Dispersion of water waves

Long waves travel faster! The speed c of water waves is given by c2 = (g/k) tanh kh

where g is the acceleration due to gravity, h is the water depth, the wavenumber k = 2 π / λ and λ is the wavelength. The speed c is a monotonically decreasing function of k and hence a monotonically increasing function of λ. In other words, the larger the wavelength of a wave the faster it will travel.

In general, an elevation of the free surface can be thought of as being made up of components with all wavelengths. As an initial elevation collapses the long-wave components are seen first and move rapidly away. Shorter waves follow on behind.

The three animations below are for three different initial elevations of the free surface; in each case there is symmetry about the peak of the elevation and only the right-hand half is shown. The three initial elevations have increasingly peaked profiles. The more sharply peaked the initial profile, the larger the amplitude of the short-wavelength components.

(Click on the bottom-left button to play each animation.)