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Lecture 19 Purdue University, Physics 220 1

Lecture 19

Waves

PHYSICS 220


What is a Wave

• A wave is a disturbance that travels away from its

source and carries energy.

• A wave can transmit energy from one point to

another without transporting any matter between

the two points.

• Examples:

– Stadium waves (people move up & down)

– Water waves (water moves up & down)

– Sound waves (air moves back & forth)

– Seismic waves (earth moving up & down)

– Electromagnetic waves (what moves ??)


Why are Waves Important?

• Transport energy from one place to another

– Electromagnetic waves transport energy

(electromagnetic energy in the form of light) from the

Sun to the Earth.

– Sound waves transport energy from speakers to our

ear drums.

• In waves, energy is transported over large

distances, but matter is not.

Power

• The power carried by a wave equals the energy

emitted by the source (and carried away by the

wave) per unit time. [watts]=[joules/second]



Intensity

• Average power per unit area carried by the wave

past a surface which is perpendicular to the

direction of propagation of the wave.

• For spherical waves, the intensity decreases with

distance:

• Unit: W/m2

• I ! A2

I =P

4!r2

Plane wave

• Light produced by laser good example

• Intensity does not change with distance from the

source.


Demo 1S - 16

Circular waves --- water waves

Longitudinal waves --- sound

Transverse -- light

For frequency watch

one point

For wavelength

look at one time

For velocity watch

one peak


• Longitudinal: The medium oscillates in the same

direction as the wave is moving.

– Sound

– Slinky

• Transverse: The medium oscillates perpendicular to the

direction the wave is moving.

– Water (more or less)

– Slinky

Types of Waves


•http://www.youtube.com/watch?v=UHcse1jJAto&feature=

related

•http://www.youtube.com/watch?v=aguCWnbRETU&featur

e=related

•http://www.youtube.com/watch?v=XA5XW0sGN_I&featu

re=related

•http://www.youtube.com/watch?v=g49mahYeNgc&feature

=related


Waves on a String

velocity =T

µ

µ = linear mass density =m

L

T = Tension


Question

Suppose that a longitudinal wave moves along a Slinky

at a speed of 5 m/s. Does one coil of the slinky move

through a distance of five meters in one second?

A) Yes

B) No


Harmonic Wavesy(x,t) = A sin("t – kx)

A = Amplitude= Maximum

displacement of a point on the

wave

# =Wavelength: Distance

between identical points on the

wave

T=Period: Time for a point on thewave to undergo one completeoscillation.

f = frequency =1/T

" = angular frequency= 2$/T

k = wave number = 2$/#

v =

!

T= ! f =

"

k


! Period: The time T for a point on the wave to undergo one complete oscillation.

! Speed: The wave moves one wavelength # in one period T so its speed is v = # / T.

Period and Velocity

v =

!

T= ! f = 2" f( )

!

2"=#

kLecture 19 Purdue University, Physics 220 14

Harmonic Waves Exercisey(x,t) = A cos("t –kx)

Label axis and tic marks for a graph

showing an observation of the wave

y(x,t) = 2 cos(4t –2x) at x=0.

Recall: T = 2 $ /"

t

+2

-2

T = 2 $ / " = 2 $/ 4 = 1.58 s

$ / 2$/4 3$/4

What is the period of this wave?

What is the

velocity of this

wave?

v = "/k= 4/2 m/s=2m/s


• The speed of a wave is a constant that depends only on the

medium, not on amplitude, wavelength or period (similar to

SHM)

# and T are related !

# = v T or # = 2$ v / " (since T = 2$ / " )

or # = v / f (since T = 1/ f )

• Recall f = cycles/sec or revolutions/sec

" = 2$f = radians/second

• Intensity I ! A2

v = # / T

Wave Properties


# – wavelength: distance between crests (meters)

T – period: the time between crests passing fixed location (seconds)

v – speed: the distance one crest moves in a second (m/s)

f – frequency: the number of crests passing fixed location in one

second (1/s or Hz)

" – angular frequency: =2$f (rad/s)

Wave Description

v =

!

Tf =

1

Tv = ! f


Direction

%t

A wave y = A cos("t - kx) travels in +x direction

A wave y = A cos("t + kx) travels in -x direction

y(x,t) = Acos(!t " kx)

k !1

"

! "1

T

!t " kx = const !#t " k#x = 0

#x

#t=!

k


The wavelength of microwaves generated by a microwaveoven is about 3 cm. At what frequency do these wavescause the water molecules in your burrito to vibrate?

A) 1 GHz B) 10 GHz C) 100 GHz

1 GHz = 109 cycles/sec

The speed of light is c = 3x108 m/s

Exercise


! Recall that v = #f

1 GHz = 109 cycles/sec

The speed of light is c = 3x108 m/s

H H

O

Makes water molecules wiggleMakes water molecules wiggle

Exercise

f =

v

!=

3 " 108 m s

.03m= 1010 Hz = 10GHz


Superposition Principle

• When two or more waves pass through the same

region the actual displacement is the sum of the

separate displacements.

• If two waves pass through the same region they

continue to move independently.

!y (x,t) = y

1(x,t) + y

2(x,t)


Superposition


Wavelength: The distance # between identical points on the wave.

Amplitude: The maximum displacement A of a point on the wave.

Amplitude A

A

y(x,t) = A cos("t –kx)

Angular Frequency ": " = 2 $ f

x

y

Wave Number k: k = 2 $ / #

Recall: f = v / #

Mathematical Description

#

Wavelength


Wave Speed

A wave y = A cos("t - kx) travels in +x direction

A wave y = A cos("t + kx) travels in -x direction

Phase:

y(x,t) = Acos(!t " kx)

!t " kx = const !#t " k#x = 0

#x

#t=!

k= v

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