College of Science & Engineering Alumni Newsletter
Soliton Wave Trains - the Non-Linear Boat Wake
Everyone is familiar with the waves which trail behind a fast-moving boat in a narrow V-shaped wake, rocking other craft and providing sport for water skiers. But what about the rest of the water surface, to the side and in front of the boat? Does the boat arrive unheralded, ahead of its wake, like a supersonic jet aircraft? Recent research carried out at SFSU's Romberg Center for Environmental Studies in Tiburon indicates that this is far from true. The commuter ferries operating in the North San Francisco Bay (see figure 1) have provided an ideal laboratory for the study of boat wakes. It now appears that a different type of wake, consisting of very long, non-linear waves (solitons), nearly invisible to the eye, extends for hundreds of meters in front and to the side of the ship. A sensitive instrument, conceived, designed and built by SFSU students and faculty, has detected these waves for the first time.
The birth of soliton theory took place in England in 1842, when engineer John Scott Russell observed "a large solitary wave" produced by an English canal boat. He galloped on horseback alongside the wave for over a mile, until the canal took a sharp 90-degree bend and the wave disappeared. He was amazed both at the speed of the wave and by the fact that it maintained its shape over such a long distance. Why was this???
Water waves are well known to physicists as an example of "dispersive" waves; a single localized wave pulse, whether it is caused by a stone thrown into a pond or by a tsunami, rapidly disperses into its component waves of different wavelengths, each traveling at its own speed. The simple theory of linear waves predicts a maximum speed for waves in water of depth h of vmax = (gh)1/2, where g = 9.8 m/s2 is the acceleration of gravity, and h is the depth of the water in meters. This "speed limit" is only achieved in the long-wavelength limit; the shorter the wavelength, the slower the wave travels. The Fourier representation of a pulse as a superposition of a spectrum of wavelengths thus explains why a localized pulse is a transient phenomenon; as the wave propagates, the long waves outrun the short waves, and the wave "comes apart" into its component waves, losing its initial shape and eventually dispersing. But the soliton observed by Russell broke all the rules, traveling faster than the "speed limit" for linear waves, and perfectly maintaining its shape.
Several years ago a research project began at the Romberg center to study the motion of water in the Bay using an acoustic technique developed by oceanographers to study the temperature of the Pacific ocean. Physics students Kip Laws, Andrea Grant and Mike Ugawa, working with professor Roger Bland, designed an apparatus to measure the speed of moving water. Sound waves of frequency 200 kHz are generated by underwater transducers (adapted from fish finders used by small-boat owners). The travel time for sound from the transmitter to the receiver is measured with extremely high precision. By comparing the travel times from south to north and from north to south, the speed of the medium (the water) is extracted. (The technique is reminiscent of the famous Michaelson-Morely experiment to detect the motion of the ether.) This apparatus was initially intended to study current flow in the Sacramento-San Joaquin River Delta. However, preliminary tests in the Bay revealed unexpected and interesting results.
Figure 2 shows one of the ferries of the Larkspur-San Francisco line, with its wake trailing behind the ship. This V-shaped train of waves is the predominant feature of the linear theory of boat wakes, first described by William Thompson (otherwise known as Lord Kelvin, after whom the Kelvin temperature scale is named). This is not the whole story, though. Figure 3 graphs the measured water speed observed as a similar ferry passed. The rapid variations near 14:18 mark the arrival of the classical Kelvin wake. This wake rocks boats at anchor in Sausalito, stirs up mud in the Corte Madera Ecological Reserve, and in general makes its presence known in the North Bay. The new discovery, first noted by master's student Daniel Neuman, is the train of eleven precursors, arriving well in front of the ferry. (Other SFSU students involved in analysis of boat-wake data are Daryl Haggard and Eric Tapio.) The precursors form a train of triangular pulses (see inset), not a smooth modulated sinusoid as in the case of the Kelvin wake. Linear waves are limited in water of depth 10 m (the average depth along the path of the approaching ferry) to a speed of c0 = ? 10 m/s, where g is the acceration of gravity. The diagram in figure 3 illustrates a hypothetical pattern of waves produced by the ferry. Since the ferry itself is traveling at 10 to 12 m/s over this part of its route, it is clear that the precursors, traveling faster than the ferry, are violating the "speed limit" for conventional, linear waves.
The practical importance of these long-wavelength solitons remains to be determined. The long wavelength makes the waves hard to see; in figure 2, the water is apparently smooth and motionless in front of the boat. However, about 20% of the energy in the wave of figure 3 is emitted in the form of solitons, and the solitons may travel long distances throughout the Bay before reaching a vulnerable sloping shoreline where they deposit their energy, rather than reflecting. Besides their contribution to erosion processes, they may constitute one of the principle sources of turbidity during summer and fall, when the Bay waters are at their clearest. The effect of this increased turbidity on survival of fingerling fish and other organisms remains to be evaluated.
Another possible interest of soliton emission in ship wakes concerns ship detection from space. In a paper published recently in the Canadian Journal for Remote Sensing (Neuman et al., CJRS 27, 361, August 2001) we proposed detection by synthetic-aperture radar (SAR), taking advantage of the long wavelengths which could be resolved from space. It remains to see, however, if ships in deep water also produce soliton wave trains, large enough to detect.
The solitons emitted by ferries in SF Bay are the most dramatic evidence for non-linear waves so far observed in boat wakes. Many questions remain to be answered. What is the relation of velocity waves to surface waves? How does a non-linear wave spread? Do the waves really propagate indefinitely without changing form? How do the waves react to changes in depth? Theoretical results for restricted geometries (ships moving in canals, typically) suggest that solitons may react to changes in depth or other boundary conditions by dividing or shedding. We hope in the future to obtain funding to place instruments on the bottom of the bay, directly under the ferries in the shipping lane, to answer some of these questions.
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