7/24/2019 Topic 4: Waves http://slidepdf.com/reader/full/topic-4-waves 1/18 Topic 4: Waves 4.1 – Oscillations Simple harmonic oscillations Oscillations 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$oes SHM i! it e#periences a !orce which is proportional and opposite o! the displacement !rom its equilibrium position. *ewton+s Second ,aw& SHM can be de!ined as the !ollowin$ equations http:--ima$e.slidesharecdn.com-4%simpleharmonicmotion%/01220/4/% phpapp02-31-4%simple%harmonic%motion%%5/6.p$7cb8/5356514 where #0 is the amplitude (ma#imum displacement)& # is the displacement& v is the velocit& and a is the acceleration. The an$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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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$equation
onditions !or simple harmonic motionDn a SHM& there is an interchan$e between G= and C= throu$hout the motion.However& the total ener$ remains constant.
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
' 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$ion where particles are !urthestapart !rom each other is called arare!action.
'mplitude and intensitThe amplitude and intensit o! a wave depends on its ener$.The intensit o! a wave is proportional to the square o! its amplitude (D∝ '2).
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.
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$le& the re!lected ra is totall
polari9ed and the re!lected ra is perpendicular to the re!racted ra.
http:--2.bp.blo$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$le between the transmission a#is and the anal9er.
"ast%to%slow: towards normalQ slow%to%!ast: awa !rom normal
Dn addition& the re!ractive inde# n and n2 are related b the !ollowin$ 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$th o! the waves o! their respective mediums
Total internal re!lection onl occurs when the li$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 $iven b
Dnter!erence patternsMa#imums !orm at constructive inter!erence (the ma#imum is shown b %2) andminimums !orm at deconstructive inter!erence (the !irst minimum is shown b /%4).
Aouble%slit inter!erence,i>e sin$le%slit di!!raction& double%slit di!!raction occurs via the same methods o!inter!erence and has a similar di!!raction pattern.
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 $reenwaves.
oundar conditions 'ir particles can oscillate and create standin$ 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).
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.