C
HEMICAL
E
NGINEERING 132A
Professor Todd Squires
Spring Quarter, 2007
Review for Final
FOURIER ANALYSIS
Remember, the basic idea of Fourier Analysis is to take a function and represent it in a different form. It is all
the same information, just a different way to present it. I had used an analogy with language: if you wanted to read
Homer’s Iliad, you have two ways to do it: either learn ancient Greek, or buy a translation. In a variety of situations
in Mathematics/Physics/Engineering, it turns out to be far easier to transform, or ‘translate’ functions than to use
the standard
f
(
x
) form. We have already seen this with the heat / diffusion equation – we very naturally wind up
with a solution to the PDE in the form of a Fourier series, or a Fourier-Bessel Series.
There are a variety of transforms we have used. In the first half of class, we used Laplace Transforms, which we
used primarily to solve initial value ODE’s.
The second half of class saw Fourier Series, Fourier Sine and Cosine
Series, Fourier Transforms, and Fourier-Bessel Series.
The basic idea of the Fourier Series and its variants is to take a function and to represent it as a combination of sines
and cosines – oscillatory (wiggly) functions, each with a frequency or wavenumber
q
. High
q
means rapid oscillations,
low-
q
means slow oscillations, and
q
= 0 means a constant. The cochlea in your ear is a natural Fourier analyser –
it takes a time-dependent pressure signal and converts it into sound frequencies, each with some amplitude. X-Ray
crystallography gives Fourier-transformed structures.
As we will see in a bit, the Fourier-Bessel Series is morally very similar to the Fourier Series – the difference being
that Bessel Functions
J
n
(
qx
) and
Y
n
(
qx
) are used in place of sin(
qx
) and cos(
qx
). Bessel Functions are also wiggly
functions, and
q
functions a lot like a frequency/wavenumber, just like it does for sines and cosines. Note also the
subscript
n
– this is called the order of the Bessel function. We’ll talk about this in a bit. The short version is that
Bessel functions with different
n
’s should be thought of as totally different functions. (They are linearly independent,
for those who know what that means).
The simplest way to start is with the Fourier Series, which gives a way to represent a function on a finite domain,
defined between
−
L
and
L
.
This arises in many contexts – we used them in the one-dimensional heat/diffusion
equation (temperature in a thin bar of length 2
L
, for example). The Fourier series representation of a function
f
(
x
)
is
f
(
x
) =
A
0
2
+
∞
summationdisplay
n
=1
A
n
cos
nπx
L
+
B
n
sin
nπx
L
.
(1)
There are several things to notice here.
•
An infinite sum is taken. This involves an infinite number of cosines and sines, meaning an infinite number of
different frequencies/wavenumbers is involved. This should not be surprising, since there are infinitely many ways to
make a function between
−
L
and
L
. (Side comment: what is in fact surprising is that a
countable
(that is, discrete
n
= 1
,
2
,
3
, ...
) sum works, rather than a continuous integral. A note for the math wonks in the room.) In practice,