Beats: Interference of Sound Waves
Have you ever heard a piano, band, or orchestra play more than one note at a time? (These are called
chords.) Chances are that the music you heard was a series of pleasant-sounding chords and melodies.
Sometimes the pitches of the notes are close to each other, however, which causes the sounds to clash
in a way that is unpleasant—or sounds modern!
Two sound waves with frequencies close to each other—but not exactly the same—will sound harsh
when played together; this is called dissonance. As the frequencies of two sound waves become closer
and closer to each other, it is possible to hear a slow pattern of loud and soft intensities called beats.
The beating pattern is a consequence of the interference between two sound waves of slightly different
frequency.
Watch Video in
eBook
Definitions
Investigation
Video: Beats and Dissonance
[insert: VIDEO1 demonstrating:
1. consonance (major interval)
2. dissonance (minor interval)
3. beat patterns for a few Hz to
few tenths of Hz
4. beat frequency increasing
until dissolves into
dissonance]
When two sounds of slightly different frequency are heard simultaneously, we hear a
slow pattern of interference between them called beats. The beat pattern produces an
alternation between loud and soft sounds. When the beats become faster than 10 Hz or
so, the human ear can no longer recognize the loud-soft pattern and the two sounds are
harsh to the ear; this is called dissonance. If the two sounds have widely-spaced
frequencies and are pleasant to the ear, then it is called consonance.
Use the audio tool in the ebook to explore beat frequencies.
Beat Frequency
The frequency of loud and soft sounds that is heard between two simultaneous sound
waves is called the beat frequency. The beat frequency is the difference between the
frequencies of the two sound waves, f1 and f2:
Beat Frequency: fB = f1 - f2
As the two frequencies become closer and closer to one another, the beat frequency fB
will become smaller—which means that the alternating loud-soft beats will be spaced
further apart in time. The beat pattern therefore sounds slow for two close frequencies.
Likewise, as the two frequencies get further and further apart, the beat frequency
becomes larger and the beating pattern sounds faster.
oldpianomusic, CC By 2.0
What Causes
Beats?
When a piano tuner adjusts two strings to sound the same note, he will
play them simultaneously and listen for the beat frequency. He then
adjusts one string slightly—either by tightening or loosening the screw
—in order to slow down the beat frequency. He knows he is turning the
screw in the correct way when the beat frequency slows down, rather
than speeds up. The two strings are in tune with each other when the
beat pattern disappears completely.
What creates the beat pattern when two
sounds are heard together? The
superposition of the two sound waves
causes interference between them, and
this interference pattern is the origin of the
beat pattern.
When two sound waves of the same
frequency are heard simultaneously, they
superpose and can either create
constructive or destructive interference,
depending on the relative phase between
the two waves. If the peaks of the waves
are in phase, then the amplitudes of the
two waves add together in constructive
interference; the resultant wave is larger
in amplitude. If the peaks of the two
waves are out-of-phase, then the peak of
one overlaps with the trough of the other;
the result is that the two wave amplitudes
partially or fully cancel each other, which
is destructive interference.
When two sound waves with different frequencies are heard simultaneously, however,
there will be an alternating pattern of constructive and destructive interference. At
certain times, as shown in the diagram above, the two sound waves are in phase with
each other and interfere constructively—i.e., the peak amplitudes of the two waves add
to make a higher intensity peak. At other times the two sound waves are out-of-phase
with each other and interfere destructively—the peak of one wave lines up with the
trough of the other wave, canceling each other out. The resulting sum of the two
waves—the “superposition of waves one and two”—shows a variable amplitude due to
a variable pattern of constructive and destructive interference.
The outermost “envelope” of the sum of the two waves in the figure above is the
alternating loud-soft sound that we hear as beats. Notice in the diagram how the
frequency of the outer envelope—the beats—has a significantly lower frequency than
either of the two original sound waves.
A frequency of 440 Hz is called “concert pitch,” an internationally agreed-upon
standard frequency for the note A above middle C on the keyboard. Before an
orchestra begins a rehearsal or concert, the oboist will usually play the concert pitch
to allow the other musicians to tune their instruments to the same frequency. The
oboist often has an electronic tuner on her stand in order to ensure the accuracy of the
pitch of her tuning note.
[Video Insert]
Beats: Interference of Sound Waves
Image
Sound
Text
plain background
In music, we will often play two or
more different notes at the same time.
This is called a chord.
C–E–G
chord C – E – G
C – E – G [fade out]
262 Hz – 330 Hz – 392 Hz
consonant
Middle C (261.6 Hz),
then E (329.6 Hz),
then G (392.0 Hz),
then C-E-G together.
C, E, G
[repeat] C, E, G
chord C, E, G
In physics, each of these musical notes,
or pitches, corresponds to a frequency
for the sound waves.
Some musical note combinations
sound pleasant to the ear, or
consonant.
Other combinations of notes sound
harsh to the ear, or dissonant.
dissonant
chord C (261.6 Hz), C# (277.2
Hz), D (293.7 Hz)
What makes some of these sounds
have consonance or dissonance?
consonance vs. dissonance
Widely-spaced musical pitches, when
played together, can create a consonant
sound, like playing C and G on the
keyboard.
Image: keyboard with C and
G highlighted
C – G [widely spaced]
C–G
262 Hz – 392 Hz
C (261.6 Hz) – G (329.0 Hz)
The frequencies of these two sound
waves are far apart, which means that
they can have a more pleasant sound
when heard together.
C–G
Image: keyboard with C and
D highlighted
But closely-spaced musical pitches,
when played together, can create a
dissonant sound that is harsh to the ear.
C–D
C (261.6 Hz) – D (293.7 Hz)
C – D [closer spaced]
262 Hz – 294 Hz
C-D
Image: keyboard with C and
C# highlighted
C – C# [closely spaced]
C (261.6 Hz) – C# (277.2 Hz)
262 Hz – 277 Hz
C – C#
As the frequencies become closer
together, the sounds clash even more.
As those two frequencies become
nearly, but not quite, the same, we can
hear the physical phenomenon that is
giving rise to the unpleasant, dissonant
sound.
262 Hz – 263 Hz
262 Hz and 263 Hz
We begin to hear a slow, pulsing
sound, alternating loud and soft. Can
you hear it?
This phenomenon is called beats. It
arises when we hear simultaneously
two frequencies of sound that are very
close—but not exactly the same as
each other. The sound of the beats
arises because of the interference
between the two sound waves.
Beats are caused by the
interference between sound
waves of slightly different
frequencies.
These beats can be very slow...
slow beats
262 Hz and 262.25 Hz
with a low frequency...
low frequency
or the beats can be very rapid...
fast beats
262 Hz and 265 Hz
with a higher frequency.
high frequency
If the beat frequency becomes so fast
that we can no longer distinguish the
pattern, then we merely hear the sound
as dissonance.
Faster Beat Frequency ->
Dissonance
Keep 262 Hz fixed;
simultaneous 265 Hz
increasing to 293.7 Hz
Ebook Investigation: Measuring Beat Frequencies
Part 1:
1. Set the first frequency to 440 Hz. Set the second frequency to 441 Hz—one hertz higher than
the first frequency. Set the timer for 10 seconds. Play the first note by itself and listen closely.
When it is finished, play the second note by itself and listen. Can you hear any difference in
pitch between them?
2. Play both notes simultaneously and listen carefully. What do you hear?
3. Play both notes simultaneously again and count the number of beats you hear—the number of
loud sounds resulting from the interference between the two sound waves. How many beats did
you hear in 10 seconds?
4. While keeping the first frequency at 440 Hz, set the second frequency to 442 Hz. Play the two
notes again and count the number of beats you hear in 10 seconds. How many beats did you
count? Was it more or fewer than in the first exercise?
5. Try the exercise one more time by setting the second frequency to 443 Hz. Can you predict
how many beats you will hear in 10 seconds? Play the two notes again, count the number of
beats in 10 seconds, and compare with your prediction.
6. Based on your investigation, how is the frequency of the beats related to the frequencies of the
two notes?
f1
f2
f2 - f 1
Nbeats =
Beats per second =
Beats in 10 seconds
Nbeats / 10 s
(Hz)
(Hz)
(Hz)
(Hz)
440
441
Part 2:
7. Play the mystery sound and its comparison sound. [plays two sounds with 2 Hz beating.] Do
you hear the beats? Press the “+” button to adjust the pitch—i.e., increase the frequency—of
the comparison frequency, and then play the sounds again. Based on how the beat frequency
changed, are the two sounds now closer or further apart in frequency? Try adjusting the pitch
until you eliminate the sound of the beats.
Technology Application: AM Radio
When two sound waves with slightly different frequencies are superposed—heard simultaneously—the
resulting sound has the alternating pattern of loud-and-soft sounds called beats. What happens when
we superpose two waves with vastly different frequencies?
One illustration of a technology that relies on superposition of waves of very different frequencies is
amplitude-modulated radio—commonly known as AM radio. The AM radio band in the United States
is broadcast at frequencies between 520 kHz (or 520,000 Hz) and 1610 kHz, which are frequencies
much higher than what the human voice can produce or the ear can hear (such as 440 Hz for the concert
pitch note A). The AM radio frequency is typically around one thousand times higher than the
frequency of the sounds the radio station is broadcasting.
The idea behind the technology of AM radio is that the broadcast radio signal's amplitude is modulated
—varied up and down—by the lower-frequency sound waves that are being broadcast. Let's take an
example of an audio sound at 440 Hz—the “concert pitch” of the note A—that is broadcast over AM
radio at 950 kHz. The electronics at the radio station combine the low-frequency of the sound wave
with the high frequency radio wave and broadcast this from their radio tower. In the diagram, you can
see the amplitude of the envelope of the high-frequency broadcast radio signal modulating up and
down at the lower frequency of the audio sound wave. The radio receiver in your home or car then
detects the radio broadcast signal and removes the radio broadcast frequency (say, at 950 kHz), leaving
just the audio signal at 440 Hz to be amplified through your speakers.
Laboratory Investigation
Vibrating String: Frequency, Dependence on Length and Tension, and Beats
[Note: this is a more general lab investigation suitable for the entire sound unit. An exploration of
beats would be included as one element of the overall lab investigation.]
Equipment Needed:
Plywood board (approx. 1 m x 1 m)
Two sturdy screws
Two lengths of piano wire or strong fishing line, at least 1.5 m each
Two wedges (wood or metal) to act as a “bridge” for the strings
Two scale pans
Two sets of identical masses
Meter ruler
Stopwatch (or other timer with a second hand)
[optional] Tuning fork
Note to instructor: experiment ahead of time to establish a position of the wedge and mass on the scale
that will produce a tone roughly 220 Hz (the note A below middle C).
1. Determine the masses M1 and M2 of each scale pan loaded with the masses.
2. Set up the two weighted wires with identical masses in the scale pans. One movable wedge
should be placed at 50.0 cm from its screw. Sound the wire by plucking it gently.
3. ...investigation for how the sound varies when wedge is moved to shorten the resonating part of
the wire... how the sound varies when the mass is increased...
4. Place the wedge under the second wire. Adjust it by small amounts until it makes the same
sound as the first wire when the two are plucked at the same time. You should hear no beats
between the two wires.
5. ...exercises for dependence on length [halve the length], weight [double the mass]....
6. [beats exercise] Using a sharp pencil, mark on the wooden board the location of the second
wedge. Move the second wedge approximately 2 mm closer to the screws and mark its location
again on the wooden board.
a. Gently pluck the two wires at the same time. What do you hear?
b. Using the timer, stopwatch, or the seconds hand on the wall clock, measure then number of
beats during a 10 second interval.
c. Adjust the second wedge again, moving it another 2 mm closer to the screw. Mark its
location on the wooden board. Measure the number of beats during a 10 second interval.
d. Repeat for two more positions, approximately 6 and 8 mm from the initial position.
e. Remove the second wedge. Measure the change in the wedge's position—i.e., the
separation between the wedge's initial position and the three other locations.
f. Calculate the number of beats per second based on the number of beats you measured in 10
seconds for each position.
g. Do you notice a trend between the position of the wedge and the number of beats per second
that you measured? What do you think causes the beat frequency to vary in this way with
the length of the wire?
Change in wedge's
position
(mm)
Nbeats =
Beats in 10
seconds
Beats per second =
Nbeats / 10 s
(Hz)
2.0
6. Return the second wedge to its initial position, and verify that you cannot hear beats between
the two wires when plucked at the same time. Add a ____ g mass to the second scale pan.
h. Gently pluck the two wires at the same time. What do you hear?
i. Using the timer, stopwatch, or the seconds hand on the wall clock, measure then number of
beats during a 10 second interval.
j. Add another ___ g mass to the second scale pan. Measure the number of beats during a 10
second interval.
k. Repeat for another two added masses, measuring the number of beats during a 10 second
interval each time.
l. Calculate the total mass suspended by the wire for each added mass. (You will need to add
the.)
m. Calculate the ratio of the added mass to the initial mass M2 of the second scale pan and its
contents
n. Calculate the number of beats per second.
o. Graph the ratio of the masses m/M2 against the number of beats per second.
Added
Mass
m
(g)
Ratio of
masses
m/M2
(g)
Nbeats =
Beats in 10
seconds
Beats per second =
Nbeats / 10 s
(Hz)
10.0
For the instructor:
When the wedge is moved 2 mm in step 5, the frequency sounded by the wire should change by (2
mm / 50.0 cm) = 1/250. If the masses were set to produce an original sounding frequency of 220 Hz,
then the frequency of the second wire should increase by slightly more than 1 Hz; students should hear
beats at the rate of approximately one loud-soft pair per second. A similar increase is found each time
the wire is shortened by 2 mm.
Chapter 22 Review
Some References
“6.4.5 Laws of Strings,” in Cunningham, J., & Herr, N. Hands-On Physics Activities with Real-Life
Applications (San Francisco: Wiley Bass, 1994), 414-415.
1. Massachusetts Curriculum Frameworks (2006):
Grades 3-5: Sound Waves:
Learning Standard
Ideas for Developing
Investigations and Learning
Experiences
11. Recognize that sound is
Use tuning forks to demonstrate
produced by vibrating objects
the relationship between
and requires a medium through vibration and sound.
which to travel. Relate the rate of
vibration to the pitch of the
sound.
Suggested Extensions to
Learning in
Technology/Engineering
Design and construct a simple
telephone (prototype) using a
variety of materials (e.g., paper
cups, string, tin cans, wire).
Determine which prototype
works best and why. (T/E 1.1,
1.2, 2.2, 2.3)
Introductory Physics, High School:
4. Waves
Central Concept: Waves carry energy from place to place without the transfer of matter.
4.1.Describe the measurable properties of waves (velocity, frequency, wavelength, amplitude,
period) and explain the relationships among them. Recognize examples of simple harmonic
motion.
4.2.Distinguish between mechanical and electromagnetic waves.
4.3.Distinguish between the two types of mechanical waves, transverse and longitudinal.
4.4.Describe qualitatively the basic principles of reflection and refraction of waves.
4.5.Recognize that mechanical waves generally move faster through a solid than through a liquid
and faster through a liquid than through a gas.
4.6.Describe the apparent change in frequency of waves due to the motion of a source or a receiver
(the Doppler effect).
2. SAT Subject Test: Physics
IV. Waves and optics (15%–19%)
A) General Wave Properties, such as wave speed, frequency, wavelength, superposition,
standing wave diffraction, and Doppler effect
B) Reflection and Refraction, such as Snell’s law and changes in wavelength and speed
C) Ray Optics, such as image formation using pinholes, mirrors, and lenses
D) Physical Optics, such as single-slit diffraction, double-slit interference, polarization, and
color
3. Advanced Placement Physics (B)
IV. WAVES AND OPTICS
A. Wave motion (including sound)
1. Traveling waves
Students should understand the description of traveling waves, so they can:
a) Sketch or identify graphs that represent traveling waves and determine the
amplitude, wavelength and frequency of a wave from such a graph.
b) Apply the relation among wavelength, frequency and velocity for a wave.
c) Understand qualitatively the Doppler effect for sound in order to explain why there
is a frequency shift in both the moving-source and moving-observer case.
d) Describe reflection of a wave from the fixed or free end of a string.
e) Describe qualitatively what factors determine the speed of waves on a string and the
speed of sound.
2. Wave propagation
a) Students should understand the difference between transverse and longitudinal
waves, and be able to explain qualitatively why transverse waves can exhibit
polarization.
b) Students should understand the inverse-square law, so they can calculate the
intensity of waves at a given distance from a source of specified power and compare
the intensities at different distances from the source.
3. Standing waves
Students should understand the physics of standing waves, so they can:
a) Sketch possible standing wave modes for a stretched string that is fixed at both ends,
and determine the amplitude, wavelength and frequency of such standing waves.
b) Describe possible standing sound waves in a pipe that has either open or closed
ends, and determine the wavelength and frequency of such standing waves.
4. Superposition
Students should understand the principle of superposition, so they can apply it to
traveling waves moving in opposite directions, and describe how a standing wave may
be formed by superposition.