Whether spraying graceful arcs of water into the air or letting water tumble down steep slopes, outdoor fountains draw attention. They can startle the eye and soothe the ear. They can offer cool respite from a day's travails.
Andrew J. Simoson of King College has pondered the factors that contribute to the visual impact of water fountains. What makes some fountains more spectacular than others? To Simoson, one ideal is the mathematical answer to the question: "For a given initial speed of water from a spigot or jet, what angle of the jet maximizes the visual impact of the water spray in the fountain?"
Simoson presents his analysis in the article "Maximizing the Spectacle of Water Fountains," published in the September College Mathematics Journal.
Jets of water spray out from the center of a fountain at Chevy Chase Circle in Washington, D.C. Photos by I. Peterson.
In many fountains, water sprayed from a jet or spewed from a spigot tends to follow a parabolic path. The particular arc is determined by the jet's angle and the speed at which water leaves the jet.
Simoson visualized the surfaces suggested by two arrays of water jets: one in which the jets are in a row and another in which they form a ring. For a circular array, the suggested surface is a surface of revolution of any of the parabolic streams about an appropriate vertical axis.
Simoson proposed two criteria for visual impact. In each of the two types of arrays that he examined, he posited that the visual impact is greatest when the volume enclosed by the suggested surface is a maximum with respect to the angle of the jets or when the surface area of the suggested surface is a maximum with respect to the angle of the jets.
In his analysis, Simoson assumed that the water streams are smooth parabolic arcs. He also ignored any friction effects. In the linear case, the problem simplifies to finding the angle that gives the maximum area and the maximum arc length for a single parabolic stream.
Jets spray water along paths that approximate parabolas.
Simoson's analysis shows that, for a linear array, the most spectacular fountains with respect to enclosed volume occur when the jets are inclined at 60°. However, with respect to surface area, the critical angle to achieve the most spectacular effect can be as low as 52.8°, depending on the difference in height between the start and end of an arc. For streams in which the end point is at half the height to which the water rises, the angle is about 53.5°.
For circular arrays of jets, the critical angles are less than 60° for both maximum volume and maximum surface area and differ from the results for linear arrays, going as low as 49.05°.
"Intuition may have suggested that the angles for the volume-spectacular fountain and surface-spectacular fountain might be the same as the angles that give a maximum area and a maximum arc length for solitary parabolic stream," Simoson notes. That doesn't happen.
"The reason must lie in the difference between a linear array of jets versus a circular array of jets," he explains. "Evidently, the advantage of having jets set at 60° in as far as accruing area under the streams and hence volume under the surface of revolution is eroded by the fact that adjacent streams get closer together as they near the fountain center, hence contributing less volume than intuition may have at first expected."
Simoson's analysis opens up the possibility of deliberately designing fountains that meet his criteria for spectacle—or checking existing fountains to see how close they come to the relevant criteria.
Portland, Ore., where MathFest 2009 took place earlier this month, has a variety of outdoor water fountains. Unfortunately, my photos of several of these fountains were not precise enough for me to check how closely they matched Simoson's criteria.
Water streams in parabolic arcs from a tubular fountain in Portland, Ore.
At the same time, fountains don't have to feature water jets in any sort of array to be spectacular or even aesthetically pleasing.
Water cascades down stone blocks in this striking Portland fountain.
And water jets can be combined with other mathematics—the Fibonacci sequence, for instance—to create intriguing effects, as seen in Helaman Ferguson's Fibonacci Fountain in Bowie, Md.
Helaman Ferguson's Fibonacci Fountain in Bowie, Md. Photo courtesy of Helaman Ferguson.
Simoson, A.J. 2009. Maximizing the spectacle of water fountains. College Mathematics Journal 40(September):263-274.