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7/24/2019 Topic 4: Waves
http://slidepdf.com/reader/full/topic-4-waves 1/18
Topic 4: Waves
4.1 – Oscillations
Simple harmonic oscillationsOscillations are periodic motions which center around an equilibrium position.Simple harmonic motion (SHM) is a special tpe o! oscillation. "or e#ample: The simple pendulum
The vibration o! strin$s in a violin
The sprin$%mass sstem& where the mass is initiall displaced to produce a
periodic motion around the equilibrium position
'n obect under$oes SHM i! it e#periences a !orce which is proportional and oppositeo! the displacement !rom its equilibrium position.
*ewton+s Second ,aw& SHM can be de!ined as the !ollowin$ equations
http:--ima$e.slidesharecdn.com-4%simpleharmonicmotion%/01220/4/%phpapp02-31-4%simple%harmonic%motion%%5/6.p$7cb8/5356514where #0 is the amplitude (ma#imum displacement)& # is the displacement& v is thevelocit& and a is the acceleration.
The an$ular !requenc (w) is related to the period o! the SHM b the !ollowin$equation
https:--classconnection.s/.ama9onaws.com-23-!lashcards-1/46023-pn$-equation%41'14/0532136453.pn$
The period is independent o! the amplitude o! the SHM and can be $iven b the!ollowin$ equation
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http:--.bp.blo$spot.com-%%;<='>dWih?-@"%aA6BC5D-''''''''0ic-wHbi6"dtA'-s500-2%2%SHM.0.$i! !or a pendulum
and
http:--.bp.blo$spot.com-%h1m/'wA,eE-@"%h0bmiD-''''''''0iw->uFrEMO4c4c-s500-2%2%SHM.04.$i! !or a sprin$%mass sstem
Time period& !requenc& amplitude& displacement and
phase di!!erenceAisplacement(#)
'mplitude Ceriod (T) "requenc (!) Chasedi!!erence
Aisplacemento! theoscillatin$
obect at aspeci!ic time!rom itsequilibriumposition
Ma#imumdisplacemento! the
oscillatin$obect
Time ta>en !or one completeoscillation (in
seconds)
*umber o!times theobect
oscillates perunit time(usuall onesecond)!8-T
The di!!erencebetween twoSHMs with the
same!requenc interms o! theirrelativeposition in acclemeasured inradian
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https:--upload.wi>imedia.or$-wi>ipedia-commons-4-44-Simple;harmonic;motion.pn$
onditions !or simple harmonic motionDn a SHM& there is an interchan$e between G= and C= throu$hout the motion.However& the total ener$ remains constant.
https:--tap.iop.or$-vibration-shm-/01-im$;!ull;4550/.$i!
Summar: 't ma#imum displacement& C= is at ma# while G=80
't 9ero displacement& G= is at ma# while C=80
't minimum displacement& C= is at ma# while G=80
Total ener$ (G=C=) remains constant throu$hout the motion
http:--4.bp.blo$spot.com-%5;5MvOW"n"o-@"%@o;vhD-''''''''0h4-w"'I>AJ54'-s500-2%2%SHM.00.pn$
4.2 – Travelling waves
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Travellin$ waves ' travellin$ wave is a continuous disturbance in a medium characteri9ed b repeatin$oscillations. "or e#ample: ' rope that is !lic>ed up and down continuousl creates a repeatin$ disturbance
similar to the shape o! a sine-cosine wave.
=ner$ is trans!erred b waves.Matter is not trans!erred b waves.The direction o! a wave is de!ined b the direction o! the ener$ trans!er.
Wavelen$th& !requenc& period and wave speed
http:--2.bp.blo$spot.com-%s239DM#sS0-@"S</EWTD-''''''''0q?-rsCo?Mw>r'-s500-waveparameters.$i! Wavelen$th& !requenc& and period !ollow the same rules o! SHM.Wave speed can be calculated b the !ollowin$ equation
http:--www.one%school.net-Malasia-Eniversitandolle$e-SCM-revisioncard-phsics-wave-ima$es-velocit!ormula.pn$
Transverse and lon$itudinal waves
Transverse wave ,on$itudinal wave
The direction o! oscillation is perpendicular tothe direction o! the wave
http:--www.everthin$maths.co.9a-science-$r ade%0-06%transverse%waves-pspictures-503e21/ddd3a2e5ea32beda0602.pn$
The direction o! oscillation is parallelto the direction o! the wave
http:--www.everthin$maths.co.9a-science-$rade%0-03%lon$itudinal%waves-pspictures-a/c0e40e/2d2524dc6cb6eeaeed.pn$
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http:--/.bp.blo$spot.com-%GCIBqt!udb>-E!h'?D!i0qD-''''''''K5'-BA4?%<!<B=-s500-2%/%waves%transverse.$i!
http:--/.bp.blo$spot.com-%Da5DhSE4F6-E!h'*eKCbD-'''''' ''K1E-teDwl$Ds4-s500-2%/%waves%lon$itudinal.$i!
=#ample: Water waves
Wave in a strin$ !lic>ed up and down
,i$ht (electroma$netic waves)
=#ample: Wave produced in a sprin$
Sound waves
=arthqua>e C%waves
"JD
Transverse wave ,on$itudinal wave
' point with ma#imum positivedisplacement is called a crest. ' point with minimum displacement iscalled a trou$h.
' re$ion where particles are closed toeach other is called a compression. ' re$ion where particles are !urthestapart !rom each other is called arare!action.
http:--www.everthin$maths.co.9a-science-$
rade%0-06%transverse%waves-pspictures-6/cd161e/e56/b/10255e/1ca3dc/1.pn$
http:--www.everthin$maths.co.9a-science-$rade%0-03%lon$itudinal%waves-pspictures-12e/c35ab35a64ba05c3cce!.pn$
The nature o! electroma$netic waves 'll =M waves travel in vacuum at the same speed o! /L06m-s.=M waves are transverse waves.
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http:--en.es%static.us-upl-202-01-em;spectrum.pn$
The nature o! sound wavesThe speed o! sound in 20 de$rees elsius dr air is appro#imatel /4/.2m-s.Sound waves are lon$itudinal waves.
http:--hperphsics.ph%astr.$su.edu-hbase-sound-im$sou-lwav2.$i!
4.3 – Wave characteristics
Wave!ronts and rasWave!ronts: ,ines oinin$ points which vibrate in phase.
an be strai$ht lines or curves.
The distance between successive wave!ronts is the wavelen$th o! the wave.
Ias: ,ines which indicate the direction o! wave propa$ation.
Ias are perpendicular to wave!ronts.
https:--www.ualberta.ca-Npo$osan-teachin$-CHJS;/0-ima$es-ras.pn$
'mplitude and intensitThe amplitude and intensit o! a wave depends on its ener$.The intensit o! a wave is proportional to the square o! its amplitude (D∝ '2).
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http:--www.schoolphsics.co.u>-a$e5%3-Wave20properties-Wave20properties-te#t-Dntensit;and;amplitude-ima$es-.pn$
Transverse and lon$itudinal wavesSee previous section with the same title.
SuperpositionThe principle o! superposition states that the net displacement o! the underlin$
medium !or a wave is equal to the sum o! the individual wave displacements.
http:--paleocave.sciencesorto!.com-wp%content-uploads-200-0-SuperCosition.p$The le!t shows constructive inter!erence (superposition) where the two waves add up(e.$. 82). The ri$ht shows deconstructive inter!erence (superposition) where thetwo waves cancel each other (e.$. (%)80).
Colari9ation,i$ht is a transverse wave (polari9ation onl occur to transverse waves).The polari9ation o! li$ht re!ers to the orientation o! the oscillation in the underlin$electric !ield.,i$ht is plane polari9ed i! the electric !ield oscillates in one plane.
http:--pediaa.com-wp%content-uploads-201-03-Ai!!erence%etween%Colari9ed%and%Enpolari9ed%,i$ht%How;a;polari9in$;!ilter;wor>s.p$,e!t shows unpolari9ed li$ht and ri$ht shows polari9ed li$ht.
Polarization by reflection
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When li$ht is transmitted across a boundar between two mediums with di!!erentre!ractive inde#es& part o! the li$ht is re!lected and the remainin$ part is re!racted (!or!urther e#planation& see section 4.4).
The li$ht re!lected is partiall polari9ed& meanin$ that it is a mi#ture o! polari9ed li$htand unpolari9ed li$ht.
The e#tent to which the re!lected li$ht is polari9ed depends on the an$le o! incidenceand the re!ractive inde# o! the two mediums.The an$le o! incidence at which the re!lected li$ht is totall polari9ed is called therewster+s an$le ( ) $iven b the equationϕ
http:--www.diracdelta.co.u>-science-source-b-r-brewsters20law-ima$e00.$i! where n and n2 are the re!ractive inde#es !or their respective mediumsWhen the an$le o! incidence is equal to rewster+s an$le& the re!lected ra is totall
polari9ed and the re!lected ra is perpendicular to the re!racted ra.
http:--www.as>iitians.com-onlinetest-studmaterial;ima$es-40;brewsters20law.p$
Polarizers and AnalyzersColari9er: ' polari9er is a sheet o! material which polari9es li$ht.
When unpolari9ed li$ht passes throu$h a polari9er& its intensit is reduced b
10.
'nal9er: When polari9ed li$ht passes throu$h a polari9er& its intensit will be reduced b
a !actor dependent on the orientation o! the polari9er. This propert allows us todeduce the polari9ation o! li$ht b usin$ a polari9er.
' polari9er used !or this purpose is called an anal9er.
Malus+ ,aw relates the incident intensit and transmitted intensit o! li$ht passin$throu$h a polari9er and an anal9er.
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http:--2.bp.blo$spot.com-% ;*Bo1MTJ>o-EOswO?lv=tD-''''''''SIo-ti*OwH/OE?-s500-%2%polari9ation%a.06.pn$where D is the transmitted intensit& D0 is the initial li$ht intensit upon the anal9er& P
is the an$le between the transmission a#is and the anal9er.
https:--lh5.$oo$leusercontent.com-%q/?4CJ*pw>-EOpqE><#%uD-''''''''SG4-cA2B>2,WS20-s600-%2%polari9ation.03.$i!
https:--lh/.$oo$leusercontent.com-%oAu#*HDmp?-EOpKw'C4rhD-''''''''SF?-uIu0#B;TD@o-s600-%2%polari9ation.002.$i!
When li$ht passes throu$h an opticall active substance& the plane o! polari9ationrotates.
4.4 – Wave behavior
Ie!lection and re!raction!eflection
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http:--www.qr$.northwestern.edu-proects-vss-docs-media-ommunications-re!lection.$i! 'n$le o! incidence 8 'n$le o! re!lection
Ie!lection o! waves !rom a !i#ed end is inverted.
http:--labman.phs.ut>.edu-phs222core-modules-m3-ima$es-!i#edrope.$i!
Ie!lection o! waves !rom a !ree end is not inverted.
http:--labman.phs.ut>.edu-phs222core-modules-m3-ima$es-looserope.$i!
!efractionIe!raction is the chan$e in direction o! a wave when it transmits !rom one medium toanother.
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http:--ms>!atima.weebl.com-uploads-4--5-1-45101/3-603305;ori$.pn$
The an$le o! incidence and the an$le o! re!raction can be determined b Snell+s law$iven b the !ollowin$ !ormula
http:--hperphsics.ph%astr.$su.edu-hbase-$eoopt-im$$o-snell2.$i! where n and n2 are the re!ractive inde#es !or their respective mediums
http:--.bp.blo$spot.com-%qbrrIw4M6-T#I#sGbev?D-''''''''<@$-lrKOJ#!SF9?-s500-%2%re!lectionre!raction.04.pn$
http:--2.bp.blo$spot.com-%
M1/,,J=C>-T#I#r><F9GD-''''''''<@D-=s#6q>J">-s500-%2%re!lectionre!raction.05.pn$
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"ast%to%slow: towards normalQ slow%to%!ast: awa !rom normal
Dn addition& the re!ractive inde# n and n2 are related b the !ollowin$ equation
http:--www.studphsics.ca-newnotes-20-unit04;li$ht-chp3;li$ht-ima$es-snells;law.pn$where v and v2 are the speed o! the waves in their respective mediums and R andR2 are the wavelen$th o! the waves o! their respective mediums
https:--lh5.$oo$leusercontent.com-%@b1F"!phMK4-EC#/<v9#eD-''''''''T*4-a5p%K!wiWcJ-s500-han$in$Media%new.$i!
Snells law& critical an$le and total internal re!lectionSee previous section (Ie!lection and re!raction) !or Snell+s law.
http:--www.allin!o.or$.u>-revision%$cse-Dma$es-internal.$i!
The re!ractive inde# and the critical an$le are related b the !ollowin$ equation
http:--2.bp.blo$spot.com-%
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a;BrbrO,>D-Ee'0"IMuJ5D-''''''''T4-viHI9CdKpE-s500-!ormula.pn$
Total internal re!lection onl occurs when the li$ht ra propa$ates !rom a opticalldenser medium to an opticall less dense medium.
Ai!!raction throu$h a sin$le%slit and around obectsSpecial di!!raction patterns appear when li$ht is di!!racted b a sin$le slit which iscomparable to the wavelen$th o! the li$ht in si9e.We can represent this di!!raction pattern b plottin$ the li$ht intensit a$ainst thean$le o! di!!raction.The an$le o! di!!raction !or the !irst minimum P can be $iven b
https:--www.patana.ac.th-secondar-science-anrophsics-ntopic-ima$es-sin$le
20slit20eqn.FC<where R is the wavelen$th and b is the si9e-len$th o! the slit
http:--hperphsics.ph%astr.$su.edu-hbase-phopt-im$pho-sinslit.$i! http:--hperphsics.ph%astr.$su.edu-hbase-phopt-phopic-sinslitv.p$
Dnter!erence patternsMa#imums !orm at constructive inter!erence (the ma#imum is shown b %2) andminimums !orm at deconstructive inter!erence (the !irst minimum is shown b /%4).
http:--hperphsics.ph%astr.$su.edu-hbase-phopt-im$pho-sinslitwid.$i!
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https:--lh4.$oo$leusercontent.com-% ''rHCB#6c$-E??d"'1SD-''''''''TJ=->9,4aE#S?-s600-%1%inter!erence.00.$i!
https:--lh4.$oo$leusercontent.com-%vd3/vutbvD-E??d26lM#iD-''''''''TJ=-,6CWm1W;TnD-s600-%1%
inter!erence.003.$i!
http:--.bp.blo$spot.com-%W?%*Ibo3Aw-@O#A4dIa,D-''''''''/lc-*v%>@>3i;2$-s500-%1%inter!erence%labels.02.pn$
Aouble%slit inter!erence,i>e sin$le%slit di!!raction& double%slit di!!raction occurs via the same methods o!inter!erence and has a similar di!!raction pattern.
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http:--hperphsics.ph%astr.$su.edu-hbase-phopt-im$pho-dslit.$i! http:--hperphsics.ph%astr.$su.edu-hbase-phopt-phopic-dslit.p$
Cath di!!erence
http:--.bp.blo$spot.com-% 'D?JH;eT956-@O#O!5C6l5D-''''''''/mo-KT,Eu0viB*$-s500-pathlen$ths.$i!
http:--2.bp.blo$spot.com-%WAsMB4q55h>-@O#HuJSd'1D-''''''''/mJ-BKMG,'<s10-s500-pathdi!!erence.$i!
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http:--.bp.blo$spot.com-%uFs=F<2'<1$-@O#=$KT2MD-''''''''/l6-sbr=uK?E-s500-%1%doubleslitrevised.003.pn$
4." – #tanding waves The nature o! standin$ wavesStandin$ waves (stationar) waves result !rom the superposition o! two oppositewaves which are otherwise identical.=ner$ is not trans!erred b standin$ waves.
http:--www.phsicsclassroom.com-lass-waves-u0l4b.$i! ' wave hits a wall and is re!lected identicall opposite.
http:--www.phsicsclassroom.com-lass-waves-sw!.$i! The blac> wave shows the wave created b the superposition o! the blue and $reenwaves.
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http:--www.phsicsclassroom.com-lass-waves-u0l4eani.$i!
oundar conditions 'ir particles can oscillate and create standin$ waves in pipes with open or closedends. 'ntinodes are positioned at open ends and nodes are positioned at closed ends.Standin$ waves on a strin$ is equivalent to that in a pipe which is closed on bothends (nodes%node).
http:--/.bp.blo$spot.com-%@=!aS1,CpM-@H=c@3BFD4D-''''''''01o-hM0"KwFDpeM-s500-2%/%sound42.0.pn$
The !ollowin$ table summari9es the behavior o! standin$ waves in pipes and strin$s:
One closed end and oneopen end
Two closed ends Two open ends
http:--www.phsicsclassroom.com-lass-sound-ul1d.$i!
st Harmonic
http:--www.phsicsclassroom.com-lass-waves-u0l4e.$i!
http:--www.phsicsclassroom.com-lass-sound-ul1c.$i!
http:--www.phsicsclassroom.com-lass-sound-ul1d2.$i!
2nd Harmonic
http:--www.phsicsclassroom.com-lass-waves-u0l4e2.$i!
http:--www.phsicsclassroom.com-lass-sound-ul
1c2.$i!
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http:--www.phsicsclassro
om.com-lass-sound-ul1d/.$i!
/rd Harmonic
http:--www.phsicsclassroom.com-lass-waves-u0l4
e/.$i!
http:--www.phsicsclassro
om.com-lass-sound-ul1c/.$i!
nth HarmonicR84,-n(*ote that even harmonicsdo not e#ist !or pipes withone closed end and oneopen end)
nth HarmonicR82,-n
nth HarmonicR82,-n
*odes and antinodesCositions alon$ the wave which are !i#ed are called nodes (minimum) and those with
the lar$est displacement are called antinodes (ma#imum)."or standin$ waves& the distance between adacent nodes 8 the distance betweenadacent antinodes 8 R-2.
http:--www.phsicsclassroom.com-lass-waves-u0l4d2ani.$i!
"JDAi!!erence between standin$ waves and travellin$ waves
Standin$ waves Travellin$ waves
*o ener$ is propa$ated alon$ a
standin$ wave ' standin$ wave has nodes and
antinodes The amplitude o! the standin$ wave
varies alon$ the wave Carticles between two adacent
nodes oscillate in phase and
particles separated b e#actl onenode oscillate in antiphase.
=ner$ is propa$ated in a travellin$
wave ' travellin$ wave has neither nodes
nor antinodes The amplitude o! the standin$ wave
is constant throu$hout the wave The phase di!!erence between two
particles o! a travellin$ wave can
ta>e an value between 0 and 2