Wave Interference - 2D Map and Section

The 3D sea surface is a sum of three sinusoids with the exact same formula used here: eta(x,z,t) = H [0.48 sin(0.42 x + 1.4 w t) + 0.34 sin(0.29 z - 0.86 w t) + 0.28 sin(0.19 (x+z) + 0.57 w t)], with H the wave-height slider and w the wind slider. Each component is a plane wave sin(k.r + omega t): wavelength 2 pi / |k| and phase speed omega / |k| (about 3.3 w, 3.0 w and 2.9 w metres per second here, so nearly non-dispersive).

Because the components have different frequencies the surface never repeats exactly: crests add where phases coincide (up to 1.10 H peak) and cancel elsewhere. The mean-square elevation is the sum of the component powers, <eta^2> = H^2 (0.48^2 + 0.34^2 + 0.28^2)/2 = 0.2122 H^2, which is the basic statistics used for significant wave height H_s = 4 sqrt(<eta^2>).

The buoy follows the surface in heave and tilts with the local slope (grad eta). Simplifications: kinematic surface only; no wave-buoy interaction, no breaking, no dispersion relation omega^2 = g k tanh(k h) (shown for comparison in the checks panel only).

• Switch off two components and watch a single plane wave.
• Raise wind to speed everything up; wavelength stays the same.
• Move the buoy marker and compare the heave record.
• Look for the highest crest on the map: it is where all three phases line up.