1
BMET2960 Week 9 Tutorial: Fourier Integrals & Transforms
In this week's tutorial we will use the
Fourier Integral
method to analytically solve for the di
ff
usion of an
aerosol released along an unbounded space – a problem we cannot solve using Fourier series. We will also
introduce the
Fourier Transform
in Matlab to analyse synthetic and real signals in the frequency domain.
Part 1 – Fourier Integral Solution to the heat/diffusion equation
Dolce & Gabbana have recently set up a new shop mid-way along a long and busy alleyway in Paris. They have
a marketing idea for a new women’s fragrance called “
B-Me
” but aren’t sure how effective the strategy will be.
The idea is to release a large quantity of the new fragrance right outside the shop front and have it waft along the
alleyway in both directions during the day (Figure 1). D&G want to know how far the fragrance odor will reach
along the alleyway, and how many people are therefore likely to come under the influence of their marketing
campaign. This will help them determine if it’s a cost effective marketing approach. And they need your help!
Assuming no wind, the di
ff
usion of the fragrance along the alleyway is governed by the di
ff
usion equation:
)
1
(
xx
t
DY
Y
=
where
Y
is the mass fraction of the fragrance and
D
is the diffusivity. Since the alleyway is long and narrow we
will model it as a one dimensional problem. The initial condition describing the fragrance ‘bomb’ released
directly outside the shop-front is:
)
2
(
elsewhere
zero
and
m
1
m
1
for
1
)
(
)
0
,
(
<
<
−
=
=
x
x
f
x
Y
There are no boundary conditions.