51. Two adjacent natural frequencies of an organ pipe are
determined to be 550 Hz and 650 Hz. Calculate (a) the
fundamental frequency and (b) the length of this pipe.
52. Why is the following situation impossible? A student is lis-
tening to the sounds from an air column that is 0.730 m
long. He doesn’t know if the column is open at both
ends or open at only one end. He hears resonance
from the air column at frequencies 235 Hz and 587 Hz.
53. A student uses an audio oscillator of adjustable fre-
quency to measure the depth of a water well. The
student reports hearing two successive resonances at
51.87 Hz and 59.85Hz. (a) How deep is the well?
(b) How many antinodes are in the standing wave at
Section 18.6 Standing Waves in Rods and Membranes
54. An aluminum rod is clamped one-fourth of the way
along its length and set into longitudinal vibration by
a variable-frequency driving source. The lowest fre-
quency that produces resonance is 4 400 Hz. The speed
of sound in an aluminum rod is 5 100 m/s. Determine
the length of the rod.
55. An aluminum rod 1.60 m long is held at its center. It
is stroked with a rosin-coated cloth to set up a longi-
tudinal vibration. The speed of sound in a thin rod
of aluminum is 5 100 m/s. (a) What is the fundamen-
tal frequency of the waves established in the rod?
(b) What harmonics are set up in the rod held in this
manner? (c) What If? What would be the fundamental
frequency if the rod were copper, in which the speed of
sound is 3 560 m/s?
Section 18.7 Beats: Interference in Time
56. While attempting to tune the note C at 523 Hz, a piano
tuner hears 2.00 beats/s between a reference oscillator
and the string. (a) What are the possible frequencies
of the string? (b) When she tightens the string slightly,
she hears 3.00 beats/s. What is the frequency of the
string now? (c)By what percentage should the piano
tuner now change the tension in the string to bring it
57. In certain ranges of a piano keyboard, more than one
string is tuned to the same note to provide extra loud-
ness. For example, the note at 110 Hz has two strings
at this frequency. If one string slips from its nor-
mal tension of 600 N to 540 N, what beat frequency
is heard when the hammer strikes the two strings
58. Review. Jane waits on a railroad platform while two
trains approach from the same direction at equal
speeds of 8.00m/s. Both trains are blowing their whis-
tles (which have the same frequency), and one train is
some distance behind the other. After the first train
passes Jane but before the second train passes her,
she hears beats of frequency 4.00 Hz. What is the fre-
quency of the train whistles?
59. Review. A student holds a tuning fork oscillating at
256Hz. He walks toward a wall at a constant speed
of 1.33m/s. (a)What beat frequency does he observe
when the piston is at a distance d
5 22.8 cm from the
open end and again when it is at a distance d
5 68.3 cm
from the open end. (a) What speed of sound is implied
by these data? (b)How far from the open end will the
piston be when the next resonance is heard?
44. A tuning fork with a frequency
of f5 512 Hz is placed near the
top of the tube shown in Figure
P18.44. The water level is low-
ered so that the length L slowly
increases from an initial value
of 20.0 cm. Determine the next
two values of L that correspond
to resonant modes.
45. With a particular fingering,
a flute produces a note with
frequency 880 Hz at 20.0°C.
The flute is open at both ends.
(a) Find the air column length.
(b) At the beginning of the
halftime performance at a late-
season football game, the ambient temperature is
25.00°C and the flutist has not had a chance to warm
up her instrument. Find the frequency the flute pro-
duces under these conditions.
46. A shower stall has dimensions 86.0 cm 3 86.0 cm 3
210cm. Assume the stall acts as a pipe closed at both
ends, with nodes at opposite sides. Assume singing
voices range from 130 Hz to 2 000 Hz and let the speed
of sound in the hot air be 355 m/s. For someone sing-
ing in this shower, which frequencies would sound the
richest (because of resonance)?
47. A glass tube (open at both ends) of length L is posi-
tioned near an audio speaker of frequency f 5 680 Hz.
For what values of L will the tube resonate with the
48. A tunnel under a river is 2.00 km long. (a) At what fre-
quencies can the air in the tunnel resonate? (b) Explain
whether it would be good to make a rule against blow-
ing your car horn when you are in the tunnel.
49. As shown in Figure P18.49,
water is pumped into a tall,
vertical cylinder at a volume
flow rate R 5 1.00 L/min.
The radius of the cylinder is
r 5 5.00 cm, and at the open
top of the cylinder a tuning
fork is vibrating with a fre-
quency f 5 512 Hz. As the
water rises, what time interval
elapses between successive
50. As shown in Figure P18.49,
water is pumped into a tall,
vertical cylinder at a volume
flow rate R. The radius of the cylinder is r, and at the
open top of the cylinder a tuning fork is vibrating with
a frequency f. As the water rises, what time interval
elapses between successive resonances?
Problems 49 and 50.
chapter 18 Superposition and Standing Waves
66. A 2.00-m-long wire having a mass of 0.100 kg is fixed
at both ends. The tension in the wire is maintained at
20.0N. (a)What are the frequencies of the first three
allowed modes of vibration? (b) If a node is observed at
a point 0.400 m from one end, in what mode and with
what frequency is it vibrating?
67. The fret closest to the bridge on a guitar is 21.4 cm
from the bridge as shown in Figure P18.67. When the
thinnest string is pressed down at this first fret, the
string produces the highest frequency that can be
played on that guitar, 2 349 Hz. The next lower note
that is produced on the string has frequency 2 217 Hz.
How far away from the first fret should the next fret
68. A string fixed at both ends and having a mass of 4.80g,
a length of 2.00 m, and a tension of 48.0 N vibrates in
its second (n 5 2) normal mode. (a) Is the wavelength
in air of the sound emitted by this vibrating string
larger or smaller than the wavelength of the wave on
the string? (b) What is the ratio of the wavelength in
air of the sound emitted by this vibrating string and
the wavelength of the wave on the string?
69. A quartz watch contains a crystal oscillator in the form
of a block of quartz that vibrates by contracting and
expanding. An electric circuit feeds in energy to main-
tain the oscillation and also counts the voltage pulses
to keep time. Two opposite faces of the block, 7.05 mm
apart, are antinodes, moving alternately toward each
other and away from each other. The plane halfway
between these two faces is a node of the vibration. The
speed of sound in quartz is equal to 3.70 3 103m/s.
Find the frequency of the vibration.
70. Review. For the arrangement shown in Figure P18.70,
the inclined plane and the small pulley are frictionless;
the string supports the object of mass M at the bottom
of the plane; and the string has mass m. The system
is in equilibrium, and the vertical part of the string
has a length h. We wish to study standing waves set up
in the vertical section of the string. (a) What analysis
model describes the object of mass M? (b) What analy-
sis model describes the waves on the vertical part of the
between the tuning fork and its echo? (b) How fast
must he walk away from the wall to observe a beat fre-
quency of 5.00 Hz?
Section 18.8 nonsinusoidal Wave Patterns
60. An A-major chord consists of the notes called A, C#,
and E. It can be played on a piano by simultaneously
striking strings with fundamental frequencies of
440.00 Hz, 554.37 Hz, and 659.26 Hz. The rich con-
sonance of the chord is associated with near equality
of the frequencies of some of the higher harmonics of
the three tones. Consider the first five harmonics of
each string and determine which harmonics show near
61. Suppose a flutist plays a 523-Hz C note with first har-
monic displacement amplitude A
5 100 nm. From Fig-
ure 18.19b read, by proportion, the displacement ampli-
tudes of harmonics 2 through 7. Take these as the values
in the Fourier analysis of the sound and
5 ??? 5 B
5 0. Construct a graph of
the waveform of the sound. Your waveform will not look
exactly like the flute waveform in Figure 18.18b because
you simplify by ignoring cosine terms; nevertheless, it
produces the same sensation to human hearing.
62. A pipe open at both ends has a fundamental frequency
of 300 Hz when the temperature is 0°C. (a) What is the
length of the pipe? (b) What is the fundamental fre-
quency at a temperature of 30.0°C?
63. A string is 0.400 m long and has a mass per unit length
of 9.00 3 10–3 kg/m. What must be the tension in the
string if its second harmonic has the same frequency as
the second resonance mode of a 1.75-m-long pipe open
at one end?
64. Two strings are vibrating at the same frequency of
150Hz. After the tension in one of the strings is
decreased, an observer hears four beats each second
when the strings vibrate together. Find the new fre-
quency in the adjusted string.
65. The ship in Figure P18.65 travels along a straight line
parallel to the shore and a distance d 5 600 m from
it. The ship’s radio receives simultaneous signals of the
same frequency from antennas A and B, separated by
a distance L 5 800m. The signals interfere construc-
tively at point C, which is equidistant from A and B.
The signal goes through the first minimum at point D,
which is directly outward from the shore from point B.
Determine the wavelength of the radio waves.
76. A nylon string has mass 5.50 g and
length L 5 86.0 cm. The lower end
is tied to the floor, and the upper
end is tied to a small set of wheels
through a slot in a track on which
the wheels move (Fig. P18.76). The
wheels have a mass that is negli-
gible compared with that of the
string, and they roll without fric-
tion on the track so that the upper
end of the string is essentially free.
At equilibrium, the string is vertical
and motionless. When it is carrying a small-amplitude
wave, you may assume the string is always under uni-
form tension 1.30 N. (a) Find the speed of transverse
waves on the string. (b) The string’s vibration pos-
sibilities are a set of standing-wave states, each with
a node at the fixed bottom end and an antinode at
the free top end. Find the node–antinode distances
for each of the three simplest states. (c)Find the fre-
quency of each of these states.
77. Two train whistles have identical frequencies of
180 Hz. When one train is at rest in the station and
the other is moving nearby, a commuter standing on
the station platform hears beats with a frequency of
2.00 beats/s when the whistles operate together. What
73. Review. Consider the apparatus shown in Figure 18.11
and described in Example 18.4. Suppose the number
of antinodes in Figure 18.11b is an arbitrary value n.
(a) Find an expression for the radius of the sphere in
the water as a function of only n. (b) What is the mini-
mum allowed value of n for a sphere of nonzero size?
(c) What is the radius of the largest sphere that will
produce a standing wave on the string? (d) What hap-
pens if a larger sphere is used?
74. Review. The top end of a yo-yo string is held stationary.
The yo-yo itself is much more massive than the string. It
starts from rest and moves down with constant accelera-
tion 0.800 m/s2 as it unwinds from the string. The rub-
bing of the string against the edge of the yo-yo excites
transverse standing-wave vibrations in the string. Both
ends of the string are nodes even as the length of the
string increases. Consider the instant 1.20 s after the
motion begins from rest. (a) Show that the rate of change
with time of the wavelength of the fundamental mode of
oscillation is 1.92 m/s. (b) What if? Is the rate of change
of the wavelength of the second harmonic also 1.92 m/s
chapter 18 Superposition and Standing Waves
(b) Determine the amplitude and phase angle for this
84. A flute is designed so that it produces a frequency of
261.6Hz, middle C, when all the holes are covered and
the temperature is 20.0°C. (a) Consider the flute as a
pipe that is open at both ends. Find the length of the
flute, assuming middle C is the fundamental. (b) A sec-
ond player, nearby in a colder room, also attempts to
play middle C on an identical flute. A beat frequency
of 3.00 Hz is heard when both flutes are playing. What
is the temperature of the second room?
85. Review. A 12.0-kg object hangs in equilibrium from a
string with a total length of L 5 5.00 m and a linear mass
density of m 5 0.001 00 kg/m. The string is wrapped
around two light, frictionless pulleys that are separated
by a distance of d 5 2.00 m (Fig. P18.85a). (a) Deter-
mine the tension in the string. (b) At what frequency
must the string between the pulleys vibrate to form the
standing-wave pattern shown in Figure P18.85b?
Problems 85 and 86.
86. Review. An object of mass m hangs in equilibrium
from a string with a total length L and a linear mass
density m. The string is wrapped around two light,
frictionless pulleys that are separated by a distance d
(Fig. P18.85a). (a)Determine the tension in the string.
(b) At what frequency must the string between the pul-
leys vibrate to form the standing-wave pattern shown in
87. Review. Consider the apparatus shown in Figure
P18.87a, where the hanging object has mass M and the
string is vibrating in its second harmonic. The vibrat-
ing blade at the left maintains a constant frequency.
The wind begins to blow to the right, applying a con-
are the two possible speeds and directions the moving
train can have?
78. Review. A loudspeaker at the front of a room and an
identical loudspeaker at the rear of the room are being
driven by the same oscillator at 456 Hz. A student
walks at a uniform rate of 1.50 m/s along the length
of the room. She hears a single tone repeatedly becom-
ing louder and softer. (a) Model these variations as
beats between the Doppler-shifted sounds the student
receives. Calculate the number of beats the student
hears each second. (b) Model the two speakers as pro-
ducing a standing wave in the room and the student as
walking between antinodes. Calculate the number of
intensity maxima the student hears each second.
79. Review. Consider the copper object hanging from
the steel wire in Problem 32. The top end of the wire
is fixed. When the wire is struck, it emits sound with a
fundamental frequency of 300 Hz. The copper object is
then submerged in water. If the object can be positioned
with any desired fraction of its volume submerged, what
is the lowest possible new fundamental frequency?
80. Two wires are welded together end to end. The wires
are made of the same material, but the diameter of one
is twice that of the other. They are subjected to a ten-
sion of 4.60N. The thin wire has a length of 40.0 cm
and a linear mass density of 2.00 g/m. The combina-
tion is fixed at both ends and vibrated in such a way
that two antinodes are present, with the node between
them being right at the weld. (a)What is the frequency
of vibration? (b) What is the length of the thick wire?
81. A string of linear density 1.60 g/m is stretched between
clamps 48.0 cm apart. The string does not stretch
appreciably as the tension in it is steadily raised from
15.0 N at t 5 0 to 25.0 N at t 5 3.50 s. Therefore, the
tension as a function of time is given by the expression
T 5 15.0 1 10.0t/3.50, where T is in newtons and t is
in seconds. The string is vibrating in its fundamental
mode throughout this process. Find the number of
oscillations it completes during the 3.50-s interval.
82. A standing wave is set up in a string of variable length
and tension by a vibrator of variable frequency. Both
ends of the string are fixed. When the vibrator has a
frequency f, in a string of length L and under tension
T, n antinodes are set up in the string. (a) If the length
of the string is doubled, by what factor should the fre-
quency be changed so that the same number of anti-
nodes is produced? (b) If the frequency and length are
held constant, what tension will produce n 1 1 anti-
nodes? (c) If the frequency is tripled and the length of
the string is halved, by what factor should the tension be
changed so that twice as many antinodes are produced?
83. Two waves are described by the wave functions
(x, t) 5 5.00 sin (2.00x 2 10.0t)
(x, t) 5 10.0 cos (2.00x 2 10.0t)
where x, y
, and y
are in meters and t is in seconds.
(a)Show that the wave resulting from their super-
position can be expressed as a single sine function.
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