Physics in
Entertainment
and the Arts
The Superposition Principle
Arithmetic of Waves
Chapter VI
The Superposition Principle
• This is why when you listen to an
orchestra playing
– you can separate out the individual
instrument’s
instrument
s sounds
• You can hear the clarinets vs the trumpets
vs the violins
• Two or more waves can coexist in a
medium without having any effect on each
other
– The amplitude of the combined wave at any
point in the medium is just the sum of the
waves’ displacements at that point
The Superposition Principle
• The superposition principle is also why we
can send TV, FM, AM, cell phone, etc…
radio signals through the same air and
separate them out at the receiver
– They don’t get mixed together; they just get
added temporarily
The Superposition Principle
• Let’s start by studying two pulses with
positive amplitudes traveling along a long
slinky
– One travels to the left; one to the right
• The pulses will meet temporarily in the
center of the slinky
– the sounds are simply added together not
jumbled together!
The Superposition Principle
• Since the pulses both
have positive amplitudes
– they both add in the center
• but only temporarily…
• The pulses pass through
each other
The Superposition Principle
• Now let’s see what happens when the
pulses have opposite amplitudes
– One with a positive amplitude; one with a
negative amplitude
The Superposition Principle
• Now the pulses both
subtract in the center
• but only temporarily…
• The pulses once again
pass through each other
– with no lasting effect
– with no lasting effect
Figure from Physics, Cutnell & Johnson, 7th ed.
Figure from Physics, Cutnell & Johnson, 7th ed.
1
Continuous Waves Phase Difference
The Superposition Principle
• Note that in this subtraction case
– even though the slinky is temporarily flat at
the moment of interaction
– the waves have not been destroyed!
• The waves just temporarily add to give a
zero displacement at that instant
• Let’s now move on to the superposition of
two continuous waves
• Assume that the waves have the same
frequency…
– If the waves are of different frequency, it’s a
lot more complicated!
Phase Difference
• If the two waves vibrate up and down at
the same time
– we say that the waves are “in phase” with
each
h other
th
Phase Difference
• Note that when continuous waves travel
down a string
– the resulting wave is the sum (superposition)
of both waves
Atotal
• In phase means that each wave reaches a
crest together, and then a trough together
– as they move down the string
A1
A2
The total wave
amplitude is the sum
of the individual wave
amplitudes
A1 + A2 = Atotal
Phase Difference
• Note that when continuous waves travel
down a string
– the resulting wave is still the sum (superposition)
of both waves, even though they are 180o out of
phase
A1
A2
Atotal
The waves
destroy each
other!
The total wave
amplitude is the sum
of the individual wave
amplitudes
A1 − A2 = Atotal
Phase Difference
• “In phase” and “180o out of phase” are the
two extremes
– the waves can be any number of degrees out
of phase with each other (the math just gets
harder to do then)
The “red” wave is 110o
out of phase with the
“blue” wave. The
“yellow” wave is the
superposition (sum) of
the two waves
Phase Difference
• Suppose we generate two waves with the
same frequency
– moving in the same direction down the same
string
• How we do this is fairly easy, but not
important for this discussion
Phase Difference
• Alternatively, if one wave vibrates up while
the other wave vibrates down at the same
time
– we say that the waves are “180
180o out of phase
phase”
with each other
• 180o out of phase means that one wave
reaches a crest and the other a trough
together
– as they move down the string
Phase Difference
• Note that the combined “yellow” wave has
an amplitude much greater than zero, but
much less than the sum of the “red” and
“blue” waves
These waves are
referred to as simply
“out of phase”
2
Phase Difference
Phase Difference
• Why is the phase difference important?
– Let’s look at a typical example
Phase Difference
• If you wire up the speakers correctly
– the sound from each speaker will reinforce
each other at your ear
• S
Suppose you have
h
a stereo
t
system
t
that
th t
you wish to wire up for home use
S
t
e
r
e
o
• However, even with correct speaker wiring
– there could still be potential problems with
wave cancellation
• Suppose you don’t stand equal distances
from each speaker…
Wave Interference
• If d 2 > d1 , we can define the path length
difference for the two waves as
PLD = d 2 − d1
• If the PLD equals a half a wavelength
– the waves will reach your ear 180o out of
phase and hence cancel each other out!
– the sound from each speaker will oppose
each other at your ear
S
t
e
r
e
o
In phase speakers: waves add at your ears
Wave Interference
• If you wire up the speakers incorrectly
Wave Interference
Wave Interference
• Despite the speakers producing sound in
phase
– the distance traveled by each sound wave is
different!
d 2 > d1
– because they traveled different distances!
• The waves’ crests and troughs no longer
match up at your ear
d1
Wave Interference
Wave Interference
• This effect is called destructive interference
– and the waves are said to be “destructively
interfering” with each other
• Mathematically:
PLD Destructive Interference = d 2 − d1 =
• Even though each wave starts out in
phase, they are out of phase when they
reach your ear
d 2 > d1
d2
S
t
e
r
e
o
180o out of phase speakers: waves cancel at your ears
λ
2
• Note that complete wave cancellation only
occurs if
– both waves have the same frequency
– both waves have the same amplitude
– the waves start out in phase
– the waves maintain the same phase relationship
for at least a short period of time
• This is called “coherence”, and the waves are said to
be “coherent”
• Any other conditions will only give partial
cancellation
3
Wave Interference
• Also note that the PLD for destructive
interference is different
– for different frequencies
• Example #1: 2 sound waves at 1 kHz
destructively interfere at a PLD = 0.55 ft =
6.6 in
PLD =
λ
2
=
v / f (1100 ft /s ) / 1000 Hz
=
= 0.55 ft
2
2
Wave Interference
• In a “real” stereo system
– The speakers don’t produce identical waves
• That would be monaural sound, not stereo!
– The waves reflect off the walls, the floor, the
ceiling, furniture, etc…
• complicating the PLD geometry
– You have two listening locations (ears)
separated by 6 – 7 inches of dead space
• Your head is about half a wavelength thick at 1 kHz
– Different PLD’s for each ear!
• Example #2: 2 sound waves at 5 kHz
destructively interfere at a PLD = 0.11 ft =
1.3 in
PLD =
λ
2
=
v / f (1100 ft /s ) / 5000 Hz
=
= 0.11 ft
2
2
• Increasing the frequency by a factor of 5
• The constructive and destructive points
are actually more general than this
• There are multiple points in reality:
where n = 0,1, 2, 3, K
1

PLD Destructive Interference = d 2 − d 1 =  n +  λ
2

where n = 0,1, 2, 3, K
Wave Interference
• Time for a reality check…
• The conditions for complete destructive
i t f
interference
(complete
(
l t wave cancellation)
ll ti )
are difficult to meet in a real situation
– for several reasons
– decreased the PLD for destructive interference
by the same amount
Wave Interference
• Of course, if the PLD is a whole wavelength
rather than a half wavelength
– the two waves will arrive at your ear in phase
• This
Thi iis called
ll d constructive
t ti interference
i t f
– and the two waves are said to be
“constructively interfering” with each other
• Mathematically:
PLDConstructi ve Interference = d 2 − d 1 = λ
Wave Interference
PLDConstructi ve Interference = d 2 − d 1 = nλ
Wave Interference
Beat Frequency
Wave Interference
• Finally, wave interference occurs for all
types of waves
– not just sound waves as we’ve been
discussing
• If you listen to AM radio while driving,
you’ve probably heard interference effects
– It’s that fading in and out of the signal as you
pass in and out of the partial cancellation
points (caused by multiple reflections of the
radio wave)
Beat Frequency
• One other interference effect is called
“beats” or the “beat frequency” between
two waves
• Because of this, the waves alternate
between constructive and destructive
interference as they pass a point
• If the two interfering wave have slightly
different frequencies
• Your ears will hear both waves plus a third
wave
– their phase difference is not constant
– having a frequency equal to the difference
between the original waves’ frequencies
4
Beat Frequency
Beat Frequency
• This difference frequency is called the
beat frequency
Beat Frequency = f 2 − f1
• The 2 Hz beat frequency will be heard as
a pulsation in the loudness of the average
of the two frequencies
– The pulsations having a frequency of 2 pulses
per second at a 441 Hz frequency
440 Hz
– usually expressed as a positive number
• Example: Pure tones of 440 Hz and
442 Hz produce a beat frequency of 2 Hz
Standing Waves
• If a pulse on a string
reflects off a fixed endpoint
(or a change in medium)
– the pulse is inverted
• It undergoes a 180o change of
phase
442 Hz
CH6_beat_demo.ds
441 Hz w/beats
• The same is true for a
continuous wave
Figure from Physics, Cutnell & Johnson, 7th ed.
Figure from Physics, Cutnell & Johnson, 7th ed.
Standing Waves
• For continuous wave reflection, you end up
with two waves traveling in opposite
directions with opposite phase
Standing Waves
• The standing wave has regions of
constructive and destructive interference
– destructive interference creates a “node”
– constructive interference creates an “antinode”
antinode
Standing Waves
• A distance of half a wavelength separates
one node from the next
– as well as one antinode from the next
λ
• The superposition of these two waves is
called a standing wave
2
– because the added wave appears to be
standing still
Standing Waves
• An important point about standing waves:
– Unlike “real” waves, standing waves carry no
energy
– Standing waves store energy however
Standing Waves
• The previous discussion assumes only one
end of the string is fixed
– and therefore a standing wave of any
frequency can be generated
• But if we fix both ends (such as in a guitar)
– only very specific standing wave frequencies
are generated
• Let’s discuss those specific frequencies
now…
String Harmonics
• Some musical instruments are based on
standing waves on strings
– guitars
– cellos
– pianos
– violins
– harps
– banjos
• Let’s examine how waves are produced on
stringed instruments
5
String Harmonics
String Harmonics
• When a string fixed at both ends is plucked
String Harmonics
• This is what our string would look like:
• The 1st harmonic frequency, f1, is related to
the length of the string (L) and the speed of
wave on the string (v):
– It has 2 nodes and 1 antinode
– many different specific standing wave
frequencies are produced
λ1 / 2
• and they are all related!
f1 =
• If we pluck the string “just right”
– the pluck will produce a standing wave of
frequency f1, the lowest possible frequency
v
2L
• And since v = f1 λ1 :
• Frequency f1 is called the “1st harmonic” or
“fundamental” frequency of the string
λ1 = 2 L or L =
Figure from Physics, Cutnell & Johnson, 7th ed.
String Harmonics
• If we pluck the string differently, we would
get a different specific standing wave
pattern:
λ2 / 2
String Harmonics
• Continuing to the next specific standing
wave frequency:
• f 2 is called the “second harmonic” or “1st
overtone” of the string
λ3 / 2
L = 2×
• When f 2 is measured, it is found to be
exactly
λ2
2
f 3 = 3 f1
String Harmonics - Summary
String Harmonics
String Harmonics - Summary
λ1 / 2
• f 3 is called the “third harmonic” or “2nd
overtone” of the string
L = 3×
λ3
f3
• When f 3 is measured, it is found to be
exactly
f 2 = 2 f1
• In this second case
case, the length of the string
is
2
String Harmonics
• In this second case
case, the length of the string
is
f2
λ1
L=
λ1
• So, the length of the string is always an
integer number of half wavelengths
2
• The length of the string determines
– the specific wavelengths allowed
λ2 / 2
f2
2
L = 2×
λ3 / 2
f3
L = 3×
λ2
2
λ3
2
• The tension and mass of the string
determine
– the speed of the wave on the string and hence
the specific wavelengths
v
f =
λ
6
String Harmonics - Summary
v
•
harmonic: f 1 =
2L
– fundamental
1st
String Harmonics - Summary
• The pitch of the wave can be changed in
two ways
• 2nd harmonic: f 2 = 2 f1
• 3rd harmonic:
– changing
λ by changing the string length
• As when you hold the string flat against the frets on
a guitar
f 3 = 3 f1
– changing
v by changing the string tension
• This is how musicians tune their instruments!
• The fundamental frequency determines the
pitch of the wave produced
Pipe Harmonics
– pipes with both ends open
• For a pipe open at both ends, the standing
waves have antinodes at each end
– This is a slight approximation, but a good one
• Example: a flute
• Some musical instruments are based on
standing waves in pipes
– pipe organs
– saxophones
– trumpets
– clarinets
– flutes
– wind chimes
• Let’s examine how waves are produced in
piped instruments
Pipe Harmonics – Open Ended
• There are two types of piped instruments
Pipe Harmonics
Pipe Harmonics – Open Ended
• The standing wave frequencies for an open
ended pipe have the same mathematical
form as those for a string fixed at both ends
– The wave’s
a e’s physical
ph sical form is different
however…
– pipes with one end open
• Example: a clarinet
1×
• Each type of pipe produces waves with
different characteristics
λ1
= f1
2×
2
= 2 f1
λ2
• sound wave versus mechanical wave
2
• Open ended pipes produce all harmonics
– 1st, 2nd, 3rd, 4th, etc…
A = antinode
N = node
Figure from Physics, Cutnell & Johnson, 7th ed.
Pipe Harmonics – Closed End
• For a pipe closed at one end, the standing
waves have an antinode at the open end
and a node at the closed end
1×
λ1
= f1
3×
4
λ2
= 3 f1
Pipe Harmonics – Closed End
• The standing wave frequencies for a pipe
closed at one end are different!
• Due to wave reflections off of the closed
end, the even harmonics undergo complete
destructive interference
– and do not survive!
4
• Only the odd harmonics are produced
A = antinode
– 1st, 3rd, 5th, 7th, etc…
N = node
Pipe Harmonics – Summary
• The fundamental frequencies for each type
of pipe are determined by the length of the
pipe
– open ended pipe
L = 1×
λ1
⇒ λ1 = 2 L
2
f1 =
v
λ1
=
v
2L
– closed end pipe
L = 1×
λ1
4
f1 =
⇒ λ1 = 4 L
v
λ1
=
v
4L
Figure from Physics, Cutnell & Johnson, 7th ed.
7
Pipe Harmonics – Summary
• The higher harmonic frequencies are
– open ended pipe
fn =
v
λn
 v 
= n × 

 2L
for n = 1, 2, 3, 4,K
– closed end pipe
 v 

fn =
= n × 
λn
4L
v
for n = 1, 3, 5, 7, K
Pipe Harmonics – Examples
• Example #1
Pipe Harmonics – Summary
• The fundamental wavelengths are
• What if the lengths aren’t the same?
• Example #1
– open ended pipe:
λ1 = 2 L
– A 2 ft long open pipe versus a 2 ft long closed
pipe
– closed end pipe:
λ1 = 4 L
– Open:
• For an open ended pipe, the fundamental
wavelength is twice the length of the pipe;
for a closed end pipe, its four times
Pipe Harmonics – Examples
• Example #2
– So an open ended pipe produces a
fundamental frequency twice as large as a
closed end pipe of the same length
Pipe Harmonics – Examples
– A 2 ft long open ended pipe versus a 1 ft long
closed end pipe
– Open:
 v 
 1100 ft / s 
 = 1 × 
 = 275 Hz
f 1 = 1 × 
 2 × 2 ft 
2L
 v 
 1100 ft / s 
 = 1 × 
 = 275 Hz
f1 = 1 × 
 2 × 2 ft 
 2L
 v 
 1100 ft / s 
 = 1 ×  4 × 2 ft  = 137.5 Hz


4L
– Closed: f1 = 1 × 
Pipe Harmonics – Examples
• Example #2
– So an open ended pipe twice as long as a
closed end pipe produces the same
fundamental frequency!
 v 
 1100 ft / s 
 = 1 × 
 = 275 Hz
– Closed: f 1 = 1 × 
 4 × 1 ft 
4L
Pipe Harmonics – Examples
• Example #3
– Sound from an open ended pipe twice as long
as a closed end pipe with the same
fundamental frequency (275 Hz)
Hz), but different
higher harmonics
CH6_open_pipe.ds
CH6_closed_pipe.ds
Standing Waves – Finale?
• Of course, real musical instruments are
much more complicated than these easy
examples
– Real instruments have finger holes, valves,
slides different density strings
slides,
strings, resonant
cavities etc…
• While we can usually get a rough estimate
of the instrument’s operation with what
we’ve learned so far
– it’s not nearly enough to get it exactly correct!
8