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Physics 111: Lecture 26, Pg 1
Physics 111:Physics 111:
Conceptual discussion of wavemotion
Wave properties
Mathematical description
Waves on a string
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Physics 111: Lecture 26, Pg 2
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#
$!amples:
%ound waves &air moves "ac' ( forth)
%tadium waves &people move up ( down)
Water waves &water moves up ( down)
Light waves&what moves**)
%tadium waves
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Physics 111: Lecture 26, Pg +
Types of WavesTypes of Waves
Transverse:he medium oscillates perpendicular to the direction the wave is moving# &water)
Longitudinal: he medium oscillates in the same directionas the wave is moving &%ound)
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Physics 111: Lecture 26, Pg -
Wave PropertiesWave Properties
Wavelength
Wavelength:he distance "etween identical points on the wave#
mplitudeA
mplitude:he ma!imum displacementAof a point on the wave#
A
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Physics 111: Lecture 26, Pg .
Wave Properties...Wave Properties...
Period:he time Tfor a point on the wave to undergo one complete oscillation#
%peed:he wave moves one wavelength in one period Tso its speed is v = / T#
Tv
=
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Physics 111: Lecture 26, Pg 6
Wave Properties...Wave Properties...
We will show that the speed of a wave is a constantthatdepends only on the medium, not on amplitude, wavelength, or period#
and Tare related/
= vT or = 2v/ &sinceT = 2/ )
or = v/ f &sinceT = 1 / f ) 0ecall f = cycles/sec orrevolutions/sec
= rad/sec = 2f
v = / T
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7/73Physics 111: Lecture 26, Pg
WAVE P!PA"AT#!$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$T
.
.
.
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8/73Physics 111: Lecture 26, Pg
'!(E !& WAVE P!PA"AT#!$
2!(0 WAVE
- *!'PE#!$AL
- ,EA
%&A*E WAVE
WAV 3
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9/73Physics 111: Lecture 26, Pg 3
Table 2: Seismic Waves
Type (andnames)
Particle Motion Typical Velocity Other Characteristics
P,Compressional, Primary,Longitudinal
Alternatingcompressions(pushes!) anddilations (pulls!)"hich are directed inthe same direction asthe "a#e ispropagating (alongthe raypath)$ 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$& 9.'32.1 *m+s in(solid) inner core
63"a#es do not tra#el through%luids, so do not e:ist in arth-s
outer core (in%erred to ;eprimarily li5uid iron) or in air or"ater or molten roc* (magma). 6"a#es tra#el slo"er than P "a#esin a solid and, there%ore, arri#ea%ter the P "a#e.
Characteristics of %eismic Waves
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10/73Physics 111: Lecture 26, Pg 14
L, Lo#e,6ur%ace "a#es,Long "a#es
Trans#erse hori7ontalmotion, perpendicularto the direction o%propagation andgenerally parallel tothe arth-s sur%ace
VL& 9.1 3 8.'*m+s in the arthdepending on%re5uency o% thepropagating "a#e
Lo#e "a#es e:ist ;ecause o% thearth-s sur%ace. They are largestat the sur%ace and decrease inamplitude "ith depth. Lo#e"a#es are dispersi#e, that is, the"a#e #elocity is dependent on%re5uency, "ith lo" %re5uenciesnormally propagating at higher
#elocity.
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11/73Physics 111: Lecture 26, Pg 11
*!'PE#!$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$E"0
WAV 6
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12/73Physics 111: Lecture 26, Pg 12WAV 7
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13/73Physics 111: Lecture 26, Pg 1+WAV 8
%VA*E WAVE 4 #$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
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Physics 111: Lecture 26, Pg 2+
Lecture =6>Lecture =6>Act 1Act 1olutionolution
What are these fre
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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
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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
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Physics 111: Lecture 26, Pg 26
'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
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Physics 111: Lecture 26, Pg 2
'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
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Physics 111: Lecture 26, Pg 2
'ath uary'ath uary
he formuladescri"es a harmonic wave ofamplitudeAmoving in theE0direction#
( ) ( )tk0cosAt$0y = y
0
A
$ach point on the wave oscillates in the ydirection withsimple harmonic motion of angular fre
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Physics 111: Lecture 26, Pg 23
Lecture =6>Lecture =6>Act 2Act 2Wave 'otionWave 'otion
harmonic wave moving in the positive ! direction can "edescri"ed "y the e
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Physics 111: Lecture 26, Pg +4
Lecture =6>Lecture =6>Act 2Act 2olutionolution
0ecall y0$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!
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Physics 111: Lecture 26, Pg +1
Lecture =6>Lecture =6>Act 2Act 2olutionolution
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$t" = A cos k0 +t" = A cos k0 t""
= A cos k0 t" also moving toward E!
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Physics 111: Lecture 26, Pg +2
Waves on a stringWaves on a string
What determines the speed of a wave*
Consider a pulse propagating along a string:
v
>%nap? a rope to see such a pulse ow can you ma'e it go faster*
Movie &string1)
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Physics 111: Lecture 26, Pg ++
Waves on a string...Waves on a string...
he tension in the string is
he mass per unit length of the string is &kg/m)
he shape of the string at the pulse7s ma!imum iscircular and has radius (
(
uppose:uppose:
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Physics 111: Lecture 26, Pg +-
Waves on a string...Waves on a string...
v
0
y
Consider moving along with the pulse & )
pply = mato the small "it of string at the >top? of the pulse
Movie &string2)
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Physics 111: Lecture 26, Pg +.
Waves on a string...Waves on a string...
0
y
he total force B$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 )
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Physics 111: Lecture 26, Pg +6
Waves on a string...Waves on a string...
2
m =(2
(
0
y
he mass mof the segment is its length &(! 2) timesits mass per unit length #
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Physics 111: Lecture 26, Pg +
Waves on a string...Waves on a string...
(
v
0
y
he acceleration aof the segment is v 2H ( ¢ripetal)in the Dydirection#
a
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Physics 111: Lecture 26, Pg +
Waves on a string...Waves on a string...
%o 3!T= ma"ecomes: (
v2(2
2
=
2v =
= v
T4T m a
v
tension mass per unit length
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Physics 111: Lecture 26, Pg +3
Waves on a string...Waves on a string...
%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
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Physics 111: Lecture 26, Pg -4
Lecture =6>Lecture =6>Act 3Act 3Wave 'otionWave 'otion
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
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Physics 111: Lecture 26, Pg -1
Lecture =6>Lecture =6>Act 3Act 3olutionolution
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/
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Physics 111: Lecture 26, Pg -2
ecap of todays lectureecap of todays lecture
Conceptual discussion of wave motion
Wave properties
Mathematical description
Waves on a string
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Physics 111: Lecture 26, Pg -+
Todays AgendaTodays Agenda
Wave power
=low of energy
%uperposition ( Anterference
he wave e
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Physics 111: Lecture 26, Pg --
Lecture =7>Lecture =7>Act 1Act 1Wave 'otionWave 'otion
"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
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Physics 111: Lecture 26, Pg -.
Lecture =7>Lecture =7>Act 1Act 1olutionolution
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
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Physics 111: Lecture 26, Pg -6
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
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Physics 111: Lecture 26, Pg -
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
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Physics 111: Lecture 26, Pg -
,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
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Physics 111: Lecture 26, Pg -3
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!
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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
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Physics 111: Lecture 26, Pg .1
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
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Physics 111: Lecture 26, Pg .2
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
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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 ,-
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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
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Physics 111: Lecture 26, Pg ..
Power EBaple...Power EBaple...
%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
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Physics 111: Lecture 26, Pg .6
Power EBaple...Power EBaple...
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 ==
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Physics 111: Lecture 26, Pg .
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
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Physics 111: Lecture 26, Pg .
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
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Physics 111: Lecture 26, Pg .3
Lecture =7>Lecture =7>Act 2Act 2olutionolution
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
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Physics 111: Lecture 26, Pg 64
Lecture =7>Lecture =7>Act 2Act 2olutionolution
ut since v = f = / 2is constant,
'i
'f
f
i
i
f
=
i#e# dou"ling the wavelength halves the fre
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Physics 111: Lecture 26, Pg 61
Lecture =7>Lecture =7>Act 2Act 2olutionolution
'
'
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
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Physics 111: Lecture 26, Pg 62
uperpositionuperposition
/:/:What happens when two waves >collide*?
A:A:hey 55 together/
We say the waves are >superposed#?
Movie &superJpulse)
Movie &superJpulse2)
%hive model
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Physics 111: Lecture 26, Pg 6+
Aside: Why superposition worDsAside: Why superposition worDs
s we will see in the ne!t lecture, the e
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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
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Physics 111: Lecture 26, Pg 6.
uperposition C #nterferenceuperposition C #nterference
Consider two harmonic wavesAand ;meeting at0=+#
%ame amplitudes, "ut 2F 1#1. ! 1# he displacement versus time for each is shown "elow:
What does
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Physics 111: Lecture 26, Pg 66
uperposition C #nterferenceuperposition C #nterference
Consider two harmonic wavesAand ;meeting at0 = +#
%ame amplitudes, "ut 2F 1#1. ! 1# he displacement versus time for each is shown "elow:
A1t"
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Physics 111: Lecture 26, Pg 6
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
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Physics 111: Lecture 26, Pg 6
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$t"=f0 vt"
ow do we 'now a wave of thisform really satisfiesBewton7s2nd Law**
We will now prove thisisthe case#
v
wherek
v
=
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Physics 111: Lecture 26, Pg 63
The Wave [email protected] Wave E@uation...
%uppose we have the pluc'ed string shown "elow:
he displacement is greatly e!aggerated in the p icture###1and 2are "oth close to +#
dm F
F
12
0
y
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Physics 111: Lecture 26, Pg 4
The Wave [email protected] Wave E@uation...
=F cos&2) cos&1) + &no net force in ! direction)
ut =;F sin&2) sin&1)&2 D 1) F d
F
0
ynd:
d0
dyslopetan ==
d0d
d
d0d0
d y
d0d0
= =
2
2
%o:
sin cos 1 for small tan
dmF
12
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Physics 111: Lecture 26, Pg 1
The Wave [email protected] Wave E@uation...
ut =;F dma;F dm a;
dm
d0 Nse and
2
2
2
2
dt
yd
d0
yd =
d d y
d0d0=
2
2
=;
%o =;F
ut for a string we found that = v2&Lecture 26), so:
d y
d0 v
d y
dt
2
2 2
2
2
1=
d0dm = 22
> dt
yd
a =
d:d:
ydd =
2
2
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Physics 111: Lecture 26, Pg 2
&inally:&inally:
heis calledthe>Wave$
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ecap of todays lectureecap of todays lecture
Wave power and intensity
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