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Ch. 11: waves
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11. Waves
Waves and Energy Transport
What determines speed of wave?
Mathematical form of wave: example
demos
Superposition of waves
HITT quiz
Waves in nature
Energy transport
When a stone is dropped into a pond, the water is disturbed from its equilibrium
positions as the wave passes; it returns to its equilibrium position after the wave
has passed.
Intensity is a measure of the amount of energy/sec that passes through a square
meter of area perpendicular to the wave’s direction of travel.
I=
The water moves up and
down as the disturbance
moves outward.
Power
P
=
4π r 2 4π r 2
Intensity has units of
watts/m2 .
Sphere with
radius r
Source PS
r
This is an inverse square law. The intensity drops as the inverse square of the
distance from the source. (Light sources appear dimmer the farther away from
them you are.)
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Example
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Wave on a string
Example: At the location of the Earth’s upper atmosphere, the intensity of the
Sun’s light is 1400 W/m2. What is the intensity of the Sun’s light at the orbit of
the planet Mercury?
Ie =
Psun
4π res2
Im =
d
Psun
4π rms2
v
Divide one equation by the other:
Psun
2
2
2
 r   1.50 × 1011 m 
I m 4π rms
 = 6.57
=
=  es  = 
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P
Ie
sun
 rms   5.85 ×10 m 
4π res2
I m = 6.57 I e = 9200 W/m 2
A transverse wave, i.e. displacement, d is perpendicular to
direction of wave motion, v!
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Wave on a string
A wave traveling on this string will have a speed of
Example
v=
F
Example (text problem 11.8): When the tension in a cord is 75.0 N, the wave
speed is 140 m/s. What is the linear mass density of the cord?
µ
• Speed of Propagation: The speed of any mechanical wave depends on both
µ=
(wave speed)
F
µ
Solving for the linear mass density:
the inertial property of the medium (stores kinetic energy) and the elastic
property (stores potential energy).
elastic
v=
inertial
v=
The speed of a wave on a string is
where F is the force applied to the string (tension) and µ is the mass/unit
length of the string (linear mass density).
F
75.0 N
=
= 3.8 ×10 −3 kg/m
v 2 (140 m/s )2
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Transverse vs. longitudinal waves
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Special case: periodic waves
(kx − ωt )
Units
is called the phase.
Wave speed
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General form of a wave
Example
Example (text problem 11.21): A wave on a string has an equation:
Convince yourself that
(before we used
moves to left.
y ( x, t ) = (4.00 mm )sin ((600 rad/sec ) t − (6.00 rad/m ) x )
which moves to right )
Compare this to
y ( x, t ) = A sin (ωt − kx ) = − A sin( kx − ωt ) = A sin( kx − ωt + π )
(a) What is the amplitude of the wave?
A = 4.00 mm
(b) What is the wavelength?
The wave number k is 6.00 rad/m.
λ=
Example continued:
2π
ω
=
14
Example continued (SHM):
(c) What is the period?
T=
2π
2π
=
= 1.05 m
k
6.00 rad/m
(f) What is the maximum speed of a point on the string, take x=0
y (0, t ) = (4.00 mm )sin ((600 rad/sec)t ) = A sin ωt
2π
= 1.05 ×10− 2 sec
600 rad/sec
SHM
vmax = Aω = 4.00mm × 600 rad/sec = 2.4 m/ sec
(d) What is the wave speed?
(g) What is maximum acceleration of a point on the string?
ω 600 rad/sec
 λ 
v = λf = 
= 100 m/s
(2πf ) = =
k 6.00 rad/m
 2π 
amax = Aω 2 = 1440 m/sec 2
(e) What direction is the wave traveling.
Along the +x direction.
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