Presets
Playback
Parameters
Model checks
The science
Linear water waves
A small-amplitude wave of angular frequency ω = 2π/T and wave number k on water of depth h obeys ω² = g k tanh(k h). In deep water (kh > π) this gives L = g T² / 2π and c = gT/2π; in shallow water (kh < π/10) it gives c = √(g h). Water particles move in closed orbits whose horizontal and vertical semi-axes are a cosh k(z+h) / sinh kh and a sinh k(z+h) / sinh kh: circles in deep water, flattened ellipses near the bed.
Shoaling and breaking
Wave energy travels at the group speed cg = ½ c (1 + 2kh / sinh 2kh). With no losses the energy flux P = ρ g H² cg / 8 is constant, so as the water gets shallower cg falls and H must rise: H / H0 = √(cg0 / cg), which tends to Green's law H ∝ h-1/4 in shallow water. When H reaches about 0.78 h the crest becomes unstable and the wave breaks. The Iribarren number ξ = tanβ / √(H0/L0) tells whether it spills, plunges or surges.
Honest simplifications: waves arrive straight on, so there is no refraction; linear theory is used even near breaking; there is no wave reflection, bed friction or current; after breaking the height is simply held at 0.78 h; the picture exaggerates wave height and orbit size (the factor is printed on the image).
What to try
- Move the probe from deep water to the shore and read how wavelength and speed fall while height rises.
- Choose Steep beach and Gentle swell and compare where the wave breaks and the breaker type.
- Choose Long swell, deep and see the orbits become flattened ellipses when the wave feels the bed.
- Open the second graph: the yellow shoaling curve follows Green's law in shallow water and stops at the red breaking limit.