PHY138 – Waves Lecture 9
Quarter Review, including:
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Error Propagation, Standard Deviation
Simple Harmonic Motion: Force, Energy
Mass on spring / Pendulum
Damped Oscillations, Resonance
Traveling Waves, Power and Intensity
Standing Waves, Interference, Beats
Ray Model of Light, Ray-Tracing
Reflection, Refraction of Light
Thin Lens equation; Magnification
Tomorrow evening, 6:00 PM
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It is mandatory that you go to the room
assigned to your tutorial group.
You should have no communication device
(phone, pager, etc.) within your reach or
field of vision during the test.
The test has nine equally weighted
multiple-choice questions (7 marks each).
The test has one multi-part problem
counting for 37 marks; you must show your
work.
Don’t forget…
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Your student card.
A non-programmable calculator without
text storage and communication capability.
A single, 2-sided original, handwritten 22
× 28 cm sheet of paper on which you have
written anything you wish. We will supply
any numerical constants you might need.
A dark-black, soft-lead 2B or 2HB pencil
with an eraser.
Some more words to the wise…
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A good aid-sheet is well organized, easy to
read, and contains all the major equations
from the assigned sections from the reading.
Copies of detailed specific problem solutions
are unlikely to help.
Be ready to think; get a good night’s sleep
tonight.
Keep in mind: Your best 3 out of 4 tests will
count for 30% of your mark in the course.
f
What will happen to the rays emerging to the
right of the lens if the face is moved a little
closer to the lens?
A. They will remain parallel.
B. They will diverge (spread out).
C. They will converge (toward a focus).
f
What will happen to the rays emerging to the
right of the lens if the face is moved a little
further away from the lens?
A. They will remain parallel.
B. They will diverge (spread out).
C. They will converge (toward a focus).
Propagation of Errors: good to have on
aid sheet!!
z = A x
Δz = A Δx
Standard Deviation and error of the
mean, good to have on your aid sheet!
Mean:
Estimated standard
deviation:
Error of the mean:
xavg
1

N
 est 
N
x
i 1
i
1 N
2
(
x

x
)

i
avg
N  1 x 1
( xavg ) 
 est
N
Mass on Spring versus Pendulum
Condition for
S.H.M.
Natural frequency
[rad/s]
Period
Mass on a
Spring
Pendulum
Small oscillations
(spring obeys
Hooke’s Law)
Small angles
(sinθ ≈ θ)
k

m
m
T  2
k

g
L
T  2
L
g
14.7 Damped Oscillations
x  Ae
 bt / 2 m
cos(t  0 )
Snapshot Graph
History Graph
Sinusoidal Wave Snapshot Graph
k = 2π/λ is the wave number
Sinusoidal Wave History Graph
ω=2π/T is the angular frequency
Sound Waves can be described either by the
longitudinal displacement of the individual
particles, or by the air or fluid pressure.
D( xeq , t )  x  xeq  A sin( kxeq  t  0 )
D( x, t )  P  Patm  Pmax cos(kx  t  0 )
Electric and Magnetic
fields, when oscillated,
can create waves which
carry energy. At certain
frequencies, we see
electro-magnetic waves
as Light.
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E( x, t )  E0 sin( kx  t  0 )
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B( x, t )  B0 sin( kx  t  0 )
Power and Intensity
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The Power, P, of any wave source is how
much energy per second is radiated as
waves [units = Watts]
The Intensity, I, is the energy rate per area.
This determines how loud (sound) or bright
(light) the wave is.
I=P/a, where a is an area perpendicular to the
wave direction.
At a distance r from a small source, the
intensity is I=P/(4πr2)
Doppler Effect
Principle of Superposition
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If two or more waves combine at a given
point, the resulting disturbance is the sum
of the disturbances of the individual
waves.
Two traveling waves can pass through
each other without being destroyed or
even altered!
Standing Wave:
The superposition of two 1-D sinusoidal
waves traveling in opposite directions.
Harmonic frequencies of Standing Waves
Transverse standing wave on a string
clamped at both ends: there are nodes in
displacement at both ends.
v
fm  m
2L
(m  1,2,3,...)
Standing sound wave in a tube open at both
ends: there are nodes in pressure both ends.
v
fm  m
2L
(m  1,2,3,...)
Wave Interference
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Two waves moving in the same direction
with the same amplitude and same
frequency form a new wave with
amplitude:
  
A  2a cos

 2 
where a is the amplitude of either of
the individual waves, and  is their
phase difference.
Beat frequency
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Beats are loud sounds separated by soft sounds
The beat frequency is the difference of the
frequencies of the two waves that are being
added:
beat
mod
1
2
f
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2f
 f f
The frequency of the actual sound is the
average of the frequencies of the two waves
that are being added:
f avg
f1  f 2

2
The Law of Reflection
1  1
Snell’s Law of Refraction
n1 sin 1  n2 sin 2
Total Internal Reflection
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Can only occur when n2<n1
θc = critical angle.
When θ1 ≥ θc, no light is transmitted
through the boundary; 100% reflection
n2
sin  c 
n1
Virtual Image Formation by
Reflection
s'  s
Virtual Image Formation by Refraction
n2
s'  s
n1
Diverging rays through a Converging Lens
Focal length, f
s
s’
1 1 1
Thin Lens Equation:  
s s' f
Magnification
h
M 
h
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s
M 
s
The absolute magnitude of the magnification |M | is
defined to be the ratio of image height to object
height.
A positive value of M indicates that the image is
upright relative to the object. A negative value of M
indicates the image is inverted relative to the object.
Note that when s and s’ are both positive, M is
negative.