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1/73

Physics 111: Lecture 26, Pg 1

Physics 111:Physics 111:

Conceptual discussion of wavemotion

Wave properties

Mathematical description

Waves on a string


2/73


What is a wave ?What is a wave ?

ccording to our te!t:

wave is a traveling distur"ance that transports energy "ut not matter#

$!amples:

%ound waves &air moves "ac' ( forth)

%tadium waves &people move up ( down)

Water waves &water moves up ( down)

Light waves&what moves**)

%tadium waves


3/73

Physics 111: Lecture 26, Pg +

Types of WavesTypes of Waves

Transverse:he medium oscillates perpendicular to the direction the wave is moving# &water)

Longitudinal: he medium oscillates in the same directionas the wave is moving &%ound)


4/73

Physics 111: Lecture 26, Pg -

Wave PropertiesWave Properties

Wavelength

Wavelength:he distance "etween identical points on the wave#

mplitudeA

mplitude:he ma!imum displacementAof a point on the wave#

A


5/73

Physics 111: Lecture 26, Pg .

Wave Properties...Wave Properties...

Period:he time Tfor a point on the wave to undergo one complete oscillation#

%peed:he wave moves one wavelength in one period Tso its speed is v = / T#

Tv

=


6/73


Wave Properties...Wave Properties...

We will show that the speed of a wave is a constantthatdepends only on the medium, not on amplitude, wavelength, or period#

and Tare related/

= vT or = 2v/ &sinceT = 2/ )

or = v/ f &sinceT = 1 / f ) 0ecall f = cycles/sec orrevolutions/sec

= rad/sec = 2f

v = / T


7/73Physics 111: Lecture 26, Pg

WAVE P!PA"AT#!$WAVE P!PA"AT#!$TAVEL#$" #$ A %$#&!' 'E(#%'TAVEL#$" #$ A %$#&!' 'E(#%'

WAV )

!*+WAVE

'!(E

LE$"T, -

! &E/%E$*0 - f

VEL!*#T0 -V

A'PL#T%(E -A

.

(E$#T0 -

#$*!'PE#2#L#T0 -+

#"#(#T0 -

P!!#T0 -

&L%#( *!$TE$T

.

.

.


8/73Physics 111: Lecture 26, Pg

'!(E !& WAVE P!PA"AT#!$

2!(0 WAVE

- *!'PE#!$AL

- ,EA

%&A*E WAVE

WAV 3


9/73Physics 111: Lecture 26, Pg 3

Table 2: Seismic Waves

Type (andnames)

Particle Motion Typical Velocity Other Characteristics

P,Compressional, Primary,Longitudinal

Alternatingcompressions(pushes!) anddilations (pulls!)"hich are directed inthe same direction asthe "a#e ispropagating (alongthe raypath)$ andthere%ore,perpendicular to the"a#e%ront

VP& ' *m+sin typical arth-scrust$ & / *m+s inarth-s mantleand core$ 0.'*m+s in "ater$ 1.2

*m+s in air

P motion tra#els %astest inmaterials, so the P3"a#e is the%irst3arri#ing energy on aseismogram. 4enerally smallerand higher %re5uency than the 6and 6ur%ace3"a#es. P "a#es in ali5uid or gas are pressure "a#es,including sound "a#es.

6, 6hear,6econdary,

Trans#erse

Alternating trans#ersemotions

(perpendicular to thedirection o%propagation, and theraypath)$ commonlypolari7ed such thatparticle motion is in#ertical or hori7ontalplanes

V6& 2 8 *m+s

in typical arth-s

crust$ & 8.' *m+s inarth-s mantle$& 9.'32.1 *m+s in(solid) inner core

63"a#es do not tra#el through%luids, so do not e:ist in arth-s

outer core (in%erred to ;eprimarily li5uid iron) or in air or"ater or molten roc* (magma). 6"a#es tra#el slo"er than P "a#esin a solid and, there%ore, arri#ea%ter the P "a#e.

Characteristics of %eismic Waves



L, Lo#e,6ur%ace "a#es,Long "a#es

Trans#erse hori7ontalmotion, perpendicularto the direction o%propagation andgenerally parallel tothe arth-s sur%ace

VL& 9.1 3 8.'*m+s in the arthdepending on%re5uency o% thepropagating "a#e

Lo#e "a#es e:ist ;ecause o% thearth-s sur%ace. They are largestat the sur%ace and decrease inamplitude "ith depth. Lo#e"a#es are dispersi#e, that is, the"a#e #elocity is dependent on%re5uency, "ith lo" %re5uenciesnormally propagating at higher

#elocity.



*!'PE#!$AL WAVE

*!'PE#!$AL 4 L!$"#T%(#$AL 4 P5WAVE 4 P#'A0WAVE WAVE WAVE

PAT#*LE '!T#!$ PAALLEL T! WAVE -E$E"0 (#E*T#!$

PAT#*LE

WAVE E$E"0

WAV 6


12/73Physics 111: Lecture 26, Pg 12WAV 7


13/73Physics 111: Lecture 26, Pg 1+WAV 8

%VA*E WAVE 4 #$TE&A*E WAVE

9"!%$( !LL:A0LE#", WAVE

!T,E &!'

P!#2LEVA04 !.; V

Lecture =6>Act 1Act 1olutionolution

We have shown that v= / T= f &since f = 1 / T)

%o f v=

%ince is the same in "oth cases, andv

v1$+++$+++

light

sound

f

f 1$+++$+++light

sound


23/73

Physics 111: Lecture 26, Pg 2+

Lecture =6>Lecture =6>Act 1Act 1olutionolution

What are these fre


24/73

Physics 111: Lecture 26, Pg 2-

Wave &orsWave &ors

%o far we have e!amined>continuous wavescontinuous waves? that goon forever in each direction/

v

v We can also have >pulses?caused "y a "rief distur"anceof the medium:

v nd >pulse trains? which are

somewhere in "etween#

0ope on floor


25/73

Physics 111: Lecture 26, Pg 2.

'atheatical (escription'atheatical (escription

%uppose we have some function y = f0":

0

y

f0 a"is @ust the same shape moveda distance ato the right:

0

y

0 = a+

+

Let a = vt hen

f0 vt"will descri"e the sameshape moving to the right withspeed v. 0

y

0 = vt+

v


26/73


'ath...'ath...

Consider a wave that is harmonicin0and has a wavelength of #

( )

= 02

cosA0y

Af the amplitude is ma!imum at

0 = +this has the functional form:

y

0

A

Bow, if this is moving to

the right with speed v it will "edescri"ed "y:

y

0

v

( ) ( )

= vt02

cosAt$0y


27/73


'ath...'ath...

( ) ( )

= vt02cosAt$0y

yusing vT

= = 2

from "efore, and "y defining k2

%o we see that a simple harmonicwave moving with speed vin the0direction is descri"ed "y the e


28/73


'ath uary'ath uary

he formuladescri"es a harmonic wave ofamplitudeAmoving in theE0direction#

( ) ( )tk0cosAt$0y = y

0

A

$ach point on the wave oscillates in the ydirection withsimple harmonic motion of angular fre


29/73


Lecture =6>Lecture =6>Act 2Act 2Wave 'otionWave 'otion

harmonic wave moving in the positive ! direction can "edescri"ed "y the e


30/73

Physics 111: Lecture 26, Pg +4


0ecall y0$t" = A cos k0 t"came from

( ) ( )

= vt0

2At0y

cos,

he sign of the term containing the tdetermines thedirection of propagation#

We change the sign to change the direction:

y0$t" = A cos k0 t" moving toward E!

y0$t" = A cos k0 +t" moving toward D!


31/73



0ecall y0$t" = A cos k0 t "came from

( ) ( ) = vt02

At0y

cos,

ctually , it7s the relative sign "etween the term containingthe0and the term containing the v:

y0$t" = A cos k0 t" moving toward E!

y0$t" = A cos k0 +t" = A cos k0 t""

= A cos k0 t" also moving toward E!


32/73


Waves on a stringWaves on a string

What determines the speed of a wave*

Consider a pulse propagating along a string:

v

>%nap? a rope to see such a pulse ow can you ma'e it go faster*

Movie &string1)


33/73

Physics 111: Lecture 26, Pg ++

Waves on a string...Waves on a string...

he tension in the string is

he mass per unit length of the string is &kg/m)

he shape of the string at the pulse7s ma!imum iscircular and has radius (

(

uppose:uppose:


34/73

Physics 111: Lecture 26, Pg +-


v

0

y

Consider moving along with the pulse & )

pply = mato the small "it of string at the >top? of the pulse

Movie &string2)


35/73

Physics 111: Lecture 26, Pg +.


0

y

he total force B$is the sum of the tension at

each end of the string segment#

he total force is in the ydirection#

B$F 2

&since is small, sin G )


36/73



2

m =(2

(

0

y

he mass mof the segment is its length &(! 2) timesits mass per unit length #


37/73



(

v

0

y

he acceleration aof the segment is v 2H ( &centripetal)in the Dydirection#

a


38/73



%o 3!T= ma"ecomes: (

v2(2

2

=

2v =

= v

T4T m a

v

tension mass per unit length


39/73



%o we find:

= v

Ma'ing the tension "igger increases the speed#

Ma'ing the string heavier decreases the speed#

As we asserted earlier> thisAs we asserted earlier> this depends only on the nature ofdepends only on the nature ofthe ediuthe ediu> not on aplitude> fre@uency> etc. of the wave.> not on aplitude> fre@uency> etc. of the wave.

v

tension mass per unit length


40/73

Physics 111: Lecture 26, Pg -4


heavy rope hangs from the ceiling, and a smallamplitude transverse wave is started "y @iggling therope at the "ottom#

s the wave travels up the rope, its speed will:

&a) increase

&") decrease

&c) stay the same

v

0ope

5rop slin'y


41/73



he speed at any point will "e determined "y

v

v = at that point

he tension in the rope near the top is greater than thetension near the "ottom since it has to support theweight of the rope "eneath it/

he speed of the wave will "e greater at the top/


42/73


ecap of todays lectureecap of todays lecture

Conceptual discussion of wave motion

Wave properties

Mathematical description

Waves on a string


43/73

Physics 111: Lecture 26, Pg -+

Todays AgendaTodays Agenda

Wave power

=low of energy

%uperposition ( Anterference

he wave e


44/73

Physics 111: Lecture 26, Pg --


"oat is moored in a fi!ed location, and waves ma'e it move upand down# Af the spacing "etween wave crests is 2+ metersandthe speed of the waves is 5 m/s, how long tdoes it ta'e the"oat to go from the top of a crest to the "ottom of a trough*

&a) 2 sec &") 6 sec &c) sec

t

t 7 t


45/73

Physics 111: Lecture 26, Pg -.


We 'now that v = / T, hence T =/ v

t

t 7 t

An this case = 2+ mand v = 5 m/s, so T = 6 sec

he time to go from a crest to a trough is T/2&half a period)

%ot = 2 sec


46/73


Wave PowerWave Power

wave propagates "ecause each part of the medium communicates its motion to ad@acent parts#

$nergy is transferred since wor' is done/

ow much energy is moving down the string per unit time#

&i#e# how muchpower*)

P


47/73


Wave Power...Wave Power...

hin' a"out gra""ing the left side of the string and pulling it up and down in the ydirection# ;ou are clearly do ing wor' since F#dr 8 +as your hand moves up and down# his energy must "e mo ving away from your hand &to the right) since the 'inetic energy &motion) of the string stays the same#

P

Movie &pump)

%lin'y


48/73


,ow is the energy oving?,ow is the energy oving?

Consider any position0on the string# he string to the left of0does wor' on the string to the right of0, @ust as your hand did:

0

0

FPower 'F F#v

v


49/73


Power along the stringPower along the string

%ince vis along the ya!is only, to evaluate 'owerF F#vwe only need to find y= sin if is small#

We can easily figure out "oth thevelocityvand the angle at anypoint on the string:

Af

0

F v

y

( ) ( )tk0sinAdt

dy

t$0vy ==

( ) == tk0sinkAd0

dytan

0ecall

sin cos 1

for small

tan

"tk0cosA"t$0y =

=y

dy

d!


50/73

Physics 111: Lecture 26, Pg .4

Power...Power...

%o:

ut last time we showed that andk

v = v= 2

( ) ( )tk0sinAvt$0' 222 =

( )tk0sin2

( )tk0cos

)tk0sinkAvt"'0$ 22y =

( ) ( )tk0Asint0$vy =( )tk0kAsin


51/73


Average PowerAverage Power

We @ust found that the power flowing past location0on the string at time tis given "y:

( ) ( )tk0Avt0' 222 = sin,

We are often @ust interested in the average power movingdown the string# o find this we recall that the averagevalue of the function sin2 k0 t"is 1/2and find that:

' v A=1

2

2 2

At is generally true that wave power is proportional to thespeed of the wave vand its amplitude s


52/73


Energy of the WaveEnergy of the Wave

We have shown that energy >flows? along the string#

he source of this energy &in our picture) is the hand that is sha'ing the string up and down at one end#

$ach segment of string transfers energy to &does wor' on) the ne!t segment "y pulling on it, @ust li'e the hand#

' A v=1

2

2 2 We found that

d!

dt

A d0

dt

=1

2

2 2 d! A d0 =1

2

2 2

%o is the average energy per unit length#d!

d0A=

1

2

2 2


53/73

Physics 111: Lecture 26, Pg .+

Power EBaple:Power EBaple:

rope with a mass of F+.2 kg/mlays on a frictionlessfloor# ;ou gra" one end and sha'e it from side to sidetwice per secondwith an amplitude of +.15 m# ;ou noticethat the distance "etween ad@acent crests on the wave youma'e is +.95 m#

What is the average power you are providing the rope*

What is the average energy per unit length of the rope*

What is the tension in the rope*

A = +.15 m

F+.95 mf = 2 ,-


54/73

Physics 111: Lecture 26, Pg .-

Power EBaple...Power EBaple...

We'nowA, and F 2f# Weneedtof ind v/

0ecallv = f= .95m"2s1" = 1.5m/s #

%o:

' v A=1

2

2 2

' #=+ 5**.

( ) ( )22

m15+,-22s

m51

m

kg2+

2

1' ###

=

verage power


55/73

Physics 111: Lecture 26, Pg ..


%o:

d!

d0A=

1

2

2 2

d!

d0:/m=+ *55.

( ) ( )d!d0kgm

,- m= 12 + 2 2 2 +15 2 2

. .

verage energy per unit length


56/73



We also 'now that the tension in the rope is related tospeed of the wave and the mass density:

ension in rope: = +.65 3

2

2

sm5.1

mkg2.+v ==


57/73


ecap C %seful &orulas:ecap C %seful &orulas:

d!

d0A=

1

2

2 2

' v A=12

2 2

y

0

A

v =

Waves on a string

( ) ( )tk0cosAt$0y =

k=2

v fk

= =

= =2 2fT

Ieneral harmonic waves

tension

mass H length


58/73


Lecture =7>Lecture =7>Act 2Act 2Wave PowerWave Power

wave propagates on a string# Af "oth the amplitude and thewavelength are dou"led, "y what factor will the average powercarried "y the wave change* &he velocity of the wave isunchanged)#

&a) 1 &") 2 &c) 6


59/73



We have shown that the average power ' A v=1

2

2 2

''

A v

A v

AA

f

i

f f

i i

f f

i i

= =

1

21

2

2 2

2 2

2 2

2 2

%o


60/73



ut since v = f = / 2is constant,

'i

'f

f

i

i

f

=

i#e# dou"ling the wavelength halves the fre


61/73



'

'

A

A

A

Af

i

f f

i i

i

f

f

i

= =

2 2

2 2

2 2

%o

=

=

1

2

2

11

2 2

same power


62/73


uperpositionuperposition

/:/:What happens when two waves >collide*?

A:A:hey 55 together/

We say the waves are >superposed#?

Movie &superJpulse)

Movie &superJpulse2)

%hive model


63/73

Physics 111: Lecture 26, Pg 6+

Aside: Why superposition worDsAside: Why superposition worDs

s we will see in the ne!t lecture, the e


64/73

Physics 111: Lecture 26, Pg 6-

uperposition C #nterferenceuperposition C #nterference

We have seen that when colliding waves com"ine &add) the result can either "e "igger or smaller than the original waves#

We say the waves add >constructively? or >destructively? depending on the relative sign of each wave#

Movie &super)

will add constructively

will add destructively

An general, we will have "oth happening

ead

model


65/73

Physics 111: Lecture 26, Pg 6.


Consider two harmonic wavesAand ;meeting at0=+#

%ame amplitudes, "ut 2F 1#1. ! 1# he displacement versus time for each is shown "elow:

What does


66/73



Consider two harmonic wavesAand ;meeting at0 = +#

%ame amplitudes, "ut 2F 1#1. ! 1# he displacement versus time for each is shown "elow:

A1t"


67/73


2eats2eats

( ) ( )tcostcosA2"tcosA"tcosA ,)21 =+

( ) )= 12

1 2 ( )21,21 +=

Can we predict this pattern mathematically* Kf course/

ust add two cosines and remem"er the identity:

where and

cos)t"

%ound

generator


68/73


The Wave E@uationThe Wave E@uation

armonic waveshave the form y0$t"=A cosk0 t".

An general, a wave traveling to the right with velocity visgiven "y y0$t"=f0 vt"

ow do we 'now a wave of thisform really satisfiesBewton7s2nd Law**

We will now prove thisisthe case#

v

wherek

v

=


69/73


The Wave [email protected] Wave E@uation...

%uppose we have the pluc'ed string shown "elow:

he displacement is greatly e!aggerated in the p icture###1and 2are "oth close to +#

dm F

F

12

0

y


70/73



=F cos&2) cos&1) + &no net force in ! direction)

ut =;F sin&2) sin&1)&2 D 1) F d

F

0

ynd:

d0

dyslopetan ==

d0d

d

d0d0

d y

d0d0

= =

2

2

%o:

sin cos 1 for small tan

dmF

12


71/73



ut =;F dma;F dm a;

dm

d0 Nse and

2

2

2

2

dt

yd

d0

yd =

d d y

d0d0=

2

2

=;

%o =;F

ut for a string we found that = v2&Lecture 26), so:

d y

d0 v

d y

dt

2

2 2

2

2

1=

d0dm = 22

> dt

yd

a =

d:d:

ydd =

2

2


72/73


&inally:&inally:

heis calledthe>Wave$


73/73

ecap of todays lectureecap of todays lecture

Wave power and intensity

%uperposition

Anterference

he Wave $

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