Abstract: A meander is a curve which occurs naturally when a winding object, such a river, does the least amount of work in turning. Although its exact representation requires the solution of an elliptic integral, it has been found that a sine-generated curve serves as a good approximation. This article will suggest some activities for students of trigonometry to gain insight into such approximations.
A class of application problems that are common in algebra texts involves calculating distances, times, and rates of objects moving along a "straight" river. However, rivers in nature only exhibit such straightness in short stretches; more realistically, they traverse in extremely curved patterns. Such curves are called meanders and they develop from an underlying structure which requires them to do the least amount of work in turning. Such meandering can also be observed in the movement of snakes, the bending of spring steel, and the stabilizing configuration of a chain which is perturbed between fixed endpoints. Mathematical analysis of such curvature involves finding the most probable path taken by a "random walk" of fixed length, and requires the solution of an elliptic integral which is certainly beyond the scope of high school mathematics. Fortunately, however, the meandering shape can be closely approximated by a sine-generated curve and the investigation of this provides a good activity for students of trigonometry.
Before we can begin our investigation, we need to establish the following definitions:
mean down valley direction:
A line which simulates the general direction of river flow. It can be approximated by bisecting the graph half-way between the high and low points of the curve's amplitude.
distance along channel:
The length of the curve, measured from starting point to ending point.
angular direction of the channel:
The measure of the angle of the tangent to the curve with respect to the mean down valley direction. Measure it in degrees needed to rotate a line which starts parallel to the mean down valley direction line and ends parallel to the tangent in the direction of the flow of the channel. If the rotation is clockwise (or to the right), consider the angle negative. If the rotation is counterclockwise (or to the left), consider the angle positive.
The plot of the relationship between the angular direction of the channel (on the Y axis) and the distance along the channel (on the X axis) is approximately a sine function. Thus, given the graph of a meander, one can produce the corresponding sine function which generates it, with equation, and, given a sine function, one can generate the corresponding meander. We will do an example to demonstrate our concept of measuring the angular direction of the channel, then leave three exercises for the student to work out.
Produce a sine function whose graph fits the plot of angular direction vs. distance along the channel for this meander.
Sketch the meander generated by y = 110 sin 20(x-7), for 0<x<30
Try to create your own meander. A good device to use is a jewelry chain, like the type worn around the neck. Place the chain over a piece of graph paper on top of a notebook or some type of object that can be "jiggled." Have a friend hold the ends of the chain fixed on the graph paper while you gently tap the notebook from below until the chain seems to have stabilized into a meandering configuration. Trace a portion of the curve which looks most like the other meandering curves you have seen. Take measurements and see how closely a sine curve fits the data.
Think about how the period, phase shift, and amplitude parameters affect the curvature of the meander. What would be the visual appearance of a meander generated by a sine curve of low amplitude, say 45? What would be a feature of a meander which was generated by a sine curve of amplitude 180? How would two meanders differ if their generating sine curves had the same amplitude but different periods? Where would you have to start measuring channel length to have a generating sine curve with a zero phase shift?
For a detailed analysis of the mathematics of meanders and a variety of graphic illustrations, please refer to the article, "River Meanders," by Luna Leopold and W.B. Langheim, Scientific American, June, 1966.