jw fundamentals of physics 1 chapter 14 waves - i 1.waves & particles 2.types of waves...
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jw Fundamentals of Physics 1
Fundamentals of Physics
Chapter 14 Waves - I
1. Waves & Particles2. Types of Waves3. Transverse & Longitudinal Waves4. Wavelength & Frequency5. Speed of a Traveling Wave6. Wave Speed on a Stretched String7. Energy & Power in a Traveling String Wave8. The Wave Equation9. The Principle of Superposition for Waves10. Interference of Waves11. Phasors12. Standing Waves13. Standing Waves & Resonance
jw Fundamentals of Physics 2
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)
jw Fundamentals of Physics 3
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.
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.
He moves his hand once.
jw Fundamentals of Physics 4
Traveling Waves
Transverse Wave:The displacement (and velocity) of every point along the medium carrying the wave is perpendicular to the direction of the wave.
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 vibrating string e.g. a sound wave
They oscillate their hand in SHM.
Longitudinal, Transverse and Mixed Type Waves
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Transverse Wave
Displacement versus position
not versus time
Each point along the string just moves up and down.
transverse wave applet
jw Fundamentals of Physics 6
• amplitude - maximum displacement
• wavelength - distance between repetitions of the shape of the wave.
• angular wave number
• period - one full oscillation
• frequency - oscillations per unit time
Wave Length & Frequency
2T
2k
21 Tf
jw Fundamentals of Physics 7
The phase, kx – wt , changes linearly with x and t, which causes the sine function to oscillate between +1 and –1.
Wave Length & Frequency
timespace
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Each point on the wave, e.g. Point A, retains its displacement, y;
hence:
Speed of a Traveling Wave
Consider a wave traveling in the positive x direction; the entire wave pattern moves a distance x in time t:
vx
t
y x t y kx tm( , ) sin constant
kx t constant Note: both x and t are changing!
dxdt k
v • Differentiating:
jw Fundamentals of Physics 9
Direction of the Wave
y = f (x + v t) traveling towards -x
All traveling waves are functions of (kx + t) = k(x + vt) .
Consider an unchanging pulse traveling along positive x axis.
traveling towards +x y = f (x’) = f (x - v t)
jw Fundamentals of Physics 10
Descriptions of the phase of a Traveling Wave
( , ) sinmy x t y kx t
v fk T
1
2f
T
2 vvT
k f
. .x
kx t k x vt t etc etcv
( , ) sin 2m
x ty x t y
T
jw Fundamentals of Physics 11
Wave Speed on a Stretched String
A single symmetrical pulse moving along a string at speed vwave.
stringwave F
v
In general, the speed of a wave is determined by the properties of the medium through which it travels.
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Wave Speed on a Stretched String
Consider a single symmetrical pulse moving along a string at speed v:
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.
= tension in the string = the string’s linear density
Moving along with the pulse on the string.
2 sin 2r
lF
R
m l
2va
R
2
2
netF m a
l vl
R R
v
String moving
to the left.
(Roughly a circular arc)
jw Fundamentals of Physics 13
Energy of a Traveling String Wave
Energy
Driving force imparts energy to a string, stretching it.
The wave transports the energy along the string.
jw Fundamentals of Physics 14
• Driving force imparts energy to a stretched string.
• The wave transports energy along the string.
– Kinetic energy - transverse velocity of string mass element, m = x– Potential energy - the string element x stretches as the wave passes.
Energy & Power of a Traveling String Wave
dK dmdy
dt
dK dx y kx t
dK
dtv y
dU
dt
v y
m
avg
m
avg
m
12
2
12
2
14
2 2
12
2 2
cos
Average Power
sinmy y kx t
(See Section 14-3)
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The Principle of Superposition for Waves
ytotal(x,t) = y1(x,t) + y2(x,t)
Overlapping waves algebraically add to produce a resultant wave:
y1(x,t) y2(x,t)Add the amplitudes:
Overlapping waves do not alter the travel of each other!
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The Principle of Superposition for Waves
Interference of waves traveling in opposite directions.
Constructive
Interference
Destructive
Interference
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Interference of Waves Traveling in the Same Direction
It is easy to show that:
= “phase difference”
sin sin sin cos 2 12
12
1( , ) sinmy x t y kx t
2 ( , ) sinmy x t y kx t
my
1 2( , ) ( , ) ( , )y x t y x t y x t
jw Fundamentals of Physics 18
Interference of Waves
The magnitude of the resultant wave depends on the relative phases of the combining waves - INTERFERENCE.
Constructive Interference Destructive Interference Partial Interference
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Standing Waves
y’(x,t) = y1(x,t) + y2(x,t)
y1(x,t) = ym sin (k x - t)
y2(x,t) = ym sin (k x + t)
Two sinusoidal waves of the same amplitude and wavelength travel in opposite directions along a string:
For a standing wave, the amplitude, 2ymsin(kx) , varies with position.
sin sin sin cos 2 12
12
positive x direction
Their interference with each other produces a standing wave:
negative x direction
It is easy to show that:
jw Fundamentals of Physics 20
Standing Wave
The amplitude of a standing wave equals zero for:
minimums @ x = ½ n n = 0, 1, 2, . . . NODES
0sin xk
3,2,1,0nnxk
2
k
jw Fundamentals of Physics 21
Standing Waves
2
k
minimums @ x = ½ n n = 0, 1, 2, . . . NODES
maximums @ x = ½(n + ½) n = 0, 1, 2, . . . ANTINODES
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Reflections at a Boundary
Reflected pulse has opposite
sign
Reflected pulse has same sign
Newton’s 3rd Law
“soft” reflection“hard” reflection
Tie the end of the string to the wall
End of the string is free to move
Antinodeat boundary
Nodeat boundary
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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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Standing Waves & Resonance
Resonance for certain frequencies for a string with both ends fixed.
This can only be true when:
v
f
v is the speed of the traveling waves on the string.
,3,2,12
nL
vnfn
,3,2,12
nnL
Consider a string with both ends fixed; it has nodes at both ends.
Only for these frequencies will the waves reflected back and forth be in phase.
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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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Standing Waves & Resonance
The Harmonic Series
both ends fixed
,3,2,1
2 1
n
fnL
vnfn
jw Fundamentals of Physics 27
String Fixed at One End
fixed end
Prenault’s applets
free end
Resonance:
,5,3,1
4
n
nL
,5,3,1
4 1
n
fnL
vnfn
Standing wave applet
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Fundamentals of Physics
Waves - II
1. Introduction2. Sound Waves3. The Speed of Sound4. Traveling Sound Waves5. Interference6. Intensity & Sound Level
The Decibel Scale7. Sources of Musical Sound8. Beats9. The Doppler Effect
Detector Moving; Source StationarySource moving; Detector StationaryBat Navigation
10. Supersonic Speeds; Shock Waves
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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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Elastic property of the medium– Strings - tension ( 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 ( in kg/m3)
The Speed of Sound
vB
v
p BV
V
property inertial
property restoring velocity
The speed of waves depends on the medium, not on the motion of the source.
jw Fundamentals of Physics 31
The Speed of Sound
For an ideal gas, B/ can be shown to be proportional to absolute temperature; hence, the speed of sound depends on the square root of the absolute temperature.
vB
vR T
M
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)
p B
V
V Equation for the speed of sound:
343 760m mis hv
jw Fundamentals of Physics 32
Traveling Sound Waves
tkxstxs m cos,
Pressure-Variation Function:(pressure change as wave passes x)
Displacement Function:(of the air element about x) SHM
tkxptxp m sin,
jw Fundamentals of Physics 33
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
jw Fundamentals of Physics 34
Traveling Sound Waves
mm svp
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.
jw Fundamentals of Physics 35
Interference
Consider two sources of waves S1 and S2, which are “in phase”:
L
2
1L
2L
12 LLL
,3,2,,0
,3,2,1,02
L
mm
“Constructive Interference”
2L
is the “phase difference” @ P1
“arrive in phase”
jw Fundamentals of Physics 36
Interference
Two sources of sound waves S1 and S2:
L
2
1L
2L
12 LLL
,,,
,3,2,1,012
25
23
21
L
mm
“Destructive Interference”
,,, 25
23
21 L
arrive “out of phase”
jw Fundamentals of Physics 37
Psychological dimensions of sounds
Pitch
Loudness
1500 Hz
150 Hz
150 Hz with twice the amplitude of 1500 Hz
A healthy young ear can hear sounds between 20 - 20,000 Hz.
- age reduces our hearing acuity for high frequencies.
300-Hz sound
500-Hz sound
2.3soundvm
f
0.23m
jw Fundamentals of Physics 38
Intensity of a Sound Wave:
The power of the wave is time rate of energy transfer.
The area of the surface intercepting the sound.
Intensity & Sound Level
Area
PowerI
24 R
PI source
All the sound energy from the source spreads out radially and must pass through the surface of a sphere:
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:
2221
msvI
2
1~
RI
2~ msI
2221 AvP
jw Fundamentals of Physics 39
Intensity & Sound Level
The Decibel Scale: - Sound Level
Mammals hear over an enormous range:
where I0 is the approximate threshold of human hearing.
(pain) 110 :humans 2212
mW
mW
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.
0
log10I
I (decibel)
010 212
0 m
WI
Alexander Graham Bell
jw Fundamentals of Physics 40
Intensity & Sound Level
Every 10dB is a factor 10 change in intensity; 20 dB is a factor 100 change in intensity
Human Perception of Sound
0
log10I
I
010 212
0 m
WI
~3dB is a factor 2 change in intensity
See Table 17-2.
jw Fundamentals of Physics 41
Intensity & Sound Level
jw Fundamentals of Physics 42
Sources of Musical Sound
Closed End– Molecules cannot move
• Displacement node
• Pressure Antinode
Open End– Molecules free to move
• 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.
Standing Waves in a Pipe
jw Fundamentals of Physics 43
Sources of Musical Sound
Fundamental Frequency
“1st Harmonic”
“Fundamental mode”
nodes or antinodes at the ends of the resonant structure
jw Fundamentals of Physics 44
Sources of Musical Sound
Both Ends Open One End Open
Harmonic Number
L
vnvf
nn
L
2
,3,2,12
1n
L
vnvf
nn
L
4
,5,3,14
jw Fundamentals of Physics 45
Sources of Musical Sound
length of an instrument fundamental frequency
jw Fundamentals of Physics 46
Sources of Musical Sound
Fundamental & Overtones
Overtones
jw Fundamentals of Physics 47
Beats
2 waves with slightly different frequencies are traveling to the right.
ttss m coscos2
2121
2121
The "beat" wave oscillates with the average frequency, and its amplitude envelope varies according to the difference frequency.
The superposition of the 2 waves travels in the same direction and with the same speed.
jw Fundamentals of Physics 48
Beats
tss
tss
m
m
22
11
cos
cos
21
ttss
ttss
sss
m
m
2121
2121
21
21
coscos2
coscos
Consider two similar sound waves:
Superimpose them:
2121
2121
“Beat Frequency”:
2121 2 beatandthenIf
21 fffbeat
ttss m coscos2
jw Fundamentals of Physics 49
Interference:
Standing Wave 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.
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.
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]
Beats created from two traveling waves:
Beats demo
jw Fundamentals of Physics 50
The Doppler Effect
Doppler Effect
http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Flash/#sound_waves
Applet: Doppler Effect
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The Doppler Effect
S - Wave Source D - Detector (ear)
Sv
Dv
Case 0: both are stationary 0 DS vv
frequency detected: ff
v
f
sound theoffrequency
sound theof wavelength
f
v
mediumthe in sound of speed
jw Fundamentals of Physics 52
The Doppler Effect
Case 1: Detector is moving towards the source
0Dv
frequency detected: ff
0Sv
jw Fundamentals of Physics 53
The Doppler Effect
vD t v t
Case 1: Detector is moving towards the source
Number of wavefronts intercepted
tvtv D
“Rate of Interceptions”
v
vvf
t
tvtvf DD
f
v
vf
A higher frequency is detected
See section 14.8
jw Fundamentals of Physics 54
The Doppler Effect
Case 1: Detector is moving towards the source
0Dv
Applet: Doppler Effect
0Sv
v
vvff D
Case 2: Detector is moving away from the source v
vvff D
The detected frequency is less than the source frequency.
jw Fundamentals of Physics 55
The Doppler Effect
Case 3: Source is moving towards the detector
Svv
vff
S@ emissions between time the isT
D@ h wavelengtdetected the is
Tf
1
The detected frequency is greater than the source frequency.
Case 4: Source is moving away from the detector
Svv
vff
TvTv
vvf
S
jw Fundamentals of Physics 56
Supersonic Speeds
Applet: Doppler Effect
jw Fundamentals of Physics 57
Supersonic Speeds & Shock Waves
No waves in front of the source.
Waves pile up behind the source to form a shock wave.
The “Mach Cone” narrows as vS goes up.
sound
vMach Number
v
sv
vsin
jw Fundamentals of Physics 58
Supersonic Speeds
sv
vsin
o33
vS = Mach 1.8
You won’t hear it coming!
jw Fundamentals of Physics 59
Supersonic
sin sound
source
v
v