Topic 4: Waves

7/24/2019 Topic 4: Waves

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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



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



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



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$itudinal%waves-pspictures-a/c0e40e/2d2524dc6cb6eeaeed.pn$



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.



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).



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



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.



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



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.



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$



"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-%



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!



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.



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!



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.



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!



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

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Topic 4: Waves