Physics 123C Waves

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Physics 123C Waves
Lecture 10 (TM 15.4)Waves Barriers April 23,
2008 (27 Slides)

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

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Sound Wave Intensity
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Sound Intensities
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Hearing Response of the Ear
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Waves in an Open-Open Pipe
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Waves in an Open-Closed Pipe
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Pipes and Modes
Open-Open or Closed-Closed
Open-Closed
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ExampleThe Length of an Organ Pipe
An organ pipe open at both ends sounds its
2nd harmonic at a frequency of 523 Hz (one octave
above middle C). What is the length of the
pipe from sounding hole to end?
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Clicker Question 1
An open-open tube of air supports standing
waves of frequencies of 300 Hz and 400 Hz, with
no frequencies between these two. The
second harmonic (m2) of this tube has frequency
(a) 100 Hz (b) 200 Hz (c) 400 Hz
(d) 600 Hz (e) 800 Hz.
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Woodwinds vs. Strings
Many woodwind instruments are effectively an
open-closed pipe. This means they have only odd
harmonics. Their fundamental frequency will be
The vibrating string of a stringed
instrument is the equivalent of a closed-closed
pipe. This means it will have both odd and even
harmonics.Its fundamental frequency is
Note that for wind instruments, Lis the
only adjustable parameter, while for stringed
instruments, L, Ts and m can, in principle, be
varied. However, wind instruments can be played
at relatively pure harmonic frequencies, while
strings cannot.
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ExampleThe Notes of a Clarinet
A clarinet (an open-closed instrument) is 66
cm long. The speed of sound in warm air is 350
m/s. What are the frequencies of the lowest
note on a clarinet and of the next highest
harmonic?
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Reflection from a Boundary
When a traveling wave encounters a
terminating discontinuity in the medium (mR),
there is a complete negative reflection at the
discontinuity. All of the wave energy is
reflected as the negative of the incoming wave.
While the wave is inverted in displacement
direction, its amplitude is unchanged. At the
boundary point the wave and its reflection always
subtract to produce zero deflection. The
situation can be simulated as an un-terminated
string with positive and negative amplitude waves
moving in opposite directions and meeting at the
boundary.
Note that the reflected wave has the same
speed and wavelength (and energy) as the incident
wave.
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Creating Standing Waves
Plucking a Standing Wave Considering the
reflections at boundaries, it is easy to see how
string vibration occur. When a string is
plucked in the middle, waves travel in both
directions to the boundaries, where they are
reflected and propagate back and forth along the
string. The net result is a superposition of
right and left moving traveling waves that
produce a standing wave. The waves so produced
must have nodes at both boundaries.
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Standing Wave Normal Modes
Standing waves have the form D(x,t) (2a
sin kx)cos wt The two string boundary conditions
are D(x0, t) 0 and D(xL, t) 0.
Therefore, 2a sin kL 0 , which implies
that kL mp, where m is an integer. Butk
2p/l, so
The frequency f is related to the wavelength
l by f v/l, so the allowed waves on a string
of length L will have frequencies
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About Normal Modes

1. The integer m is the number of antinodes of the
   standing wave. The number of nodes of the wave
   is m 1.
2. The fundamental mode, with m 1, has wavelength
   l1 2L (not L). Half a wavelength fits on the
   string, because the spacing between nodes is l/2.
3. The frequencies of the normal modes of a string
   form an arithmetic series f, 2f, 3f, 4f,
   Therefore, the fundamental frequency f1 can be
   found as the difference between the frequencies
   of any two adjacent modes, i.e.,Df fm1 - fm
   f1.

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Clicker Question 1
A standing wave on a string vibrates as
shown. If the tension is quadrupled while
the frequency and distance between boundaries
remain the same, which diagram represents the new
vibration?
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Waves, Power, and Energy
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Reflection and Transmission
Slow to Fast Transition When a
traveling wave encounters a speed-up
discontinuity in the medium (mLgtmR), there is a
positive reflection at the discontinuity. Part
of the wave energy is reflected and part is
transmitted.
Fast to Slow Transition When a
traveling wave encounters a slow-down
discontinuity in the medium (mLltmR), there is a
negative reflection at the discontinuity. Again,
part of the wave energy is reflected and part is
transmitted.
Slow Fast Þ positive reflection
Fast Slow Þ negative reflection
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Transmission Coefficients
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ExampleTwo Soldered Wires
Two wires with different linear mass
densities are soldered end-to-end and then
stretched to a tension FT. The wave speed v1 on
the first wire is twice the wave speed v2 on the
second wire

1. If the incident wave amplitude is A, what are
   the amplitudes Ar and At of the reflected and
   transmitted waves?
2. What is linear mass density ratio m1/m2 of the
   wires?
3. What fraction of the incident average power is
   reflected at the junction, and what fraction is
   transmitted?

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Transparent Optical Media
Rather surprisingly, there are typesof
matter, solids, liquids, and gasses,that are
transparent and that transmitlight almost
unimpeded. When youconsider that such matter is
made ofatoms, electrically charged
nucleiorbited by clouds of electrically
chargedelectrons, it is quite remarkable
thatelectromagnetic radiation, the carrierof
electric fields that interact stronglywith these
charged particles, is not immediately absorbed.
Instead, within the transparent medium the
bound electrons vibrate together at the frequency
of the incoming electric field to help along
the incident light without absorbing its energy.
This usually reduces its speed through the
material as it is transmitted.
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The Index of Refraction
Light travels through transparent media at a
speed less than its speed c in vacuum. We
define the index of refraction in a transparent
medium as
Is n always greater than 1? Almost
always. There are a few media in which the phase
velocity of light waves is greater than c.
However, this super-luminal speed cannot be used
to send signals or energy at a speed greater than
c.
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Waves vs. Particles
If twopitchingmachinessimultane-ously
throwbaseballs,they willcollide
andbounce.Twoparticlescannotoccupy the same
space point at the same time.
On theother hand,if two loud-speakersmake
soundwaves atthe sametime, theywill
passthrougheach otherwithout collision. Two
waves can occupy the same space point at the same
time.
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Diffraction