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Flowing Along

To the Editors:

As a physicist-turned-software engineer, I always enjoy Brian Hayes's Computing Science column, and his essay in the November-December issue ("Up a Lazy River") was no exception. But as a physicist-turned-software engineer, I noticed something important was missing. In the equation θ = ω sins, s is described as the distance along the stream centerline. But one can't take the sine of one inch, one foot, or one mile; the argument to the trigonometric function has to be dimensionless. What is the scale factor? The width or the depth of the river, or the square root of the cross-sectional area? And how does one account for the variation of the scale factor, whatever it is, with distance along the stream centerline?

Steve Chessin
Mountain View, CA

Mr. Hayes responds:

My decision to omit the scale factor from that equation was not a careless error but a deliberate one. At the time, I thought that ignoring all questions of dimensions and units would make the mathematical model easier to understand; complaints from several readers have persuaded me I was wrong.

It's common practice when dealing with abstract equations to pretend that all quantities are dimensionless. In a graph of the sine curve, y = sinx, we don't ask about the units of x and y. The simplest of the meander models falls into the same category. There is no natural scale factor; neither the width nor the depth of the river can serve in this role because the model river has no width or depth but only length. There's nothing more to the model than a curve in the plane, defined by the equation θ = ω sins. In this case, however, the argument to the sine function is distance along the curve (s) rather than distance along a coordinate axis (x or y). Apparently, it's harder to view s as a dimensionless quantity than it is x or y.

I certainly could have written the equation with a scale factor, θ = ω sink s. The value of k is arbitrary. Indeed, we can always set k equal to 1 by adopting appropriate units of measure fors, much as physicists often adopt "natural units" in which Planck's constant and the speed of light are numerically equal to 1. Nevertheless, mentioning k explicitly would have caused no harm and would have avoided confusion for some readers.

I should add that the equation really needs a second constant as well: Strictly speaking, it should be written θ = ωsink s + Φ, where Φ defines the phase of the wave. Curiously, no one has objected to the omission of Φ.

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