physics 116 - university of washington
TRANSCRIPT
•! Guest lecturer today: Michael Dziomba •! Wilkes will be back at 2:45 today, for office hour
(until 3:15)
Announcements
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Today
Lecture Schedule (up to exam 1)
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Waves on strings •! End effects:
–! If end of rope is fixed in wall and can’t move,
•! reaction force from wall (Newton’s 3rd Law) opposes motion: upward pulse coming toward wall becomes downward pulse leaving wall
•! Pulse is inverted
–! If end of rope is unconstrained and can move vertically,
•! no reaction force
•! pulse is reflected without inversion
•! Wave speed on a rope or string depends on
–! Tension in string : if F=0 wave does not propagate
–! Mass of string : really, mass per unit length
•! Speed of wave on a string
Justify this result with dimensional analysis:
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Example •! Rope has 92 N tension and is 12 m long
•! Pulse takes 0.45 s to travel length of rope
•! What is total mass of rope?
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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
A = 2 m
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Space and time pictures of waves
time, seconds
(Here: period T=1sec)
variation with
time at a fixed point in space
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Last 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)
•! Wavelength ! = length of one full cycle (distance between peaks)
•! Amplitude A = maximum displacement (height)
! = 1 m
A = 2 m
snapshot =
picture of rope, frozen at one
instant of time:
configuration
in space at a fixed point in
time
At one instant: a snapshot in time
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(Here: wavelength ! =1 m)
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Harmonic functions to describe wave motion •! To describe waves, we need to give {disturbance} = y (x, t)
•! From our “snapshot” plot we see
•! From our “standing in one place” plot, we see that the position of the peak that was at x=0 at t=0, has moved to the right after time t, by a distance
•! So the value of the wave function at time t is equal to the value at time t=0 for any combination of x and t such that
•! So we can describe the wave with the “harmonic function” *
* The circular functions sin/cos are “harmonic” because they can describe sound waves that are multiples of some base frequency – next time…
x !t
T" = 0
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Example •! A transverse wave has ! = 2.6 m, and moves in the + x direction
(“to the right”) with speed 14.3 m/s
•! Its amplitude is 0.11 m and it has y=0.11 m at t=0
•! Give an equation describing y(x,t) for this wave
General form for a wave moving in +x direction* is (for y=A at t=0 *)
For this wave,
So
* What if it were moving to the left (-x)?
what if it had y=0 at t=0, with y increasing at that time?
what if it had y= -1 at t=0, with y increasing?
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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
•! Sound = compression wave in material medium (air, water, iron)
•! Sound speed depends on material properties and density (so,
temperature, humidity etc)
www.kettering.edu/~drussell/Demos/waves/wavemotion.html
Sound waves Last time:
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Speed of sound •! Speed of sound c is about 343 m/s in air (depends on air density,
temperature and humidity) = 1235 km/hr = 770 mph
–! So sound travels 1 mile in about 5 sec (lightning and thunder)
At 0°C, c = 331 m/s
At 15°C, c = 340 m/s
At 20°C, c = 343 m/s
At 25°C, c = 346 m/s
•! Speed is faster in denser materials:
•! Example:
Pirate sees cannon on pursuing
ship flash, and counts 8 seconds
before he hears the boom – how
far away is the Royal Navy?
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Sound frequency; pitch and musical tones •! Frequency ranges
–! Audible – nominally 20 Hz to 20 kHz (actual range is closer to 50Hz-15kHz)
–! Infrasonic - below audible (below about 0.1 Hz we call it “vibration” !)
–! Ultrasonic - Above 20 kHz
•! Speed of sound does not vary much with f
–! If v depended on f, sound signals would change significantly depending upon
how far away you are
•! This is called “dispersion”
–! Small f dependence can be observed, for example in undersea sound
transmission
•! A pulse with many frequencies in it will spread out in time as it travels
•! Pitch will vary – pulse becomes a “chirp”
•! Perception of sound –! Pitch = perceived frequency of sound
–! Associated with musical tones by our brain
–! JND = “just noticeable difference” in frequency ~0.4 Hz
–! Harmonic scales: eg in western music, “A above middle C” = 440 Hz, next A
(one octave higher pitch) = 880 Hz - octave = doubling of base frequency)
–! “Equal temperament” scale: 12 tones per octave, each is 1.06 f of previous
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