Did you know that
every sufficiently smooth function on an interval
can be expressed as an infinite sum of sines and cosines
of various frequencies and amplitudes?
Yes, it's true! It is analogous to the fact that
you can approximate irrational numbers
as the sum of a bunch of rational numbers
(that's what a decimal expansion of
an irrational number is, after all)! Writing
a function as a sum of sines and cosines is called
a Fourier series.
In fact, your ears do Fourier series automatically!
There are little hairs (cilia) in you ears which
vibrate at specific (and different) frequencies.
When a wave enters your ear, the cilia will vibrate
if the wavefunction "contains"
any component of the correponding frequency!
Because of this, you can distinguish sounds
of various pitches!
Presentation Suggestions:
This is a great Fun Fact to
reinforce the connection of mathematics with other
disciplines. You can show students how to find
Fourier series by working examples into a
homework on integration.
The Math Behind the Fact:
You can learn about Fourier series in an advanced
differential equations course, one which covers
boundary value problems, or an advanced course in
analysis. Fourier series are used to
find solutions to partial differential equations, such
as problems involving heat flow.
Fourier series can be used to construct
some pathological functions such as one which is
continuous but nowhere differentiable.
By the way, one type of "sufficiently smooth" function
(as mentioned in the first sentence above) is a function
that is piecewise differentiable.
Being a continuous function is not enough; see the reference.
How to Cite this Page:
Su, Francis E., et al. "Fourier Ears Only."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
