| | Ivars Peterson's MathTrek |
July 19, 1999
In varsity and Olympic competition, races may involve boats with one, two, four, or eight rowers. Interestingly, although a shell with eight rowers is much larger than one with a single rower, all the boats have roughly the same proportions (at least for the surface area over which the shell makes contact with water).
Data from 2000-meter world and Olympic championship races show that the larger boats go faster than the smaller ones. In the late 1960s, that fact caught the eye of Thomas A. McMahon (1944-1999), a professor of applied mechanics and an expert in animal locomotion at Harvard. He wondered why that might be true. How does the speed depend on the number of rowers?
Shell dimensions and performances
| No. of rowers | Length, l (m) | Width, b (m) | Ratio, l/b | Time for 2,000 m (min) | |||
| I | II | III | IV | ||||
| 18.28 | 0.610 | 30.0 | 5.87 | 5.92 | 5.82 | 5.73 | |
| 11.75 | 0.574 | 21.0 | 6.33 | 6.42 | 6.48 | 6.13 | |
| 9.76 | 0.356 | 27.4 | 6.87 | 6.92 | 6.95 | 6.77 | |
| 7.93 | 0.293 | 27.0 | 7.16 | 7.25 | 7.28 | 7.17 | |
Race data are from the 1964 Olympics in Tokyo, the 1968 Olympics in Mexico City, the 1970 World Rowing Championships in Ontario, and the 1970 Lucerne International Championships.
McMahon recognized that because rowing shells seating one, two, four, and eight rowers happen to be built with roughly the same proportions, it may be possible to use a relatively simple mathematical model to predict the speed as a function of boat size, even though the physics, in all its gory detail, is quite complex.
The total mass of the rowers plus that of the boat equals the mass of the water displaced by the boat. Because the boats are geometrically similar, this displacement is proportional to the volume, or the cube of the boat's length. That length is, in turn, proportional to the number or rowers.
Two main effects produce the drag experienced by a boat moving through water: wave generation and friction between hull and water. The long, thin shapes of the boats minimize the part of the drag due to wave-making, so that component can be neglected. Skin-friction drag is proportional to the product of the wetted area and the square of the speed. The wetted area itself is proportional to the square of the boat's length. So the drag force is proportional to speed squared times length squared.
McMahon then made the additional assumption that the power available to drive a boat is proportional to the number of rowers. That power is used to overcome the drag force and provide speed. Hence, the power is proportional to the drag force times the speed, or the speed cubed times the length squared.
Putting those proportionalities together algebraically allows you deduce a relationship between speed and the number of rowers. This model predicts that the speed should be proportional to the number of rowers raised to the one-ninth power. McMahon found that plotted data from various races fit that theoretical relationship quite nicely. It would be interesting to see if it still holds for more recent sculling events.
McMahon's study of rowing is a striking example of how a relatively simple mathematical model can capture essential features of a complex physical phenomenon and yield insights into what is going on. The trick is to come up with an appropriate model. That task requires not only a firm grasp of the relevant areas of mathematics but also an understanding of physical law and behavior.
The summer graduate program in "dynamics of low-dimensional continua," held recently at the Mathematical Sciences Research Institute in Berkeley, Calif., introduced mathematics students to tools and concepts for developing and analyzing mathematical models applicable to fluid flow, crystal growth, and many other phenomena encountered in materials science, chemistry, biology, engineering, and physics.
Conducted by Lakshminaryanan Mahadevan and Anette Hosoi of the Massachusetts Institute of Technology, the two-week course gave the students some insights into the art of building mathematical models relevant to the real world. The students ended up doing computational projects on such topics as fluid mixing, vortex formation, turbulent convection, and dendritic crystal growth.
McMahon's rowing study was just one of a number of examples Mahadevan cited to illustrate how scaling arguments and dimensional analysis can provide a good starting point. You get good guesses for extremely complicated problems, he says.
Once the basic framework is in place, you can then focus on details and study deviations from an initially derived theoretical relationship. In the rowing example, boats with more rowers actually perform a little better than the one-ninth power relationship suggests. It's possible, for instance, that the larger, more heavily laden boats are better at overcoming wave-making drag, which was neglected in the initial model.
In a 1915 Nature paper, Lord Rayleigh (1852-1919) extolled the value of the principle of similtude (now called dimensional analysis) in providing physical insight and deducing physical laws. "It happens not infrequently that results in the form of 'laws' are put forward as novelties on the basis of elaborate experiments, which might have been predicted a priori after a few minutes' consideration," he wrote.
Copyright 1999 by Ivars Peterson
References:
Lord Rayleigh. 1915. The principle of similitude. Nature 45(March):66.
McMahon, T.A. 1971. Rowing: A similarity analysis. Science 173:349.
McMahon, T.A., and J.T. Bonner. 1983. On Size and Life. New York: Scientific American Books.
An online video of L. Mahadevan's lecture on dimensional analysis and scaling is available at http://www.msri.org/publications/video/.
A brief biography of Lord Rayleigh can be found at http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Rayleigh.html.
Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.