Author: Claudia Erhart

Dispersion is an effect which occurs due to the separation of waves caused by their differing rates of travel, it is the variation of phase velocity with wavelength. The dispersion equation for gravity waves c= √(gL/2π) indicates that waves of longer wavelength (L) propagate faster than those of shorter wavelengths. Compared to that, capillary waves (only forced by surface tension) travel faster for shorter wavelengths.

One of the characteristics defining the behavior of a wave is its dispersion relation. The dispersion relation describes the relationship between the wavelength (L) or wavenumber (k=2π/L) of a wave and its angular frequency (ω=2π/T). The dispersion relation of water waves is affected by the two different forces: gravity and surface tension. For long wavelengths or small wavenumbers gravity is the dominant force (=gravity waves). However for short wavelengths where the wavenumber is large, the surface tension (σ) is the main restoring force (capillary wave). For intermediate wavenumbers, both forces may have an effect simultaneously. The figure shows an overview of the dispersion relations for different equatorial wave types.

\begin{equation*} \omega^2 = gk* \frac{e^{kH}-e^{-kH}}{e^{kH}+e^{-kH}} = gk* tanh(kH)} \end{equation*}

With Buoyancy frequency N and where Φ is the angle between the wave vector k and the horizontal plane ( also see internal waves).

This dispersion relation is valid for any wavenumber (for deep water waves).

• 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

The dispersion relation is the same as for shallow water waves.

Münnich, M. (2014): Introduction to Physical Oceanography:Lecture 12
Ocean waves (2014):http://atoc.colorado.edu/~whan/ATOC5051/Class_Notes/chapter4.pdf
Open university (1989): Waves, Tides & Shallow-water processes, Chapter 1
Wikipedia (2014): Dispersion Relation: http://en.wikipedia.org/wiki/Dispersion_relation
Wikipedia (2014): Dispersion water waves: http://en.wikipedia.org/wiki/Dispersion_%28water_waves%29
Wikipedia (2014): Capillary wave waves: http://en.wikipedia.org/wiki/Capillary_wave