Dispersion (water waves)

In fluid dynamics, dispersion of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds.

As a result, water with a free surface is generally considered to be a dispersive medium.

This section is about frequency dispersion for waves on a fluid layer forced by gravity, and according to linear theory.

A sine wave with water surface elevation η(x, t) is given by:[2] where a is the amplitude (in metres) and θ = θ(x, t) is the phase function (in radians), depending on the horizontal position (x, in metres) and time (t, in seconds):[3] where: Characteristic phases of a water wave are: A certain phase repeats itself after an integer m multiple of 2π: sin(θ) = sin(θ+m•2π).

The dispersion relation will in general depend on several other parameters in addition to the wavenumber k. For gravity waves, according to linear theory, these are the acceleration by gravity g and the water depth h. The dispersion relation for these waves is:[6][5]

An initial wave phase θ = θ0 propagates as a function of space and time.

According to linear theory for waves forced by gravity, the phase speed depends on the wavelength and the water depth.

In the left figure, it can be seen that shallow water waves, with wavelengths λ much larger than the water depth h, travel with the phase velocity[2] with g the acceleration by gravity and cp the phase speed.

Since this shallow-water phase speed is independent of the wavelength, shallow water waves do not have frequency dispersion.

Since the phase speed satisfies cp = λ/T = λf, wavelength and period (or frequency) are related.

As a result, the group velocity is, for the limit k1 → k2 :[10][11] Wave groups can only be discerned in case of a narrow-banded signal, with the wave-number difference k1 − k2 small compared to the mean wave number ⁠1/2⁠ (k1 + k2).

The complete theory for linear water waves, including dispersion, was derived by George Biddell Airy and published in about 1840.

A similar equation was also found by Philip Kelland at around the same time (but making some mistakes in his derivation of the wave theory).

[15] The shallow water (with small h / λ) limit, ω2 = gh k2, was derived by Joseph Louis Lagrange.

[16] For two homogeneous layers of fluids, of mean thickness h below the interface and h′ above – under the action of gravity and bounded above and below by horizontal rigid walls – the dispersion relationship ω2 = Ω2(k) for gravity waves is provided by:[17] where again ρ and ρ′ are the densities below and above the interface, while coth is the hyperbolic cotangent function.

For the case ρ′ is zero this reduces to the dispersion relation of surface gravity waves on water of finite depth h. When the depth of the two fluid layers becomes very large (h→∞, h′→∞), the hyperbolic cotangents in the above formula approaches the value of one.

To the third order, and for deep water, the dispersion relation is[19] This implies that large waves travel faster than small ones of the same frequency.

The dot product k•V is equal to: k•V = kV cos α, with V the length of the mean velocity vector V: V = |V|.

Sinusoidal wave.
Dispersion of gravity waves on a fluid surface. Phase and group velocity divided by shallow-water phase velocity √ gh as a function of relative depth h / λ .
Blue lines (A): phase velocity; Red lines (B): group velocity; Black dashed line (C): phase and group velocity √ gh valid in shallow water.
Drawn lines: dispersion relation valid in arbitrary depth.
Dashed lines (blue and red): deep water limits.
Dispersion of gravity waves on a fluid surface. Phase and group velocity divided by deep-water phase velocity √ gλ / (2 π ) as a function of relative depth h / λ .
Blue lines (A): phase velocity; Red lines (B): group velocity; Black dashed line (C): phase and group velocity √ gh valid in shallow water.
Drawn lines: dispersion relation valid in arbitrary depth.
Dashed lines (blue and red): deep water limits.
Frequency dispersion in bichromatic groups of gravity waves on the surface of deep water. The red square moves with the phase velocity , and the green circles propagate with the group velocity.
The number of waves per group as observed in space at a certain moment (upper blue line), is different from the number of waves per group seen in time at a fixed position (lower orange line), due to frequency dispersion.
North Pacific storm waves as seen from the NOAA M/V Noble Star, Winter 1989.
Frequency dispersion of surface gravity waves on deep water. The superposition (dark blue line) of three sinusoidal wave components (light blue lines) is shown.
Dispersion of gravity-capillary waves on the surface of deep water. Phase and group velocity divided by as a function of inverse relative wavelength .
Blue lines (A): phase velocity, Red lines (B): group velocity.
Drawn lines: dispersion relation for gravity-capillary waves.
Dashed lines: dispersion relation for deep-water gravity waves.
Dash-dot lines: dispersion relation valid for deep-water capillary waves.
Wave motion on the interface between two layers of inviscid homogeneous fluids of different density, confined between horizontal rigid boundaries (at the top and bottom). The motion is forced by gravity. The upper layer has mean depth h and density ρ , while the lower layer has mean depth h and density ρ . The wave amplitude is a , the wavelength is denoted by λ .