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53. We refer to the points where the rope is attached as
A
and
B
, respectively. When
A
and
B
are not
displaced horizontal, the rope is in its initial state (neither stretched
(under tension)
nor slack). If they are
displaced away from each other, the rope is clearly stretched. When
A
and
B
are displaced in the same
direction, by amounts (in absolute value)

ξ
A

and

ξ
B

,theni
f

ξ
A

<

ξ
B

then the rope is stretched,
and if

ξ
A

>

ξ
B

the rope is slack. We must be careful about the case where one is displaced but the
other is not, as will be seen below.
(a) The standing wave solution for the shorter cable, appropriate for the initial condition
ξ
=0a
t
t
= 0, and the boundary conditions
ξ
=0at
x
=0and
x
=
L
(the
x
axis runs vertically here),
is
ξ
A
=
ξ
m
sin(
k
A
x
) sin(
ω
A
t
). The angular frequency is
ω
A
=2
π/T
A
,andthewav
enumbe
ri
s
k
A
=2
π/λ
A
where
λ
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 Fall '08
 SPRUNGER
 Physics

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