Wave Flume Lab (2D)

Waves in a flume

A paddle at the left end of a 12 m flume makes regular waves of height H and period T. The water is h deep over a flat section and then rises along a beach of slope 1:n. The page uses linear (Airy) wave theory with a slowly varying depth.

Dispersion relation

The angular frequency w = 2 pi / T and the wavenumber k = 2 pi / L are linked by w^2 = g k tanh(k h). In deep water (h > L/2) this gives w^2 = g k and the phase speed c = g T / (2 pi): long waves outrun short ones. In shallow water (h < L/20) it gives c = sqrt(g h), independent of the period. The energy travels at the group speed c_g = 1/2 (1 + 2 k h / sinh 2 k h) c.

Energy, shoaling and breaking

The wave energy density is E = 1/8 rho g H^2 and the power per metre of crest is P = E c_g. Without losses P is constant, so as the water shallows and c_g falls the height rises: H ~ h^(-1/4) in shallow water (Green's law). When H reaches about 0.78 h the wave breaks; the page caps the height at that value and marks the breaking point.

Particle motion

The water particles move in closed orbits with velocities u = a w cosh(k(z+h))/sinh(kh) cos(theta) and w = a w sinh(k(z+h))/sinh(kh) sin(theta), where a = H/2: circles in deep water, flatter ellipses in shallow water, and back-and-forth sliding at the bed. Following a particle through the wave field gives a small net drift, the Stokes drift a^2 w k exp(2 k z) in deep water.

Numerical checks on this page

(1) The phase speed measured by following a crest in the simulated surface equals w/k from the dispersion relation. (2) For a long wave on the beach the computed height growth equals Green's law. (3) A marker followed for 10 periods drifts at the Stokes drift speed.

Honest simplifications

The model is linear theory with a WKB (slowly varying depth) amplitude, so there is no reflection from the beach, no bottom friction and no real breaking physics (only a height cap); crests are sinusoidal rather than peaked, and the flume is two-dimensional. The vertical scale is exaggerated in the 2D view so that centimetre waves can be seen.

• Deep-water swell and Short ripples: compare the speeds in graph 3; the shorter wave is slower.
• Shallow-water waves: the phase and group speeds both approach sqrt(g h).
• Shoaling on a gentle beach: move the probe towards the shore and watch the height grow in graph 2.
• Steep waves break: the height is capped at 0.78 of the depth and a red line marks the breaking point (foam in 3D).
• Follow a yellow marker: it moves in an orbit that shrinks with depth (graph 4).