Deep-Water Dispersion

Plate 9 speaks for itself: “The crests outrun the wave.”

The Physics

On deep water, longer waves travel faster. The dispersion relation makes the group, which carries the energy, move at half the phase speed, so individual crests appear at the back of a group, run forward and vanish at the front.

ω² = g k, c_g = ½ c_p

• ω: angular frequency, rad s⁻¹
• g: gravitational acceleration, m s⁻²
• k: wavenumber, rad m⁻¹
• c_p: phase speed √(g/k), m s⁻¹
• c_g: group speed, m s⁻¹

c_p = √(g/k): crests run twice as fast as the group.

Honest caveat: Deep water means depth larger than about half a wavelength.

Sources

• H. Lamb, Hydrodynamics, 6th ed., §§229–236.

On the poster

Plate 009, “Deep-Water Dispersion”: the pixel-art plate with its slogan “The crests outrun the wave.”, the key equation and the facts panel shown above. Pixelated Physics. Physics. Personified.