Chapter 18
Superposition and
Standing Waves
Quick Quiz 18.1
Two pulses are traveling toward each other at 10 cm/s on a
long string, as shown in the figure below. Sketch the shape
of the string at t = 0.6 s.
Quick Quiz 18.1
Answer: The shape of the string at t = 0.6 s is shown below.
Quick Quiz 18.2
Two pulses move in opposite directions on a string and are
identical in shape except that one has positive displacements
of the elements of the string and the other has negative
displacements. At the moment that the two pulses
completely overlap on the string,
(a) the energy associated with the pulses has disappeared
(b) the string is not moving
(c) the string forms a straight line
(d) the pulses have vanished and will not reappear
Quick Quiz 18.2
Answer: (c). The pulses completely cancel each other in
terms of displacement of elements of the string from
equilibrium, but the string is still moving. A short time later,
the string will be displaced again and the pulses will have
passed each other.
Quick Quiz 18.3
Consider a standing wave on a string as shown in the figure below. Define the velocity of
elements of the string as positive if they are moving upward in the figure. At the moment
the string has the shape shown at the bottom of part (a), the instantaneous velocity of
elements along the string
(a) is zero for all elements
(b) is positive for all elements
(c) is negative for all elements
(d) varies with the position of the element
Quick Quiz 18.3
Answer: (a). The pattern shown at the bottom of Figure
18.9a corresponds to the extreme position of the string. All
elements of the string have momentarily come to rest.
Quick Quiz 18.4
Continuing with the scenario in question 3, at the moment
the string has the shape shown at the bottom of part b of the
figure above, the instantaneous velocity of elements along
the string
(a) is zero for all elements
(b) is positive for all elements
(c) is negative for all elements
(d) varies with the position of the element
Quick Quiz 18.4
Answer: (d). Near a nodal point, elements on one side of the
point are moving upward at this instant and elements on the
other side are moving downward.
Quick Quiz 18.5
When a standing wave is set up on a string fixed at both
ends,
(a) the number of nodes is equal to the number of antinodes
(b) the wavelength is equal to the length of the string divided
by an integer
(c) the frequency is equal to the number of nodes times the
fundamental frequency
(d) the shape of the string at any time is symmetric about the
midpoint of the string.
Quick Quiz 18.5
Answer: (d). Choice (a) is incorrect because the number of
nodes is one greater than the number of antinodes. Choice
(b) is only true for half of the modes; it is not true for any
odd-numbered mode. Choice (c) would be correct if we
replace the word nodes with antinodes.
Quick Quiz 18.6
(a) 1
(b) 2
(c) 3
(d) 4
(e) 5
Courtesy of Professor Thomas D. Rossing, Northern Illinois University
A wine glass can be shattered through resonance by
maintaining a certain frequency of a high-intensity sound
wave. The figure below shows a side view of a wine glass
vibrating in response to such a sound wave. If an integral
number of waves "fit" around the circumference of the
vibrating rim, how many wavelengths fit around the rim in
part (a)?
Quick Quiz 18.6
Answer: For each natural frequency of the glass, the
standing wave must “fit” exactly around the rim. In Figure
18.17a we see three antinodes on the near side of the glass,
and thus there must be another three on the far side. This
corresponds to three complete waves. In a top view, the
wave pattern looks like this (although we have greatly
exaggerated the amplitude):
Quick Quiz 18.7
A pipe open at both ends resonates at a fundamental
frequency fopen. When one end is covered and the pipe is
again made to resonate, the fundamental frequency is fclosed.
Which of the following expressions describes how these two
resonant frequencies compare?
(a) fclosed = fopen
(b) fclosed = 1/2 fopen
(c) fclosed = 2 fopen
(d) fclosed = 3/2 fopen
Quick Quiz 18.7
Answer: (b). With both ends open, the pipe has a
fundamental frequency given by Equation 18.11: fopen = v /
2L. With one end closed, the pipe has a fundamental
frequency given by Equation 18.12:
f closed
v
v
1


 1 f open
2 2L
2
4L
Quick Quiz 18.8
Balboa Park in San Diego has an outdoor organ. When the
air temperature increases, the fundamental frequency of one
of the organ pipes
(a) stays the same
(b) goes down
(c) goes up
(d) is impossible to determine
Quick Quiz 18.8
Answer: (c). The increase in temperature causes the speed of
sound to go up. According to Equation 18.11, this will result
in an increase in the fundamental frequency of a given organ
pipe.
Quick Quiz 18.9
You are tuning a guitar by comparing the sound of the string
with that of a standard tuning fork. You notice a beat
frequency of 5 Hz when both sounds are present. You
tighten the guitar string and the beat frequency rises to 8 Hz.
In order to tune the string exactly to the tuning fork, you
should
(a) continue to tighten the string
(b) loosen the string
(c) impossible to determine
Quick Quiz 18.9
Answer: (b). Tightening the string has caused the
frequencies to be farther apart, based on the increase in the
beat frequency.