooth sponge layer, not a physical slope. Barrier walls reflect perfectly (zero flux). Time runs in slow motion by default.

• Choose Two-source interference and widen the separation: count the nodal lines and compare with d sin(theta) = m lambda.
• Switch to Opposite-phase pair: the centre line becomes a node instead of an antinode.
• Run Double slit, then narrow the slit width until the pattern spreads: diffraction needs a comparable to lambda.
• Pick Refraction at a shelf and tilt the edge: wavelength shrinks by sqrt(h2/h1) and the fronts bend.
• Turn the beach off with the solid wall: a standing wave with nodes lambda/2 apart forms.
• Use the Lamp projection view and compare with what the real classroom screen shows.