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. <https://www.math.hmc.edu/funfacts>.
References:
Rudin, W. Real and Complex Analysis, 1987, item 5.11.
Fun Fact suggested by:
Francis Su