Physics 123C Waves

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Physics 123C Waves
Lecture 3Traveling Waves April 1, 2005

• John G. Cramer
• Professor of Physics
• B451 PAB
• cramer_at_phys.washington.edu

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Lecture 3 Announcements

• No Labs until next week, but Tutorials start
  this week. Buy your 123Z Lab Manual at the
  Communications Copy Center before next week.
• Obtain a Clicker from the Univ. Book Store and
  register it (using the link on the 123C Syllabus
  page) as soon as possible.
• Lecture Homework 1 has been posted on the
  Tycho system. It is due at 900 PM on
  Wednesday, April 6.

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Lecture Schedule (Weeks 1-3)
We are here
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Example A Damped Pendulum

• A 500 g mass on a 50 cm long string oscillates as
  a pendulum. The amplitude of the pendulum is
  observed to decay to ½ of its initial value after
  35 s.
• What is the time constant t of the damped
  oscillator?
• At what time t1/2 will the energy of the
  system have decayed to ½ of its initial value?

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Driven Oscillations
Now, suppose we drive a damped mechanical
oscillator with an external force F(t) Fd
cos(wdt). Then the equation of motion is
The system will show the property of
resonance. The oscillation amplitude will depend
on the driving frequency wd, and will have its
maximum value when
i.e., when the system is driven at its resonant
frequency w0.
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Resonance
The solution of this equation for driven
oscillations is
When w2k/m, the first term in G vanishes
and the amplitude of the oscillation is a
maximum. This is the resonance condition.
The width of the resonance curves depends on b,
i.e., on the amount of damping. Wider curves
with smaller resonance curves correspond to more
damping and larger values of b.
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The Wave Model
We will focus on the basic properties of
waves using the wave model, which emphasizes the
aspects on wave behavior common to all waves
(e.g., water waves, sound waves, light waves,
etc.) The wave model is built around the idea of
traveling waves, wave disturbances that travel
with a well-defined speed.

• We will begin by distinguishing threetypes of
  waves
• Mechanical waves can travel onlywithin a medium,
  such as air or water.Examples sound waves,
  water waves.
• Electromagnetic waves are self-sustaining
  oscillations that require no medium and can
  travel through a vacuum.Examples radio waves,
  microwaves, light, x-rays, gamma rays, etc.
• Matter waves also can travel in vacuum and are
  the basis for quantum physics (i.e. quantum
  mechanics).Examples quantum wave functions for
  electrons, photons, atoms, etc.

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Two Types of Wave Motion
A transverse wave is a wave in which the
particles of the medium move perpendicular to the
direction of wave motion. They can be
polarized.Examples waves on a string,
electromagnetic waves.
A longitudinal wave is a wave in which the
particles of the medium move parallel to the
direction of wave motion. They cannot be
polarized.Example sound waves.
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Water Waves
Water waves are a combination of transverse
and longitudinal motion, because each particle of
water participating in the wave motion travels in
a circular path as the wave propagates. The
particles stay in the same average position as
the waves move to the right.
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Traveling Waves
Wave speed depends on the restoring forces
in the medium. It does not depend on pulse size,
shape, generation method, or distance traveled.
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Waves on a String
String density m m/L
Example
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Waves on a String
Waves on a string are produced by
transverse motion of each particle of the string,
participating in the wave motion by moving in a
vertical path as the wave propagates. Note
that although the wave moves to the right, the
individual particles of the string return to
their original positions.
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ExampleSpeed of a Wave Pulse
A L2.0 m long string with a mass of m4.0 g
is tied to a wall at one end, stretched
horizontally by a pulley 1.5 m away, then tied to
a physics book hanging from the string.
Experiments find that a wave pulse travels at
v40 m/s. What is the mass Mb of the book?
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Snapshot Graphs
We can make a series of snapshots of wave
motion showing wave displacement Dy at all
positions at a given instant. Each snapshot
shows the wave displacement vs. position for one
instant of time.
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History Graphs
We can also make a history graph of wave
motion showing the time-dependent wave
displacement Dy at a given position vs. time.
Each history graph shows the wave displacement
vs. time for a particular particle of the medium.
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Example Finding a History Graph from a Snapshot
Graph
The graph at the right shows a snapshot at t0 of
a wave moving to the right at a speed of 2.0
m/s. Draw a history graph of the same wave
for position x8 m.

• The leading edge at 4 m will require 2 s to reach
  the point of interest.
• The negative peak arrives 0.5 s later.
• The positive peak arrives 1.0 s after that.
• The trailing edge arrives 0.5 s after that.

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Clicker Question 1
The figure to the right shows the history
graph at x 4m of a wave traveling to the right
at a speed of 2 m/s. Which history graph
represents the same wave at a position of x0?
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Longitudinal Waves
Longitudinal waves (e.g., sound) are
produced in a compressible medium by longitudinal
motion of each particle of the medium,
participating in the wave motion by moving in a
horizontal path as the wave propagates. This
produces moving regions of compression and
rarefaction in the medium. Note that
although the wave moves to the right, the
individual particles return to their original
positions.
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Visualizing Longitudinal Waves
maxpos
maxneg
0
0
0
low
low
high
Note that the positions of maximum
compression (high) and rarefaction (low) do not
correspond to the positions of maximum
displacement.
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Moving Displacement
Suppose we wish to move a parabolic function
f(x) x2 so that it is centered at greater and
greater values of the independent variable x.
The we change x to x-2, x-4, etc. Similarly,
if we wish to displace some arbitrary function
f(x) by a distance d, we replace f(x) by
f(x-d). If we wish to make the displacement
increase with time, i.e., move the curve with
velocity v, we make dvt, so that the function
becomes f(x-vt).
Similarly, to displace the function in the
negative x direction, we use f(xvt).
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ExampleA Traveling Wave Pulse
Draw snapshot graphs at t0, 1 s, and 2s to
show the displacement function D(x,t)
2m3/x-(2 m/s)t2 1 m, where x is in m and t
is in s, represents a traveling wave pulse that
travels in the x direction without changing
shape. That is, it is a traveling wave with
speed v 2 m/s.
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Sinusoidal Waves
The waves produced in SHM are sinusoidal,
i.e., they can be described by a sine or cosine
function with appropriate amplitude, frequency,
and phase. The figure at the right shows a
sinusoidal wave moving along the x axis.

• The history plot and snapshot plot of
  sinusoidal waves shown show an interesting
  feature
• In the history plot (D vs. t), the distance
  between adjacent sinusoid maxima is the period T.
• In the snapshot (D vs. x), the distance between
  adjacent sinusoid maxima is the wavelength l.

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The Fundamental Relationshipof Sinusoidal Waves
Consider a sinusoidal wave moving along the
x axis with velocity v. In the time interval of
one period T, the crest of the traveling
sinusoidal wave moves forward by one wavelength l.
Note that while this relation works for
periodic waves, a non-periodic pulse will have a
definite velocity (speed of its peak) but has
neither a frequency nor a wavelength.
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Clicker Question 2
What is the frequency of this traveling wave?

• 0.10 Hz
• 0.20 Hz
• 2.0 Hz
• 5.0 Hz
• 10.0 Hz.

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The Mathematicsof Sinusoidal Waves
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Angular Frequency wand Wave Number k
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ExampleAnalyzing a Sinusoidal Wave

• A sinusoidal wave with amplitude A 1.0 m at and
  frequency f 100 Hz travels at v 200 m/s in the
  x direction. At t0, the point at x1.0 m is on
  the crest of the wave.
• Find A, v, l, k, f, w, T, and f0 for this wave.
• Write the wave equation.
• Draw a snapshot graph at t-0.

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Velocity of Waves on a String
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ExampleGenerating a Sinusoidal Wave

• A very long string with m 2.0 g/m is stretched
  along the x axis with a tension of Ts 5.0 N.
  At position x0 it is tied to a 100 Hz simple
  harmonic oscillator that vibrates perpendicular
  to the string with an amplitude of 2.0 mm. The
  oscillator is at its maximum positive
  displacement at t0.
• Write the displacement equation of waves on the
  string.
• At t 5.0 ms, what is the strings displacement
  at a point 2.7 m from the oscillator?

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End of Lecture 3

• Before the next lecture, read Knight, Chapters
  20.4 through 20.6
• Lecture Homework 1 has been posted on the Tycho
  system and is due at 900 PM on Wednesday, April
  6.