Wednesday, April 04, 2007

Singularities on the surface of the sea

One Sunday in the Summer of '98, I drove down to Alum Chine, parked my car at the top of the headland, and walked down to the beach. As I strode down the hill, the entire expanse of Poole Bay was before me, stretching from Hengistbury Head to the left, and Sandbanks to the right. The sky was pure azure, the sun was bright and hot, and the surface of the sea was a billion glittering points of reflected sunlight, criss-crossed by a seething foment of multi-coloured yachts, motorboats, speedboats, water-skiers, and jet-skis.
In the popular imagination, singularities are primordial or destructive things, found either at the beginning of the universe, or within the interiors of black holes. These, however, are merely singularities of space-time. There is a quite distinct type of singularity, which belongs to a mathematical discipline called singularity theory, and the sparkling points of light upon the surface of the sea are singularities in this sense of the term.

To understand this, freeze the surface of sea at a moment in time, and fix the position of the Sun (an extended light source) and the eye of the observer. According to Fermat's principle, the light rays between a source and an observer must be extrema of something called the optical length. (Extrema are either maxima or minima). The optical length of a path in space is determined by its geometrical length, and the index of refraction of any media through which it passes. In the case of interest to us here, the only medium is the air, so we can equate the optical length with the geometrical length.

For light to reach the eye via a reflective surface, such as the surface of the sea, it must follow an extremum of the optical length L. If we let x denote an arbitrary point on the surface of the sea, then, of all the possible paths from the Sun to the eye via point x, the actual path taken by the light is typically the one which minimizes the geometrical length.

Whilst reflected light will reach the eye from all the points upon the surface of the sea, only some points reflect a bright image of the Sun. If one treats the optical length as a function defined upon the surface of the sea, then these bright points are (i) singularities of the optical length, in the sense that the gradient vanishes, ∂ L(x)/∂ x = 0; and (ii) caustics, in the sense that the Hessian of the optical length is singular. (The Hessian, ∂2 L(x)/∂ xi ∂ xj, is the matrix of second-order derivatives, and if a matrix has a zero determinant, then it is deemed to be singular).

The caustic singularities upon the surface of the reflective medium, relative to the fixed observer, are responsible for focusing the light rays for that fixed observer. Each momentary bright image of the Sun upon the surface of the sea, is responsible for a focal plane which intersects the eye of the observer for that brief moment of time. As the surface of the sea chops and shifts, the caustic singularities change for a fixed observer, and if the surface of the sea is frozen, the caustic singularities will be different for different observers.

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