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Gone with the wind - The maths of optics explains what every old sea dog knows

点击量:   时间:2017-05-20 05:01:03

By Robert Matthews AN AMATEUR yachtsman who is also a physicist has found an ingenious way to work out the fastest route for sailing from port A to port B. He’s done it by using the mathematics of optics. One of the cornerstones of optical physics is Fermat’s principle, which states that rays of light always follow the path that takes the least time. In a vacuum, this is a straight line. But when light passes through a medium with a variable refractive index, the quickest route becomes a curve that can be worked out mathematically. Now John Kimball of the University of Albany in New York has shown that the same mathematics can be used when setting the course of a yacht. Contrary to the beliefs of landlubbers, sailing boats are not compelled to travel with the wind. Clever orientation of the boat and its sails can exploit the wind to move it in a much wider range of directions. This range is summed up in a “speed diagram” for each boat—an apple-shaped graph showing how fast the vessel can travel in different directions relative to the wind. The skill of sailing lies in setting a course that makes the most of the wind. This can be tricky if the destination is upwind—and worse still if the wind direction keeps changing. Kimball realised that the situation was similar to that faced by light rays passing through optically variable media, and sought an answer in the mathematics governing the path taken by such rays. First worked out by the Irish mathematical physicist William Rowan Hamilton, the fastest path turns out to be governed by a vector equation—a relationship between two mathematical quantities having direction as well as size. One of these vectors is the direction and speed of the rays, and the other is a quantity known for technical reasons as “slowness”. In the current issue of the European Journal of Physics (vol 19, p 15), Kimball and his colleague Harold Story turn Hamilton’s vector equation into a recipe for working out the best route to sail. First, draw the boat’s speed diagram with its axis of symmetry lined up with the direction of the wind at that instant. Next, draw a line from the starting point A towards the destination B: this is the same direction as the slowness vector. At right angles to that draw a line that just touches the speed diagram. The best initial direction is then found by connecting the starting point to this point of contact. By continually recalculating as the boat moves, this recipe can be used to define the ideal course. According to Kimball, the mathematics of light rays explains some subtle sailing tips that old sea dogs pick up through years of experience. “The most surprising discovery was the simplicity of the least-time sailing path. In principle, it means sailors should be able to sail a perfect race,” he says. “My sailing has certainly been helped by knowing I should try to get the best boat direction using this method,