Fourier analysis

In the previous section we found that the displacement of a particle an be described as a linear combination of normal modes, i.e. a superposition of sinusoidal terms. This decomposition of the motion into various frequencies is more general. It can be shown that any arbitrary periodic function $f(x)$ of period $T$ can be written as a Fourier series of sines and cosines:

$$f(t)=\frac{1}{2}a_{0}+\sum_{n=1}^{\infty }(a_{n}\cos {n\omega t}+b_{n}\sin {% n\omega t}),$$

where $\omega _{0}$ is the fundamental angular frequency given by

$$\omega _{0}=\frac{2\pi }{T}.$$

The sine and cosine terms represent the harmonics. The Fourier coefficients $% a_{n}$ and $b_{n}$ are given by

$$a_{n} = \frac{2}{T}\int_{-T/2}^{T/2}{f(t)\cos{n\omega _0 t}dt}$$ (52)

$$b_{n} = \frac{2}{T}\int_{-T/2}^{T/2}{f(t)\sin{n\omega _0 t}dt}$$ (53)

The constant term $1/2a_{0}$ is the average value of $f(t)$. (all the oscillating contributions vanish in average)

In general, an infinite number of terms is needed to represent an arbitrary function $f(t)$. In practice, a good approximation can usually be obtained by including a relatively small number of terms.