Applied Natural Sciences - Porous media · 2019-10-01 · 1 P ave F A • When a duck swims, it...
Transcript of Applied Natural Sciences - Porous media · 2019-10-01 · 1 P ave F A • When a duck swims, it...
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Applied Natural Sciences
Leo Pele‐mail: [email protected]://tiny.cc/3NAB0
Het basisvak Toegepaste Natuurwetenschappen
http://www.phys.tue.nl/nfcmr/natuur/collegenatuur.html
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Copyright © 2012 Pearson Education Inc.
PowerPoint® Lectures forUniversity Physics, Thirteenth Edition
– Hugh D. Young and Roger A. Freedman
Lectures by Wayne Anderson
Chapter 15
Mechanical Waves
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LEARNING GOALS
• What is meant by a mechanical wave, and the different varieties of mechanical waves
• How to use the relationship among speed, frequency, and wavelength for a
periodic wave.
• How to interpret and use the mathematical expression for a sinusoidal periodic wave.
• How to calculate the speed of waves on a rope or string.
• How to calculate the rate at which a mechanical wave transports energy.
• What happens when mechanical waves overlap and interfere.
• The properties of standing waves on a string, and how to analyze these waves.
• How stringed instruments produce sounds of specific frequencies.
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Simple Harmonic oscillator
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Harmonic oscillator and waves
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Depending on the direction of the displacement relative to the direction of propagation, we can define wave motion as:• Transverse – if the direction of displacement is perpendicular to the direction of propagation
Types of waves
Earth quake wavesAnimation courtesy of Dr. Dan Russell, Grad. Prog. Acoustics, Penn State 7
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Depending on the direction of the displacement relative to the direction of propagation, we can define wave motion as:• Longitudinal – if the direction of displacement is parallel to the direction of propagation
Types of waves
Sound
Animation courtesy of Dr. Dan Russell, Grad. Prog. Acoustics, Penn State 8
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But also a combination of Transverse and Longitudinal
Types of waves
Water waves
Animation courtesy of Dr. Dan Russell, Grad. Prog. Acoustics, Penn State 9
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Transverse and longitudinal waves
Sound (part II)air
Mechanical (part I)materials
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Waves have in common:
1. In each case the disturbance travels or propagateswith a
definite speed through the medium: the wave speed.
2. The medium itself does not travel through space.
3. To set any of these systems into motion, we have to put in
energy by doing mechanical work on the system.
Waves transport energy, but not matter, from one region to
another.
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MATHEMATICAL
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Harmonic oscillator and waves
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One position: SHM One time shot:position in space
Have a description NEED ONE)cos()( tAtx14
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Wavelength (lambda), 16
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Harmonic wave
wavelength
=Wave speed
=v
Wave speed = v =distancetime
wavelengthperiod= =
T = f
but 1/T=f
V=for f=V/
Shake end ofstring up & down
with SHM period = T
this is the “Golden Rule” for waves 17
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Example: wave on a string
A wave moves on a string at a speed of 4 cm/sA snapshot of the motion shows that the wavelength, is 2
cm, what is the frequency, ?v = , so = v / = (4 cm/s ) / (2 cm) = 2 Hz
T = 1 / f = 1 / (2 Hz) = 0.5 s
2 cm 2 cm 2 cm
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Look at:
x=5
x=10
)cos()( tAtx
Waves
)0(2cos)( xAxy
)5(2cos)( xAxy
)10(2cos)( xAxy
position = 5 t= vt
t
fAtx 2cos)(
shm
)(2cos),( vtxAtxy
move right
)(2cos),( vtxAtxy
move left
Movement of wave
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)(2cos),( vtxAtxy
vtxAtxy
22cos),( fv
tfxAtxy 22cos),(
tkxAtxy cos),(2k
Wave number
Alternate notation
“+” if wave travels toward –x
“-” if wave travels toward +x 20
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Definitions
Amplitude ‐ (A, ym) Maximum value of the displacement of a particle in a medium
(radius of circular motion).
Wavelength ‐ () The spatial distance between any two points that behave identically,
i.e. have the same amplitude, move in the same direction (spatial period)
Wave Number ‐ (k) Amount the phase changes per unit length of wave travel. (spatial
frequency, angular wavenumber)
Period ‐ (T) Time for a particle/system to complete one cycle.
Frequency ‐ (f) The number of cycles or oscillations completed in a period of time
Angular Frequency ‐ Time rate of change of the phase.
Phase ‐ kx ‐ t Time varying argument of the trigonometric function.
Phase Velocity ‐ (v) The velocity at which the disturbance is moving through the
medium 21
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The Wave Equation
If we see an equation
that looks like:
... we can write down the
amplitude, frequency,
velocity, and
wavelength of the
wave it describes.
xty m 5796.4 s 8.1570cosm15.0 -1-1
direction.x in the m/s, 343
m 1.372 m 5796.42Hz 250 s 1570.8 2 m 15.0
22cos
1-
1-
fv
ffA
xftAy
)cos()cos(
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General description SHM
• The force described by Hooke’s Law is the net force in Newton’s Second Law.
)()(2
2
tkxdt
txdm
1. The acceleration is proportional to the displacement of the block
2. The force is conservative. In the absence of friction, the motion will continue forever.
mk
tAtx
= where
)cos()(
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General description waves
2y(x, t)x2 1
v2
2y(x, t)t2
Wave Equation:
position time
Partial Differential Equation (PDE)
tkxAtxy cos),(A solution is:
We will not try to solve this equation
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General description waves
2y(x, t)x2 1
v2
2y(x, t)t2
Wave Equation:
tkxAtxy cos),(
tkxAty sin
tkxAty
cos2
2
2
tkxkAxy sin
tkxAkxy
cos2
2
2
tkxAv
tkxAk coscos 2
22
And 2k
fv kv25
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Sinusoidal Wave on a String, final
The maximum magnitudes of the transverse speed and
transverse acceleration are
vy, max = A
ay, max = 2 A
The speed v is constant for a uniform medium,
whereas vy varies sinusoidally.
Tgolf‐getal k
v
vy
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Rope: mass + springs
mktAtx = where)cos()(
1) Mass rope: m larger > slower
2) More tension > k larger > faster
How fast will it be transmitted
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Speed of a Wave on a String
The speed of the wave depends on the physical characteristics of the string and the tension to which the string is subjected.
•This assumes that the tension is not affected by the pulse.•This does not assume any particular shape for the pulse.
tension force
(string mass / string length)
F
lengthmasstensionv
/
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• The average power:
Energy in a Wave
22
21 AFPave
• When a duck swims, it necessarily produces waves on the surface of the water. • The faster the duck swims, the larger the wave amplitude and the more power the duck must supply to produce these waves.• The maximum power available from their leg muscles limits the maximum swimming speed of ducks to only about 0.7 m/s (2.5 km/h)
Duck swimming
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Waves Intensity
Wave intensity for a three dimensional wave from a point source:
22 W/mofunitsin
4 rPI
22
212
1 44 IrIr
21
22
2
1
rr
II
Example: earthquake, sound waves33
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Wave boundary/
interference
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The principle of superpositionWhen two waves overlap, the actual displacement of anypoint at any time is obtained by adding the displacementthe point would have if only the first wave were present andthe displacement it would have if only the second wave werepresent:
),(),(),( 21 txytxytxy
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Superposition of equal amplitude waves
Constructive Destructive
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Reflection (from a fixed end)
Animations courtesy of Dr. Dan Russell, Kettering University 38
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Reflection (from a loose end)
Animations courtesy of Dr. Dan Russell, Kettering University 39
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Superposition of equal amplitude waves
Constructive Destructive
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Standing Wave
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Adding waves
harmonic waves in opposite directions
incident wave
reflected wave
resultant wave
(standing wave)
Animations courtesy of Dr. Dan Russell, Kettering University 43
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Standing Waves ‐ Resonance
tkxAtxy cos),(1
tkxAtxy cos),(2
tkxAtkxAtxy coscos),(
cos x - cos y = -2 sin ((x - y)/2) sin( (x + y)/2 )
)sin()sin(2),( tkxAtxy 45
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Nodes and Antinodes
Node – position of no displacement
Antinode – position of maximum displacement
y 2Asin kx cos t
2kx x 0, , 2 ,3 ...
3x 0, , , ,.....2 2
Nodes
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Confined waves
Only waves with wavelengths that just fit in survive(all others cancel themselves out)
nL
n2
2nL 12
nfLvnfn
2kx x 0, , 2 ,3 ...
3x 0, , , ,.....2 2
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Allowed frequencies
=(2/3)L
f0=V/ = V/2L
f1=V/ = V/L=2f0
= 2L
=L
=(2/5)L
=L/2
f2=V/=V/(2/3)L=3f0
f3=V/=V/(1/2)L=4f0
f4=V/=V/(2/5)L=5f0
Fundamental tone
1st overtone
3rd overtone
4th overtone
2nd overtone
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VIDEO + UITWERKING
HznLvnfn 7.4
2
L = 1.8 mString mass= 188 grTension = 3N
µ =0.00810 kg/m
smTv /92.16
4.7 9.4 14.1 18.8 23.5 28.2 32.9 36.6 42.3 47.0 51.7 56.4 61.1 65.81 2 3 4 5 6 7 8 9 10 11 12 13 14
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4.7 9.4 14.1 18.8 23.5 28.2 32.9 36.6 42.3 47.0 51.7 56.4 61.1 65.81 2 3 4 5 6 7 8 9 10 11 12 13 14
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Fundamental, 2nd, 3rd... Harmonics
Fig 14.18, p. 443
Slide 25
Fundamental (n=1)
2nd harmonic
3rd harmonic
n 2 L
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Loose Ends
(Organ pipes open at one end)
L 2n 1 4 L
4
L 34
L 5 4
)12(4
n
Ln
1)12(4
)12(
fnLvnfn
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Harmonics with a bridge
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Stringed instruments
Three types
Plucked: guitar, bass, harp, harpsichord
Bowed: violin, viola, cello, bass
(Bowing excites many vibration modes
simultaneously mixture of tones (richness)
Struck: piano
All use strings that are fixed at both ends
• Use different diameter strings (mass per unit
length is different)
• The string tension is adjustable – tuning T
Lf
21
1
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A 1.00 m string fixed at both ends vibrates in its fundamental mode at 440 Hz. What is the speed of the waves on this string?
1. 220 m/s
2. 440 m/s
3. 660 m/s
4. 880 m/s
5. 1.10 km/s
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A 1.00 m string fixed at both ends vibrates in its fundamental mode at 440 Hz. What is the speed of the waves on this string?
1. 220 m/s
2. 440 m/s
3. 660 m/s
4. 880 m/s
5. 1.10 km/s
12nf
Lvnfn
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Lfv 21
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A stretched string is fixed at points 1 and 5. When it is vibrating at the second harmonic frequency, the nodes of the standing wave are at points
1. 1 and 5.
2. 1, 3, and 5.
3. 1 and 3.
4. 2 and 4.
5. 1, 2, 3, 4, and 5. 57
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A stretched string is fixed at points 1 and 5. When it is vibrating at the second harmonic frequency, the nodes of the standing wave are at points
1. 1 and 5.
2. 1, 3, and 5.
3. 1 and 3.
4. 2 and 4.
5. 1, 2, 3, 4, and 5.
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A stretched string of length L, fixed at both ends, is vibrating in its third harmonic. How far from the end of the string can the blade of a screwdriver be placed against the string without disturbing the amplitude of the vibration?
1. L/6
2. L/4
3. L/5
4. L/2
5. L/3 59
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A stretched string of length L, fixed at both ends, is vibrating in its third harmonic. How far from the end of the string can the blade of a screwdriver be placed against the string without disturbing the amplitude of the vibration?
1. L/6
2. L/4
3. L/5
4. L/2
5. L/3 60
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Drum: 2D standing waves
Mode (0,1) Mode (1,1)
Mode (2,1) Mode (0,2)Animations courtesy of Dr. Dan Russell, Kettering University 61
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Applied Natural Sciences
Thank you for your attention
Merry Christmas
Have a happy holiday and see you next year
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Summary
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Summary
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