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°

Today, we discuss waves

Can you think of some

examples of waves?

What kinds of waves do we

experience while I play guitar?

When I was your age…

College

High School

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 ==

What do you think are the most dangerous kinds of waves?

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

How to change the pitch (f) of your voice

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)