Ocean waves - 2D wave flume

Linear waves

For small waves on water of depth h the surface is η = (H/2) cos(kx − ωt), and the frequency follows the dispersion relation ω² = gk tanh(kh), solved here by Newton iteration. In deep water (kh > π) the speed is c = gT/2π, long waves; in shallow water (kh < π/10) it tends to √(gh). Energy travels at the group speed c_g = (c/2)(1 + 2kh/sinh 2kh).

Orbits, shoaling and breaking

Water particles move on ellipses with horizontal and vertical semi-axes a cosh k(z + h)/sinh kh and a sinh k(z + h)/sinh kh: circles in deep water that flatten to back-and-forth motion near the bed. As the wave runs into shallower water its energy flux E c_g with E = ρgH²/8 stays constant, so the height grows as H = H₀√(c_g0/c_g) (Green's law); the wavelength shortens, and the wave breaks when H exceeds about 0.78 h (or a steepness of 1/7). The second-order Stokes drift u_s = a²ωk cosh 2k(z + h)/(2 sinh² kh) is a slow net forward transport.

Honest simplifications: linear (Airy) theory throughout, with Stokes drift added only as a formula; the seabed is a smooth ramp in a flume drawn over five deep-water wavelengths, so for short waves the slope is steeper than the slowly-varying approximation allows; no refraction, reflection, bottom friction or currents; breaking just caps the height at the limit and draws foam instead of dissipating energy through a surf-zone model; the surface height is exaggerated in the drawing (the factor is shown) while depth is to scale.

• Pick Shoaling and breaking: the crests grow and foam appears where the blue line meets the red dashed limit.
• Pick Wind waves, deep water: the orbits are circles that shrink rapidly with depth.
• Pick Tsunami-like long wave: the orbits become flat, the wave never feels deep water.
• Raise the period and compare phase and group speed on the second graph.