LETTERS TO THE EDITORS

# 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
sin*s, 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* = sin*x*, 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 sin*s*. 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 sin*k s.* The value of *k* is
arbitrary. Indeed, we can always set *k* equal to 1 by
adopting appropriate units of measure for*s,* 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 =
ωsin*k s* + Φ, where Φ defines the phase of
the wave. Curiously, no one has objected to the omission of Φ.

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