Water Waves on a Sloping Beach

To the landperson waves are most encountered at the beach, or swimming near the beach. Far from the beach the waves are nearly uniform progressive waves with wavelength from a few metres to a few hundred metres and speeds ranging from 4m/s to 20m/s. These waves can be modelled as simple sinusoidal waves approaching the shore as plane waves. As the waves approach the beach, the wavefronts are not necessarily parallel to the line of the beach.

As the waves approach the beach the depth of water is reduced and three features become apparent:

  1. The waves bend towards the beach, or are refracted.

  2. The waves move slower and the distance between successive crests – the wavelength – is reduced.

  3. The amplitude of the wave increases.

As the wave rides up the beach, the amplitude grows more rapidly, the peaks steepen and the wave no longer takes a sinusoidal form. The effect of the beach is to slow most the deepest water, so the top of the wave travels more quickly than the bottom. As the speed is reduced the energy is re - expressed in the increasing amplitude of the wave. Eventually the water curls over the crest and the wave breaks. During this breaking, the energy of the wave is partially lost through turbulence and partially . For waves of initial wavelength 10m, the wave speed falls from around 4m/s to less than 1m/s when the depth is 0.1m. After a wave has broken, the water is projected up the beach where gravity will tend to return it to the main body of water, but before this can happen, the next wave crashes down, sending water further up the beach, including water from the earlier wave. After several occurrences of this, the water projected up the beach rushes back down in a torrent large enough to possibly rush bathers off their feet.

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