1451
Chapter 15
Traveling Waves
Conceptual Problems
1
•
[SSM]
A rope hangs vertically from the ceiling.
A pulse is sent up
the rope. Does the pulse travel faster, slower, or at a constant speed as it moves
toward the ceiling? Explain your answer.
Determine the Concept
The speed of a transverse wave on a uniform rope
increases with increasing tension. The waves on the rope move faster as they
move toward the ceiling because the tension increases due to the weight of the
rope below the pulse.
2
•
A pulse on a horizontal taut string travels to the right. If the rope’s
mass per unit length decreases to the right, what happens to the speed of the pulse
as it travels to the right? (
a
) It slows down. (
b
) It speeds up. (
c
) Its speed is
constant. (
d
) You cannot tell from the information given.
Determine the Concept
The speed
v
of a pulse on the string varies with the
tension
F
T
in the string and its mass per unit length
μ
according to
μ
T
F
v
=
.
Because the rope’s mass per unit length decreases to the right, the speed of the
pulse increases.
( )
b
is correct.
3
•
As a sinusoidal wave travels past a point on a taut string, the arrival
time between successive crests is measured to be 0.20 s. Which of the following is
true? (
a
) The wavelength of the wave is 5.0 m. (
b
) The frequency of the wave is
5.0 Hz. (
c
) The velocity of propagation of the wave is 5.0 m/s. (
d
) The
wavelength of the wave is 0.20 m. (
e
) There is not enough information to justify
any of these statements.
Determine the Concept
The distance between successive crests is one
wavelength and the time between successive crests is the period of the wave
motion. Thus,
T
= 0.20 s and
f
= 1/
T
=
5.0 Hz.
)
(
b
is correct.
4
•
Two harmonic waves on identical strings differ only in amplitude.
Wave A has an amplitude that is twice that of wave B’s. How do the energies of
these waves compare? (
a
)
E
A
=
E
B
. (
b
)
E
A
=
2
E
B
. (
c
)
E
A
=
4
E
B
. (
d
) There is
not enough information to compare their energies.
Picture the Problem
The average energy transmitted by a wave on a string is
proportional to the square of its amplitude and is given by
(
)
x
A
E
Δ
Δ
2
2
2
1
av
μω
=
where
A
is the amplitude of the wave,
μ
is the linear density (mass per unit length)
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Chapter 15
1452
of the string,
ω
is the angular frequency of the wave, and
Δ
x
is the length of the
string.
Because the waves on the strings
differ only in amplitude, the energy
of the wave on string A is given by:
2
A
A
cA
E
=
Express the energy of the wave on
string B:
2
B
B
cA
E
=
Divide the first of these equations by
the second and simplify to obtain:
2
B
A
2
B
2
A
B
A
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
=
A
A
cA
cA
E
E
Because
A
A
= 2
A
B
:
4
2
2
B
B
B
A
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
A
A
E
E
⇒
( )
c
is correct.
5
•
[SSM]
To keep all of the lengths of the treble strings (unwrapped
steel wires) in a piano all about the same order of magnitude, wires of different
linear mass densities are employed. Explain how this allows a piano manufacturer
to use wires with lengths that are the same order of magnitude.
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 Fall '08
 oshea
 Physics, Energy density, Substitute numerical values, Pav

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