Physics of waves and oscillations

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THEPHYSICSOFWAVES HOWARDGEORGI HarvardUniversity Originallypublishedby PRENTICEHALL EnglewoodCliffs,NewJersey07632Chapter1 HarmonicOscillation Oscillators are the basic building blocks of waves. We begin by discussing the harmonic oscillator. We will identify the general principles that make the harmonic oscillator so spe- cial and important. To make use of these principles, we must introduce the mathematical device of complex numbers. But the advantage of introducing this mathematics is that we can understand the solution to the harmonic oscillator problem in a new way. We show that the properties of linearity and time translation invariance lead to solutions that are complex exponentialfunctionsoftime. Preview Inthischapter,wediscussharmonicoscillationinsystemswithonlyonedegreeoffreedom. 1. We begin with a review of the simple harmonic oscillator, noting that the equation of motionofafreeoscillatorislinearandinvariantundertimetranslation; 2. We discuss linearity in more detail, arguing that it is the generic situation for small oscillationsaboutapointofstableequilibrium; 3. We discuss time translation invariance of the harmonic oscillator, and the connection betweenharmonicoscillationanduniformcircularmotion; 4. Weintroducecomplexnumbers,anddiscusstheirarithmetic; 5. Usingcomplexnumbers,wefindsolutionstotheequationofmotionfortheharmonic oscillator that behave as simply as possible under time translations. We call these solutions“irreducible.” Weshowthattheyareactuallycomplexexponentials. 6. We discuss anLC circuit and draw an analogy between it and a system of a mass and springs. 12 CHAPTER1. HARMONICOSCILLATION 7. Wediscussunits. 8. Wegiveonesimpleexampleofanonlinearoscillator. 1.1 TheHarmonicOscillator When you studied mechanics, you probably learned about the harmonic oscillator. We will beginourstudyofwavephenomenabyreviewingthissimplebutimportantphysicalsystem. 1 Considerablockwithmass,m,freetoslideonafrictionlessair-track,butattachedtoalight Hooke’s law spring with its other end attached to a fixed wall. A cartoon representation of thisphysicalsystemisshowninfigure1.1.  - ....... ....... ....... ....... ....... ....... ....... ....... ...... ...... ...... ...... ...... ...... ...... ...... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ........ ........ ........ ........ ........ ........ ........ ........ .......... .......... .......... .......... .......... .......... .......... .......... ........ ................ ................ ................ ................ ................ ................ ................ ........ Figure1.1: Amassonaspring. This system has only one relevant degree of freedom. In general, the number of de- grees of freedom of a system is the number of coordinates that must be specified in order to determine the configuration completely. In this case, because the spring is light, we can assumethatitisuniformlystretchedfromthefixedwalltotheblock. Thentheonlyimportant coordinateisthepositionoftheblock. Inthissituation, gravityplaysnoroleinthemotionoftheblock. Thegravitationalforce is canceled by a vertical force from the air track. The only relevant force that acts on the block comes from the stretching or compression of the spring. When the spring is relaxed, there is no force on the block and the system is in equilibrium. Hooke’s law tells us that theforcefromthespringisgivenbyanegativeconstant,−K,timesthedisplacementofthe block from its equilibrium position. Thus if the position of the block at some time isx and itsequilibriumpositionisx ,thentheforceontheblockatthatmomentis 0 F =−K(x−x ). (1.1) 0 1 “Light” here means that the mass of the spring is small enough to be ignored in the analysis of the motion of the block. We will explain more precisely what this means in chapter 7 when we discuss waves in a massive spring.1.1. THEHARMONICOSCILLATOR 3 The constant, K, is called the “spring constant.” It has units of force per unit distance, or −2 MT in terms ofM (the unit of mass),L (the unit of length) andT (the unit of time). We canalwayschoosetomeasuretheposition,x,oftheblockwithouroriginattheequilibrium position. Ifwedothis,thenx = 0in(1.1)andtheforceontheblocktakesthesimplerform 0 F =−Kx. (1.2) HarmonicoscillationresultsfromtheinterplaybetweentheHooke’slawforceandNew- ton’s law,F =ma. Letx(t) be the displacement of the block as a function of time,t. Then Newton’slawimplies 2 d m x(t) =−Kx(t). (1.3) 2 dt Anequationofthisform,involvingnotonlythefunctionx(t),butalsoitsderivativesiscalled a“differentialequation.” Thedifferentialequation, (1.3), is the“equation of motion” for the system of figure 1.1. Because the system has only one degree of freedom, there is only one equation of motion. In general, there must be one equation of motion for each independent coordinaterequiredtospecifytheconfigurationofthesystem. The most general solution to the differential equation of motion, (1.3), is a sum of a constanttimescosωtplusaconstanttimessinωt, x(t) =acosωt+bsinωt, (1.4) where √ K ω≡ (1.5) m −1 isaconstantwithunitsofT calledthe“angularfrequency.” Theangularfrequencywillbe a very important quantity in our study of wave phenomena. We will almost always denote it bythelowercaseGreekletter,ω (omega). Because the equation involves a second time derivative but no higher derivatives, the most general solution involves two constants. This is just what we expect from the physics, because we can get a different solution for each value of the position and velocity of the block at the starting time. Generally, we will think about determining the solution in terms of the position and velocity of the block when we first get the motion started, at a time that we conventionally take to bet = 0. For this reason, the process of determining the solution in terms of the position and velocity at a given time is called the “initial value problem.” The values of position and velocity att = 0 are called initial conditions. For example, we ′ canwritethemostgeneralsolution,(1.4),intermsofx(0)andx (0),thedisplacementand velocityoftheblockattimet = 0. Settingt = 0in(1.4)givesa =x(0). Differentiatingand ′ thensettingt = 0givesb =ωx (0). Thus 1 ′ x(t) =x(0)cosωt+ x (0)sinωt. (1.6) ω4 CHAPTER1. HARMONICOSCILLATION For example, suppose that the block has a mass of 1 kilogram and that the spring is 0.5 2 meters long with a spring constant K of 100 newtons per meter. To get a sense of what this spring constant means, consider hanging the spring vertically (see problem (1.1)). The gravitationalforceontheblockis mg≈ 9.8 newtons. (1.7) In equilibrium, the gravitational force cancels the force from the spring, thus the spring is stretchedby mg ≈ 0.098 meters = 9.8 centimeters. (1.8) K Forthismassandspringconstant,theangularfrequency,ω,ofthesysteminfigure1.1is √ √ K 100N/m 1 ω = = = 10 . (1.9) M 1kg s If, for example, the block is displaced by 0.01 m (1 cm) from its equilibrium position and releasedfromrestattime,t = 0,thepositionatanylatertimetisgiven(inmeters)by x(t) = 0.01×cos10t. (1.10) Thevelocity(inmeterspersecond)is ′ x (t) =−0.1×sin10t. (1.11) The motion is periodic, in the sense that the system oscillates — it repeats the same motion overandoveragainindefinitely. Afteratime 2π τ = ≈ 0.628 s (1.12) ω the system returns exactly to where it was at t = 0, with the block instantaneously at rest with displacement 0.01 meter. The time, τ (Greek letter tau) is called the “period” of the oscillation. However, the solution, (1.6), is more than just periodic. It is “simple harmonic” motion,whichmeansthatonlyasinglefrequencyappearsinthemotion. Theangularfrequency,ω,istheinverseofthetimerequiredforthephaseofthewaveto change by one radian. The “frequency”, usually denoted by the Greek letter, ν (nu), is the inverse of the time required for the phase to change by one complete cycle, or 2π radians, and thus get back to its original state. The frequency is measured in hertz, or cycles/second. Thustheangularfrequencyislargerthanthefrequencybyafactorof2π, ω (inradians/second) = 2π (radians/cycle) · ν (cycles/second). (1.13) 2 Thelengthofthespringplaysnoroleintheequationsbelow,butweincludeittoallowyoutobuildamental pictureofthephysicalsystem.1.2. SMALLOSCILLATIONSANDLINEARITY 5 Thefrequency,ν,istheinverseoftheperiod,τ,of(1.12), 1 ν = . (1.14) τ Simpleharmonicmotionlike(1.6)occursinaverywidevarietyofphysicalsystems. The question with which we will start our study of wave phenomena is the following: Why do solutionsoftheformof(1.6)appearsoubiquitouslyinphysics? Whatdoharmonically oscillatingsystemshaveincommon? Of course, the mathematical answer to this question is that all of these systems have equations of motion of essentially the same form as (1.3). We will find a deeper and more physical answer that we will then be able to generalize to morecomplicatedsystems. Thekeyfeaturesthatallthesesystemshaveincommonwiththe mass on the spring are (at least approximate) linearity and time translation invariance of the equations of motion. It is these two features that determine oscillatory behavior in systems fromspringstoinductorsandcapacitors. Each of these two properties is interesting on its own, but together, they are much more powerful. They almost completely determine the form of the solutions. We will see that if the system is linear and time translation invariant, we can always write its motion as a sum ofsimplemotionsinwhichthetimedependenceiseitherharmonicoscillationorexponential decay(orgrowth). 1.2 SmallOscillationsandLinearity A system with one degree of freedom is linear if its equation of motion is a linear function ofthecoordinate,x,thatspecifiesthesystem’sconfiguration. Inotherwords,theequationof motionmustbeasumoftermseachofwhichcontainsatmostonepowerofx. Theequation of motion involves a second derivative, but no higher derivatives, so a linear equation of motionhasthegeneralform: 2 d d α x(t)+β x(t)+γx(t) =f(t). (1.15) 2 dt dt Ifallofthetermsinvolveexactlyonepowerofx,theequationofmotionis“homogeneous.” Equation (1.15) is not homogeneous because of the term on the right-hand side. The “in- homogeneous” term, f(t), represents an external force. The corresponding homogeneous equationwouldlooklikethis: 2 d d α x(t)+β x(t)+γx(t) = 0. (1.16) 2 dt dt In general,α,β andγ as well asf could be functions oft. However, that would break thetimetranslationinvariancethatwewilldiscussinmoredetailbelowandmakethesystem6 CHAPTER1. HARMONICOSCILLATION much more complicated. We will almost always assume thatα,β andγ are constants. The equation of motion for the mass on a spring, (1.3), is of this general form, but withβ andf equal to zero. As we will see in chapter 2, we can include the effect of frictional forces by allowingnonzeroβ,andtheeffectofexternalforcesbyallowingnonzerof. The linearity of the equation of motion, (1.15), implies that if x (t) is a solution for 1 externalforcef (t), 1 2 d d α x (t)+β x (t)+γx (t) =f (t), (1.17) 1 1 1 1 2 dt dt andx (t)isasolutionforexternalforcef (t), 2 2 2 d d α x (t)+β x (t)+γx (t) =f (t), (1.18) 2 2 2 2 2 dt dt thenthesum, x (t) =Ax (t)+Bx (t), (1.19) 12 1 2 forconstantsAandB isasolutionforexternalforceAf +Bf , 1 2 2 d d α x (t)+β x (t)+γx (t) =Af (t)+Bf (t). (1.20) 12 12 12 1 2 2 dt dt The sum x (t) is called a “linear combination” of the two solutions, x (t) and x (t). In 12 1 2 thecaseof“free”motion,whichmeansmotionwithnoexternalforce,ifx (t)andx (t)are 1 2 solutions,thenthesum,Ax (t)+Bx (t)isalsoasolution. 1 2 The most general solution to any of these equations involves two constants that must be fixedbytheinitialconditions,forexample,theinitialpositionandvelocityoftheparticle,as in (1.6). It follows from (1.20) that we can always write the most general solution for any externalforce,f(t),asasumofthe“generalsolution”tothehomogeneousequation,(1.16), andany“particular”solutionto(1.15). Nosystemisexactlylinear. “Linearity”isneverexactly“true.” Nevertheless,theideaof linearity is extremely important, because it is a useful approximation in a very large number ofsystems,foraverygoodphysicalreason. Inalmostanysysteminwhichthepropertiesare smoothfunctionsofthepositionsoftheparts,thesmalldisplacementsfromequilibriumpro- duce approximately linear restoring forces. The difference between something that is “true” andsomethingthatisausefulapproximationistheessentialdifferencebetweenphysicsand mathematics. In the real world, the questions are much too interesting to have answers that are exact. If you can understand the answer in a well-defined approximation, you havelearnedsomethingimportant. To see the generic nature of linearity, consider a particle moving on thex-axis with po- tential energy,V(x). The force on the particle at the point,x, is minus the derivative of the potentialenergy, d F =− V(x). (1.21) dx1.2. SMALLOSCILLATIONSANDLINEARITY 7 A force that can be derived from a potential energy in this way is called a “conservative” force. At a point of equilibrium, x , the force vanishes, and therefore the derivative of the 0 potentialenergyvanishes: d ′ F =− V(x) =−V (x ) = 0. (1.22) 0 x=x 0 dx We can describe the small oscillations of the system about equilibrium most simply if we redefinetheoriginsothatx = 0. Thenthedisplacementfromequilibriumisthecoordinate 0 x. WecanexpandtheforceinaTaylorseries: 1 ′ ′ ′′ 2 ′′′ F(x) =−V (x) =−V (0)−xV (0)− x V (0)+··· (1.23) 2 Thefirsttermin(1.23)vanishesbecausethissystemisinequilibriumatx = 0, from(1.22). ThesecondtermlookslikeHooke’slawwith ′′ K =V (0). (1.24) The equilibrium is stable if the second derivative of the potential energy is positive, so that x = 0isalocalminimumofthepotentialenergy. Theimportantpointisthatforsufficientlysmallx,thethirdtermin(1.23),andall subsequenttermswillbemuchsmallerthanthesecond. Thethirdtermisnegligibleif ′′′ ′′ xV (0) ≪V (0). (1.25) Typically,eachextraderivativewillbringwithitafactorof1/L,whereListhedistanceover whichthepotentialenergychangesbyalargefraction. Then(1.25)becomes x≪L. (1.26) There are only two ways that a force derived from a potential energy can fail to be approxi- matelylinearforsufficientlysmalloscillationsaboutstableequilibrium: 1. Ifthepotentialisnotsmoothsothatthefirstorsecondderivativeofthepotentialisnot well defined at the equilibrium point, then we cannot do a Taylor expansion and the argument of (1.23) does not work. We will give an example of this kind at the end of thischapter. 2. Even if the derivatives exist at the equilibrium point, x = 0, it may happen that ′′ ′′′ V (0) = 0. In this case, to have a stable equilibrium, we must have V (0) = 0 as well, otherwise a small displacement in one direction or the other would grow with time. Then the next term in the Taylor expansion dominates at smallx, giving a force 3 proportionaltox .8 CHAPTER1. HARMONICOSCILLATION .. . ... .... .... .... ... ... .... .... ... ... ... .... ... .... ... ... ... .... 5E ... .... .. ... ... .... ... ... .. .... ... .... .. ... ... .... .. .... .. ... ... .... .. .... .. ... ... .... .. .... .. ... .. .... ... .... .. ... .. .... .. .... .. ... ... .... .. .... .. ... .. .... .. .... .. ... .. .... 4E ... .... .. .... .. ... .. .... .. .... .. ... ... .... .. .... .. .... .. ... .. .... ... .... .. .... .. ... ... .... .. .... .. .... ... ... .. .... ... .... .. .... ... .... ... ... .. .... ... .... ... .... ... .... .... .... ... ... ... .... 3E ... .... ... .... ... .... ... .... .. .... ... .... .. .... ... .... .. .... .. .... ... .... .. .... .. ..... ... .... .. .... ... .... .. ..... ... .... ... ..... ... .... ... ..... .... ..... ... ..... ... ...... .... ...... ..... ....... ....... .......... ................ 2E E 0 0 L 2L 3L 4L 5L Figure1.2: Thepotentialenergyof(1.27). Both of these exceptional cases are very rare in nature. Usually, the potential energy is a ′′ smoothfunctionofthedisplacementandthereisnoreasonforV (0)tovanish. Thegeneric situationisthatsmalloscillationsaboutstableequilibriumarelinear. Anexamplemaybehelpful. Almostanypotentialenergyfunctionwithapointofstable equilibrium will do, so long as it is smooth. For example, consider the following potential energy ( ) L x V(x) =E + . (1.27) x L This is shown in figure 1.2. The minimum (at least for positivex) occurs atx = L, so we firstredefinex =X +L,sothat ( ) L X +L V(X) =E + . (1.28) X +L L Thecorrespondingforceis ( ) L 1 F(X) =E − . (1.29) 2 (X +L) L wecanlooknearX = 0andexpandinaTaylorseries: ( ) ( ) 2 E X E X F(X) =−2 +3 +··· (1.30) L L L L Now,theratioofthefirstnonlineartermtothelineartermis 3X , (1.31) 2L1.3. TIMETRANSLATIONINVARIANCE 9 whichissmallifX≪L. Inotherwords,thecloseryouaretotheequilibriumpoint,theclosertheactualpotential energyistotheparabolathatwewouldexpectfromthepotentialenergyforalinear,Hooke’s lawforce. Youcanseethisgraphicallybyblowingupasmallregionaroundtheequilibrium point. In figure 1.3, the dotted rectangle in figure 1.2 has been blown up into a square. Note that it looks much more like a parabola than figure 1.3. If we repeated the procedure and again expanded a small region about the equilibrium point, you would not be able to detect thecubictermbyeye. 2.1E . .... ... ... .... ... .... ... .... .... ... .... .... .... .... ... .... .... ..... .... ..... .... ...... ..... ..... .... ..... .... ..... .... ...... ..... ..... ..... ...... .... ...... ..... ...... ..... ...... ...... ....... ..... ...... ...... ....... ...... ....... ....... ........ ....... ........ ....... .......... ......... .......... .......... ........... .............. .............. ....................... ....................... ............................ 2E 0.9L L 1.1L Figure1.3: Thesmalldashedrectangleinfigure1.2expanded. 2 Often, the linear approximation is even better, because the term of orderx vanishes by symmetry. For example, when the system is symmetrical about x = 0, so that V(x) = 3 n V(−x), the orderx term (and allx forn odd) in the potential energy vanishes, and then 2 thereisnoorderx termintheforce. For a typical spring, linearity (Hooke’s law) is an excellent approximation for small dis- placements. However, there are always nonlinear terms that become important if the dis- placementsarelargeenough. Usually, inthisbookwewillsimplysticktosmalloscillations and assume that our systems are linear. However, you should not conclude that the subject of nonlinear systems is not interesting. In fact, it is a very active area of current research in physics. 1.3 TimeTranslationInvariance 1.3.1 UniformCircularMotion ............................ ........................... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ............................... ................................ .................................1-1 ................ .................................................10 CHAPTER1. HARMONICOSCILLATION Whenα,β andγ in (1.15) do not depend on the time,t, and in the absence of an external force,thatisforfreemotion,timeentersin(1.15)onlythroughderivatives. Thentheequation ofmotionhastheform. 2 d d α x(t)+β x(t)+γx(t) = 0. (1.32) 2 dt dt Theequationofmotionfortheundampedharmonicoscillator, (1.3), hasthisformwithα = m,β = 0andγ =K. Solutionsto(1.32)havethepropertythat Ifx(t)isasolution,x(t+a)willbeasolutionalso. (1.33) Mathematically,thisistruebecausetheoperationsofdifferentiationwithrespecttotimeand replacingt→t+acanbedoneineitherorderbecauseofthechainrule d d d d ′ ′ x(t+a) = (t+a) x(t) = x(t) . (1.34) ′ ′ dt dt dt ′ dt ′ t =t+a t =t+a The physical reason for (1.33) is that we can change the initial setting on our clock and the physicswilllookthesame. Thesolutionx(t+a)canbeobtainedfromthesolutionx(t)by changing the clock setting bya. The time label has been “translated” bya. We will refer to theproperty,(1.33),astimetranslationinvariance. Most physical systems that you can think of are time translation invariant in the absence ofanexternalforce. Togetanoscillatorwithouttimetranslationinvariance,youwouldhave todosomethingratherbizarre,suchassomehowmakingthespringconstantdependontime. For the free motion of the harmonic oscillator, although the equation of motion is cer- tainlytimetranslationinvariant, themanifestationoftimetranslationinvarianceonthesolu- tion, (1.6) is not as simple as it could be. The two parts of the solution, one proportional to cosωtandtheothertosinωt,getmixedupwhentheclockisreset. Forexample, cosω(t+a) = cosωa cosωt−sinωa sinωt. (1.35) It will be very useful to find another way of writing the solution that behaves more simply underresettingoftheclocks. Todothis,wewillhavetoworkwithcomplexnumbers. Tomotivatetheintroductionofcomplexnumbers,wewillbeginbyexhibitingtherelation between simple harmonic motion and uniform circular motion. Consider uniform circular motioninthex-y planearoundacirclecenteredattheorigin,x =y = 0,withradiusR and withclockwisevelocityv =Rω. Thexandy coordinatesofthemotionare x(t) =Rcos(ωt−ϕ), y(t) =−Rsin(ωt−ϕ), (1.36) whereϕisthecounterclockwiseangleinradiansofthepositionatt = 0fromthepositivex axis. Thex(t)in(1.36)isidenticaltothex(t)in(1.6)with ′ x(0) =Rcosϕ, x (0) =ωRsinϕ. (1.37)1.3. TIMETRANSLATIONINVARIANCE 11 Simple harmonic motion is equivalent to one component of uniform circular motion. This relation is illustrated in figure 1.4 and in program 1-1 on the programs disk. As the point movesaroundthecircleatconstantvelocity,Rω,thexcoordinateexecutessimpleharmonic motion with angular velocityω. If we wish, we can choose the two constants required to fix ′ thesolutionof(1.3)tobeR andϕ, insteadofx(0) andx (0). In thislanguage, the actionof resettingoftheclockismoretransparent. Resettingtheclockchangesthevalueofϕwithout changinganythingelse. q q q q q q q q q q q q q q q qq qq q q q q q q q R q q q q q q q q Rω qqqqqqqqqqqqqqqqq Figure1.4: Therelationbetweenuniformcircularmotionandsimpleharmonicmotion. But we would like even more. The key idea is that linearity allows us considerable freedom. We can add solutions of the equations of motion together and multiply them by constants, and the result is still a solution. We would like to use this freedom to choose solutionsthatbehaveassimplyaspossibleundertimetranslations. Thesimplestpossiblebehaviorforasolutionz(t)undertimetranslationis z(t+a) =h(a)z(t). (1.38) That is, we would like find a solution that reproduces itself up to an overall constant, h(a) when we reset our clocks by a. Because we are always free to multiply a solution of a homogeneous linear equation of motion by a constant, the change from z(t) to h(a)z(t) 3 doesn’t amount to much. We will call a solution satisfying (1.38) an “irreducible solution” with respect to time translations, because its behavior under time translations (resettings of theclock)isassimpleasitcanpossiblybe. It turns out that for systems whose equations of motion are linear and time translation invariant, as we will see in more detail below, we can always find irreducible solutions that 3 The word “irreducible” is borrowed from the theory of group representations. In the language of group theory,theirreduciblesolutionisan“irreduciblerepresentationofthetranslationgroup.” Itjustmeans“assimple aspossible.”12 CHAPTER1. HARMONICOSCILLATION havetheproperty,(1.38). However,forsimpleharmonicmotion,thisrequirescomplexnum- bers. Youcanseethisbynotingthatchangingtheclocksettingbyπ/ω justchangesthesign ofthesolutionwithangularfrequencyω,becauseboththecosandsintermschangesign: cos(ωt+π) =−cosωt, sin(ωt+π) =−sinωt. (1.39) Butthenfrom(1.38)and(1.39),wecanwrite −z(t) =z(t+π/ω) =z(t+π/2ω +π/2ω) (1.40) 2 =h(π/2ω)z(t+π/2ω) =h(π/2ω) z(t). Thuswecannotfindsuchasolutionunlessh(π/2ω)hastheproperty 2 h(π/2ω) =−1. (1.41) 4 The square of h(π/2ω) is−1 Thus we are forced to consider complex numbers. When we finish introducing complex numbers, we will come back to (1.38) and show that we can alwaysfindsolutionsofthisformforsystemsthatarelinearandtimetranslationinvariant. 1.4 ComplexNumbers The square root of−1, calledi, is important in physics and mathematics for many reasons. Measurable physical quantities can always be described by real numbers. You never get a reading ofi meters on your meter stick. However, we will see that wheni is included along with real numbers and the usual arithmetic operations (addition, subtraction, multiplication and division), then algebra, trigonometry and calculus all become simpler. While complex numbers are not necessary to describe wave phenomena, they will allow us to discuss them inasimplerandmoreinsightfulway. 1.4.1 SomeDefinitions Animaginarynumberisanumberoftheformitimesarealnumber. Acomplexnumber,z,isasumofarealnumberandanimaginarynumber:z =a+ib. Therealand“imaginary”parts,Re(z)andIm(z),ofthecomplexnumberz =a+ib: Re(z) =a, Im(z) =b. (1.42) 4 TheconnectionbetweencomplexnumbersanduniformcircularmotionhasbeenexploitedbyRichardFeyn- maninhisbeautifullittlebook,QED.1.4. COMPLEXNUMBERS 13 Notethattheimaginarypartisactuallyarealnumber,therealcoefficientofiinz =a+ib. ∗ Thecomplexconjugate,z ,ofthecomplexnumberz,isobtainedbychangingthesign ofi: ∗ z =a−ib. (1.43) ∗ ∗ NotethatRe(z) = (z +z )/2andIm(z) = (z−z )/2i. The complex plane: Because a complex numberz is specified by two real numbers, it can be thought of as a two-dimensional vector, with components (a,b). The real part ofz, a = Re(z),isthexcomponentandtheimaginarypartofz,b = Im(z),isthey component. The diagrams in figures 1.5 and 1.6 show two vectors in the complex plane along with the correspondingcomplexnumbers: Theabsolutevalue,z,ofz,isthelengthofthevector(a,b): √ √ 2 2 ∗ z = a +b = z z. (1.44) Theabsolutevaluezisalwaysareal,non-negativenumber. 6 2+i↔ (2,1)         θ = arg(2+i) = arctan(1/2)   - Figure1.5: Avectorwithpositiverealpartinthecomplexplane. Theargumentorphase,arg(z),ofanonzerocomplexnumberz,istheangle,inradians, ofthevector(a,b)counterclockwisefromthexaxis:    arctan(b/a)fora≥ 0, arg(z) = (1.45)   arctan(b/a)+π fora 0.14 CHAPTER1. HARMONICOSCILLATION ◦ Like any angle, arg(z) can be redefined by adding a multiple of 2π radians or 360 (see figure1.5and1.6). 6 . . . θ = arg(−1.5−2i) . . . . = arctan(4/3)+π . . ............ .. . ......... . .. .. . ..... ..... ... . ..... arctan(4/3) .... . .. .. ... . ... . ... .. .. ... . ... ... .. . ... . ... .. ... . . .. . ... ... .. ... . . . - . .. ... ... .. ..  .. ... ... ... .. .... .        / −1.5−2i↔ (−1.5,−2) Figure1.6: Avectorwithnegativerealpartinthecomplexplane. 1.4.2 Arithmetic ............................ ........................... ..... ..... ..... ..... ...... ...... ..... ..... ..... ..... ............................... .............................. .................................1-2 .................. ............................................... The arithmetic operations addition, subtraction and multiplication on complex numbers are defined by just treating the i like a variable in algebra, using the distributive law and the 2 ′ ′ ′ relationi =−1. Thusifz =a+ibandz =a +ib,then ′ ′ ′ z +z = (a+a )+i(b+b ), ′ ′ ′ (1.46) z−z = (a−a )+i(b−b ), ′ ′ ′ ′ ′ zz = (aa−bb )+i(ab +ba ). Forexample: (3+4i)+(−2+7i) = (3−2)+(4+7)i = 1+11i, (1.47) (3+4i)·(5+7i) = (3·5−4·7)+(3·7+4·5)i =−13+41i. (1.48) It is worth playing with complex multiplication and getting to know the complex plane. Atthispoint,youshouldcheckoutprogram1-2.1.4. COMPLEXNUMBERS 15 Divisionismorecomplicated. Todivideacomplexnumberz byarealnumberr iseasy, just divide both the real and the imaginary parts byr to getz/r = a/r +ib/r. To divide ∗ ′ ′ ′ ′ 2 by a complex number, z , we can use the fact thatz z =z is real. If we multiply the ′ ′∗ numeratorandthedenominatorofz/z byz ,wecanwrite: ′ ′∗ ′ 2 ′ ′ ′2 ′2 ′ ′ ′2 ′2 z/z =z z/z = (aa +bb )/(a +b )+i(ba−ab )/(a +b ). (1.49) Forexample: (3+4i)/(2+i) = (3+4i)·(2−i)/5 = (10+5i)/5 = 2+i. (1.50) With these definitions for the arithmetic operations, the absolute value behaves in a very simple way under multiplication and division. Under multiplication, the absolute value of a productoftwocomplexnumbersistheproductoftheabsolutevalues: ′ ′ zz =zz. (1.51) Divisionworksthesamewaysolongasyoudon’tdividebyzero: ′ ′ ′ z/z =z/z if z ̸= 0. (1.52) Mathematicians call a set of objects on which addition and multiplication are defined and for which there is an absolute value satisfying (1.51) and (1.52) a division algebra. It is a peculiar (although irrelevant, for us) mathematical fact that the complex numbers are oneofonlyfourdivisionalgebras,theothersbeingtherealnumbersandmorebizarrethings calledquaternionsandoctoniansobtainedbyrelaxingtherequirementsofcommutativityand associativity(respectively)ofthemultiplicationlaws. Thewonderfulthingaboutthecomplexnumbersfromthepointofviewofalgebraisthat 2 all polynomial equations have solutions. For example, the equation x − 2x + 5 = 0 has no solutions in the real numbers, but has two complex solutions,x = 1±2i. In general, an equation of the formp(x) = 0, where p(x) is a polynomial of degreen with complex (or real)coefficientshasn solutionsif complexnumbers are allowed, but it may not have any if xisrestrictedtobereal. Note that the complex conjugate of any sum, product, etc, of complex numbers can be obtained simply by changing the sign ofi wherever it appears. This implies that if the poly- nomialp(z)hasrealcoefficients,thesolutionsofp(z) = 0comeincomplexconjugatepairs. ∗ Thatis,ifp(z) = 0,thenp(z ) = 0aswell. 1.4.3 ComplexExponentials Consider a complex number z = a +ib with absolute value 1. Becausez = 1 implies 2 2 a +b = 1,wecanwriteaandbasthecosineandsineofanangleθ. z = cosθ+isinθ for z = 1. (1.53)16 CHAPTER1. HARMONICOSCILLATION Because sinθ b tanθ = = (1.54) cosθ a theangleθ istheargumentofz: arg(cosθ+isinθ) =θ. (1.55) Let us think about z as a function of θ and consider the calculus. The derivative with respecttoθ is: ∂ (cosθ+isinθ) =−sinθ+icosθ =i(cosθ+isinθ) (1.56) ∂θ A function that goes into itself up to a constant under differentiation is an exponential. In ∂ particular,ifwehadafunctionofθ,f(θ),thatsatisfied f(θ) =kf(θ)forrealk,wewould ∂θ kθ concludethatf(θ) =e . Thusifwewantthecalculustoworkinthesamewayforcomplex numbersasforrealnumbers,wemustconcludethat iθ e = cosθ+isinθ. (1.57) We can check this relation by noting that the Taylor series expansions of the two sides areequal. TheTaylorexpansionoftheexponential,cos,andsinfunctionsare: 2 3 4 x x x x e = 1+x+ + + +··· 2 3 4 2 4 x x (1.58) cos(x) = 1− + ··· 2 4 3 x sin(x) =x− +··· 3 ThustheTaylorexpansionoftheleftsideof(1.57)is 2 3 1+iθ+(iθ) /2+(iθ) /3+··· (1.59) whiletheTaylorexpansionoftherightsideis 2 3 (1−θ /2+···)+i(θ−θ /6+···) (1.60) The powers ofi in (1.59) work in just the right way to reproduce the pattern of minus signs in(1.60). Furthermore,themultiplicationlawworksproperly: ′ iθ iθ ′ ′ e e = (cosθ+isinθ)(cosθ +isinθ ) ′ ′ ′ ′ (1.61) = (cosθ cosθ −sinθ sinθ )+i(sinθ cosθ +cosθ sinθ ) ′ ′ ′ i(θ+θ ) = cos(θ+θ )+isin(θ+θ ) =e .1.4. COMPLEXNUMBERS 17 Thus (1.57) makes sense in all respects. This connection between complex exponentials and trigonometric functions is called Euler’s Identity. It is extremely useful. For one thing, the logic can be reversed and the trigonometric functions can be “defined” algebraically in termsofcomplexexponentials: iθ −iθ e +e cosθ = 2 (1.62) iθ −iθ iθ −iθ e −e e −e sinθ = =−i . 2i 2 Using(1.62),trigonometricidentitiescanbederivedverysimply. Forexample: 3iθ iθ 3 3 2 cos3θ = Re(e ) = Re((e ) ) = cos θ−3cosθ sin θ. (1.63) Anotherexamplethatwillbeusefultouslateris: ′ ′ ′ ′ ′ ′ i(θ+θ ) −i(θ+θ ) i(θ−θ ) −i(θ−θ ) cos(θ+θ )+cos(θ−θ ) = (e +e +e +e )/2 (1.64) ′ ′ iθ −iθ iθ −iθ ′ = (e +e )(e +e )/2 = 2cosθ cosθ . Every nonzero complex number can be written as the product of a positive real number (itsabsolutevalue)andacomplexnumberwithabsolutevalue1. Thus iθ z =x+iy =Re where R =z, and θ = arg(z). (1.65) In the complex plane, (1.65) expresses the fact that a two-dimensional vector can be written √ eitherinCartesiancoordinates,(x,y),orinpolarcoordinates,(R,θ). Forexample, 3+i = √ iπ/6 iπ/4 3iπ/2 −iπ/2 2e ; 1+i = 2e ;−8i = 8e = 8e . Figure 1.7 showsthe complex number √ iπ/4 1+i = 2e . Therelation,(1.65),givesanotherusefulwayofthinkingaboutmultiplicationofcomplex numbers. If iθ iθ 1 2 z =R e and z =R e , (1.66) 1 1 2 2 then i(θ +θ ) 1 2 z z =R R e . (1.67) 1 2 1 2 In words, to multiply two complex numbers, you multiply the absolute values and add the arguments. Youshouldnowgobackandplaywithprogram1-2withthisrelationinmind. Equation(1.57)yieldsanumberofrelationsthatmayseemsurprisinguntilyougetused iπ iπ/2 2iπ to them. For example: e =−1;e = i;e = 1. These have an interpretation in the iθ complexplanewheree istheunitvector(cosθ,sinθ),18 CHAPTER1. HARMONICOSCILLATION 6 √ iπ/4 1+i= 2e 1  π/4 - 1 Figure1.7: Acomplexnumberintwodifferentforms. ◦ which is at an angleθ measured counterclockwise from thex axis. Then−1 is 180 orπ ◦ radians counterclockwise from the x axis, while i is along the y axis, 90 or π/2 radians ◦ fromthexaxis. 2π radiansis360 ,andthusrotatesusallthewaybacktothexaxis. These relationsareshowninfigure1.8. 1.4.4 Notation It is not really necessary to have a notation that distinguishes between real numbers and complex numbers. The reason is that, as we have seen, the rules of arithmetic, algebra and calculus apply to real and complex numbers in exactly the same way. Nevertheless, some readers may find it helpful to be reminded when a quantity is complex. This is probably particularly useful for the quantities likex that represent physical coordinates. Therefore, at leastforthefirstfewchaptersuntilthereaderisthoroughlycomplexified,wewilldistinguish betweenrealandcomplex“coordinates.” Iftheyarereal, wewilluselettersxandy. Ifthey arecomplex,wewillusez andw. 1.5 ExponentialSolutions We are now ready to translate the conditions of linearity and time translation invariance into mathematics. What we will see is that the two properties of linearity and time translation invariance lead automatically to irreducible solutions satisfying (1.38), and furthermore that1.5. EXPONENTIALSOLUTIONS 19 6 iπ/2 i=e 6 iπ 2iπ −1=e 1=e  - - −iπ/2 3iπ/2 ?−i=e =e Figure1.8: Somespecialcomplexexponentialsinthecomplexplane. these irreducible solutions are just exponentials. We do not need to use any other details about the equation of motion to get this result. Therefore our arguments will apply to much morecomplicatedsituations,inwhichthereisdampingormoredegreesoffreedomorboth. Solongasthesystemhastimetranslationinvarianceandlinearity,thesolutionswillbe sumsofirreducibleexponentialsolutions. We have seen that the solutions of homogeneous linear differential equations with con- stantcoefficients,oftheform, 2 d M x(t)+Kx(t) = 0, (1.68) 2 dt have the properties of linearity and time translation invariance. The equation of simple har- monic motion is of this form. The coordinates are real, and the constantsM andK are real because they are physical things like masses and spring constants. However, we want to al- low ourselves the luxury of considering complex solutions as well, so we consider the same equationwithcomplexvariables: 2 d M z(t)+Kz(t) = 0. (1.69) 2 dt Note the relation between the solutions to (1.68) and (1.69). Because the coefficients ∗ M andK are real, for every solution,z(t), of (1.69), the complex conjugate,z(t) , is also a solution. The differential equation remains true when the signs of all thei’s are changed.

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