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Graphical
Representation
of MotionWhen x is a maximum or
minimum, velocity is zero
When x is zero, the speed is
a maximum (slope of x)
Acceleration vs. time is the
slope the of velocity graph.
When x is max in the
positive direction, a is max
in the negative direction
x = A cos ωωωωt
a = -Aωωωω2 cos ωωωωt
v = -A ωωωω sin ωωωωt
Calculator Warning!
•What are the units of ω t ?
Thus, your calculator will either need to
be in radians to give the correct answer,
or you need to convert ω t to degrees.
tAx ωcos=tAv ωω sin−=
tAa ωω cos2−=
2π radians = 360°
The World is Full of Waves
• Sound waves
• Waves on strings
• Seismic waves
• Electromagnetic waves (visible light, radio
waves, television signals, x-rays)
The source
of waves
• All of these waves are caused by some vibration
(a plucked string or vibration of electrons in an antenna)
• So we describe waves by simple harmonic motion
• Mechanical waves require
–A medium that can be disturbed (water, air, string)
• Light waves (like sunlight) do not require a
medium (can travel through space; starlight)
Main Ideas in Class Today
After class today, you should be able to:
• Identify different types of waves
• Calculate wave velocity, period and frequency.
• Calculate tension or velocity for a wave on a string.
• Understand wave interference and some applications of it
Extra Practice: 13.49, 13.51, 13.53, 13.57, 13.61
What does it look like this disturbance is doing?
When you flip one end of a long rope that is
under tension and fixed at one end, the pulse
travels at a fixed speed
Traveling Waves
http://phet.colorado.edu/sims/wave-
on-a-string/wave-on-a-string_en.html
What does the wave do after it
hits the wall (section 13.11)?
Types of Waves – Transverse
In a transverse wave, each element that is
disturbed moves in a direction perpendicular to
the wave motion
Can you think of an example?
Transverse: Doing “The Wave”
Synchronized standing and sitting is
an example of a transverse wave.
People move up and down but wave
moves horizontally.
It takes ~ a few dozen fans leaping to their
feet with their arms up to trigger a wave.
It usually travels at a speed of ~20 seats/s.
Using a Sine Curve to Describe a Wave
• The brown curve is a
“snapshot” of a wave at t=0
• The blue curve is later in
time
• The high points are crests
• The low points are troughs
• The distance ∆x traveled by
the wave is = v t (Ch. 2)
Sinusoidal curves
If you move your hand up and down to
create a transverse wave, what would
you do to create a longitudinal wave?https://www.youtube.com/watch?v=aguCWnbRETU
Transverse
Longitudinal
Red spots here look
closer together or
more dense
Perpendicular
disturbance Transverse
Types of Waves – Longitudinal
In a longitudinal wave, the elements of the
medium undergo displacements parallel to the
motion of the wave
Also called a compression wave
Transverse
Longitudinal
Longitudinal Wave Represented as a
Sine Curve
• A longitudinal wave can also be represented as a sine curve
• Crests = Compressions, Troughs = stretches
• Also called density waves or pressure waves
Mechanical Wave Definitions
Amplitude: maximum height of crest or depth of trough (not both!)
Wavelength (λ): distance over which wave repeats itself
Frequency (f): Number of crests that pass a given point per unit time
Wave velocity: a crest travels one wavelength in one period T:
Tv
λ=
Note: this is the velocity of the wave, not of the particles in the medium
fλ=
6 m/s 2 m
If the water wave below has a velocity of 6 m/s, how many times per second does a crest pass by the sitting bird?
fT
v λλ
==
A. 0.33 times per second
B. 0.5 times per second
C. 2 times per second
D. 3 times per second
E. 12 times per second
Q124
Waves on Strings
What happens when I
tune the guitar?
fv λ=F = tension in the string
µ = linear density = mass/Length
If I wanted to change the sound of my
guitar string, what variable would I be
changing?
µ
F=
How could I change f ?
String Wave
ExampleHow could we find the tension in
one of my guitar strings?
µλ
Ffv ==
Side view
Length
The 6 strings on a guitar have
frequencies of 82 Hz, 110 Hz, 147
Hz, 196 Hz, 247 Hz, 330 Hz.
If we had
the guitar
string, how
could we
figure this
out?
If you reduce by half the wavelength λof a wave on a string, what happens to
the wave speed v and the wave
frequency f?
A. v is doubled and f is doubled.
B. v is doubled and f is unchanged.
C. v is unchanged and f is halved.
D. v is unchanged and f is doubled.
E. v is halved and f is unchanged.
Q125
µλ
Ffv ==
A certain FM radio station broadcasts
music at a frequency of 101.9 MHz.
Find the wave’s period and its wavelength.
(Radio waves are electromagnetic waves that
travel at the speed of light, 3.00 x 108 m/s.
How long would
it take these
waves to get to
you if you were
1 mile away
from the tower?
Interference of Waves
• Traveling waves can meet and pass through each other without being destroyed or even altered
• Waves obey the Superposition Principle
– Meaning when 2 or more waves encounter each other, the resulting wave is found by adding together the displacements of the individual waves
Whenever a
crest coincides
with a trough,
the water
surface is
flattened
Constructive Interference
• Two waves (a and b) have the same frequency and crests at the same times
– Thus, they are in phase
• If you add them together, the combined wave (c) has the same frequency and a greater amplitude
Constructive Interference in a String
• Two pulses are traveling in opposite directions
• The net displacement when they overlap is the sum of the displacements of the pulses
• Note that the pulses are unchanged after the interference
Destructive Interference in a
String
• The net displacement when they overlap is decreased since the displacements of the pulses subtract
• Note that the pulses are unchanged after the interference
Noise-Canceling Headphones
Noise-canceling headphones use a microphone that
listens for noise and a speaker that produces the
same noise but out of phase (cancellation by
destructive interference)
External Noise
Canceling Sound
Radar misleading
fighter jets
• A French fighter plane called the Rafale uses
destructive interference to avoid Radar.
• Radar (radio detection and ranging) can detect
objects by sending electromagnetic waves out
and measuring the time and frequency of the
reflected wave (~ Xbox Kinect).
• When the jet receives an incoming wave, it sends
out the direct opposite pattern of the wave.
A spring stretches 3.0 cm from its relaxed length
when a force of 7.5 N is applied.
A 0.50 kg object rests on a frictionless horizontal
surface and is attached to the free end of that
spring. The object is stretched x = 5.0 cm and
released from rest at t = 0.
a) What is the force constant of the spring?
b) What are the angular frequency, frequency, and period?
c) What is the total energy of the system?
d) What are the max velocity and acceleration?
e) Find the displacement, velocity and acceleration at t=0.5 s.
6.00
3.00
-6.00
-3.00
Determine the amplitude, period, and frequency of this
wave. Find it’s velocity and acceleration at a specific time.
Damped Oscillations
Why does a child stop swinging
if not continuously pushed?
When work is done by a dissipative force (friction or air
resistance), not all of the mechanical energy is conserved.
This means not all of her potential energy
at the top of each swing is converted into
kinetic energy so her next swing is not
as high.
The period of oscillations stays the same.
The amplitude decreases with time.
Two waves can interfere:
A. Only when traveling in the same direction
B. Only when the frequencies are the same
C. Only when both are sinusoidal
D. Only when the phase difference is constant
E. None of the above
Q126
The Donald Duck Effect
Sounding ~2.7 octaves higher makes you sound a bit
like Donald Duck.
It is not dangerous to briefly breathe helium. At
worst, you'll pass out. Like anything, don’t overdo it.
If you exhale
before breathing in,
you’ll get more
helium.
And the rockets' red glare, the bombs bursting in air,
Gave proof through the night that our flag was still there.
O say, does that star-spangled banner yet wave
O'er the land of the free and the home of the brave?
Clicker Answers1=D, 2=E, 3=D, 4=B, 5=C, 6=A, 7=C, 8=E, 9=A, 10=C, 11=B,
12=C, 13=E, 14=A, 15=B, 16=C, 17=B, 18=D, 19=A, 20=C, 21=B,
22=B (hard), 23=A, 25=A, 26=B, 27=E, 28=C, 29=C, 30=D, 31=C,
32=D, 33=C, 34=B, 35=D, 36=A, 37=C, 38=B, 39=E, 40=B, 41=B,
42=C, 43=A, 44=B, 45=A, 46=A, 47=C, 48=C, 49=A, 50=C, 51=E,
52=D, 53=E, 54=D, 55=A, 56=D, 57=A, 58=A, 59=B, 60=B, 61=C,
62=D, 63=B, 64=A, 65=B, 66=B, 67=A, 68=B, 69=A, 70=B,
71=A, 72=A, 73=D, 74=A, 75=B, 76=C, 77=B, 78=B, 79=B, 80=B,
81=B, 82=C, 83=B, 84=A, 85=D, 86=B, 87=B, 88=B, 89=B, 90=C,
91=C, 92=A, 93=D, 94=2, 95=2, 96=D, 97=B, 98=B, 99=B, 100=A,
101=C, 102=B, 103=A, 104=C, 105=A, 106=B, 107=A, 108=B,
109=B, 110=B
Ch.9A: 111=D, 112=B, 113=A, 114=D, 115=D (sorry for the skip)
Ch.13A: 120=D, 121=A, 122=B, 123=D
Ch.13B: 124=D, 125=D, 126=E (waves can always interfere)
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