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No. 847:
Subtle Math

Today, mathematics tells us more than we want to know. The University of Houston's College of Engineering presents this series about the machines that make our civilization run, and the people whose ingenuity created them.

You young people out there -- study mathematics. Do study mathematics. But when you do, be careful. Forty years ago I worked on refitting a big construction tractor so it could lay natural gas pipe. We meant to swing a long boom from one side and mount a great counterweight on the other. The boom would pick up heavy pipe sections and lay them in their trench.

The boom would pivot from the base of the tractor. The cables would exert a bending force on it. I was to calculate the bending stress. That meant solving a differential equation.

I soon found I could do the problem two ways and get two different answers. That didn't make sense. For days I turned the problem around in the light. No mistakes, two answers! Crazy! Shouldn't physical problems have unique answers?

I finally saw that I'd found the maximum stress in two places. One was in the boom where it should be. The other was in a fictitious place in the empty air beyond the end of the boom.

Math is a way we figure things out that would otherwise lie beyond our understanding. When we use math, we necessarily function beyond our understanding. We must stay alert.

That beam gave me an early lesson in the subtlety of math. It was not my last such lesson. Ten years later I did something far more complex. A colleague and I solved a non-linear equation on a computer to learn how viscosity slows a spreading liquid sheet.

Our computer solution behaved nicely for a while. Then it diverged. We refined my starting conditions. The solution went a little further -- then it diverged again. Change a digit in the fifth decimal place and a solution that diverged off to minus infinity now diverged to plus infinity.

I didn't know it then, but meteorologists were seeing the same thing when they solved non-linear equations for the weather. Remember the Jeff Goldblum character in Jurassic Park talking about the Butterfly Effect? Well, if a butterfly's wings brushed my starting number, the result went as mad as the weather.

You young people -- study mathematics. But don't study it if you want only answers. Study it if you like questions. For that's what math will give you. It'll open up questions. It won't just give you answers. Math will also say, "Yes, but!"

Today I know the stress in that beam. I know how fast liquid sheets spread. But I also know that real beams buckle in surprising ways. I know that we can predict only a little way into our own future. I know that, whatever seems certain, the world still holds far more mystery and beauty yet to be found. And that is what I really learned -- when I studied mathematics.

I'm John Lienhard, at the University of Houston, where we're interested in the way inventive minds work.

(Theme music)

Paulos, J.A., Beyond Numeracy: Ruminations of a Numbers Man. New York: Alfred A. Knopf, 1991.

J.H. Lienhard and T.A. Newton, Effect of Viscosity upon Liquid Velocity in Axi-Symmetrical Sheets, Zeit. f. Ang. Math. u. Phys (ZAMP), Vol. 17, No. 2, 1966, pp. 348-353.

For more on chaos and the Butterfly Effect, see Episodes 652657, and 829.

A note on the spreading sheet problem for the cognoscenti:

Two liquid jets, initially moving at velocity U, and of radius R, collide axially (head on.) The liquid forms a sheet that spreads radially (in the r-direction) with a gently diminishing velocity, u. Call the kinematic viscosity n and define dimensionless values of the radial position, y = r/R, and the local velocity, x = u/R. The equation of motion for the sheet is then the nonlinear equation:

y'' + y' - y/(x*x) - yy' = 0

with boundary conditions:

y(1) = RU/n   and   y between 0 and 1   for  x greater or equal 0


To solve this numerically we used y(1) = 0, and we assumed a starting value of y'. Then we marched forward, checking to see if y stayed bounded. We kept correcting y' until it did. After many trials, we found that for y'(1) = -0.7081 the solution behaved well up to x = 6. Then it diverged to plus infinity. For y'(1) = -0.70818 it diverged to minus infinity after x = 6.

After we had done the numerical solution we found the equation could be transformed into a Riccatti equation, then into an Euler equation, then solved exactly. We also found that we could closely approximate the non-linear term as yy' =RUy/n, whence the equation reduced to the linear Frobenius equation which admitted a simple solution.

The exact solution gave an exact value of y'(1) in the form of a fairly complicated ratio of Bessel functions. Its value, of course, was
y'(1) = -0.7081+ .

This is just one more example of a nonlinear equation showing nice behavior at first and then reflecting latent instability. In that respect it truly is like the weather.