Chapter 16: WavesI
HRW 16.14
The tension in a wire clamped at both ends is doubled without appreciably changing the wire’s length
between the clamps. What is the ratio of the new to the old wave speed for transverse waves traveling
along the wire.
Answer:
The wave speed of transverse waves on a string is given by
v
=
r
τ
μ
where
τ
is the tension on the string, and
μ
is the mass density. If the length of the wire doesn’t
change,
μ
is constant. Therefore the ratio of the velocities would be (
τ
new
= 2
τ
)
v
new
v
old
=
q
2
τ
μ
q
τ
μ
v
new
v
old
=
√
2
The wave speed increases by a factor of
√
2.
1
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View Full DocumentHRW 16.25
A uniform rope of mass
m
and length
L
hangs from a ceiling.
y
τ
mg
mass m =
μ
y
Figure 1: The forces on a section of rope of lenght
y
(a) Show that the speed of a transverse wave on the rope is a function of
y
, the distance from the
lower end, and is given by
v
=
√
gy
.
Answer:
If we draw a freebodydiagram of the piece of rope of length
y
, we can see that the
piece is in static equilibrium due to the downward force of gravity on the section below point
y
, and an upward tension force
τ
(See ﬁgure 1). The mass of the rope below
y
is related to
the linear mass density.
μ
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 Winter '08
 BROWN
 HRW

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