COMPUTING SCIENCE

# Up a Lazy River

Meandering through a classic theory of why rivers meander

Around the Bend

Leopold brought a distinctively quantitative and mathematical style to the study of rivers. For example, he formulated scaling laws that describe how the cross section of a natural channel changes as a function of the volume of water flowing through it. He even did some computer simulations—without a computer! Using shuffled decks of cards or tables of random numbers, he carried out probabilistic studies of landform features such as the branching of a drainage network.

The serpentine shapes of meanders certainly invite mathematical analysis. Although in nature the curves are highly irregular—no two alike, perhaps—Leopold argued that they all derive from a specific underlying form, which he called a sine-generated curve.

Imagine you are canoeing down a meandering river with a compass in hand, making note of your heading at regular intervals. According to Leopold, your direction should vary sinusoidally as a function of the distance you have traveled along the river centerline. This is not to say that the shape of the river itself is a sine curve; rather, the sine function specifies the heading. The governing equation is:

θ=ωsin*s.*

Here q is the heading angle, measured with respect to the mean
down-valley direction (the path the river would follow if it did not
meander at all); *s* is distance along the stream centerline;
and w is the maximum angle that the path makes with the down-valley
axis. For small values of w, less than 90 degrees, the
sine-generated curve has gentle undulations, so that the river
weaves back and forth but at all times maintains a down-valley
component of motion. At w=90 degrees, the path of the stream crosses
perpendicular to the valley axis. At still larger values of w, the
lobes of the curve become horseshoe-shaped, and for part of each
meander cycle the river's course takes it back up the valley. A
little beyond w=120 degrees, adjacent lobes of the curve begin to
overlap. On graph paper the lines merely cross, but in a river this
event signals the development of a "cutoff," diverting the
flow and leaving behind a stranded oxbow lake.

The sine-generated curve looks like a plausible candidate for
describing meanders, at least within a limited parameter range. But
what made Leopold so sure it was the one *right* candidate?
His argument goes as follows. Take two points *a* and
*b* connected by a stretch of river of length *L*,
where *L* is greater than the straight-line distance from
*a* to *b*. Now think of all the ways of bending and
folding this segment of river into a smooth curve without changing
its length or detaching it from its end points. Among all such
paths, the sine-generated curve has three interesting properties: It
is the path of minimal bending stress, it is the path of minimal
variance in direction, and it is the path representing the most
likely random walk. I shall first discuss the two minimization
principles and return later to the random walks.

The bending stress of a river is the work or energy that has to be
expended to make its path deviate from a straight line. At each
point along the route, the bending stress is proportional to the
square of the curvature at that point. For a straight segment,
bending stress and curvature are both zero; they increase as a turn
gets sharper. Among all smooth, length-*L* curves from
*a* to *b*, the sine-generated curve has the smallest
squared curvature summed over the entire path.

Directional variance is a similar concept. As you follow the river
from *a*to *b*, measure at each point along the way
how much your heading deviates from the mean down-valley direction,
then compute the sum of the squares of these angles. Again, the
sine-generated curve yields the smallest possible total.

These properties of the sine-generated curve are mildly surprising. I would have guessed that an arc of a circle—the most symmetrical curve—would have the lowest squared curvature and directional variance, but that is not the case. (Of course a straight line is superior, but that solution is forbidden by the length constraint.)

Leopold offers a simple demonstration of how the sine-generated curve emerges as a natural solution to a problem of minimizing work or energy. If you hold the ends of a strip of spring steel so that it forms a horseshoe-shaped loop, the metal spontaneously adopts the form of a sine-generated curve. I couldn't resist trying this myself. I found that it works reliably only for single loops. If you try to fold the spring into multiple meanders, the configuration is unstable.

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**Of Possible Interest**

**Feature Article**: Twisted Math and Beautiful Geometry

**Perspective**: Searching for Great Adventures

**Letters to the Editors**: Prime Beef