8.4 Resonance in Air Columns
This section builds on the students’ knowledge of standing waves and mechanical resonance gained in Chapter 6 with the
experimental discovery of the resonant lengths for closed air columns (Investigation 8.4.1) and the measurement of the speed
of sound in a closed air column (Investigation 8.4.2). The resonant lengths for open columns are then discussed. Numerous
problems are provided to apply the principles of resonance on air columns.
Achievement Chart
Categories
Assessment Opportunities/Specific
Expectation Addressed
Assessment Tools
Knowledge/Understanding
Practice Questions
Understanding Concepts, q. 1–7
WS1.01, WS1.08
Section 8.4 Questions
Understanding Concepts, q. 1–9
WS1.01, WS1.08
Investigation 8.4.1
Investigation 8.4.2
WS1.02, WS1.03, WS1.08, WS2.03
Rubric1: Knowledge/Understanding
Inquiry
Rubric 2: Inquiry Skills
Expectations Addressed
Overall Expectations—WSV.02
Overall Skills Expectation—SIS.06, SIS.07
Specific Expectations:
• WS1.02 describe and illustrate the properties of
transverse and longitudinal waves in different media,
and analyze the velocity of waves travelling in those
media in quantitative terms
• WS1.03 compare the speed of sound in different
media, and describe the effect of temperature on the
speed of sound
• WS1.08 analyze, in quantitative terms, the conditions
needed for resonance in air columns, and explain how
resonance is used in a variety of situations (e.g.,
analyze resonance conditions in air columns in
quantitative terms, identify musical instruments using
such air columns, and explain how different notes are
produced)
BACKGROUND INFORMATION
Longitudinal waves travelling along a tube of definite
length are reflected at the ends of the tube in much the
same way that transverse waves in a string are reflected at
its ends (see text page 289). Interference between the
waves travelling in opposite directions gives rise to
standing waves.
If the reflection takes place at a closed end, the
displacement of the particles of air at that end will always
© 2002 Nelson Thomson Learning
• WS2.01 draw, measure, analyze, and interpret the
properties of waves (e.g., reflection, diffraction, and
interference, including interference that results in
standing waves) during their transmission in a
medium and from one medium to another, and during
their interaction with matter
• WS2.02 design and conduct an experiment to
determine the speed of waves in a medium, compare
theoretical and empirical values, and account for
discrepancies
• WS2.03 analyze, through experimentation, the
conditions required to produce resonance in vibrating
objects and/or in air columns (e.g., in string
instruments, tuning forks, wind instruments), predict
the conditions required to produce resonance in
specific cases, and determine whether the predictions
are correct through experimentation
be zero (fixed-end reflection). Thus a node always forms
at the fixed end. If the end of the tube is open, the nature
of the reflection is more complex and depends on whether
the tube is wide or narrow compared with the wavelength.
If the tube is narrow, which is the case in most musical
instruments, the reflection in the column of air is similar to
a free-end reflection in a transverse wave. Thus, a loop
forms at the open end. But what does a loop look like in a
longitudinal standing wave pattern? It is probably
sufficient to say that successive double compressions and
Unit 3 Waves and Sound 205
rarefactions are created by the constructive interference at
the open end, and that one should not even try to draw a
sketch to illustrate the phenomenon.
A detailed explanation of resonance in a closed air
column has been omitted on page 289 because the authors
feel that such an explanation would not be entirely based
on previous student knowledge and experience. Also, it is
difficult to conceptualize longitudinal standing waves.
The reflections at the opening where the instrument is
blown are found to have a loop located at or near the
opening. The effective resonant length of the air column of
a wind instrument is thus less definite than the length of a
string fixed at its ends.
ADDRESSING ALTERNATIVE
CONCEPTIONS
In looking at reference material dealing with air columns,
students may find an explanation of resonance in which
pressure rather than displacement is described. Note that
where a minimum displacement occurs—for example, at
the closed end of an air column—the pressure is
maximum, and where a maximum displacement occurs,
the pressure is minimum. Thus, pressure diagrams are the
opposite of displacement or amplitude diagrams.
Related Background Resources
Nelson Web site:
www.science.nelson.com
for specific Web links
PLANNING
Suggested Time
Narrative/Practice25 to 30 minutes
Investigation 8.4.120 to 30 minutes
Investigation 8.4.220 to 30 minutes
Section Questions20 to 25 minutes
Core Instructional Resources
• Solutions Manual
Supplemental Resources
• Lab and Study Blackline Masters
TEACHING SUGGESTIONS
• This section relies on the understanding of periodic
waves from Chapter 6, especially the study of standing
waves. It is a good idea to review section 6.8 before
starting this section, especially Figures 1 and 2, pages
228 and 229, respectively.
206 Chapter 8 Music, Musical Instruments, and Acoustics
• You could begin this section with a demonstration of
resonance in sound. Find an inexpensive long-stemmed
wine glass that “sings” when you rub a moist finger
around the rim. (Moisten your finger with vinegar if
water doesn’t work.) Have the students hypothesize
what effect will be observed when the amount of water
is changed, and then demonstrate the concept. Also,
allow the students to observe from close by the
vibrations on the water surface.
• For added fun, fill the glass half full with water, get it
ringing, and quickly pour out the water while you listen
for the change in pitch. Practise this demonstration
before your classroom debut.
• As suggested in the text, the wooden resonance boxes
attached to a tuning fork have a specific length for a
specific tuning fork. This can be demonstrated by
detaching the fork from its resonance box, striking it
once, and then striking it again, the second time
touching it to its box and then to another box with a
different resonant length.
• You can demonstrate the resonant sounds produced in
the air column so that students will know what to listen
for in the investigation that follows. A simple but
effective demonstration of the difference between open
and closed air columns can be performed using a rigid,
hollow, rubber tube, such as a plumbing tube from a
hardware store. The tube should be about 5 cm in
diameter and 50 cm to 100 cm long. Hold the tube
tightly in one hand, slap one end of it with the open
palm of your other hand, and leave your palm resting on
the end of the tube. This approximates the sound that
comes from a column closed at one end, giving a
frequency (f). Repeat the procedure, this time bouncing
the palm of your hand quickly off the end of the tube.
This approximates the sound coming from an open air
column. Its frequency is 2f.
• Stationary longitudinal waves can be demonstrated in a
column of gas using the Kundt’s tube apparatus, if
available. Longitudinal standing waves can be
demonstrated easily with the Kundt’s tube using cork
dust.
• The important teaching point in this section is that air
columns of proper length enhance the intensity (and
quality) of the original sound.
• Resonating air columns should not be used to find the
temperature of the classroom. Such irrelevant problems
should be avoided. They are only mathematical
exercises, not good physics.
Investigation 8.4.1
• This student investigation is required in the Ontario
curriculum.
• Traditionally, this topic has been demonstrated by the
teacher using an air column whose length is increased or
decreased by a water reservoir connected by rubber
tubing to the bottom of the glass column.
© 2002 Nelson Thomson Learning
• This investigation uses graduated cylinders and plastic
pipe. The plastic pipe suggested is 1.25 in. (9.25 cm)
rigid PVC pipe, which can be purchased in 8 ft. (2.4 m)
lengths and cut into 80 cm lengths with a wood saw or a
hacksaw. Large glass or plastic cylinders (1000 mL) are
required.
• It is difficult for students to perform this investigation if
they all work in the same room. Try to separate the
groups by using the hall, the prep room, and so on.
• A correction that can be used for “end error” in closed
air columns is 0.4 multiplied by the diameter of the
column.
Investigation 8.4.2
• This student investigation is required in the Ontario
curriculum.
• Using their experience from the previous investigation,
the students should be left completely on their own to do
this one.
• Note that it is assumed that the students will find the
speed of sound using resonance. The alternative is using
reflection, but this was done in Investigation 7.3.1.
• Pairs of students could do the investigation in separate
areas away from other students (see Assessment below).
• This investigation challenges the students to design and
perform a procedure to locate the resonant lengths for an
open air column, reinforcing the concepts in this section.
It is also an excellent opportunity to assess inquiry
skills.
• The resonant lengths for open columns can be
demonstrated using two 80 cm tubes. PVC or copper
pipe could be used, choosing the sizes carefully so one
slides inside the other. The procedure is the same as in
Investigation 8.4.1.
• The mathematical calculations that continue through to
the end of this section should pose no difficulty for your
students if you stress the ½λ concept.
large glass cylinder (1000 mL) (graduated is not required;
ungraduated is much cheaper)
at least two tuning forks (e.g., 512 Hz and 1024 Hz)
metre stick
thermometer
Safety and Disposal:
• The main safety issue is the possibility of the large glass
cylinders toppling and the resulting broken glass.
Placing the cylinder on the floor instead of on a lab desk
minimizes the risk.
• Students should be cautioned that the apparatus has a
high centre of gravity and is easy to knock over.
(The cost of 1000 mL cylinders is rather high!)
Assessment:
• This investigation should be assessed for all of the
inquiry skills listed on page 287.
Student Preparation
• Students need to understand the properties of standing
waves, which are a combination of resonance and
interference. For details, they can review section 6.8,
pages 226–30.
DURING
• It is a good idea to demonstrate what a loud “resonance
“ sound is like so students will recognize it. That way, as
several resonances are heard throughout the classroom,
students will recognize which one is coming from their
own apparatus.
AFTER
• Resonance was noted for both tuning forks at each of
the first three resonant lengths. The intensity of the
sound at these lengths was notably louder. The resonant
lengths are recorded in Table 1.
• The air temperature was measured and recorded as
20°C.
Table 1 Resonant Lengths
INVESTIGATION 8.4.1
Resonance in Closed Air Columns
• The objective of this investigation is to have students
measure resonance and write a formal report for
assessment.
Resonant
point
first
second
third
BEFORE
Teacher Preparation
Time: 20 to 30 minutes
Materials and Equipment:
Each group of three or four students will need
80 cm of plastic pipe
© 2002 Nelson Thomson Learning
Resonant
point
first
second
third
Tuning fork 1
(f = 512 Hz)
Length (cm)
Length
(wavelengths)
17.0
0.253
49.8
0.742
84.2
1.25
Tuning Fork 2
(f = 1024 Hz)
Length (cm)
Length
(wavelengths)
8.40
0.250
25.4
0.756
42.3
1.26
Unit 3 Waves and Sound 207
(b) The speed of sound was determined using the accepted
method:
m/s
vsound = 332 m/s + 0.59
above 0°C
°C
At 20°C, vsound = 332m/s + (0.59
m/s
)(20°C) = 344 m/s
°C
(c) The wavelengths of the sounds emitted by the tuning
forks were determined using the universal wave equation,
solved for wavelength:
v = fλ
(d)–(f) For the 512 Hz tuning fork:
v
λ=
f
434.8 m/s
=
512 Hz
λ = 0.671 m
The wavelength of the 512-Hz tuning fork was 0.671 m, or
67.1 cm.
For the 1024 Hz tuning fork:
v
λ=
f
434.8 m/s
=
1024 Hz
λ = 0.336 m
The wavelength of the 1024-Hz tuning fork was 0.336 m,
or 33.6 cm.
To determine the relationship between a resonant length
and the wavelength of the sound producing the resonance,
the following ratio calculation was made. Using the first
resonant length for the 512 Hz tuning fork as an example:
17.0 cm
=0.253
67.1 cm
As a fraction, this is close to one-quarter (0.250).
Similarly, the first resonant lengths were both found to be
roughly one-quarter of a wavelength. The second resonant
lengths were found to be roughly three-quarters (0.750) of
a wavelength, and the third resonant lengths were found to
be roughly five-quarters (1.25) of a wavelength.
As a general rule, the first three resonant lengths for a
closed air column are one-quarter, three-quarters, and fivequarters of the wavelength of the sound producing the
resonance.
Extensions/Modifications:
• Many schools have the resonance tube apparatus
designed specifically for this type of investigation. If
you intend to use it, be sure to check for leaks before
class. (The rubber or plastic tubing and connectors tend
to crack and leak after several years of intermittent use.)
208 Chapter 8 Music, Musical Instruments, and Acoustics
INVESTIGATION 8.4.2
Speed of Sound in a Closed Air Column
• This investigation fulfils expectation WS2.02 and
should be assessed.
BEFORE
Teacher Preparation
Time: 20 to 30 minutes
Materials and Equipment:
Each group of three or four students will need
80 cm of plastic pipe
large glass cylinder (1000 mL) (graduated is not required;
ungraduated is much cheaper)
two tuning forks (e.g., 512 Hz and 1024 Hz)
metre stick
thermometer
Safety and Disposal:
• The main safety issue is the possibility of the large glass
cylinders toppling and the resulting broken glass.
Placing the cylinder on the floorinstead of on a lab desk
minimizes the risk.
• Students should be cautioned that the apparatus has a
high centre of gravity and is easy to knock over. (The
cost of 1000 mL cylinders is rather high!)
Assessment:
• Note that most of the inquiry skills listed in the margin
of page 290 are addressed in this investigation.
• It is suggested that this investigation be assigned as an
assessment vehicle, as a lab test.
• If the equipment is left set up, individual students can do
the investigation on their own after you approve their
procedure (see Design on page 290).
• If the students record their observations in pairs, they
should still write up separate reports for assessment.
Student Preparation
• Students should have completed the previous part of this
section.
DURING
• If students are being assessed during this investigation,
make sure they are aware of your rules regarding
communicating with one another.
AFTER
• As the plastic pipe was raised out of the graduated
cylinder, with the vibrating tuning fork placed directly
above it, the sound got noticeably louder at the resonant
points. See Table 1.
The air temperature was recorded at 20°C.
© 2002 Nelson Thomson Learning
Table 1 Resonant Lengths of the Tuning Forks
Resonant
point
first
second
Tuning fork 1
(f = 512 Hz)
Resonant
length (cm)
16.5
20.1
Tuning fork 2
(f = 1024 Hz)
Resonant
length (cm)
8.0
25.8
Analysis
(b) In a closed air column, the first resonant length occurs
at one-quarter of a wavelength, and the second resonant
length at three-quarters of a wavelength. According to the
observed values, the wavelength can be calculated in the
following way. Using the first resonant length of the 512Hz tuning fork as an example:
λ
= 16.5 cm
4
λ = 4(16.5 cm) = 66.0 cm (0.660 m)
Using the universal wave equation, the speed of this sound
can then be determined in the following way:
v = fλ
= 512 Hz(0.660 m)
= 338 m/s
Table 2 summarizes the resulting wavelengths associated
with each of the observed resonant points and the speed of
sound associated with each of the measurements.
Table 2 Wavelengths and Speeds of Sound
Resonant
point
first
second
Resonant
point
first
second
Tuning fork 1 (f = 512 Hz)
Wavelength
Speed of Sound
(m)
(m/s)
0.660
338
0.680
348
Tuning Fork 2 (f = 1024 Hz)
Wavelength
Speed of Sound
(m)
(m/s)
0.320
328
0.344
352
Although there is some variation in the speed of sound
obtained by these results, there is still strong conformity.
The average value of the speed of sound produced by
these results is 342 m/s.
(c)–(h) The speed of sound as determined in this
investigation was found to be 342 m/s. The range of
values was from 328 m/s to 352 m/s. The percentage
difference in these values can be calculated as follows:
% difference =
difference in the two values
average of the two values
×100%
352 m/s − 328 m/s
×100%
 (352 m/s + 328 m/s) 


2


= 7.1%
=
© 2002 Nelson Thomson Learning
With an air temperature of 20°C, the theoretical value of
the speed of sound can be calculated as follows:
m/s
vsound = 332 m/s + 0.59 o above 0°C
C
m/s
At 20°C, vsound = 332m/s + (0.59 o )(20°C) = 344 m/s.
C
When this value is compared with the value obtained in
this investigation, there is close agreement. The
percentage error is:
% error =
=
accepted value − experimental value
accepted value
344 m/s − 342 m/s
344 m/s
= 0.58%
×100%
×100%
The value of the speed of sound found in this investigation
is in very good agreement with the accepted value.
Several experimental errors and uncertainties are
associated with this investigation. First, the investigator
must use the sense of hearing and express an opinion of
exactly for what length of air column the sound is the
loudest. Second, the calibrations of the metre stick must be
assumed to be accurate, and judgements are made with
regard to the measurements. Finally, the frequencies of the
tuning forks, which are stamped on their sides, must be
assumed to be accurate.
To increase the validity of the results in a subsequent
investigation, a greater number of resonant points could be
noted and a greater number of tuning forks could be used.
Instead of obtaining a value of the speed of sound from
four measurements, a greater number of measurements
would lessen the influence of spurious results.
Determining the speed of sound in this manner rather
than trying to measure it directly is far more reliable
because great speeds are not easily measured. They tend to
have significant uncertainties associated with them unless
sophisticated equipment is used.
Extensions/Modifications:
• In Synthesis question (g), the students are asked how
they would measure the speed of sound in carbon
dioxide and helium. For extra credit, students can find
the speed of sound in these gases if they are available.
• In both cases, an alternative method needs to be devised
to alter the length of the column. (See Teaching
Suggestions above.)
• With carbon dioxide, dry ice can be placed at the bottom
of the closed end and the tubes mounted vertically.
Since CO2 is more dense than air, the gas will quickly
fill the tube, forcing the air out. A flaming flint test will
ensure that the tube is filled with CO2.
• Since helium is much less dense than air, when the tubes
are inverted, with the closed end at the top, the helium
will displace the air out the bottom of the tubes. A
flaming flint test will ensure that the tube is filled with
helium.
Unit 3 Waves and Sound 209