Physics 116
Lecture 5
Waves
Oct 6, 2011
R. J. Wilkes
Email: [email protected]
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Lecture Schedule
(up to exam 1)
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Today
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Driven oscillations and resonance
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Any system that displays SHM has natural frequency = f when it
is displaced once and left alone
Driven oscillator has external agent displacing it with some
period (not necessarily same as its natural T)
If driving force has same f as system’s natural f, energy gets
transferred into the system very efficiently
– Often disastrously, if undamped!
– This is called resonance
resonant frequency = natural f
Tacoma Narrows bridge, 1940:
Classic example of driven
oscillations
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Tacoma bridge collapse
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First Tacoma Narrows bridge (“Galloping Gertie”)
– Construction started September,1938 (WPA funded: 6.4 M$!)
– Opened July 1, 1940
– Collapsed November 7, 1940
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Videos:
http://www.youtube.com/watch?v=P0Fi1VcbpAI
(more detail: http://www.youtube.com/watch?v=j-zczJXSxnw )
Cause of failure: (see http://www.wsdot.wa.gov/tnbhistory/Machine/machine3.htm)
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Vibration was due to aeroelastic torsional fluttering
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Flutter velocity = wind speed at which fluttering begins
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Wind injects more energy than the flexing of the structure can dissipate: damping is
ineffective, exponential increase in A
Occurs even with relatively low-speed, steady winds
Now designers make sure flutter v >> max expected wind v !
Interesting example of physics misinformation: textbooks say
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“Due to resonance”
Driven by “vortex shedding”
Karman vortex shedding:
Air density behind a cylinder as air blows over it
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Waves
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Many physical phenomena involve wave motion
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ripples on a rope
compression waves in a slinky
water waves on the ocean
sound waves in the air
Light waves in intergalactic space
Actually, quantum theory says everything is a wave, sometimes
– more on that later...
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As before: first step is to describe many different kinds of waves,
in unified and unambiguous terms
Example:
wave on a rope
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Take a
snapshot, or...
Physics 116
watch it move
past you
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Space and time pictures of waves
We’ve been through this already with oscillations…
• We could stand at one place and watch wave move past us vs time
• Graph of displacement vs time
• Period T = time for one cycle ("wavelength" in time units) to go past
• Frequency f = cycles passing per second (hertz, Hz) = 1/T
– This wave has 1 cycle in 1 s, so T = 1 s
– Amplitude is 2 meters
T= 1 s
2
variation with
time at a fixed
1
point in space
0
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
A=2m
-1
-2
time, seconds
Distance,
meters
(Here: period T=1sec)
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At one instant: a snapshot in time
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Previous picture was graph of displacement vs time at one location
Here: Picture of rope at one instant of time (say, t=0):
– We see rope’s displacement vs position along rope (y vs x)
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Wavelength λ = length of one full cycle (distance between peaks)
Amplitude A = maximum displacement (height)
λ=1m
2
snapshot =
1
picture of rope,
frozen at one
instant of time:
configuration
in space at a
fixed point in
time
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0
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
A=2m
-1
-2
(Here: wavelength λ =1 m)
Distance, meters
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Speed of wave, frequency, and wavelength
Wave speed connects the space and time pictures of wave motion
• Different kinds of waves move (propagate) with varying speeds
– Speed determines relationship between wavelength (from snapshot
at one t) and period or frequency (counting waves at one spot)
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Relationship between frequency, speed and wavelength:
f ·λ = v
f is frequency in cycles per second (Hz)
λ is wavelength (meters)
v is speed of propagation of wave (m/s)
So, for example
What is wavelength of signal from KPLU-FM (88.5 MHz)
λ = v /f = (speed of light)/88,500,000 Hz
= (3x108 m/s) / (8.85x107 cycles/s)
= 3.4 m
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Longitudinal waves
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Sound waves are an example of longitudinal waves
– Disturbance consists of periodic changes in density of the medium
– At any point, material is alternately compressed and rarified
– Compression peaks propagate through the medium
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Sound = compression wave in material medium (air, water, iron)
www.kettering.edu/~drussell/Demos/waves/wavemotion.html
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Sound speed depends on material properties and density (so,
temperature, humidity etc)
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Kinds of waves
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Waves move in both space and time:
– Wave = repetitive disturbance that propagates in space
• Transverse waves: on rope = displacements of material
• Longitudinal waves: sound, slinky = compression of material
These waves propagate in a material medium (water, rope, air, spring)
• Light waves = changes in electric and magnetic fields
– Is there a medium in which light waves are disturbances?
• Luminiferous ether: massless substance that fills all space (?)
• Important implication: coordinate system in which ether is at rest is
the fundamental coordinate system of the Universe ! !
– If so, Earth's motion through ether should cause light speed to change
• A. Michelson, 1890s: no difference in light speed in any direction
– Measurements were far too good to dismiss: there is no ether
Electromagnetic waves
have no medium
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Q: then, what is the rest
frame of the Universe?
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Waves in water
• Waves on water (or any surface) are a special case
• Water inside waves moves in circles
– Motion only near surface
– Submarines do not notice storms!
– Imagine we can make a video of “particles of water”
www.kettering.edu/~drussell/Demos/waves/wavemotion.html
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Water waves
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Surf is caused by interaction of surface waves with beach
– In deep ocean, waves have small amplitude
– At shore, their amplitude gets larger
kingfish.coastal.edu/physics/projects/2001_Spring/molnar/OceanofW.htm
• Near shore, friction with bottom slows wave so:
– λ gets shorter (because f remains constant: λ=v/f)
– shallow-water speed (for depth D in m) is approximately v = gD
– Amplitude A gets bigger near shore: water piles up, and waves break
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Example
shore
• Typical surf has period 10 sec and λ = 150m Deep water
⎛ 1 ⎞
What is wave speed?
fλ =v=⎜
⎟150 m = 15 m / s
⎝ 10 s ⎠
• Tsunami (tidal wave) moves with speed 750 km/hr and
wavelength 310 km in mid-ocean, where depth is 5000 m
v ( 750 km/h ) ⎛ 1 h ⎞
What is its frequency?
−4
f = =
6.7
10
Hz
=
×
⎜
⎟
310 km ⎝ 3600 s ⎠
λ
• If it reaches shallow water near shore, its frequency stays the
same but its speed gets slower, and λ gets shorter:
– Near shore where water is 10m deep, Tsunami described above has speed
v = gD =
for
( 9.8m / s ) (10m ) ≈ 10m / s
2
f = 6.7 × 10−4 Hz → λ =
v ⎛ 10m / s ⎞
=⎜
⎟ = 15 km
f ⎝ 6.7 × 10−4 Hz ⎠
– All the water in a shallow wave 310 km long gets piled up into 15 km wave
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Clicker channel programming
• Press and hold down-arrow
• When light flashes, press 02 (zero, then 2)
• When light flashes again, press downarrow
Pop quiz #1
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We’ll wait 2 minutes for everyone to answer each question
3. Which of the following is an example of a
transverse wave?
A. Sound wave in air
B. Water waves at Waikiki Beach
C. Wave on a plucked guitar string
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