Refraction/Diffraction

# Shoaling, Refraction, and Diffraction of Waves

## Introduction

Water waves are characterized by their height, their length, and their period. The wave height is the distance between the trough (lowest part) and crest (highest part) of the wave. The wave length is the distance between wave crests. The wave period is the time for two consecutive crests to pass a point.

Waves are efficient carriers of energy, imparted by the wind most commonly. They transport very little water; most of the water motion is in nearly closed elliptical paths. (See Dalrymple's Java applet: Velocities Under Water Waves for an example: try wave height of 5 m, wave period 20 seconds, and water depth of 10 m.) The energy per unit surface area in a wave is related to the square of the wave height. The speed at which the waves carry this energy is related to the wave speed, which is the wave length divided by the wave period (since the wave has to travel one wave length every wave period).

Shoaling and refraction of waves occur when the waves are in shallow water. If the water depth is less than half the wave length, then the wave is considered to be in shallow water. In the deep ocean, tsunamis (earthquake-generated waves) are considered shallow water waves (1). When the waves move into shallow water, they begin feel the bottom of the ocean.

## Shoaling and Breaking

Shoaling occurs as as the waves enter shallower water. The wave speed and wave length decrease in shallow water, therefore the energy per unit area of the wave has to increase, so the wave height increases. The wave period remains the same in shoaling. (The wave period is the time it takes for a wave crest to travel the distance of one wave length.) When the wave crest becomes too steep, it becomes unstable, curling forward and breaking. This usually happens when the height of the wave becomes about the same size as the local water depth. That is, a 10 ft high wave usually breaks in about 10 feet of water. Figure 1 shows the progression of a wave as it comes towards the shore.(2)

## Refraction

Refraction is the bending of waves because of varying water depths underneath. The part of a wave in shallow water moves slower than the part of a wave in deeper water. So when the depth under a wave crest varies along the crest, the wave bends.

An example of refraction is when waves approach a straight shoreline at an angle. The part of the wave crest closer to shore is in shallower water and moving slower than the part away from the shore in deeper water. The wave crest in deeper water catches up so that the wave crest tends to become parallel to the shore.

Wave refraction also occurs around a circular island. The wave approaching from one direction will wrap around the island so the wave crest will approach the beach close to parallel on all sides of the island (figure 3) (1). In Figure 2, the wave crests are shown (the first crest is a horizontal line at the top of the figure), and the vertical lines are wave orthogonals (lines which would be traced out by following the wave direction). Note the criss-crossed wave pattern behind the island.

## Diffraction

Diffraction usually happens when waves encounter surface-piercing obstacle, such as a breakwater or an island. It would seem that on the lee side of the island, the water would be perfectly calm; however it is not. The waves, after passing the island in Figure 3, turn into the region behind the island and carry wave energy and the wave crest into this so-called 'shadow zone.' The turning of the waves into the sheltered region is due to the changes in wave height (say along the crest) in the same wave.

If the sides of the island are sloping under the water, then refraction would also be present.

#### Figure 3. Wave Diffraction.

The pictures shown here were taken from the first two references below. For a more elaborate explanation, see Dean and Dalrymple (1991).

### References

1. Bascom, W. Waves and beaches: the dynamics of the ocean surface. Garden City, NY: Doubleday & Co., Inc., 1964.

2. Komar, P. Beach processes and sedimentation. Englewood Cliffs, NJ: Prentice-Hall, Inc., 1976.

3. Dean, R.G. and R.A. Dalrymple, Water wave mechanics for engineers and scientists. Singapore: World Scientific, 1991.