Charlie and the pooh stick: the maths of rivers

Charlie and the pooh stick

Langstrath valley, on a typical Sunday morning. Typical if you ignore the other 51 Sundays of each year, when it is pouring down there.

Charlie, our Cavalier King Charles spaniel, with my younger daughter at Angle Tarn.

Maths

Maths is all around us. Imagine we were to drop a pooh stick at the head of the Langstrath (which we found on Sunday) and Charlie was asked to chase it. The Langstrath river curves down the valley in typical S shaped fashion. Charlie is a clever dog and knows that if he runs down the valley in a straight line, he can run slower and still get the pooh stick. It's a few miles to the end of the valley, but a pooh stick is a pooh stick, so it is still running.

How fast should Charlie run, if the river runs at v miles per hour?

The answer is 2v/pi : yes, 3.14159… is in the Langstrath valley, as well as everywhere else on Earth. On average, a river flows in semi-circles: one above the horizontal diameter, one below, one above, and so on. The curved line of one semi circle is pi * d/2, half the circumference of a circle, and the Charlie runs the diameter d: divide one into the other, and Charlie well get the pooh stick by running 64% as fast as the river.

Clever Charlie.

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