Appendix H
The Fourier Decomposition
A mathematician who is not also something of a poet will never
be complete.
— Karl Weierstrass
H.1 The Fourier Series: Harmonic Analysis
In 1822, Jean Fourier showed that complicated waves which repeat in a definite
period T can always be decomposed into a superposition of simple sinusoidal
waves, each with a frequency which is a multiple of 1/T . The same kind decomposition works for waves in space, over a length of λ. The wave can be expressed
as a sum of sinusoidal waves having wave numbers which are a multiple of 1/λ.
Each such decomposition is called a Fourier series. The technique of decomposition
is called ‘harmonic analysis’. Because of the Fourier theorem, one can analyze
the behavior of any complex wave satisfying a linear differential equation by first
studying the behavior of simple sinusoidal waves.
For a wave with a period T , i.e. ξ(x, t + T ) = ξ(x, t), the Fourier decomposition
into simple sine and cosine waves looks like this:
ξ(x, t) = a o (x)/2 +
∞
j =1
a j (x) cos (2π j t/T ) +
∞
j =1
b j (x) sin (2π j t/T ) .
(H.1)
The coefficients a j (x) and b j (x) are found from
a j (x) =
2
T
T
0
dt ξ(x, t) cos (2π j t/T ) ,
(H.2)
b j (x) =
2
T
T
0
dt ξ(x, t) sin (2π j t/T ) .
(H.3)
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