Chapter 14
Superposition and
Standing Waves
1
Waves vs. Particles
Particles have zero size Waves have a
characteristic size –
their wavelength
Multiple particles must
exist at different
locations
Multiple waves can
combine at one point
in the same medium –
they can be present at
the same location
2
Superposition Principle
• If two or more traveling waves are moving through a
medium and combine at a given point, the resultant
position of the element of the medium at that point is the
sum of the positions due to the individual waves
• Waves that obey the superposition principle are linear
waves
– In general, linear waves have amplitudes much smaller than
their wavelengths
Slide 3
Superposition Example
• Two pulses are traveling in
opposite directions
– The wave function of the pulse
moving to the right is y1 and for the
one moving to the left is y2
• The pulses have the same speed
but different shapes
• The displacement of the
elements is positive for both
Slide 4
Superposition Example, cont
• When the waves start to
overlap (b), the resultant
wave function is y1 + y2
• When crest meets crest (c
) the resultant wave has a
larger amplitude than
either of the original
waves
Slide 5
Superposition Example, final
• The two pulses separate
• They continue moving in
their original directions
• The shapes of the pulses
remain unchanged
Slide 6
Superposition in a Stretch Spring
• Two equal, symmetric
pulses are traveling in
opposite directions on a
stretched spring
• They obey the
superposition principle
Slide 7
Superposition and Interference
• Two traveling waves can pass through each other
without being destroyed or altered
– A consequence of the superposition principle
• The combination of separate waves in the same
region of space to produce a resultant wave is
called interference
Slide 8
Types of Interference
• Constructive interference occurs when the
displacements caused by the two pulses are in the same
direction
– The amplitude of the resultant pulse is greater than either
individual pulse
• Destructive interference occurs when the displacements
caused by the two pulses are in opposite directions
– The amplitude of the resultant pulse is less than either
individual pulse
Slide 9
Destructive Interference Example
• Two pulses traveling in
opposite directions
• Their displacements
are inverted with
respect to each other
• When they overlap,
their displacements
partially cancel each
other
Slide 10
Superposition of Sinusoidal Waves
• Assume two waves are traveling in the same
direction, with the same frequency, wavelength
and amplitude
• The waves differ in phase
• y1 = A sin (kx - wt)
• y2 = A sin (kx - wt + f)
• y = y1+y2
= 2A cos (f/2) sin (kx - wt + f/2)
Slide 11
Superposition of Sinusoidal Waves, cont
• The resultant wave function, y, is also sinusoidal
• The resultant wave has the same frequency and
wavelength as the original waves
• The amplitude of the resultant wave is 2A cos
(f/2)
• The phase of the resultant wave is f/2
Slide 12
Sinusoidal Waves with Constructive
Interference
• When f = 0, then
cos (f/2) = 1
• The amplitude of the
resultant wave is 2A
– The crests of one wave
coincide with the crests of
the other wave
• The waves are everywhere
in phase
• The waves interfere
constructively
Slide 13
Sinusoidal Waves with Destructive Interference
• When f = p, then
cos (f/2) = 0
– Also any even multiple of p
• The amplitude of the
resultant wave is 0
– Crests of one wave coincide
with troughs of the other
wave
• The waves interfere
destructively
Slide 14
Standing Waves
• Assume two waves with the same amplitude,
frequency and wavelength, traveling in opposite
directions in a medium
• y1 = A sin (kx – wt) and y2 = A sin (kx + wt)
• They interfere according to the superposition
principle
Slide 15
Standing Waves, cont
• The resultant wave will be
y = (2A sin kx) cos wt
• This is the wave function of a
standing wave
– There is no kx – wt term, and
therefore it is not a traveling wave
• In observing a standing wave,
there is no sense of motion in the
direction of propagation of either
of the original waves
Slide 16
Note on Amplitudes
• There are three types of amplitudes used in
describing waves
– The amplitude of the individual waves, A
– The amplitude of the simple harmonic motion of the
elements in the medium,
2A sin kx
– The amplitude of the standing wave, 2A
• A given element in a standing wave vibrates within the
constraints of the envelope function 2Asin kx, where x is the
position of the element in the medium
Slide 17
Standing Waves, Particle Motion
• Every element in the medium oscillates in simple
harmonic motion with the same frequency, w
• However, the amplitude of the simple harmonic
motion depends on the location of the element
within the medium
– The amplitude will be 2A sin kx
Slide 18
Standing Waves, Definitions
• A node occurs at a point of zero amplitude
– These correspond to positions of x where
n
x
n  0,1,
2
• An antinode occurs at a point of maximum
displacement, 2A
– These correspond to positions of x where
n
x
4
n  1,3,
Slide 19
Nodes and Antinodes, Photo
Slide 20
Features of Nodes and Antinodes
• The distance between adjacent antinodes is /2
• The distance between adjacent nodes is /2
• The distance between a node and an adjacent
antinode is /4
Slide 21
Nodes and Antinodes, cont
• The diagrams above show standing-wave patterns
produced at various times by two waves of equal
amplitude traveling in opposite directions
• In a standing wave, the elements of the medium
alternate between the extremes shown in (a) and (c)
Slide 22
EXAMPLE 14.2 Formation of a Standing Wave
• Two waves traveling in opposite directions
produce a standing wave. The individual wave
functions are
y1  (4.0 cm) sin(3.0 x  2.0t )
y2  (4.0 cm) sin(3.0 x  2.0t )
where x and y are measured in centimeters and t
is in seconds.
Slide 23
EXAMPLE 14.2 Formation of a Standing Wave
• A. Find the amplitude of the simple harmonic
motion of the element of the medium located at x
= 2.3 cm.
• B. Find the positions of the nodes and antinodes
if one end of the string is at x = 0.
Slide 24
Solution - A
• The waves described by the given equations are
identical except for their directions of travel, so
they indeed combine to form a standing wave as
discussed in this section. We can represent the
waves graphically by the blue and green curves in
Figure 14.8.
Slide 25
Solution - A
• We will substitute values in to equations
developed in this section, so we categorize this
example as a substitution problem.
• From the equations for the waves, we see that A =
4.0 cm, k = 3.0 rad/cm, and w = 2.0 rad/s. Use
Equation 14.3 to write an expression for the
standing wave:
Slide 26
Solution - A
• Find the amplitude of the simple harmonic
motion of the element at the x = 2.3 cm by
evaluating the coefficient of the cosine function
at this position:
ymax  (8.0 cm)sin 3.0 x 2.3 cm
 (8.0 cm)sin(6.9 rad)  4.6 cm
Slide 27
Solution - B
• Find the wavelength of the traveling waves:
2p
2p
k
 3.0 rad/cm   
cm

3.0
Slide 28
Solution - B
• Use Equation 14.5 to find the locations of the
nodes:

 p 
x  n  n
 cm n  0,1, 2,3,
2
 3.0 
• Use Equation 14.4 to find the locations of the
antinodes:

 p 
x  n  n
 cm n  1,3,5, 7,
4
 6.0 
Slide 29
Standing Waves in a String
• Consider a string fixed at
both ends
• The string has length L
• Standing waves are set up
by a continuous
superposition of waves
incident on and reflected
from the ends
• There is a boundary
condition on the waves
Slide 30
Standing Waves in a String, 2
• The ends of the strings must necessarily be nodes
– They are fixed and therefore must have zero displacement
• The boundary condition results in the string having a set
of normal modes of vibration
– Each mode has a characteristic frequency
– The normal modes of oscillation for the string can be described
by imposing the requirements that the ends be nodes and that the
nodes and antinodes are separated by /4
Slide 31
Standing Waves in a String, 3
• This is the first normal mode
that is consistent with the
boundary conditions
• There are nodes at both ends
• There is one antinode in the
middle
• This is the longest wavelength
mode
½1 = L  1 = 2L
Slide 32
Standing Waves in a String, 4
• Consecutive normal
modes add an antinode at
each step
• The second mode (c)
corresponds to to  = L
• The third mode (d)
corresponds to  = 2L/3
Slide 33
Standing Waves on a String, Summary
• The wavelengths of the normal modes for a string
of length L fixed at both ends are n = 2L / n
n = 1, 2, 3, …
– n is the nth normal mode of oscillation
– These are the possible modes for the string
• The natural frequencies are
v
n T
fn  n

2L 2L 
Slide 34
Quantization
• This situation, in which only certain frequencies
of oscillation are allowed, is called quantization
• Quantization is a common occurrence when
waves are subject to boundary conditions
Slide 35
Waves on a String, Harmonic Series
• The fundamental frequency corresponds to n = 1
– It is the lowest frequency, ƒ1
• The frequencies of the remaining natural modes are
integer multiples of the fundamental frequency
– ƒn = nƒ1
• Frequencies of normal modes that exhibit this
relationship form a harmonic series
• The various frequencies are called harmonics
Slide 36
Musical Note of a String
• The musical note is defined by its fundamental
frequency
• The frequency of the string can be changed by
changing either its length or its tension
• The linear mass density can be changed by either
varying the diameter or by wrapping extra mass
around the string
Slide 37
Harmonics, Example
• A middle “C” on a piano has a fundamental
frequency of 262 Hz. What are the next two
harmonics of this string?
– ƒ1 = 262 Hz
– ƒ2 = 2ƒ1 = 524 Hz
– ƒ3 = 3ƒ1 = 786 Hz
Slide 38
EXAMPLE 14.3 Give Me a C Note.
• The middle C string on a piano has a fundamental
frequency of 262 Hz, and the string for the first A
above middle C has a fundamental frequency of
440 Hz.
• A. Calculate the frequencies of the next two
harmonics of the C string.
• B. If the A and C strings have the same linear
mass density  and length L, determine the ratio
of tensions in the two strings.
Slide 39
Solution - A
• Remember that the harmonics of a vibrating
string have frequencies that are related by integer
multiples of the fundamental.
Slide 40
Solution - A
• This first part of the example is a simple
substitution problem. Knowing that the
fundamental frequency is f1 = 262 Hz, find the
frequencies of the next harmonics by multiplying
by integers:
f 2  2 f1  524 Hz
f3  3 f1  786 Hz
Slide 41
Solution - B
• This part of the example is more of an analysis
problem than is part A.
• Use Equation 14.9 to write expressions for the
fundamental frequencies of the two strings:
f1A
1 TA

2L 
and
1 TC
f1C 
2L 
Slide 42
Solution - B
• Divide the first equation by the second and solve
for the ratio of tensions:
2
f1A
TA
TA  f1A   440 



 
  2.82
f1C
TC
TC  f1C   262 
2
Slide 43
Solution - B
• If the frequencies of piano strings were
determined solely by tension, this result suggests
that the ratio of tensions from the lowest string to
the highest string on the piano would be
enormous. Such large tensions would make it
difficult to design a frame to support the strings.
In reality, the frequencies of piano strings vary
due to additional parameters, including the mass
per unit length and the length of the string. The
What If? below explores a variation in length.
Slide 44
WHAT IF?
• If you look inside a real piano, you’ll see that the
assumption made B is only partially true. The
strings are not likely to have the same length. The
string densities for the given notes might be
equal, but suppose the length of the A string is
only 64% of the length of the C string. What is
the ratio of their tensions?
Slide 45
Answer
• Using Equation 14.9 again, we set up the ratio of
frequencies:
f1A LC

f1C LA
2
TA
TA  LA   f1A 

  

TC
TC  LC   f1C 
2
2
TA
2  440 
 (0.64) 
  1.16
TC
 262 
• Notice that this result represents only a 16%
increase in tension, compared with the 182%
increase in B.
Slide 46
EXAMPLE 14.4
Changing String Vibration with Water
• One end of a horizontal string is attached to a
vibrating blade, and the other end passes over a
pulley as in Figure 14.10a. A sphere of mass 2.00
kg hangs on the end of the string. The string is
vibrating in its second harmonic. A container of
water is raised under the sphere so that the sphere
is completely submerged. In this configuration,
the string vibrates in its fifth harmonic as shown
in Figure 14.10b. What is the radius of the
sphere?
Slide 47
EXAMPLE 14.4
Changing String Vibration with Water
Slide 48
Solution
• Imagine what happens when the sphere is
immersed in the water. The buoyant force acts
upward on the sphere, reducing the tension in the
string. The change in tension causes a change in
the speed of waves on the string, which in turn
causes a change in the wavelength. This altered
wavelength results in the string vibrating in its
fifth normal mode rather than the second.
Slide 49
Solution
• The hanging sphere is modeled as a particle in
equilibrium. One of the forces acting on it is the
buoyant force from the water. We also apply the
waves under boundary conditions model to the
string.
Slide 50
Solution
• Apply the particle in equilibrium model to the
sphere in Figure 14.10a, identifying T1 as the
tension in the string as the sphere hangs in air:
 F  T  mg  0
1
T1  mg
Slide 51
Solution
• Apply the particle in equilibrium model to the
sphere in Figure 14.l0b, where T2 is the tension in
the string as the sphere is immersed in water:
T2  B  mg  0
(1) B  mg  T2
Slide 52
Solution
• The desired quantity, the radius of the sphere,
will appear in the expression for the buoyant
force B. Before proceeding in this direction,
however, we must evaluate T2 from the
information about the standing wave.
Slide 53
Solution
• Write the equation for the frequency of a standing
wave on a string (Eq. 14.8) twice, once before the
sphere is immersed and once after. Notice that the
frequency f is the same in both cases because it is
determined by the vibrating blade. In addition,
the linear mass density m and the length L of the
vibrating portion of the string are the same in
both cases.
Slide 54
Solution
• Divide the equations:
n1 T1
f 
2L 
n2
f 
2L
• Solve for T2:
n1
1 
n2
T2
T1
T2

2
 n1 
 n1 
T2    T1    mg
 n2 
 n2 
Slide 55
Solution
• Substitute this result into Equation (1):
2

 n1 
 n1  
(2) B  mg    mg  mg 1    
  n2  
 n2 
2
• Using Equation 15.5, express the buoyant force in
terms of the radius of the sphere:
B  water gVsphere  water g ( p r )
4
3
3
Slide 56
Solution
• Solve for the radius of the sphere and substitute
from Equation (2):
1/3
 3B 
r 

 4p water g 
 3m

 4p water
 n 
1   1 
  n2 
2
1/3
 

 
Slide 57
Solution
• Substitute numerical values:

 2
3(2.00 kg)
r
1  
3 
 4p (1 000 kg / m )   5 
 0.0737 m  7.37 cm
2
1/3
 

 
Slide 58
Solution
• Notice that only certain radii of the sphere will
result in the string vibrating in a normal mode;
the speed of waves on the string must be changed
to a value such that the length of the string is an
integer multiple of half wavelengths. This
limitation is a feature of the quantization that was
introduced earlier in this chapter: the sphere radii
that cause the string to vibrate in a normal mode
are quantized.
Slide 59
Standing Waves in Air Columns
• Standing waves can be set up in air columns as
the result of interference between longitudinal
sound waves traveling in opposite directions
• The phase relationship between the incident and
reflected waves depends upon whether the end of
the pipe is opened or closed
Slide 60
Standing Waves in Air Columns, Closed End
• A closed end of a pipe is a displacement node in
the standing wave
– The wall at this end will not allow longitudinal motion
in the air
– The reflected wave is 180o out of phase with the
incident wave
• The closed end corresponds with a pressure
antinode
– It is a point of maximum pressure variations
Slide 61
Standing Waves in Air Columns, Open End
• The open end of a pipe is a displacement antinode
in the standing wave
– As the compression region of the wave exits the open
end of the pipe, the constraint of the pipe is removed
and the compressed air is free to expand into the
atmosphere
• The open end corresponds with a pressure node
– It is a point of no pressure variation
Slide 62
Standing Waves in an Open Tube
• Both ends are displacement antinodes
• The fundamental frequency is v/2L
– This corresponds to the first diagram
• The higher harmonics are ƒn = nƒ1 = n (v/2L) where n = 1, 2, 3,
…
Slide 63
Standing Waves in a Tube Closed at One End
•
•
•
•
The closed end is a displacement node
The open end is a displacement antinode
The fundamental corresponds to 1/4
The frequencies are ƒn = nƒ = n (v/4L) where n = 1, 3,
5, …
Slide 64
Standing Waves in Air Columns, Summary
• In a pipe open at both ends, the natural
frequencies of oscillation form a harmonic series
that includes all integral multiples of the
fundamental frequency
• In a pipe closed at one end, the natural
frequencies of oscillations form a harmonic
series that includes only odd integral multiples of
the fundamental frequency
Slide 65
Resonance in Air Columns, Example
• A tuning fork is placed near the
top of the tube containing water
• When L corresponds to a
resonance frequency of the
pipe, the sound is louder
• The water acts as a closed end
of a tube
• The wavelengths can be
calculated from the lengths
where resonance occurs
Slide 66
EXAMPLE 14.5 Wind in a Culvert
• A section of drainage culvert 1.23 m in length
makes a howling noise when the wind blows
across its open ends.
Slide 67
EXAMPLE 14.5 Wind in a Culvert
• A. Determine the frequencies of the first three
harmonics of the culvert if it is cylindrical in
shape and open at both ends. Take v = 343 m/s as
the speed of sound in air.
• B. What are the three lowest natural frequencies
of the culvert if it is blocked at one end?
Slide 68
Solution - A
• The sound of the wind blowing across the end of
the pipe contains many frequencies, and the
culvert responds to the sound by vibrating at the
natural frequencies of the air column.
Slide 69
Solution - A
• This example is a relatively simple substitution
problem. Find the frequency of the first harmonic
of the culvert, modeling it as an air column open
at both ends:
v
343 m / s
f1 

 139 Hz
2 L 2(1.23 m)
Slide 70
Solution - A
• Find the next harmonics by multiplying by
integers:
f 2  2 f1  279 Hz
f3  3 f1  418 Hz
Slide 71
f3  3 f1  209 Hz
f5  5 f1  349 Hz
Solution - B
• Find the frequency of the first harmonic of the
culvert, modeling it as an air column closed at
one end:
v
343 m / s
f1 

 69.7 Hz
4 L 4(1.23 m)
• Find the next two harmonics by multiplying by
odd integers:
f3  3 f1  209 Hz
f5  5 f1  349 Hz
Slide 72
EXAMPLE 14.6
Measuring the Frequency of a Tuning Fork
• A simple apparatus for demonstrating resonance
in an air column is depicted in Figure 14.12. A
vertical pipe open at both ends is partially
submerged in water, and a tuning fork vibrating
at an unknown frequency is placed near the top of
the pipe.
Slide 73
EXAMPLE 14.6
Measuring the Frequency of a Tuning Fork
• The length L of the air column can be adjusted by
moving the pipe vertically. The sound waves
generated by the fork are reinforced when L
corresponds to one of the resonance frequencies
of the pipe. For a certain pipe, the smallest value
of L for which a peak occurs in the sound
intensity is 9.00 cm.
Slide 74
EXAMPLE 14.6
Measuring the Frequency of a Tuning Fork
Slide 75
EXAMPLE 14.6
Measuring the Frequency of a Tuning Fork
• A. That is the frequency of the tuning fork?
• B. What are the values of L for the next two
resonance conditions?
Slide 76
Solution - A
• Consider how this problem differs from the
preceding example. In the culvert, the length was
fixed and the air column was p1eseItedl with a
mixture of very many frequencies. The pipe in
this example is presented with one single
frequency from the tuning fork, and the length of
the pipe is varied until resonance is achieved.
Slide 77
Solution - A
• This example is a simple substitution problem.
Although the pipe is open at its lower end to
allow the water to enter, the water’s surface acts
like a barrier. Therefore, this setup can be
modeled as an air column closed at one end.
Slide 78
Solution - A
• Use Equation 14.11 to find the fundamental
frequency for L = 0.090 0 m:
v
343 m / s
f1 

 953 Hz
4 L 4(0.090 0 m)
• Because the tuning fork causes the air column to
resonate at this frequency, this frequency must
also be that of the tuning fork.
Slide 79
Solution - B
• Use Equation 13.11 to find the wavelength of the
sound wave from the tuning fork:
v 343 m / s
 
 0.360 m
f
953 Hz
Slide 80
Solution - B
• Notice from Figure 14.12b that the length of the
air column for the second resonance is 3/4:
L  3 / 4  0.270 m
• Notice from Figure 14.12b that the length of the
air column for the third resonance is 5/4:
L  5 / 4  0.450 m
Slide 81
Beats
• Temporal interference will occur when the
interfering waves have slightly different
frequencies
• Beating is the periodic variation in amplitude at a
given point due to the superposition of two waves
having slightly different frequencies
Slide 82
Beat Frequency
• The number of amplitude maxima one hears per
second is the beat frequency
• It equals the difference between the frequencies of
the two sources
• The human ear can detect a beat frequency up to
about 20 beats/sec
83
Beats, Final
• The amplitude of the resultant wave varies in
time according to
 f1  f 2 
Aresultant  2 A cos 2p 
t
 2 
– Therefore, the intensity also varies in time
• The beat frequency is ƒbeat = |ƒ1 – ƒ2|
Slide 84
EXAMPLE 14.7 The Mistuned Piano Strings
• Two identical piano strings of length 0.750 mare
each tuned exactly to 440 Hz. The tension in one
of the strings is then increased by 1.0%. If they
are now struck, what is the beat frequency
between the fundamentals of the two strings?
Slide 85
Solution
• As the tension in one of the strings is changed, its
fundamental frequency changes. Therefore, when
both strings are played, they will have different
frequencies and beats will be heard.
• We must combine our understanding of the
waves under boundary conditions model for
strings with our new knowledge of beats.
Slide 86
Solution
• Set up a ratio of the fundamental frequencies of
the two strings using Equation 14.7:
f 2 (v2 / 2 L) v2


f1 (v1 / 2 L) v1
• Use Equation 13.7 to substitute for the wave
speeds on the strings:
T2 / 
f2
T2


f1
T1
T1 / 
Slide 87
Solution
• Incorporate that the tension in one string is 1.0%
larger than the other; that is, T2 = 1.010T1:
f2
1.010T1

 1.005
f1
T1
Slide 88
Solution
• Solve for the frequency of the tightened string:
f 2  1.005 f1  1.005(440 Hz)  442 Hz
• Find the beat frequency using Equation 14. 14:
f beat  442 Hz  440 Hz  2 Hz
Slide 89
Solution
• Notice that a 1.0% mistuning in tension leads to
an easily audible beat frequency of 2 Hz. A piano
tuner can use beats to tune a stringed instrument
by “beating” a note against a reference tone of
known frequency. The tuner can then adjust the
string tension until the frequency of the sound it
emits equals the frequency of the reference tone.
The tuner does so by tightening or loosening the
string until the beats produced by it and the
reference source become too infrequent to notice.
Slide 90
Nonsinusoidal Wave Patterns
• The wave patterns produced by a musical instrument are
the result of the superposition of various harmonics
• The human perceptive response associated with the
various mixtures of harmonics is the quality or timbre
of the sound
• The human perceptive response to a sound that allows
one to place the sound on a scale of high to low is the
pitch of the sound
Slide 91
Quality of Sound –Tuning Fork
• A tuning fork produces
only the fundamental
frequency
Slide 92
Quality of Sound – Flute
• The same note played on
a flute sounds differently
• The second harmonic is
very strong
• The fourth harmonic is
close in strength to the
first
Slide 93
Quality of Sound –Clarinet
• The fifth harmonic is very
strong
• The first and fourth
harmonics are very
similar, with the third
being close to them
Slide 94
Analyzing Nonsinusoidal Wave Patterns
• If the wave pattern is periodic, it can be represented as
closely as desired by the combination of a sufficiently
large number of sinusoidal waves that form a harmonic
series
• Any periodic function can be represented as a series of
sine and cosine terms
– This is based on a mathematical technique called Fourier’s
theorem
Slide 95
Fourier Series
• A Fourier series is the corresponding sum of
terms that represents the periodic wave pattern
• If we have a function y that is periodic in time,
Fourier’s theorem says the function can be written
as
y (t )   ( An sin 2p f nt  Bn cos 2p f nt )
n
• ƒ1 = 1/T and ƒn= nƒ1
• An and Bn are amplitudes of the waves
Slide 96
Fourier Synthesis of a Square Wave
• Fourier synthesis of a square
wave, which is represented
by the sum of odd multiples
of the first harmonic, which
has frequency f
• In (a) waves of frequency f
and 3f are added.
• In (b) the harmonic of
frequency 5f is added.
• In (c) the wave approaches
closer to the square wave
when odd frequencies up to
9f are added.
Slide 97
Standing Waves and Earthquakes
• Many times cities may be built on sedimentary basins
• Destruction from an earthquake can increase if the
natural frequencies of the buildings or other structures
correspond to the resonant frequencies of the underlying
basin
• The resonant frequencies are associated with threedimensional standing waves, formed from the seismic
waves reflecting from the boundaries of the basin
Slide 98
Exercises
• 3, 10, 12, 19, 26, 37, 40, 50, 58, 62, 70
Slide 99