x
= 0. Therefore the slope at
x
= 0 is
∝
sin(
kx
) = 0 without restriction. At
x
=
L
, the
slope is
∝
sin(
kL
) = 0, so that
kL
=
nπ
⇒
f
=
n
2
L
·
v
as in the two Fxed endpoints case.
These considerations are important, say, for wind instruments, such as organ pipes, ±utes,
etc.
Question
: What are the boundary conditions for wind instruments for closed or open pipes?
Periodic boundary conditions
:
We now want
y
(
x
= 0) =
y
(
x
=
L
). We don’t want the spatial function to be sin(
kx
),
because it would always vanish. Rather, at
x
= 0 we should have a general value for the
wave function. We therefore take
y
(
x, t
)=
A
cos[
k
(
x

x
0
)] sin(
ωt
), where
x
0
is the location
of the Frst antinode. (Of course, we could use sine function, and then
x
0
would measure
the distance of
x
= 0 from a node.) Therefore,
y
(0) =
A
cos(
kx
0
) sin(
ωt
), and
y
(
L
)=
A
cos[
k
(
L

x
0
)] sin(
ωt
). The periodicity condition that yields cos(
kx
0
) = cos[
k
(
L

x
0
)],
whose solution is
k
(
L

x
0
) = 2
π n
±
kx
0
. We choose the negative solution, and obtain
kL
=2
πn
, or
L
=2
πn/
(2
π/λ
)=
nλ
=
nv/f
. Therefore,
f
=
n
v
L
, and the fundamental
frequency is
f
=
v
L
. We will do something very similar when we discuss the Bohr model in
atomic physics.
Beats
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 Spring '09
 LIORBURKO
 Physics, Frequency, Cos, phase velocity, ym cos

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