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 652, 657, 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.
______________________________________________________
The Engines of Our Ingenuity is
Copyright © 1988-1997 by John H.
Lienhard.
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