Lord Kelvin Wipes Out on Speed Boat Wakes?
Lord Kelvin is still making waves. In the 1880s, the great British physicist—then a commoner named William Thomson—argued that the wake of a boat fans out at the same angle regardless of how fast the boat is going. But scientists and engineers have long known that boats sometimes appear to have narrower wakes. Now two French physicists say they've explained that narrowing. Their idea may not sail smoothly into the textbooks, however, as experts in marine engineering are skeptical.
An avid seaman, Kelvin analyzed boat wakes and came to a rather curious conclusion: No matter the speed of the boat, it should produce a wake with a "wake angle" of 19.47°. (See figure.) That odd constancy arises for two reasons. First, the speed or "phase velocity," of water waves varies with their wavelength, with longer wavelengths traveling faster than shorter ones do. As the boat moves, it creates waves of all speeds slower than the boat itself. And the longer waves generally spread out behind it faster than the shorter ones.
To make a stable wake, however, the waves also have to overlap and "interfere" in the right way. For waves moving almost as fast as the boat, that interference occurs only right behind the boat. So the fastest waves also produce a narrow wake. Putting these two factors together, Kelvin figured out that the width of the wake is determined by waves traveling at a fixed fraction of the boat's speed: 81.6%. Because of that proportionality, the wake angle is always the same, as the faster the boat goes the faster the wake spreads.
At least, that's how it works in theory. In practice, scientists and engineers have long known that boats and ships sometimes appear to produce narrower wakes with smaller angles. Researchers have tried to explain that narrowing as a result of the finite depth of the water, waves already present on the surface, or complicated, "non-linear" interactions of the waves.
Now, Marc Rabaud and Frédéric Moisy of the University of Paris-Sud in Orsay, France, say they've found a simpler explanation. If a boat is going faster than a certain speed determined by its hull length, they argue, it cannot create waves with wavelengths longer than the hull. That wavelength "cutoff" then limits the speed of the waves and thus how fast the wake can spread. So as the boat picks up even more speed, its wake stretches and narrows.
The physicists tested their idea with satellite images of boats and ships from the website Google Earth, measuring the wake angle and length of the boat from the pictures and inferring the boat's speed from details of the wake. Sure enough, the wake angle remained constant up to a certain speed—actually a certain "Froude number," which is a combination of boat's speed and length—and then falls as predicted, the researchers report in a paper in press at Physical Review Letters. Simulations show the same effect.
"What seems most convincing is the agreement they have with observations," says Laurette Tuckerman, a fluid dynamicist with the French National Center for Scientific Research at the Laboratory for the Physics and Mechanics of Heterogeneous Materials in Paris. "It fits the data so well."
Not everyone buys it. Rabaud and Moisy never explain why a boat cannot excite waves longer that itself, says Yuming Liu, a hydrodynamicist with at the Massachusetts Institute of Technology in Cambridge, who says he's been studying ship wakes for 20 years. "Their argument is very simple, but you can't make that assumption," Liu says. In principle, he argues, standard theory can be used to calculate every detail of a wake. So Rabaud and Moisy should have used such calculations to check their assumption before they published, he says.
Rabaud and Moisy may not be measuring the right angle, says Robert Beck, a hydrodynamicist in the Department of Naval Architecture and Marine Engineering at the University of Michigan, Ann Arbor. They measure the angle between the most intense waves on either side of the boat. But in mathematical terms, Beck notes, the wake angle is defined as a particular crease in the wave pattern—and that crease could be fainter for faster boats. "If you want to define the wake angle as what's most visible, then that's okay, they're doing it right," Beck says.
Still, even if Rabaud and Moisy are all wet, Beck and Liu don't explain why the physicists' theory fits the data so well—other than to argue that it doesn't really.