Representing Breaking Waves In Computer Graphics / Daniel Blacker / 07/03/05 / Page 3

 

1. Abstract 2.Introduction 3.Ocean Waves 4.Waves in CG 5.Tool Development 6. Conclusion 7. References 8. Code

3.1 Overview

In order to accurately simulate breaking ocean waves in 3d, we must first look at the principles behind the propagation and dynamics of water waves.

When developing our tool we can study the swell that breaks on Bournemouth sea-front as a real life example. The principles surrounding these waves are the same that affect any breaking ocean wave regardless of scale, even those of the 20ft waves that break off the reefs of Hawaii.


Fig.2 - Waves Breaking on Bournemouth Sea Front

Looking at Figure.2 the first sign of the approaching wave we see is the long line of shadow visible sweeping towards the shore, forming an obvious ridge of water gaining height as it advances. As it approaches the beach a white foaming top is formed and curling breakers are created.

The water we see as the wave is forming is not the same water we observe breaking at the shore. The `Grandfather's axe' analogy can be applied to ocean waves:

’A man is very proud of an axe that once belonged to his grandfather, and he says it has only needed three new handles and two new heads since his grandfather’s time. The axe he owns is made from material that his grandfather never saw, what right has he to claim it is his grandfather's axe?’ [2]

We witness something travelling across the sea to the shore. If it isn't the water then what is it? You could call it a shape, a pattern of outline and water motion. The water, however, is a more permanent and real thing than the wave and cannot disappear. The wave, which is merely a pattern, can grow or shrink or fade out altogether.

`It must be emphasised that a travelling wave is a passage of motion only, not of water. The actual movement of the water particles that compose the wave is relatively very small' [3]

`Ocean waves, like all other waves, are defined as travelling disturbances that carry energy from place to place.' [4]

 

3.2 Wave Propagation

Waves are created by a combination of wind and gravity. The ocean would be a flat pool of water without any waves. The creation of waves begins in a storm area. As the winds blow, the surface of the water is pushed away from the storm area and swell is generated. The sizes of the waves decrease as they leave the storm area. The size and shape of the waves when they eventually reach the shore depends on the `fetch', which is the extent of open water between the observer and the source.


Fig.2 - Storm Source of a Wave [2]

The hydrodynamics of ocean waves is highly non-linear, and can become very complex. Here we will outline some of the underlying principles that affect the progression of a breaking wave.
After the waves are created, they must usually travel across deep ocean areas. Here the water `particles' rotate in a circle, forward at the crest and backward at the trough. This means there is no net motion of the particles as the wave passes through them. For these deep ocean waves the circular motion diminishes exponentially with depth, becoming negligible at depths greater than the wavelength of the wave (distance between two successive peaks).

Fig.4 - Motion of Water Particles[11]

Fig.5 - Wave Motion Derived from Orbital Flow[11]


3.3 Breaking Waves

As a wave approaches the shore, the depth at the crest is larger than at the trough, where shallow water wave theory shows the speed is lower, so the crest overtakes the trough, the peak sharpens and the wave eventually breaks. Waves breaking on sloping beaches may be of two kinds: plunging and spilling:

`Waves break and form `whitecaps' when: the angle between front and back side of the wave <120°, and its height is 1/7th of its wavelength'[5]

`Spilling breakers are concave on both faces, so that the wave profile resembles a cycloid. The wave ravels away at its unstable, thin crest as it advances' [6]

Small swell conditions and offshore winds lead to these spilling breakers. Such a wave gradually loses its form as water spills from its crest.

`The plunging breaker has a well rounded back and a concave front. In fact just as the crest plunges forward into the trough, a swimmer dead to prudence can look upward through a solid overhang of water' [6]

When a plunging wave reaches the shore, factors such as the shape of the beach and wind conditions affect how the wave breaks. The wave could plunge all at once or could break at various points forming numerous crests along its length.

The difference in the two types of waves as regards to accurately simulating them in 3d is substantial as we will go on to discuss.


 

Fig.6 - Plunging and Spilling Waves [6]
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