PHYS 302
: Unit 10 Viewing Notes –
Athabasca University
1
PHYS 302: Vibrations and Waves – Unit 10 Viewing Notes
You may note that this is MIT lecture 11 and that the recording date is 12 days after the previous lecture included in
PHYS 302
. Lecture 10 was the Exam Review provided as Supplementary Material in Unit 9, and presumably MIT
has a formal exam week during which there are no lectures.
The topic of this lecture is Fourier analysis. It turns out that our known ability to represent an arbitrary vibration in
terms of normal modes has already shown us the equivalent of Fourier analysis (2:10), but without calling it that.
We wish to examine how such a representation helps us to deal with initial conditions. In representing the vibratory
displacement as a superposition of normal modes we have
1
( , )
sin(
)cos
n
n
n
n
y x t
B
k x
t
, with
n
n
L
k
and
n
n
vk
. If we are working from an initial condition, then
t
=0 and
2
1
2
3
( ,0)
sin(
)
sin(
)
...
x
x
L
L
y x
B
B
B
(2:45). Fourier analysis can show us how each
B
n
corresponds to part of the initial condition, but first we need to
discuss some general considerations about Fourier series. For any singlevalued regular function a series can be
formed
0
2
1
1
( )
cos
sin
A
m
m
m
m
f x
A
mx
B
mx
, where the first, second, and third parts are singled out for
discussion (4:45). We have to find the values for the
A
m
and the
B
m
, given an initial value for the function at all
points. This will involve integral calculus again. If you do not have a deep acquaintance with integral calculus, you
should follow these viewing notes carefully. One important aspect of the definite integral is that it reflects the
average value of a function. The average value of a sequence of
N
numbers is their sum divided by
N
. Similarly, the
average value of a function is its definite integral over an interval, divided by the length of the interval. Consider the
interval 
π
to
π
. Over this interval, the second term has cos functions, and cos(
m
π
) is 1 for any positive integer
m
.
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 Spring '10
 martinconner
 Calculus, Derivative, Cos

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