Wave Motion

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Nora Craig
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1 Wave Motio Wave ad Wave motio: Wave is a carrier of eergy Wave is a form of disturbace which travels through a material medium due to the repeated periodic motio of the particles of the medium about their mea positio The disturbace is haded over from oe particle to aother particle of the medium Characteristics of wave motio: Wave motio is the disturbace travelig through the medium Whe a wave travels through a medium, its particles execute SHM about their mea positios Particles of medium had over the eergy to their ext eighbors, but their displacemet over oe time period is zero As the disturbace reaches a particle, it starts vibratig The disturbace is trasverse to the ext particle a little latter Hece there is a regular phase differece betwee cosecutive particles The wave travels with a uiform velocity where as the velocity of the particles is differet at differet positios The wave velocity depeds o the medium ad the particle velocity is the fuctio of time I wave motio, the trasfer of eergy ad mometum takes place from oe poit to aother of the medium, but ot matter The properties of medium ecessary for wave propagatio : a) The medium should have the property of iertia b) The medium should posses the property of elasticity c) The medium should have low resistace (o viscous) Types of Motio: Wave motio is of two types ) Trasverse wave motio ad ) Logitudial wave motio Trasverse wave motio: The wave motio i which the particles of the medium vibrate about their mea positios at right agles to the directio of propagatio of the wave is called trasverse waves Ex: Waves i a stretched strig, ripples o water surface, Electromagetic Waves etc The regio of elevatio of the medium through which the wave propagates is called crest ad the regio of depressio is called trough The distace betwee two cosecutive crests (or) troughs is called wavelegth ( ) These travel i a medium which has the elasticity of shape These ca travel i solids Desity of the medium does ot chage durig wave motio The phase differece betwee the particles at two cosecutive crests (or) troughs is π radias (or)360 These ca be polarized Logitudial wave motio:

2 The wave motio i which the particles of the material medium vibrate back ad forth about their mea positio alog the directio of the propagatio of wave is called logitudial waves Ex: Waves produced whe a sprig fixed at oe ed is placed ad released, soud waves i air etc The regio i which particles come close to a distace less tha the ormal distace is called compressio The regio i which the particle get apart to a distace greater tha the ormal distace betwee them is called rare fractio The distace betwee two cosecutive compressios (or) rarefactios is called wave legth ( ) These travel i a medium which it has elasticity of volume These ca travel i solids, liquids ad gases Desity of the medium chages durig wave motio I the compressio regio, the desity of the media icreases ad i the rare fractio, the desity of the medium decreases The phase differeces betwee the particles at two cosecutive compressios has rare fractio is π radias or360 These caot be polarized Properties of Progressive waves These waves propagate i the forward directio of medium with fiite velocity Eergy is propagated via these waves I these waves all the particles of the medium execute SHM with same amplitude ad same frequecy I these waves all the particles of the medium pass through their mea positio or positios of maximum displacemets oe after the other I these waves the velocity of the particle ad the strai are proportioal to each other This wave is a idepedet oe I these waves equal chage i pressure ad desity occurs at all poits of medium I these waves equal strai is produced at all poits I these waves all the particles of the medium cross their mea positio oce i oe time period I these waves the average eergy over oe time period is equal to the sum of kietic eergy ad potetial eergy The eergy per uit volume of a progressive wave is ρ A ω where ρ is the desity of the medium The equatio of a progressive wave alog the positive directio of x-axis is y = Asi( ωt -kx) t Or y = Asi π ( x ) T x Or y = Asi π (t ) where y = displacemet of a particle at a istat t; A = amplitude; ω = agular frequecy = π ; T = time period ad k = propagatio costat π or agular wave umber or wave vector ad is equal to The time take for oe vibratio of a particle is called time period or period of vibratio (T = ) ad elocity of the wave v= The maximum displacemet of a vibratig particle from its mea positio is called amplitude The phase of vibratio at ay momet is the state of vibratig particle as regards its positio ad directio of motio at that momet

3 Phase differece Δ φ = π x Path differece The distace travelled by a wave i the time i which the particles of the medium complete oe vibratio or the distace betwee two earest particles i the same state of vibratio (ie, same phase) is called wavelegth ( ) Reflectio ad Refractio of Waves Rigid ed: Whe the icidet wave reaches a fixed ed, it exerts a upward pull o the ed; accordig to Newto's law the fixed ed exerts a equal ad opposite dow ward force o the strig It result a iverted pulse or phase chage ofπ Crest (C) reflects as trough (T) ad vice-versa, Time chages by T ad Path chages by Free ed: Whe a wave or pulse is reflected from a free ed, the there is o chage of phase (as there is o reactio force) Crest (C) reflects as crest (C) ad trough (T) reflects as trough (T), Time chages by zero ad Path chages by zero Priciple of Superpositio The displacemet at ay time due to ay umber of waves meetig simultaeously at a poit i a medium is the vector sum of the idividual displacemets due each oe of the waves at that poit at the same time If y, y, y3 are the displacemets at a particular time at a particular positio, due to idividual waves, the the resultat displacemet y = y y + y Properties of Statioary waves: a) All the particles except a few (at odes) execute SHM b) The period of each particle is the same but the amplitude of vibratio varies from particle to particle c) The distace betwee ay two successive odes or atiodes is equal to / d) The distace betwee a ode ad eighborig atiodes is equal to /4 e) The wave is cofied to a limited regio ad does ot advace f) All the particles of a wave i a loop are i the same phase ad the phase differece is zero g) Statioary waves are formed by combiig two logitudial progressive waves or two trasverse progressive waves h) These waves do ot trasfer eergy i) The chage i pressure or desity or strai will be maximum at odes ad miimum at atiodes j) The particle velocity at a ode is zero ad at atiodes it is maximum k) The phase differece betwee the particles i adjacet loops i a statioary wave is π l) The equatio of a statioary wave is y = A sikxcos ωt or y = A coskxsi ωt Types of vibratios: Wheever a body, capable of vibratio, is displaced from its equilibrium positio ad the left to itself, the body begis to vibrate freely i its ow atural way called the free or atural vibratio of the body with a defiite frequecy This frequecy is called atural frequecy

4 The free vibratios of a body have a uique frequecy ad it is depedet o the elasticity ad iertia of the body ad the mode of vibratio Whe a body is set ito vibratio with the help of strog periodic force havig a frequecy differet from its atural frequecy, the the vibratios of the body are called forced vibratios If the amplitude of vibratios progressively decreases with time, the they are called damped vibratios Eg ibratios of a tuig fork Bells are made of metals ad ot of wood because wood dampes the vibratios while the metals are elastic If the atural frequecy of a vibratig body is equal to the frequecy of the exteral periodic force ad if they are i phase, the frequecies are said to be i resoace Tuig a radio or televisio receiver is a example of electrical resoace Optical resoace may also take place betwee the atoms i a gas at low pressure ibratios of a strig: a) Strig ca have oly trasverse vibratios that too whe it is uder tesio T b) The velocity of trasverse wave propagatig alog a strig or wire uder tesio is = m where T is tesio ad m is liear desity of the strig or wire M= M = Ad = πr d where M is l total mass of wire of legth l, A is area of cross-sectio of wire ad r is its radius Hece T Tl T T = = = = m M Ad πr d s YStrai c) If s is stress i the wire, S=T/A, hece = ; also = d d d) A wire held at the two eds by rigid support is just taut at temperature t The velocity of Yα(t ~ t) trasverse wave at a temperature t is = where α =co-efficiet of liear d expasio, Y=Youg s modulus, d=desity Frequecy of a vibratig strig : a) The waves formed i a strig uder tesio are trasverse statioary b) Always odes are formed at fixed eds ad atiodes at plucked poits ad free eds c) A strig ca have umber of frequecies depedig o its mode of vibratio Fudametal frequecy: Whe a strig vibrates i a sigle loop, it is said to vibrate with fudametal frequecy a) Frequecy is miimum ad wavelegth is maximum i this case b) If l is the legth of the strig l= = l T c) The fudametal frequecy, = where T=tesio, m=liear desity l m T T T s d) The fudametal frequecy is also give by = = = = Ml l Ad l πr d l d Δ ΔT e) For small chage i tesio i strig, the fractioal chage i frequecy is = T f) The fudametal frequecy is also called the first harmoic Overtoes: If strig vibrates with more umber of loops, higher frequecies are produced called overtoes

5 a) If strig vibrates i p loops, it is called p th mode of vibratio or p th harmoic or (p ) th overtoe The correspodig frequecy p p = l T m =p Hece, for a strig, p pα p; = p whe other, quatities are costat b) The fudametal ad overtoe frequecies are i the ratio ::3:4: l c) The wavelegth is above case is p = ie, wavelegths are i the ratio : : : p 3 Laws of trasverse waves alog stretched strig : a) Law of legth: The frequecy of a stretched strig is iversely proportioal to the legth of the strig α / l where T & m are costats, l=costat, l = l b) Law of tesio: The frequecy of a stretched strig is iversely proportioal to square root of tesio α T Whe l & T are costat =costat, = T T T c) Law of mass: The frequecy of a stretched strig is iversely proportioal to square root of liear desity α whe l & T are costats m =costat; m = m m Soometer is used to determie the velocity of trasverse waves i strigs ad to verify the laws of trasverse waves l RD = l l Statioary waves i Orga pipes: A orga pipe is a cylidrical tube havig a air colum The vibratio of a cylidrical air colum are made of two progressive logitudial vibratios movig i opposite directios with equal ad opposite speed superposed o each other Hece the waves are logitudial statioary waves The possible frequecies i which stadig waves are formed are called harmoics Closed pipes: A Pipe whose oe ed is closed ad the other ed is ope is called closed pipe At the closed ed of the pipe always a ode is formed If l is the legth of the pipe = This is called fudametal frequecy (or) st harmoic 4 l

6 I the st overtoe (or) d harmoic, two odes ad two atiodes are formed i the pipe 3 = = 3 4l Similarly for the d overtoe (or) 3 rd harmoic three odes ad three atiodes are formed i the pipe Ope pipes: : : : = : 3: 5 : 3 5 = = 5 4l 3 A pipe whose both eds are ope is called ope pipe At the ope eds of the pipe always atiodes are formed If l is the legth of the pipe i the simplest mode of vibratio two atiodes are formed, oe at each ed with a ode at the middle of the pipe = This is called the fudametal frequecy (or) st harmoic l I the st overtoe (or) d harmoic, two odes ad three atiodes are formed i the pipe = = l Similarly for the d overtoe (or) 3 rd harmoic, three odes ad four atiodes are formed i the pipe 3 = = 3 l 3 : : = : : 3: 3 It is observed that i a ope pipe all harmoics are formed where as i a closed pipe oly odd harmoics are formed

7 I Geeral Atiodes are formed earer to the ope ed outside the pipe The exact distace betwee the positio of the atiode ad the mouth of the pipe is measured as ed correctio (e) The ed correctio is double for ope pipe Ed correctio = e= 03d = 06r where d is the diameter ad r is the radius of the pipe The distace of the exact atiode from the brim of the pipe is called ed correctio For a closed pipe, l c + e = 4 4lc e = 4 For a ope pipe, l o + e = e = 4 l o BEATS : Whe two souds of slightly differet frequecies superimpose, the resultat soud cosists of alterate waxig ad waxig This pheomeo is called beats Oe waxig ad oe waig together is called oe beat If simple harmoic progressive waves of frequecies & travellig i same directio + superimpose, the resultat wave is represeted by y= a cos π t si π t The amplitude of resultat wave is a cos π t The maximum amplitude is a ad miimum amplitude zero + The frequecy of resultat wave is The umber of beats produced per secod or beat frequecy is equal to the differece of frequecies of odes producig beats = ~ If two soud waves of wavelegths ad produce beats per secod, the velocity of soud ca be determied by = ~ or = ( ~ ) The maximum umber of beats heard by a perso is 0, sice persistece of hearig is /0 sec The time iteral betwee two cosecutive maxima or miima is ( ~ ) The time iterval betwee cosecutive maxima ad miima is ( ~ ) Beats ca be produced by takig two idetical tuig fork ad loadig or filig either of them ad vibratig them together Whe a tuig fork is loaded its frequecy decreases ad whe it is filed frequecy icreases

8 DOPPLER EFFECT : The apparet chage i frequecy due to relative motio betwee the source ad the listeer is called Doppler Effect Let O ad s represets the velocities of a listeer ad a source respectively Let be the velocity of soud ad ad be the true ad apparet frequecies of the soud The if a) The source aloe is i motio towards the observer, = Clearly > s b) the source aloe is i motio away from the observer, = Clearly < + s + c) the observer aloe is i motio towards the source, = O Clearly > d) the observer aloe is i motio away from the source, = O Clearly < + e) the source ad the observer both are i motio towards each other, = O s f) the source ad the observer both are i motio away from each other, = O + s g) the source ad the observer both are i motio, source followig the observer, = O s + h) the source ad the observer both are i motio, observer followig the source, = O + s i) the source, observer ad the medium all are movig i the same directio as the soud, + = w O where w = velocity of wid + w s j) the source ad the observer are movig i the directio of the soud but the directio of wid is opposite to the directio of the propagatio of soud, = w O w s If the source of soud is movig towards a wall ad the observer is stadig betwee the source ad the wall, o beats are heard by the observer I Whe source ad observer are ot movig alog the same lie the = where s cos θ θ is agle betwee source velocity ad lie joiig source ad observer Whe source ad observer do ot move alog the lie joiig them, the compoets of their velocities alog the lie joiig them must be take as velocity of observer ad velocity of + 0 cos θ source i Doppler is formula = 0 s cos θ If r is uit vector alog lie joiig source ad observer, v is velocity of soud (take from the source to observer), v 0 is velocity of observer ad v s is velocity of source the Doppler s (vr v 0r ) effect i vector form is = ( vr v s r ) Doppler Effect i soud is asymmetric This meas the chage i frequecy depeds o whether the source is i motio or observer is i motio eve though relative velocities are same i both cases

9 Motio of source produces greater chage tha motio of observer eve though the relative velocities are same i both cases v Eg : I = v u v u II + = v I II > Doppler Effect i soud is asymmetric because soud is mechaical wave requirig material medium ad v, v 0, v s are take with respect to the medium Doppler effect i light is symmetric because light waves are electromagetic (do ot require medium) Doppler effect is ot applicable if ) 0 = s =0 (both are at rest) ) 0 = s =0 ad medium is aloe i motio 3) 0 = s =u ( 0, s are i same directio) 4) s is to lie of sight Doppler effect is applicable oly whe, 0 <<v ad s <<v (v=velocity of soud) a) Doppler effect i soud is asymmetric b) Doppler Effect holds good for light also A icrease of frequecy is called blue shift ad it idicates that the source is approachig the observer Red shift idicates that the source is recedig from the observer Red shift ( Δ) = c c) Doppler Effect i light is symmetric d) The red shift observed by Hubble i may stars supports the Big Bag Theory of the uiverse Uses of Doppler effect : It is used i a) SONAR b) RADAR (Radio detectio ad Ragig used to determie speed of objects i space) (Radio waves) c) To determie speeds of automobiles by traffic police d) To determie speed of rotatio of su ad to explai Satur s rigs e) Led to the discovery of double stars/biary stars f) I accurate avigatio ad accurate target bombig techiques g) I trackig earth s satellite Doppler s effect is used i the estimatio of the velocities of aero plaes ad submaries, the velocities of stars ad galaxies ad the velocities of satellites If the observer is stadig behid the source movig towards a wall with a velocity s, the the umber of beats heard is equal to ad is approximately equal to s s + s

Types of Waves Transverse Shear. Waves. The Wave Equation

Types of Waves Transverse Shear. Waves. The Wave Equation Waves Waves trasfer eergy from oe poit to aother. For mechaical waves the disturbace propagates without ay of the particles of the medium beig displaced permaetly. There is o associated mass trasport.