2.3 Fourier Analysis in Frequency and Time

2.3.1 Fourier Series, Take II

If we consider the motion of a single point in the medium as a wave travels past, we can describe a wave, a periodic train of pulses, or a single pulse, as a function of time instead of position.
For non-dispersive waves, the time interval corresponding to a single pulse of length $\lambda$ is given by^2.4

$\begin{displaymath} \tau = \frac{\lambda}{v_p} \end{displaymath}$ (2.65)
We can construct a Fourier series in time

$\begin{displaymath} g(t) = a_o + \sum_{n=1}^{\infty} a_n \cos \left( \frac{2 \p... ..._{n=1}^{\infty} b_n \sin \left( \frac{2 \pi n t}{\tau} \right) \end{displaymath}$ (2.66)
The coefficients are given by

$\displaystyle a_o$ $\textstyle =$ $\displaystyle \frac{1}{\tau} \int_0^{\tau} g(t) dt$

$\displaystyle a_n$ $\textstyle =$ $\displaystyle \frac{2}{\tau} \int_0^{\tau} g(t) \cos \left( \frac{2 \pi n t}{\tau} \right) dt$

$\displaystyle b_n$ $\textstyle =$ $\displaystyle \frac{2}{\tau} \int_0^{\tau} g(t) \sin \left( \frac{2 \pi n t}{\tau} \right) dt$ (2.67)

The square wave of Eq. 2.17 can be described as

$\begin{displaymath} g(t) = \left\{ \begin{array}{lc} A & (0 < t < \frac{\tau}{2}) \\ -A & (\frac{\tau}{2} < t < \tau) \end{array}\right. \end{displaymath}$ (2.68)
The Fourier coefficients are identical to those obtained for the series describing the spatial wave form (Eq. 2.20),

$\displaystyle a_o$ $\textstyle =$ $\displaystyle 0$

$\displaystyle a_n$ $\textstyle =$ $\displaystyle 0$

$\displaystyle b_n$ $\textstyle =$ $\displaystyle \frac{2A}{n\pi} (1 - (-1)^n).$ (2.69)

We make similar moves to find the Fourier transformation,

$\begin{displaymath} G(\omega) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} g(t) e^{-i \omega t} dt \end{displaymath}$ (2.70)
The inverse transformation is given by

$\begin{displaymath} g(t) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} G(\omega) e^{i \omega t} d\omega \end{displaymath}$ (2.71)

The square pulse of Eq. 2.49 can be described as

$\begin{displaymath} g(t) = \left\{ \begin{array}{lc} A & (\vert t\vert < \frac{T}{2})\\ 0 & (\vert t\vert > \frac{T}{2}) \end{array}\right. \end{displaymath}$ (2.72)

where

$\begin{displaymath} T = \frac{L}{v_p} \end{displaymath}$ (2.73)
The frequency spectrum (the Fourier transformation) is given by

$\begin{displaymath} G(\omega) = A \sqrt{\frac{2}{\pi}} \frac{\sin \left(\frac{\omega T}{2}\right)}{\omega} \end{displaymath}$ (2.74)

Next: 2.4 Bandwidth Limits Up: 2. Wave Packets Previous: 2.2 Localized Packets

$\displaystyle a_o$	$\textstyle =$	$\displaystyle \frac{1}{\tau} \int_0^{\tau} g(t) dt$
$\displaystyle a_n$	$\textstyle =$	$\displaystyle \frac{2}{\tau} \int_0^{\tau} g(t) \cos \left( \frac{2 \pi n t}{\tau} \right) dt$
$\displaystyle b_n$	$\textstyle =$	$\displaystyle \frac{2}{\tau} \int_0^{\tau} g(t) \sin \left( \frac{2 \pi n t}{\tau} \right) dt$	(2.67)