Understanding Engineering Mathematics

increasing number of times in this period. The above series is therefore adding together a
(possibly infinite) number of different oscillatory functions, with different amplitudes or
weighting. The hope is that by suitably choosing these weightings, one can approximate
any other function of the same period, 2π. The problem is of course to find the appropriate
‘weightings’ or coefficients,a 0 ,an,bnfor any given functionf(t).
The notation in the above series is conventional, as is the factor^12 witha 0 ,which
simplifies later results.a 0 ,an,bnare constants which will depend uponf(t)and which
are to be determined.
Particularly in the case of time dependent periodic functions, it is usual to speak of the
n=1 term in the Fourier decomposition as the ‘fundamental component’ (it is often the
dominant one) and thenth term (n>1) as thenth harmonic.
As an example of a Fourier series, consider the square wave period of 2πand amplitude
A, shown in Figure 17.7.

0

A

−pp 2 p

t

Figure 17.7

This can be described by the function

f(t)=−A −π<t< 0

A 0 <t<π

f(t+ 2 π)=f(t)

Mathematically this is a difficult functional form to handle and to process – for example
we cannot differentiate it, and if it appeared as a ‘forcing function’ (469
➤
) in a differential
equation then there is not a lot we could do with it as it is. However, the fact that it is
periodic with period 2πdoes mean that we can express it as a superposition of sinusoids
of the same period – i.e. as a Fourier series. We will show in Section 17.10 that the
corresponding Fourier series is in fact

f(t)=

4 A

π

(

sint+

1

3

sin 3t+

1

5

sin 5t+...

)

The amplitude of thenth harmonic is

4 A

nπ

. Now although this series looks nothing like

the original functional form of the square wave it is in fact equivalent to it. If you have
a graphical calculator you might like to try plotting each of the harmonics and the sum
of the first few. You will see that the result starts to look like a humpy square wave, and
gets more and more like it as you take more and more terms in the series. The whole
point of course is that the sinusoidal functions to which we have converted the wave are
continuous and far easier to handle than the original discontinuous form. There are, it is
true, now an infinite number of them – but as with all such series we can get as good an
approximation as we please by taking enough terms in the series.