Fundamentals of Physics
Chapter 14 Waves - I
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
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Waves & Particles
Types of Waves
Transverse & Longitudinal Waves
Wavelength & Frequency
Speed of a Traveling Wave
Wave Speed on a Stretched String
Energy & Power in a Traveling String Wave
The Wave Equation
The Principle of Superposition for Waves
Interference of Waves
Phasors
Standing Waves
Standing Waves & Resonance
Fundamentals of Physics
1
Waves & Particles
Particles - a material object moves from one place to another.
Waves - information and energy move from one point to another, but no material
object makes that journey.
– Mechanical waves
– Newton’s Laws rule!
– Requires a material medium
e.g. water, sound, seismic, etc.
– Electromagnetic waves
– Maxwell’s Equations & 3.0 x 108 m/s
– No material medium required
– Matter waves
– Quantum Mechanics - ~10-13 m
– Particles have a wave length - De Broglie (1924)
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Fundamentals of Physics
2
A Simple Mechanical Wave
A single up-down motion applied to a taut string generates a pulse.
The pulse then travels along the string at velocity v.
He moves his hand once.
Assumptions in this chapter:
No friction-like forces within the string to dissipate wave motion.
Strings are very long - no need to consider reflected waves from the far end.
jw
Fundamentals of Physics
3
Traveling Waves
They oscillate their hand in SHM.
Transverse Wave:
The displacement (and velocity) of
every point along the medium carrying
the wave is perpendicular to the
direction of the wave.
e.g. a vibrating string
Longitudinal Wave:
The displacement (and velocity) of the element of
the medium carrying the wave is parallel to the
direction of the wave.
e.g. a sound wave
Longitudinal, Transverse and Mixed Type Waves
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Fundamentals of Physics
4
Transverse Wave
Displacement versus position
not versus time
transverse wave applet
Each point along the string just moves up and down.
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Fundamentals of Physics
5
Wave Length & Frequency
•
amplitude - maximum displacement
•
•
wavelength l - distance between repetitions of the
shape of the wave.
k  2
l
angular wave number
•
period - one full oscillation
•
frequency - oscillations per unit time
 T  2
f  1 
T
2
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Fundamentals of Physics
6
Wave Length & Frequency
The phase, kx – wt , changes linearly with x and t, which causes the
sine function to oscillate between +1 and –1.
space
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time
Fundamentals of Physics
7
Speed of a Traveling Wave
Consider a wave traveling in the positive x direction; the entire wave
pattern moves a distance Dx in time Dt:
v 
Dx
Dt
Each point on the wave, e.g. Point A, retains its displacement, y;
hence:
y ( x , t )  ym sin  kx  t   constant
kx  t 
•
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Differentiating:
 constant
Note: both x and t are changing!
dx


 v
dt
k
Fundamentals of Physics
8
Direction of the Wave
Consider an unchanging pulse traveling along positive x axis.
y = f (x’) = f (x - v t)
traveling towards +x
y = f (x + v t) traveling towards -x
All traveling waves are functions of (kx + t) = k(x + vt) .
jw
Fundamentals of Physics
9
Descriptions of the phase of a Traveling Wave
y( x, t )  ym sin  kx  t 
f 
1


T
2
l 
2
v
  vT
k
f
v 

k

l
T
 fl
x
 kx  t   k  x  vt      t   etc. etc.
v

  x t 
y ( x, t )  ym sin  2    
  l T 
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Fundamentals of Physics
10
Wave Speed on a Stretched String
A single symmetrical pulse moving along a string at speed vwave.
v
wave

F string

In general, the speed of a wave is determined by the
properties of the medium through which it travels.
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Fundamentals of Physics
11
Wave Speed on a Stretched String
Consider a single symmetrical pulse moving along a string at speed v:
t = tension in the string
 = the string’s linear density
Fr  2t sin   2t  
(Roughly a circular arc)
a
Dm   Dl
t Dl
R
2
v
R
Fnet  m a
t Dl
v2
  Dl
R
R
String moving
to the left.
v2 
Moving along with the pulse on the string.
t

Speed of a wave along a stretched string depends only on the tension
and the linear density of the string and not on the frequency of the wave.
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Fundamentals of Physics
12
Energy of a Traveling String Wave
Driving force imparts energy to a string, stretching it.
The wave transports the energy along the string.
Energy
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Fundamentals of Physics
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Energy & Power of a Traveling String Wave
•
•
Driving force imparts energy to a stretched string.
The wave transports energy along the string.
– Kinetic energy - transverse velocity of string mass element, Dm =  Dx
– Potential energy - the string element Dx stretches as the wave passes.
dK 
1
2
dK 
1
2
 dy 
dm  
 dt 
2
y  ym sin  kx  t 
  dx    y
m cos kx  t 
2
2
 dK 
 dU 
2
   41  v  ym  

 dt  avg
 dt  avg
Average Power  21  v  2 ym
2
(See Section 14-3)
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Fundamentals of Physics
14
The Principle of Superposition for Waves
Overlapping waves algebraically add to produce a resultant wave:
y1(x,t)
y2(x,t)
Add the amplitudes:
ytotal(x,t) = y1(x,t) + y2(x,t)
Overlapping waves do not alter the
travel of each other!
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Fundamentals of Physics
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The Principle of Superposition for Waves
Interference of waves traveling in opposite directions.
Constructive
Interference
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Destructive
Interference
Fundamentals of Physics
16
Interference of Waves Traveling in the Same Direction
y1 ( x, t )  ym sin  kx  t 
f = “phase difference”
f
ym
y2 ( x, t )  ym sin  kx  t  f 
y( x, t )  y1 ( x, t )  y2 ( x, t )
It is easy to show that:
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sin   sin   2 sin 21     cos 21    
Fundamentals of Physics
17
Interference of Waves
The magnitude of the resultant wave depends on the relative phases of the
combining waves - INTERFERENCE.
Constructive Interference
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Destructive Interference
Fundamentals of Physics
Partial Interference
18
Standing Waves
Two sinusoidal waves of the same amplitude and wavelength travel in opposite
directions along a string:
y1(x,t) = ym sin (k x
-
 t)
y2(x,t) = ym sin (k x +  t)
positive x direction
negative x direction
Their interference with each other produces a standing wave:
y’(x,t) = y1(x,t) + y2(x,t)
It is easy to show that:
sin   sin   2 sin 21     cos 21    
For a standing wave, the amplitude, 2ymsin(kx) , varies with position.
jw
Fundamentals of Physics
19
Standing Wave
The amplitude of a standing wave equals zero for:
sin k x   0
k x  n
k 
minimums @
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n  0,1, 2, 3   
2
l
x = ½nl
n = 0, 1, 2, . . . NODES
Fundamentals of Physics
20
Standing Waves
k 
minimums @
x = ½nl
maximums @ x = ½(n + ½) l
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2
l
n = 0, 1, 2, . . . NODES
n = 0, 1, 2, . . . ANTINODES
Fundamentals of Physics
21
Reflections at a Boundary
End of the string is
free to move
Tie the end of the
string to the wall
“soft” reflection
“hard” reflection
Node
at boundary
Antinode
at boundary
Reflected
pulse has
opposite
sign
Reflected
pulse has
same sign
Newton’s 3rd Law
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Fundamentals of Physics
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Reflections at a Boundary
From high speed to low speed (low density to high density)
From high density to low density
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Fundamentals of Physics
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Standing Waves & Resonance
Consider a string with both ends fixed; it has nodes at both ends.
This can only be true when:
Ln
l
2
n  1, 2, 3,   
v is the speed of the traveling waves on the string.
v
nv
f 
fn 
n  1, 2, 3,   
l
2L
Resonance for certain frequencies for a string with both ends fixed.
Only for these frequencies will the waves reflected back and forth be in phase.
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Fundamentals of Physics
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Standing Waves & Resonance
A standing wave is created from two traveling waves, having the same frequency and
the same amplitude and traveling in opposite directions in the same medium.
Using superposition, the net displacement of the medium is the sum of the two waves.
When 180° out-of-phase with each other, they cancel (destructive interference).
When in-phase with each other, they add together (constructive interference).
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Fundamentals of Physics
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Standing Waves & Resonance
The Harmonic Series
fn 
nv
 n f1
2L
n  1, 2, 3,   
both ends fixed
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Fundamentals of Physics
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String Fixed at One End
Resonance:
l
L  n 
4
n  1, 3, 5,   
fn 
nv
 n f1
4L
n  1, 3, 5,   
Standing wave applet
Prenault’s applets
fixed end
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Fundamentals of Physics
free end
27
Fundamentals of Physics
Waves - II
1.
2.
3.
4.
5.
6.
Introduction
Sound Waves
The Speed of Sound
Traveling Sound Waves
Interference
Intensity & Sound Level
The Decibel Scale
7. Sources of Musical Sound
8. Beats
9. The Doppler Effect
Detector Moving; Source Stationary
Source moving; Detector Stationary
Bat Navigation
10. Supersonic Speeds; Shock Waves
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Fundamentals of Physics
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Sound Waves
A sound wave is a longitudinal wave of any frequency passing through a
medium (solid, liquid or gas).
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Fundamentals of Physics
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The Speed of Sound
The speed of waves depends on the medium, not on the motion of the source.
velocity 
restoring property
inertial property
Elastic property of the medium
– Strings - tension (t in N)
– Sound - Bulk Modulus (B in N/m2)
• Inertial property of the medium
– Strings - linear mass density ( in kg/m)
– Sound - volume mass density (r in kg/m3)
v 
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t

v 
B
r
Fundamentals of Physics
Dp   B
DV
V
30
The Speed of Sound
Equation for the speed of sound:
v 
B
r
Dp   B
DV
V
For an ideal gas, B/r can be shown to be proportional to absolute
temperature; hence, the speed of sound depends on the square
root of the absolute temperature.
v 
 RT
M
v  343 m s  760 mi h
T = absolute temperature
 = 1.4 for O2 and N2 (~air)
R = “universal gas constant” = 8.314 J/mol-K
M = molar mass of the gas = 29 x 10-3 kg/mol (for air)
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Fundamentals of Physics
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Traveling Sound Waves
Displacement Function:
(of the air element about x)
sx, t   sm coskx  t 
SHM
Pressure-Variation Function: Dpx, t   Dpm sin kx  t 
(pressure change as wave passes x)
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Fundamentals of Physics
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Traveling Sound Waves
As a sound wave moves in time, the displacement of air molecules, the
pressure, and the density all vary sinusoidally with the frequency of
the vibrating source.
Slinky Demo
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Fundamentals of Physics
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Traveling Sound Waves
As a sound wave moves in time, the displacement of air molecules, the pressure,
and the density all vary sinusoidally with the frequency of the vibrating source.
Dpm  v r  sm
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Fundamentals of Physics
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Interference
Consider two sources of waves S1 and S2, which are “in phase”:
L1
“arrive in phase”
DL  L2  L1
DL  l  Df  2 
f
DL

2
l
L2
f is the “phase difference” @ P1
“Constructive Interference”
f  m 2  
jw
Fundamentals of Physics
m  0, 1, 2, 3,   
DL  0, l , 2l , 3l ,   
35
Interference
Two sources of sound waves S1 and S2:
L1
arrive “out of phase”
DL  L2  L1
DL 
L2
2
l,
3
2
l,
5
2
l,   
f
DL

2
l
“Destructive Interference”
f  2m  1
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1
Fundamentals of Physics
m  0, 1, 2, 3,   
DL  1 2 l , 3 2 l , 5 2 l ,   
36
Psychological dimensions of sounds
Pitch
150 Hz
300-Hz sound
l
vsound
 2.3 m
f
500-Hz sound
l  0.23m
1500 Hz
A healthy young ear can hear sounds between 20 - 20,000 Hz.
- age reduces our hearing acuity for high frequencies.
Loudness
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150 Hz with twice the amplitude of 1500 Hz
Fundamentals of Physics
37
Intensity & Sound Level
Intensity of a Sound Wave:
I 
Power
Area
The power of the wave is time rate of energy transfer.
The area of the surface intercepting the sound.
All the sound energy from the source spreads out radially and must pass
through the surface of a sphere:
P
I  source2
4 R
I ~
1
R2
In terms of the parameters of the source and of the medium carrying the
sound, the sound intensity can be shown to be as follows:
I  12 r v  2 sm2
I ~ sm2
P  12  v  2 A2
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Fundamentals of Physics
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Intensity & Sound Level
The Decibel Scale:
 - Sound Level
Alexander Graham Bell
Mammals hear over an enormous range:
humans :
1012 W m2  1 W m2 (pain)
Sound level (or loudness) is a sensation in the consciousness of a human
being. The psychological sensation of loudness varies approximately
logarithmically; to produce a sound that seems twice as loud requires
about ten times the intensity.
  10 log
I
I0
(decibel)
where I0 is the approximate threshold of human hearing.
I 0  1012 W m2
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Fundamentals of Physics

 0
39
Intensity & Sound Level
Human Perception of Sound
  10 log
I
I0
I 0  1012 W m2

 0
~3dB is a factor 2 change in intensity
Every 10dB is a factor 10 change in intensity;
20 dB is a factor 100 change in intensity
See Table 17-2.
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Fundamentals of Physics
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Intensity & Sound Level
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Fundamentals of Physics
41
Sources of Musical Sound
Standing Waves in a Pipe
Closed End
– Molecules cannot move
Open End
– Molecules free to move
• Displacement node
• Pressure Antinode
• Displacement Antinode
• Pressure Node
Both ends closed  2 nodes with at least one antinode in between.
Both ends open  2 antinodes with at least one node in between.
One end closed  1 node and one antinode.
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Fundamentals of Physics
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Sources of Musical Sound
nodes or antinodes at the ends of the resonant structure
Fundamental Frequency
“1st Harmonic”
“Fundamental mode”
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Fundamentals of Physics
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Sources of Musical Sound
Both Ends Open
One End Open
n 1
2L
n 1, 2, 3,   
n
v
nv
f  
l 2L
l
4L
n 1, 3, 5,   
n
v
nv
f  
l 4L
l
Harmonic Number
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Fundamentals of Physics
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Sources of Musical Sound
length of an instrument  fundamental frequency
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Fundamentals of Physics
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Sources of Musical Sound
Overtones
Fundamental & Overtones
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Fundamentals of Physics
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Beats
2 waves with slightly different frequencies are traveling to the right.
The superposition of the 2 waves travels in the same direction and with
the same speed.
The "beat" wave oscillates with the average frequency, and its amplitude
envelope varies according to the difference frequency.
s  2 sm cos t  cos  t
 
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1
2
1  2 
 
Fundamentals of Physics
1
2
1  2 
47
Beats
Consider two similar sound waves:
s1  sm cos 1t
s2  sm cos 2t
1  2
Superimpose them:
s  s1  s2
s  sm cos 1t  cos 2t 
s  2 sm cos 12 1  2 t cos 12 1  2 t 
s  2 sm cos t cos  t 
  12 1  2 
  12 1  2 
If 1  2 then    and beat  2   1  2
“Beat Frequency”:
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f beat  f1  f 2
Fundamentals of Physics
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Interference:
Standing Wave created from two traveling waves:
2 sinusoidal waves having the same frequency (wavelength) and the same
amplitude are traveling in opposite directions in the same medium.
[one dot at an antinode and one at a node]
As the two waves pass through each other, the net result alternates between
zero and some maximum amplitude. However, this pattern simply oscillates;
it does not travel to the right or the left; it stands still.
Beats created from two traveling waves:
2 waves with slightly different frequencies are traveling in the same
direction.
The superposition is a traveling wave, oscillating with the average
frequency with its amplitude envelope varying according to the
difference frequency.
Beats demo
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Fundamentals of Physics
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The Doppler Effect
Doppler Effect
Applet: Doppler Effect
http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Flash/#sound_waves
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Fundamentals of Physics
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The Doppler Effect
S - Wave Source
D - Detector (ear)

vD

vS
v  speed of sound in the medium
l  wavelength of the sound
f  frequency of the sound
f 
Case 0:
both are stationary
l


vS  vD  0
frequency detected:
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v
f  f
Fundamentals of Physics
51
The Doppler Effect
Case 1:
Detector is moving towards the source

vD  0

vS  0
frequency detected:
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f  f
Fundamentals of Physics
52
The Doppler Effect
Case 1:
Detector is moving towards the source
vD Dt
Number of wavefronts intercepted
“Rate of Interceptions”
f 
v Dt
v Dt  vD Dt
l
f 
l 
v
l
v
f
v Dt  vD Dt
v  vD
 f
l Dt
v
A higher frequency is detected
See section 14.8
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Fundamentals of Physics
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The Doppler Effect
Case 1:
Detector is moving towards the source
f  f

vD  0

vS  0
Case 2:
Detector is moving away from the source
v  vD
v
f  f
v  vD
v
The detected frequency is less than the source frequency.
jw
Fundamentals of Physics
Applet: Doppler Effect
54
The Doppler Effect
Case 3:
Source is moving towards the detector
l  is the detected wavelength @ D
T is the time between emissions @ S
f 
v
v

l  v T  vS T
f  f
v
v  vS
f 
1
T
The detected frequency is greater than the source frequency.
Case 4:
Source is moving away from the detector
f  f
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v
v  vS
Fundamentals of Physics
55
Supersonic Speeds
Applet: Doppler Effect
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Fundamentals of Physics
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Supersonic Speeds & Shock Waves
No waves in front of the source.
Waves pile up behind the source to form a shock wave.
Mach Number 
v
vsound
sin  
v
vs
The “Mach Cone” narrows as vS goes up.
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Fundamentals of Physics
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Supersonic Speeds
v
sin  
vs
  33o
You won’t hear it coming!
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Fundamentals of Physics
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Supersonic
vsound
sin  
vsource

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Fundamentals of Physics
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