Exam One: Review Guide
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1 P12 ysjcs 2212K Exam 1 Study Guide I e PH Y 319 Ishtyaq Ponir Dr Jeremy Maune Traverse and Longitudinal Waves and Waves in General So the first thing on your mind probably is what the heck a wave is 0 To answer your question a wave is a disturbance traveling through a medium A medium is the area surround a wave while the disturbance is basically a change in what is known as commonplace The medium is almost always stationary not moving as the wave is traveling Alright our first definition a traverse wave is a wave where the displacement how much area the wave is taking from the origin is perpendicular opposite by a right angle from the motion the wave travels 0 Man that was a lot let39 s simplify If we have a wave ook at the example we see that the l direction in which its traveling in this case to the left is opposite the pulse the quotcurvequot of the wave in one motion of the wave 0 That39 5 what we mean when the displacement is opposite to the direction pretty neat huh We also have another type of wave there are two general types with the traverse being the other one called a longitudinal wave 0 This is a case of a wave where the particles in the medium move parallel to the direction of the waves motion Again like before the wave travels to the right but its displacement is influenced by the pushpull of the spring below The traverse wave39 s displacement is primarily caused by the updown l um i 39 Iquot quotlei N idll39lhu w 391 11M l39 tlI391HII lilli39 quot hillll 39I39 i l l lll y39 l CH quot1391quot motion of the string opposite of a longitudinal wave because in this wave it is caused by the leftright pushpull motion of a spring o Remember that quotspringquot and quotstringquot are two different words 62 Alright not that we got that out of the way lets focus on a traverse wave for a second In theory it states that the speed not to be confused by velocity in this case is determined the medium 0 This gives rise to a new equation 0 lSz rl39ng 7395U A means squareroot Where VString is the wave speed on the string Ts is the Tension a force opposing the motion and u is known as the linear density The linear density is the strings mass to length ratio That means to get it you have to divide the string39 s mass by its lengths as so 0 p mL It39 s is important to note that the wave speed depends on the properties of the medium not the maximum displacement of the wave 0 This means that no matter how much you quotmess withquot the length of those pulses we were talking about oh and by the way multiple pulses on a string at a given instant of position and time is known as a propagation it won39 t matter because the speed is determined by the influences outside of the displacement The Fight of the Century Snapshot V History Graphs Before we see Batman and Superman duke it out next year we have to learn the differences of these two very similar but very different graphs First off a snapshot graph is a graph that shows a wave39 s displacement as a function of position which means it39 s a graph literally of a wave and it is what the wave looks like at a single instance of time o This means that the time quot 39 5 39 remains constant as the 1 wave moves about doing what waves do 0 To put it bluntly Same place in time but position changes 0 A wave pulse moves forward AX that means the change in x relative to time in this case vAt during the time interval At This means that the change in position of the wave is equal to the speed multiplied by the change in time o In the is case the speed is constant Alright now for number 2 a history graph is a graph that shows the wave39 5 displacement as a function of time at a single position in space 0 Okay things got kinda tricky when we introduced history graphs but bear with me 0 One thing to note is that with a history graph the displacement like the snapshot graph is in the y direction with time now being in the x direction in seconds Alright now let39 5 compare and contrast the two graphs provided below the left being the snapshot and the right being the history graph 0 We see that both graphs have their displacements D in the y direction o The two also have a certain framework or constant from which the graphs seem to sort of quotstand stillquot by that constraint For the snapshot graph it39 s the little time symbol t and for the history graph it39 s the little position symbol x o This is important for converting from snapshot to history graphs which we will cover later 0 Now the only real values that are different for the two graphs are the values in the x direction for snapshot graphs its position and for history graphs its time 0 Here39 5 the fun part in order to understand their differences we have to look at how they relate to one another For one the history graph shows how the position at x meters changes through time The snapshot graph shows how the position changes at a single instance of time So you can apply what happens to a snapshot graph to a history graph 0 First see what position the history graph is in this will be important when we look at the snapshot graph 0 Now on the snapshot graph we track the position that was on the history graph or we move the snapshot graph by its wave speed c We repeat this and while we do so we see the value of the displacement of the position we are using on the history graph and plot that displacement on the history graph c That39 s really all too it and you continually do this and connect the points you plotted To do a history to snapshot graph you39 d do the same thing but instead you would see how different times would change given their positions Ax That should cover the basics of these two types of graphs make sure to check out the PracticeSet to see how to work some practice problems in the coming week The Sinusoidal Wave The Wave That Moves Like a Circle Sinusoidal waves are wave sources that oscillate with Simple Harmonic Motion SHM o SHM are basically motions that oscillate move in circle pattern that have a force that hinders them which is proportional to the motions displacement Each one of the particles in the medium undergoes SHM with frequency 9 f 17quot 0 Units 5391 Hz 0 Frequency is equal to 1 divided by the period 0 This is so because frequency is equal to the number of oscillation per second so 1 divided by the period is equal to the frequency 0 This is true because as the wavelength ends we get the result of the time it takes for that wavelength to finish so taking on over that takes the amount of oscillations it takes for that duration of time o The distance spanned by one cycle of the motion is called the wavelength A This is basically the distance between a I crest the top of a wave to another crest wavelngthl or a trough the bottom of the wave to trough This is the basic measure of length that we have for analyzing waves 0 Amplitude is defined as the maximum displacement of the wave In other words the wave39 s highest height from the source its girth so to speak It is measured from the base to either the crest of the wave the highest height in the positive y direction or from the base to the trough the lowest height in the y direction As we39 ve said before or might not have the size of the waves 3a amplitude is independent of the wave speed so quite literally size does not matter Waugh Traith o The way we measure speed yes speed not velocity remember were looking for scaerqualities not vectorones is by taking distancetime Fair enough but what are the distance and the time Well we figured out that A is the distance of the wavelength and T is the period of a wave or the time it takes to reach one wavelength So with that logic we say Vspeed A T or the wavelength divided by the pe od Note a wave moves forward one wavelength in one period of time Note remember to not confuse the motion of the wave with the motion of the medium we discussed this earlier 0 So from the equation we gathered earlier like three bullet points ago we can say that V if c This is known as the fundamental relationship of periodic waves o This means that a wave moves forward a distance of one wavelength during a time interval of one period c We get f from this equation because f is equal to lT or the 1 divided by the period 0 Through some simple rearranging we get the equation you see above see not so hard 0 We can further rearrange and say that A Vf o It is important to note that vis a property of the medium caused by it and f is a property of the source caused by that 0 So because of this we say the wavelength is a consequence of a wave of the frequency traveling through a medium in which the wave speed is v Now et39 s move on to displacement To calculate displacement you use the equa on o DXUAlh2TXZ Do D is of course the displacement taking into account its X position and t time values Dais called the phase constant which is defined by the initial conditions of a wave how the wave started its origins This equation can be altered in different was but still mean the same thing o D ZAin21tXZ Do Alh2TXZ 45021t rad Note the A s cancel out Asin21tXZ Do 0 Ai 2TXZ Do 39 000 The x A before the 2T comes from the fact that were taking the displacement from the quantity defined by D Z The 2 T rad comes from the equation ina 2n rad sina o This is a pre calculus proof that states that adding 2T to the a trigonometric equation does not change the value of the equation but its position 0 The same goes for the A added to X the value is the samejust the po onschange The disturbance at XZ is exactly the same as the disturbance at X Now that we have the equation of the displacement we have to set the quot wave in motionquot We does this by replacing 39x39 with the value X V1 0 This works because the wave moves a distance of Itduring time t 0 And like before when we said adding to any expression changes its positon not displacement that idea comes full circle literally 0 We subtract because we are taking an earlier time t0 and comparing it tojust x 0 SO 39 0061 DXVl i0 And Soagain o DXt4in27TXvtZ Do 0 DMUI4in27TXZIVT Do 0 Note we used VZZfZZ 739 to write vZ 17quot So now we have the formula lets add more variables to make to more complex 0 Because sinusoidal waves propagate using SHM we can inherit variables used by those equations 0 The first one we are going to look at is angular frequency oo2nf27TT remember how frequency and period relate o Units rads 5391 0 Next we have the wave number k 271l o Units radm m393L We can now simplify the displacement equation using these new quantities o D2 AinX cu t Do Note it repeats every t seconds Note if x is fixed then the equation gives a sinusoidal snapshot graph at one point in space t1 otherwise it is a history graph We essentially switched out values to get to this equation nothing to fancy This still accounts for a sinusoidal wave traveling in the x direction this case is positive In the negative X direction its o D 2 AinX cu t Do 0 So since we described the phase constant being the initial conditions the equation for it would be D0m05A5nltbo o The x and t cancel out the k and co so we are left with the phase constant 0 Note different values of C130 describe different initial conditions for a wave The equation we came up with DXz 4inX cu t Do is the displacement in the X direction We can do the same in the y direction o yXt4inkx cu t Do 0 The velocity of a particle which is not the same as the velocity of the wave along the string is found by taking the derivative with respect to time Vy dya tz cuACOX cu t Do o This is accomplished by taking the derivative of the equation o First do chain rule so take the derivative of the quotinsidequot multiplied by the derivative of the quotoutsidequot with the quotinsidequot intact c When taking the derivative of the quotinsidequot 0 kx and C130 disappear because they are constants we can get rid of them because they are separated with a and if it were multiplication or division it would be a completely different case 0 The co is the only variable that relates to time so we take its derivative co 0 When taking the derivative of the quotoutsidequot we get A multiplied by the cosine of whatever we had on the inside to start c We then multiply the two parts quotoutsidequot and quot insidequot and get the equation listed above o The maximum velocity of the string is VmaXCU4 0 Again we applied the rules of SHM to get this To find the expression for a wave speed along a string we need to look at Newton39 s laws specifically his second one 0 Of we use Ax to signify a small portion of length A we can use it in our equation 0 Next we measure the string39 s tension as in exerts a downward force on the piece of string which is pulling it back to equilibrium 0 Because of this fact we can use the equation for Newton sSecona Law Fnedymay From this we can substitute mwith uAX c We get this because u is an objects mass to length ration so to get mass we multiply the little piece of A A X and it gets rid of the A length on the bottom of the linear density39 5 mass to length ratio To further our discussion of displacement we can look at the point of maximum displacement to also find the point of maximum acceleration This is also a time derivative so it can be written 0 aydVydz W2A5hkXw z CDO And because of this we can say the crest of the wave is equal to o ay a24 This is fine and dandy but we have to remember that the angular frequency of which the particles of the string oscillate is related to the wave39 5 speed Ialong the string by the equation wvk 0 SO ayz w 24 v2k2A The v2 and the k2 come from the fact that we are squaring oo So that concludes sinusoidal waves for now One Two Three Dimensions Waves or Circles As you might have figured out we are looking at waves in different dimensions yay This is a very short section of the chapter but very important so make sure to really pay attention please 62 Alright to start off let39 s try to imagine that you are looking at a pond and ripples form from the source or the origin of displacement 0 As you can see from the ripples there immerge these circles that continually expand from the source in all different directions each with different velocities since they do have vector qualities ie direction 0 Let39 s say that we make waves from the source and continue them along the ripple39 s formation that is we separate the different circles of the wave by A a wavelength and say that when the wave touches the next circle it is our new crest and we get a new crest for every A or new circle of the ripple o Hopefully the picture was informative if you couldn39 t understand my babble o Anyways the lines we mentioned that symbolized the crests of the wave are called wave fronts 0 Like a snapshot graph the model we used only shows a wave at one instance of time The model is known as a circular wave it is a two dimensional wave that spreads across the surface simple enough Although the wave fronts LOOK like circles if you were to examine them from far away they would look like parallel lines Spherical waves are defined as a wave where the crests of the wave form a series of concentric circles that have the same center spherical shells separated by wavelength A o Loudspeakers and lightbulbs emit Ephe cal wavefront spherlcalwaves Now when examining spherical waves if we look at one very very very far away from the source small segments of the wave look like flat parallel planes flat two dimensional shape If the wave is crested at every point of these types of planes it is known as a plane wave 0 If you were looking at a sound wave for instance which is a longitudinal wave you would notice that if you located the particles in the instance of time where they were at their maximum displacement amplitude towards you you would locate the plane perpendicular Nto the travel direction Because a wave39 5 displacement depends on x but not on y or z coordinates the displacement function DXt describes a plane wavejust like in onedimensional waves When describing a spherical wave or circular wave we change DXtto Dr17 where r is the radius measured outward from the source 0 P o This is written as Dr 2 A rinr a tdi o o Amplitude is stated as being a function of r o This is because in circular and spherical waves the wave39 s amplitude has to decrease with the increasing distance r 0 Note this is why sound and light decrease in intensity as you go farther away from the source stay tuned to see how that works Now that we covered the different dimensions of waves et39 stalk about phases a little bit more 0 Remember the quatitity kX wtQM that is known as a phase denoted as Do o The wave fronts we say in the previous pictures are quotsurfaces of constant phasequot This is because the wave fronts have the same displacement so the phase must be the same at every point Now be come up with the phase difference ACID as showing two different point of the wave at time t The equation is derived by 0 ACID C132CI1Xal d5o Xwtd50lX239 X1kAx2pi AxA o The phase difference between two points on a wave depends on only the ratio of their separation Ax to the wavelength A Ex two points on a wave separated by Ax 12 have a phase difference of ACIgtpi rad Note we get the solution for the phase difference by plugging the value of AX into the equation for kAx and then simplify the equation It is important to note that the phase difference between two adjacent wave fronts is ACIgt2pi rad o This follows the fact that two adjacent waves are separated by AxA 0 Note moving from one crest of the wave to the next corresponds to changing the distance by A and changing the phase by 2pi rad That covers this short but very important section now we39 ll continue to sound and light and how they apply to waves If You Can See or Hear It It39 s a Wave For The Most Part As you might have known already sound and light waves are the basis of hearing and seeing 0 These are the two waves that are the most significant to humans because we use them every day First lets talk about sound waves 0 Sound waves are a sequence of compressions and rarefactions stay tuned to know what that means 0 Sound waves are not only limited to the air but can travel through gas liquids and even through solids o In sound waves individual molecules oscillate back and forth with displacement D Through this they propagate ongmdinalWaves H I forward a SpEGd IIIs Vsound Through this motion and fact that OOOOOOOngE compressions are regions of higher pressure a sound wave can be thought of as a pressure wave Higher densities and pressures are known as compressions and lower densities and pressures are known as rarefactions o In the picture areas of particles compressed together are known as compressions and areas where they are loose and spaced are called rarefactions The periodic sequence of compressions and rarefactions travels outward from the source as a longitudinal sound wave You can detect sound waves with frequencies between about 20 Hz and about 20000 Hz 0 20000 Hz20kHz Low frequencies are perceived as quotlow pitchquot and high frequency are perceived as quothigh pitchquot Note the speed of sound waves depends on the properties of the medium Liquids and solids are less compressible than air which makes the speed of sound in those media higher that air Note sound takes 3 seconds to travel 1km Ultrasonic frequencies are sound waves that exist at frequencies above 20 kHz Alright now that we covered sound waves lets cover light waves 0 A light wave is an electromagnetic wave or an oscillation of the electromagnetic field 0 Other electromagnetic waves eg radio waves microwaves have the same characteristics of light waves even though we can39 t see them Light waves are described as being a quotself sustaining oscillation of the electromagnetic fieldquot o The displacement D is an electric or magnetic field 0 Self sustaining means that electromagnetic waves require no material medium in order to travel Note this is why they are not considered mechanical waves Note the speed of light is or c is equal to 299792458 ms or 3OO108 ms for calculations The wavelengths in light waves are really small 0 Even so we perceive longer wavelengths as orange or red light and shorter wavelengths as blue or violet light 0 The visible spectrum is the spread of colors seen with a prism or seen as a rainbow This is a small slice of what is known as the electromagnetic spectrum or all the visual perception of frequencies and wavelengths for sound and light waves Light waves travel with speed c in a vacuum but they slow down as they pass through transparent materials such as water or glass 0 The speed of light in a material is characterized by the material 5 index of refraction n due to the phenomena listed above nspeed of light in a vacuumspeed of light in the material 0 ncv o The index of refraction is always greater Material IOR Material IOR Vacuum 10 Glass 1517 Air 100029 Glycerin 1472 Alcohol 1329 Ice 1309 Crystal 20 Ruby 177 Diamond 2417 Sapphire 177 Emerald 157 Water 133 than 1 because vltc A vacuum always has n1 As you can see from the picture liquids and solids have larger indices of refraction than glass Because of the equation VzM either A or f or both values will change when vchanges vis their master Note the frequency of a wave is the frequency of the source 0 It does not change as the wave moves from one medium to another 0 To understand this think about a calm wind in a steady pool When the wind moves the air oscillates back and forth periodically pushing on the surface of the water These pushes the air generates causes compressions that continue into the water Because of Newton39 5 Third Law the frequency of the air waves in the water must be exactly the same as the frequency of the air waves in the wind This same principle can be applied to electromagnetic waves 0 The frequency does not change as the wave moves from one material to another o The only thing that would change is the wavelength The wavelength in a transparent material is less than the wavelength in vacuum c When light enters a material the only way it can go slower while oscillating at the same frequency is to a have a smaller wavelength We Need More Power More Intensity Watts the Problem As we know or might not of if that case sorry traveling waves transfer energy from one point to another From that we state that the power of a wave is the rate at which the wave transfers energy 0 Units Us or Watts W 2 W is equal to 2joues per second As we know from prior experience light is much brighter when focused onto a small area ie a flashlight 0 From this we can say that the focused light is more intense than the diffused light one that goes in all directions not just one 62 Brightness and loudness depend not only on the rate of energy transfer power but also the area that receives the power 0 We define intensity as the quantity listed above IPapower to area ratio it39 s like this chapter39 s linear density 0 Units Wm2 Because intensity is a power to area ratio a wave focused into a small area will have a larger intensity than a wave of equal power that is spread out over a large area If a source of spherical waves radiates uniformly in all directions then the power at distance ris spread uniformly over the surface of a spherical radius r o This basically leads us to the surface area being 4pir2 so the intensity of a uniform spherical wave is 39 I leource439plr2 To compare two intensities at two different distances from the source without the need of power you use the equation 0 1112 r22r12 These do not pertain to indoor scenarios due to the amount of reflecting surfaces that come to question Because a wave39 5 intensity is proportional to the rate at which energy is transferred through the medium and because the oscillatory energy in the medium is proportional to the square of the amplitude E 12A2 we can infer that the intensity is proportional to A2 o If you double the amplitude you increase the intensity of a wave by a factor of 4 When we look at sound waves we define the sound intensity level expressed in decibals dB as o B 10 dB l0910IOIO lOdBloglo1 0 Db o 0 dB does not mean no sound it means that for most people sound cannot be heard 0 We start at 0 as our first threshold and continue on into bigger and greater sounds This is why it is known as the threshold of hearing Just think of it as starting at level 0 you really don39 t want to hear level 1000000 The important thing to note is that sound intensity level increase by 10 dB each time the actual intensity increases by a factor of 10 0 So to get from 70 dB to 80 dB the intensity increases from 10395Wm2 to 10394 Wm2 0 It is also said that perceived loudness of sound doubles with each increase in the sound intensity by a level of 10 dB The different thresholds provided provide us with a better understanding of how to interpret different sounds personally I think it39 s funny that going to a rock concert is almost the equivalent of someone beating up your ears The threshold of pain says it all Doppler Doppler We Got Doppler Here Only Those Who Hear Care Our next and final woo chapter of the section deals with motion relative to a wave source As you may have noticed the pitch of an ambulance siren drops as it goes past you 0 One of the reasons why this is After a wave crest leaves the source its motion is governed by the properties of the medium That is the motion of the source cannot affect a wave that has already been emitted We state that the change in frequency when a source moves relative to an observer is called the Doppler effect This is all you really have to know about the subject for the tests and quizzes but here are some important equations you might want to know 0 The is the aproaching direction while the is the receding direction Axwave vt3vt AXsource Vsz 3 VJ For the next equations the distance dspans 3 wavelength 0 A d3 AxwaveAxsource3 3vt 3Vs73 v vsT o f Vxl vv vsT vvvsf 39 f foJVsV f gt fa o This is the Doppler effect for an approaching source f foJVsV f lt f0 o This is the Doppler effct for an receding source f JNovfo o This is used for an observer approaching a source 39 f 1 Novfo o This is used for an observer receding from a source How Can a Wave Move If It39 5 Standing What39 5 So Super About Superposition Standing waves are waves that are created from superposition of traveling waves bouncing back and forth between the edges of the medium 0 They occur in well defined patterns called modes Each mode has its own distinct frequency Nodes however are points on a standing wave that do not oscillate at all Before we go further on that we need to discuss the principle of superposition 0 To conceptualize it we need to understand that when we talk about particles two cannot occupy the same space at the same time 0 But waves can pass directly through each other When two waves displace at the same position the net displacement is the sum of the two waves This idea leads us to the principle of superposition 0 When two or more waves are simultaneousypresent at a single point in space the displacement of the medium at that point is the sum of the displacements due to each individual wave 0 They are said to be superimposed a phenomenon called superposition This is the net displacement of all the particles in the medium Now that we covered superposition a standing wave is actually defined as the superposition of two waves They must have the same frequency wavelength and amplitude and travel in opposite directions The crests and troughs stand in place as the wave oscillates The medium is carrying two waves traveling in the opposite directions In standing waves we have two distinct patterns of movements nodes and antinodes o Nodes are points that never move They are spaced M2 apart Hallway between nodes are points where the particles in the medium oscillate with maximum displacement These are known as antinodes o They are also spaced M2 0 Nodes exist at points where the displacement of the two waves have equal magnitudes but opposite signs o Antinodes exist where the two displacements have equal magnitudes but the same signs 0 When the two displacements are equal the waves are said to be in phase The amplitude of these are twice that of individual standing waves o This is known as constructive interference o Antinodes 0 When the two displacements are equal but one is the negative of the other one they are said to be out ofphase There is zero amplitude These are known as destructive interference Nodes The intensity is maximum at points of constructive interference and zero at points of destructive interference If a sinusoidal wave is traveling to the right along the X axis we denote its amplitude as 0 DR asinkX wt If traveling left 0 DL aSlnlXoot The symbol a is used to represent the amplitude of each individual wave whereas A is used to denote the net wave The net displacement of the two is equal to o DXt DRDL asinkx oot asinkx oot 0 We can simplify by using the function sina 3 sinoccosB cosasinB This gives us o asinkxcosoot COSkXSlnoot asinkxcoswt COSkXSinoot o 2asinkxcosoot o It is useful to write the equation as DXt AXCOSoot The amplitude is equal to Ax 2asinkx Amax 2a at points where sinkx 1 This gives us the oscillation for a particle at position x Nodes exist where the amplitude is zero so they are located at positions x for which o Ax 2asinkx 0 It is important to know that the position Xm of the mth node is defined by the equation 0 Xm m M2 When a wave encounters a discontinuity some of the wave39 s energy is transmitted forward and some is reflected Discontinuities are essentially the places where the medium changes so the wave changes they are boundaries 0 Waves reflect from boundaries 0 A positive displacement of the incident wave becomes a negative displacement of the reflected wave 0 The amplitude of a wave reflected from a boundary is unchanged It is important to note that reflections at the ends of the string cause two waves of equal amplitudes and wavelength to travel opposite directions along the string A boundary condition is a mathematical statement of any constraint that must be obeyed at the boundary or edge of a medium o If x O and x L for two different positions of the string the start and the position of the final measurement then the condition would be Dx0t O And DxL t O Note nodes are required at the end of the strings The condition for L is true if c 2asinkL O 0 Which require that kL 2piLA mpi Standing waves can only exist if its wavelength is one of the values given by this equation 0 Am 2Lm We can get the frequency by applying Af v o so fm vAm v2Lm mv2L o The lowest allowed frequency is known as the fundamental frequency and is noted by f1 V2L 0 SO fm mfl Higher wave frequencies are known as harmonics The number of possible standing waves is called the mode or normal mode 0 Each mode has a unique wavelength and frequency 0 Each are numbered by the integer which is the number of antinodes on the standing wave 0 The fundamental mode has A1 2L not A1 L o The fundamental frequency can be found as the difference between the frequencies of any two adjacent modes I SO f1 M fm1 fm Because electromagnetic waves are transverse waves a standing electromagnetic wave is very much like a standing wave on a string 0 They can be established between two parallel mirrors boundaries that reflect light back and forth 0 They form a laser cavity If we are talking about a pressure wave they39 re nodes and antinodes are interchanged with those of the displacement wave Let39 s talk about applying superposition when a pattern arises resulted from a superposition of two waves it is called an interference To really understand interference we need to understand phases 0 They are defined as I C131 kX1ootCI10 I C132 kX2ootCIgt20 The difference between the phases is called the phase constant 0 ACID 132 C131 kX2ootCIgt2o kX1ootCIgt10 I kX2 X1 C1320 1310 o 2pi AxA Do o Axis X2 X1 which is known as the pathlength difference 0 It is the extra distance traveled by wave 2 on the way to the point where the two waves are combined Maximum constructive interference is achieved by o ACID 2 2pi AxA ACIDO m 2pi rad Identical sources which have ACDo O rad maximum constructive interference occurs when Ax mA Two identical sources produce maximum constructive interference when the pathlength difference is an integer number of wave lengths Constructive and destructive interference of two waves travel along the x axis Perfect destructive interference occurs when 0 ACID 2pi AxA ACDo m 12 2pi rad 0 Two identical sources produce perfect destructive interference when the path length difference is a half integer number of wavelengths Two waves can be out of phase because the sources are located at different positions we give three examples thatjustify every condition of destructive interference 0 Each condition creates waves with ACID 2 pi rad The sources are our of phase Identical sources are separated by half a wavelength The sources are both separated and partially out of phase Don39 t confuse the phase difference of the waves ACID with the phase difference of the sources A Do It is A CI the phase difference of the waves that governs interference Let39 s describe the net displacement as the superposition of two waves traveling along the x axis 0 We denote it by D 2 D1 D2 asinkX1oo tcD 10 aSlnlX2o0 HQ 20 o asinCIgt 1 asinCIgt 2 c We use this property to find the net displacement o D 2acosAcI2 sin anvgoo tCIoavg This is where CI avg CI 10 CI 202 The amplitude of a sound wave is o A 2acosACI2 In defining constructive and destructive interference for two waves we find their maximum by 0 ACID 2 2pi ArA ACDo m2pi For maximum constructive interference o ACID 2 2pi ArA ACDo m12pi For perfect destructive interference The waves from two identical sources interfere constructively at points where the pathlength difference is an integer number of wavelengths The waves interfere destructively at points where the pathlength difference is a half integer number of wavelengths The lines in the figure shown below in red are known as antinodal lines with constructive interference oscillation with maximum amplitude and intensity The dark ones are nodal lines destructive interference with no oscillation and the intensity is zero Let39 5 Look At Some Optics Of the Wave Variety The study of light is known as optics The spreading of waves is a phenomenon called diffraction o It39 s a sure sign that whatever is passing through the hole is a wave Now a very old man named Thomas Young announced way back when that he had produced interference between two waves of light 0 His experiment settled the debate in favor of a wave theory of light Since interference is a strictly wave like phenomenon It is now understood that light is an electromagnetic wave an oscillation of the electromagnetic field which requires no material medium to travel Light in its own way is very weird 0 Sometimes it shares qualities like that of waves other like particles 0 It is established that it sort of acts like both sort off Now three models of light were created but we only need two for the exam 0 We will learn about the Wave model 0 Responsible for the widely known fact that light is a wave Ray model 0 Light travels in straight lines tari 2 and D Aaaurn min n if infinite For distant SCEF EEH th ese source distance gitr35 a sump i n pI anewave atstlitso I r a i n es that ail amplitude 1 Elana 3 5mg a 5 elementsare phase 39 1 are My 3quot un r mr 3 I WE amma he G D ljlljilj i formatin ll Called a rightangfle quot x alarm v arid law a H llght a Eli mam 3 d rays Old Man Young s doubleslit experiment is fundamental to our understanding of light 0 It s called doubleslit because there are two openings 0 As the picture shows it uses the interference of two waves to determine places of destructive and constructive interference F l tilt iTl o It was concluded that two overlapped waves of equal wavelength produce interference 0 Those places of destructive and constructive interference are known as interference fringes by the way and the middle one that is really big is known as the central maximum The path length difference for the picture below is A r which is r2 r1 or o mA Bright fringes have constructive interference and dark fringes have destructive interference o The mth bright fringe occurs when the wave from one slit travels m wavelengths farther that the wave from the other slit and that proves the above equation 0 The extra distance pathlength difference is equal to Ar dsintheta o thetam md o In radians 0 We can find point P39 s position by the equa on y Ltantheta 39 ym mA UN 0 These are all for bright fringes The m 1 fringes occur at points on the screen where the light from one slit travels exactly one wavelength farther than the light from the other an The interference pattern is a series of equally spaced bright lines Fringe spacing equation 0 AyMJd For destructive interference we give the equa on o A r m 12A 0 Located at positions 39 ym m 120x UN 0 Dark fringes are located exactly halfway between the bright fringes When we talk about intensity in a band of light it is given by the equation 0 11 Ca2 0 If there were no interference the light intensity would be twice the intensity of one slit or I Iz 2ca2 This is FALSE though 0 Instead we find that the intensity equals Idouble 411COSZplCl Ly o For dark fringes 39 Idouble 0 o Forbnghtf nges 39 Idouble 411 o The maxima occur at points where ym mA Ld The phenomenon where multiple slits diffract behind one another is known as diffraction grating We give the maximum intensity of the phenomenon by 0 Imaxz N211 An experiment observing the diffraction through only one narrow slit is known as the singleslit experiment diffraction The below picture shows Huygens39 spherical wavelet wavefront 39 propagated PrInCIple which envelope a A of wavelets states two things Spherical wavefront gt 0 Each pomt on eachpoim b B emits a a wave front wavelet c C c gt C common 39 center IS the source who t plane D plane wave r0quot 5 wavefront r W3Veff0nt Of a Spherlcal each point propagated emits a enVelope wavelet e wavelet e E of wavelets spreads out wavelet at the wave speed 0 At a later time the shape of the wave is the line tangent to all the wavelets Every point on the wave front can be paired with another point distance a2 away We can use the equation below to find the angles of dark fringes o thetap pota Where p is an integer number Hey Let39 5 Look At Ray Optics Okay When light travels in a straight line it is used as the basis of the ray model A light ray is a line in the direction along which light energy is flowing There are 5 basic assumptions for the ray model 0 O 0 Light rays travel in straight lines Light rays can cross and do not interact A light ray travels forever unless it interacts with matter An object is a source of light rays The eyes see by focusing a diverging bundle of rays Most objects are reflective or selfluminous Light rays from an object are emitted in all direction They exist independent of whether you are seeing them Diverging rays from a point source or a definite source are emitted in all directions Parallel sources are created from the start of where the rays are parallel Reflection from a flat smooth surface such as a mirror or a piece of polished metal is called specular reflection The angle of incidencethetai is between the ray and a line perpendicular to the surface The angle of refraction thetar is the angle between the reflected ray and the normal to the surface 0 They both equal each other Diffuse reflection occurs when irregularities of the surface cause reflected rays to leave in many random reflections It is important to note two things 0 Rays from each point on the object spread out in all directions and strike everypoint on the mirror Only a very few of these rays enter your eye but the other rays are very real and might be seen by other observers 0 Rays from points P and Q enter your eye after reflecting from di erentareas of the mirror This is why you can39 t always see the full image of an object in a very small mirror Two things happen when a light ray is incident on a smooth boundary between tow transparent materials 0 Part of the light reflects from the boundary obeying the laws of reflection how you see reflections from pools of water or storefront windows 0 Part of light continues into the second medium it is transmitted rather than reflected but the transmitted ray changes direction as it crosses the boundary This is known as refraction We give an equation to figure the refraction of the light in two mediums by Snell39 slaw 0 Which is n1sintheta1 nzsintheta2 The conclusions that can be made by this law are 0 When a ray is transmitted into a material with a higher index of refraction it bends toward the normal 0 When a ray is transmitted into a material with a lower index of refraction it bens away from the normal Total internal reflection occurs when a ray is unable to refract through the boundary so it reflects from the boundary back into the source The critical angle is reached when the angle of refraction is 90 degrees We give it by this equation 0 thetac sin1n2n1 There is no critical angle and no total internal reflection if n2 gt m or gt1 Thatjust about covers the basics of ray and light optics and now everything Again sorry in advance for not posting weekly notes this week but I figured combining them all into a study guide would best suit your needs and karma points Thank you for choosing this review packet and good look on the test Neil
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