292
Chapter 14
Waves and Sound
Answers to Even-numbered Conceptual Questions
2.
Waves passing through a field of grain are longitudinal waves – the motion of each stalk
of grain is in the same direction as the motion of the wave itself.
4.
This wave is longitudinal, since each cat moves in the same direction as the wave.
6.
Referring to Equation 14-2, we see that to double the speed, the tension in the string must
be increased by a factor of 4.
8.
The thick string has more mass per length than the thin string.
Therefore, if the wave
speed is the same, the tension in the thick string must be greater than the tension in the
thin string.
10.
(a)
When the string is displaced (stretched) by its greatest amount, its potential energy is
a maximum, just as in the case of a spring.
(b)
At zero displacement, the string is like a
spring at its equilibrium position.
Therefore, the potential energy of the string is a
minimum.
12.
(a)
This wave travels in the negative
x
direction, because
Bx
+
Ct
= (constant) implies
x
= (constant)/
B
– (
C
/
B
)
t
.
(b)
The constant
A
is the amplitude of the wave.
(c)
The
speed of this wave, as can be seen from part (a), is
C
/
B
.
(d)
To be specific, let’s find the
times when the wave has zero displacement at
x
= 0.
In this case, we have
y
=
A
sin (
Ct
),
which is equal to zero at the times
t
= 0, ±
π
/
C
, ±2
π
/
C
, ±3
π
/
C
, ….
Other values of
x
can be
considered similarly.
14.
If the speed of sound depended on frequency, the sound in the first row – where the travel
time is so small – would not be affected significantly.
Farther back from the stage,
however, sounds with different frequencies would arrive at different times – the bass
would be “out of sync” with the treble.
16.
As was shown in Example 14-5 for the case of sound, there is a direct relationship
between the speed of an object and the Doppler frequency it experiences.
This effect
applies to radar waves as well, providing a one-to-one correspondence between the
observed frequency and the speed of the car.
18.
As can be seen in Figure 14-18, the speed of the observer must be equal to the speed of
sound.
20.
The sliding part of a trombone varies the length of the vibrating air column that produces
the trombone’s sound.
By adjusting this length, the player controls the resonant
frequencies of the instrument.
This, in turn, varies the frequency of sound produced by
the trombone.
22.
The thicker string is used to produce the low-frequency notes.
This follows because the
frequency of the fundamental depends directly on the speed of waves on the string.
Therefore, for a given tension, a string with a greater mass per length has a smaller wave
speed and hence a lower frequency.