Section 10.6 Fourier Series Example
Let's consider an example. Suppose \(f(x)\) describes a square wave of height \(C\text{,}\) so that
\begin{equation}
f(x) = C\, \Theta\left(\frac{L}{2}-x\right)=
\begin{cases}
C \amp (0\le x\lt \frac{L}{2}) \\
0 \amp (\frac{L}{2}\lt x\le L)
\end{cases}\tag{10.6.1}
\end{equation}
where the step function \(\Theta\) is defined in Section 11.1 (The value of \(f\) at the single point \(x=L/2\) doesn't matter).
According to the previous sections, we have
\begin{equation}
f(x) = \frac12 a_0 + \sum_{n=1}^\infty a_n
\cos\left(\frac{2\pi n x}{L}\right)
+ \sum_{n=1}^\infty b_n \sin\left(\frac{2\pi n x}{L}\right)\tag{10.6.2}
\end{equation}
where
\begin{align}
a_0 \amp = \frac{2}{L} \int_0^{\frac{L}{2}}\, C\,dx = C,\tag{10.6.3}\\
a_n \amp = \frac{2}{L} \int_0^{\frac{L}{2}}
\cos\left(\frac{2\pi nx}{L}\right) \, C\,dx
= 0 ,\tag{10.6.4}\\
b_n \amp = \frac{2}{L} \int_0^{\frac{L}{2}}
\sin\left(\frac{2\pi nx}{L}\right) \, C\,dx
=
\begin{cases}
\frac{2C}{\pi n} \amp \hbox{($n$ odd)}\\
0 \amp \hbox{($n$ even)}
\end{cases}\tag{10.6.5}
\end{align}
Putting this all together,
\begin{equation}
f(x) = C\left(1 + \sum_{\substack{n=1\\ n\hspace{1.5pt}\mathrm{odd}} }^\infty
\frac{2}{\pi n} \sin\left(\frac{2\pi nx}{L}\right)\right)\text{.}\tag{10.6.6}
\end{equation}
It is instructive to plot the first few terms of this Fourier series and watch the approximation improve as more terms are included, as shown in Figure 10.6.1.
