Solution by T.A. David Nichols
Physics 2a
Quiz #2 Solution
October 29, 2008
Physics 2a Quiz #2 Solution
A string with length
L
, tension
T
, and mass per unit length
μ
is fixed at both ends.
Part 1 [1 point]
: What is the angular frequency
ω
1
of the first mode (n=1)?
Using the now familiar expression
ω
n
= (
n
π
/
L
)
T
/
μ
, one has that
ω
1
=
π
L
T
μ
.
Part 2 [1 point]
: Write down the most general Fourier series for the transverse displace
ment
y
(
x
,
t
)
of the string.
Because both ends are fixed, the most general Fourier series is
y
(
x
,
t
) =
∞
∑
n
=
1
A
n
sin
n
π
x
L
cos
(
ω
n
t
+
δ
n
)
,
where
ω
n
=
n
π
L
T
μ
.
Part 3 [2 points]
: Suppose that the is released at time
t
=
0 with the following displace
ment:
y
(
x
,
t
=
0
) =
4
h
L
x
0
≤
x
<
L
/4
2
h

4
h
L
x
L
/4
≤
x
<
L
/2
0
L
/2
≤
x
≤
L
Write down an expression for the first four coefficients
A
n
(for
n
=
1, 2, 3, 4) in the series
expansion for
y
(
x
,
t
)
. (Assume the string is released from rest.)
Let’s calculate the coefficients of the Fourier series,
A
n
, for a general integer
n
and eval
uate them for
n
=
1, 2, 3, 4 at the end (this ultimately will require less work). Since the
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 Fall '08
 Martin
 mechanics, Fourier Series, Mass, Sin, David Nichols Physics

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