preclass notes: chapter 14, sections 14.1- 14 - u of t...
TRANSCRIPT
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PreClass Notes: Chapter 14, Sections 14.1-
14.4
• From Essential University Physics 3rd Edition
• by Richard Wolfson, Middlebury College
• ©2016 by Pearson Education, Inc.
• Narration and extra little notes by Jason Harlow,
University of Toronto
• This video is meant for University of Toronto
students taking PHY131.
Outline
A wave involves“a disturbance
that moves or propagates through
space. The disturbance carries
energy, but not matter. Air doesn’t
move from your mouth to a
listeners ear, but sound energy
does.”– R.Wolfson
• Wave Properties
• The Wave Equation
• Waves on a string
• Wave Power and Intensity
• Sound Waves
[Image of waves in a field of grass from https://threeriversdeep.wordpress.com/2014/07/ ]
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What is a Wave?
• A wave is a traveling disturbance that transports energy but not matter.
• Mechanical waves are disturbances of a material medium.
• The medium moves briefly as the wave goes by, but the medium itself isn't transported any distance.
• Electromagnetic waves,including light, do not require a medium.
[image from https://webspace.utexas.edu/cokerwr/www/index.html/waves.html ©1999 by Daniel A. Russell ]
Transverse Waves
• In a transverse wave, the disturbance is perpendicular
to the wave motion.
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Longitudinal Waves
• In a longitudinal wave, the disturbance is parallel to the
wave motion.
Sound is an
longitudinal wave.
Properties of Continuous Waves
• Wavelength λ is the
distance over which
a wave repeats in
space.
• Period T is the time
for a complete
oscillation of the
wave at a fixed
position.
• Frequency f is the
number of wave
cycles per unit time:
f = 1/T
• Amplitude A is the
maximum value of
the wave
disturbance.
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Wave Speed
• Wave speed is the
rate at which the
wave propagates.
• Wave speed,
wavelength, period,
and frequency are
related:
Simple Harmonic Waves
• A simple harmonic wave is described by a sinusoidal
function of space and time:
• y measures the wave disturbance at position x and time t.
• k = 2π/λ is the wave number, a measure of the rate at
which the wave varies in space.
• ω = 2π/T is the angular frequency, a measure of the rate
at which the wave varies in time.
• The ± is written so we can describe a wave going in the
+x direction (− sign) or the −x direction (+ sign).
• The wave speed is:
y x,t( ) = Acos kx ±wt( )
𝑣 = 𝜆𝑓 =𝜔
𝑘
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Got It? [1 of 5]
• These two waves have the same speed. Which
has the greater amplitude?
A
B
Got It? [2 of 5]
• These two waves have the same speed. Which
has the greater wavelength?
A
B
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Got It? [3 of 5]
• These two waves have the same speed. Which
has the greater period?
A
B
Got It? [4 of 5]
• These two waves have the same speed. Which
has the greater wave number?
A
B
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Got It? [5 of 5]
• These two waves have the same speed. Which
has the greater frequency?
A
B
The Wave Equation
• Many different types of media can support the
propagation of waves.
– Analysis of disturbances in different media often
result in the following equation (which is known as the
one-dimensional wave equation):
– This equation relates the space and time derivatives
of the disturbed quantity, y. Note that v is the wave
speed, where:
– It can be shown that any function of the form
satisfies the wave equation.
¶2 y
¶x2=
1
v2
¶2 y
¶t2
v =l
T= l f
y = f (x ± vt)
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Waves on Strings
• On strings, fibers, long springs,
cables, wires, etc., tension
provides the restoring force that
helps transverse waves propagate.
• Newton's second law gives
where F is the tension and μ
is the mass per unit length.
• The speed of such waves is
2Fq =mv2
R=
2qRmv2
R= 2qmv2
v =F
m
Wave Power
• The power carried by a wave is proportional to
the wave speed and to the square of the wave
amplitude.
– For waves on a string, the average
power is:
P = 1
2 mw2A
2v
• Other types of waves have similar equations for
average power, always dependent on the
square of the wave amplitude, 𝐴2.
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Wave Intensity
• Wave intensity is the
power crossing
a unit perpendicular area.
– In a plane wave, the
intensity remains
constant.
– A spherical wave
spreads in three
dimensions, so its
intensity drops
as the inverse square
of the distance
from its source:
I =P
A=
P
4p r2
Sound
• Sound waves are
longitudinal
mechanical waves
that propagate
through gases,
liquids, and solids.
• Sound waves in air
involve small
changes in air
pressure and
density, associated
with back-and-forth
motion of the air as
the wave passes.
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The Speed of Sound
• The speed of sound in a gas depends on the
background pressure P, density ρ, and a factor γ
that is determined by the number of atoms that
form a typical gas molecule:
v =g P
r
• The speed of sound is determined by the air and
is not dependent upon the amplitude, frequency,
or wavelength of the sound itself.
• In dry air at atmospheric pressure and room
temperature, the speed of sound is v = 343 m/s.
Human Hearing and the Decibel
• The human ear responds to a broad range of sound
intensities and frequencies
– The audible range extends from about 20 Hz to 20
kHz in frequency and over 12 orders of magnitude in
intensity
– The sound intensity level β is measured in decibels:
– The decibel is a logarithmic unit based on a
comparison to the nominal threshold of hearing:
b = 10 log I I0( )
I0 = 10-12 W / m2
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Human Hearing and the Decibel