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by: Mrs. Peter Toy


Mrs. Peter Toy

GPA 3.96

John Price

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John Price
Class Notes
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This 5 page Class Notes was uploaded by Mrs. Peter Toy on Friday October 30, 2015. The Class Notes belongs to PHYS 1240 at University of Colorado at Boulder taught by John Price in Fall. Since its upload, it has received 7 views. For similar materials see /class/232098/phys-1240-university-of-colorado-at-boulder in Physics 2 at University of Colorado at Boulder.




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Date Created: 10/30/15
Phys 1240 Fa 05 SJP 81 DRAFT Chapter 8 Sound Spectra We just learned that if you wiggle a string only certain special frequencies the quotharmonicsquot will resonate All other frequencies tend to damp away almost instantly so if you pluck a string you will hear the fundamental frequency fl AND the higher harmonics 2f 1 3f 1 4f 1 5fl etc all at the same time all superposed If you look back to Chapter 2 section 22 on waveforms we showed pictures of complex waves that had some basic fundamental frequency but didn t look like a simple sin wave Here s one of the most remarkable things about waves discovered by a mathematician named Fourier If you superpose a bunch of waves ALL harmonics of a given fundamental frequency fl you can generate ANY POSSIBLE waveform that repeats itself at frequency fl In other words all those complex waveforms shown back in Chapter 22 of the text can be thought of as arising from some particular simple sum of harmonics of the basic frequency fl A given string will produce a particular characteristic sum of harmonics e g the higher harmonics might die away quickly in amplitude and that will yield a quotsoundquot or quottimbrequot associated with the string You perceive the fundamental frequency fl as the quotpitchquot and all the higher harmonics add in just to change the shape of that tone from the whiny annoying pure sine wave to a richer fuller tone The point is if a string plays A440 you hear a PITCH of 440 Hz but the character of the sound depends on how many and which and how strong harmonics you add in If you don t add in any you hear a very quotelectronicquot sound If you add in higher harmonics in just the right way you can produce an quotAquot that sounds like a violin or a saxophone or just about anything The difference is not in the frequency but in the mix of higher harmonics By the way if you add in quotrandomquot frequencies that is frequencies which are not an exact multiple of the fundamental something between fl and 2f 1 for example then you get quotanharmonicquot sound It CAN be interesting some drums and bells have this kind of sound it generally doesn t have a very clear pitch associated with it It can also sound pretty bad like quotnoisequot If you play the piano and play two notes which are right NEXT to each other they quotclashquot they are dissonant In part that s because the frequencies of one are not related in the simple nice way that harmonics are One other cool thing about our hearing If I play 300 Hz and 600 Hz together they quotfitquot The 600 Hz is a higher harmonic n2 of the first It39s an octave If I play say 300 Hz and 900 Hz together it sounds nice too 900 Hz is NOT the same note it s not an octave or two octaves but someplace in between but because it s a higher harmonic n3 they still sound very nice together If I play say 300 Hz and 450 Hz together we again get something nice In this case 450 is NOT a harmonic at all But BOTH of these tones are harmonics of a common lower fundamental 150 in this case 300 is twice and 450 is three times this other fundamental Now things are getting really interesting You play 300 and 450 and your brain sort of vaguely senses 150 which isn39t there but you quotfeelquot its presence Part ofthe reason is that 150 Hz signals repeat every 1150 ofa second So do 300 Hz signals They repeat every 1300 of a second which means you re ALSO back to where you started after 1150 sec Similarly 450 Hz signals repeat every 1 450 sec which means your back to where you started for the third time after 1 150 sec For some reason we seem to like hearing sounds that repeat themselves If you play 300 and 320 Hz together it doesn t sound so nice It39s fairly dissonant This is about like playing two notes right next to each other on a piano Kind of jarring You could argue that these are still both harmonics of a common frequency but that common frequency is going to be 20 Hz VERY far away from either one Phys 1240 Fa 05 SJP 82 DRAFT Section 81 of the text starts off by showing you various wave forms in Figure 81 all of which except the last have the exact same frequency They all repeat themselves a little more than three times in the time shown in the graph The details are totally different one is a quotsquarequot the other a quotpulsequot the other a quottrianglequot the first is a pure sine but they will all SOUND like the exact same frequency They just have a different timbre They are each BUILT UP by superposing a bunch of different harmonics of that same common fundamental frequency The mathematics here is a little erce but also doesn t matter so much We will play with this in the Phet simulation go to httpwwwcoloradoeduphysicsphetwebpagessimulationsbasehtml and click on quotFourier making waves where you can see how to build up complicated waves by adding in di erent strengths of harmonics I strongly suggest you go play with this yourself None of the waves in Figure 81 of the text sound especially nice They all sound quotartificialquot computerlike A little harsh to my ears anyway Waves with quotsharp edgesquot like Fig 8lc have lots of higher frequencies lots of quothigh harmonicsquot People also call these quotovertonesquot It39s a funny thing you are adding up smooth waves and yet you can build up these sharp edges The sim will help you see how that works When you add a bunch of harmonics you need to be careful to note two things about every frequency that you add in First the AMPLITUDE That s just the strength So eg if you start with a pure sin wave of frequency fl and amplitude l in some units maybe you re talking about pressure waves so this is l NmAZ overpressure Then you could add in a sin wave of frequency 2fl and you can choose any amplitude you want If you have an amplitude much BIGGER than 1 it will tend to dominate If you add in this higher harmonic with amplitude much SMALLER than 1 it will subtly alter the timbre of the fundamental In most natural instruments this is how it goes automatically higher harmonics tend to have smaller and smaller amplitudes There is another thing you can alter the quotphasequot of the wave Remember that just means shifting the sin wave left or right So if two waves are quotin synchquot they both start at zero at a common time But if you shift one of them if you change its quotphasequot then one wave might e g be zero when the other is not This makes for subtle shifts which we re not going to pay too much more attention to But for real sound analysis you do have to be aware of this too So if you want to describe a periodic wave you should tell me the fundamental frequency and then tell me the amplitude of EACH of all the harmonics you add in We could represent this in a simple graph which is shown in a box at the bottom of Figure 84 You need to stare at it and make sense of it Each of the vertical bars in that graph which is called the quotspectrumquot represents the strength of a harmonic Going to the right on that graph is going up in frequency The lines occur only at discrete values of frequency fl 2fl 3fl 4fl etc the allowed harmonics The height of the line is just quothow muchquot of that frequency you39re adding When you see a complex waveform you can always think of it as a unique spectrum There is one way and only one to add all the harmonics to get to this wave So the spectrum gives you a quick and simple way of seeing how much of each harmonic is present What it doesn t show you is the phase that makes life more complicated and is why I39m not bothering with it Phys 1240 Fa 05 SJP 83 DRAFT Look carefully at gure 84 Look first at the upper left That shows from the top down the rst 6 harmonics It draws each one above each other just to separate them So I see that the fundamental low frequency long wavelength is not very STRONG it has a small amplitude Now go down below to the spectrum it s in the box with an xaxis labeled quotfquot and yaxis labeled quotdBquot for strength Notice that the very rst vertical line is not so big that represents a notsobig amplitude Now look back at the original wiggles The second one down has TWICE the frequency of the fundamental it wiggles 4 times for every two of the wave above it It must be n2 And it has a nice big amplitude That s shown below in the box again as the SECOND line being quite tall And so on Do you see how each wiggly graph represents the next harmonic and the STRENGTH of that wiggle is represented by the HEIGHT of the line in the spectrum If you can understand this figure you39ve gone a long way to making sense of interpreting the spectrum The simulator will also help the quotbar graphquot at the top is the spectrum The spectrum is kind of a ngerprint it uniquely identi es and de nes the timbre of a sound When you quotbuild upquot a sound by adding up a bunch of harmonics we might call this quotsynthesisquot or quotFourier synthesisquot Nowadays it s quite simple to do this electronically and so you can in principle artificially generate ANY possible timbre at a given frequency This is the power of computer synthesized music What39s remarkable is also how often it doesn39t sound all that nice unless you reproduce the quotpatternquot spectrum of a real instrument You can go the other way if you play music into a quotFourier analyzerquot or quotspectrum analyserquot you will get an output that looks very much like the graph at the bottom of Fig 84 You can find software for computers that will do this for you I found quotAudiXplorerquot for my Mac and will show it in class so you can try it out for yourself Phys 1240 Fa 05 SJP 84 DRAFT In Figure 86 you see the spectrum of each of the waves we looked at in Fig 8 The first one a should make total sense it39s a pure sin wave so there is only ONE frequency The square wave is demonstrated in the Phet Sim and also in gure b You need lots of high harmonics dying off slowly to generate this shape Also you only want every OTHER harmonic This can be understood in terms of symmetry Here39s the basic idea Imagine taking a sin wave and adding in a sin wave with exactly twice the frequency I sketch that here first I just draw nl the fundamental as a solid and n2 the first harmonic as dashes Now if you ADD these two curves together what do you get Over on the left half they39re both positive and add up But over on the RIGHT half one is plus and the other is minus they tend to cancel So you get a weird wave which has lost the lovely symmetry of the fundamental I tried to draw it here not perfect but you get the idea The fundamental is the same to the left and right of the peak but the sum is not So if we39re trying to quotsquare upquot the sin wave we still want it to be symmetrical around the middle and so we don39t want any n2 added in But if you go to the next wave n3 that39s cool It is once again symmetrical about the midpoint So when we add some nl to n3 we change the shape but do not lose the lovely symmetry we want I didn39t try to draw the sum this time but think about what it will look like On the LEFT third you are adding some dash to the solid bringing the solid UP a little In the middle third you are CANCELLING some of the solid bringing the top of the peak down In the right you are again lifting it UP So in the end what happens is that you create a new curve still symmetric that is quot atterquot You39ve lifted UP the low parts of the fundamental and brought DOWN the highest parts Here you can better do this yourself with the Phet sim and see for yourself how it looks Try adding some n1 and n3 Vary the relative amplitude of n3 see if you can make it look more quotsquare wavequot like That39s the reason why the square wave in 8 lb contains only every OTHER frequency We39ll discover that there are woodwind instruments that naturally do this unlike strings you CAN build a wind instrument that tends to avoid every other harmonic And the sound of say an oboe does sound a little bit quotsquare wavequot like We39ll play some of these arti cial sounds in class so you can hear them If you look at reallife instruments you will find that there are indeed quotanharmonicquot notharmonic frequencies so the spectrum is not a bunch of spikes at given frequencies but instead itself becomes a smooth curve Figure 87 shows some realistic spectra of real instruments The general features of this graph will be You get strong quotbumpsquot or spikes at the harmonic frequencies The size of the spike tends to get smaller as you go up in frequency Can you make sense of those two features If you go back to Chapter 10 and look at Figure 104c you39ll see a spectrum for a plucked violin string It39s quotideaquot rather than real so it39s not smooth you get perfect spikes only at the harmonics The f1rst2 harmonics are strong then they die off pretty strongly There are a couple of missing harmonics because of where exactly we plucked When you pluck a string somewhere you know that NO harmonic which has a NODE or zero at that particular point will contribute It39s very much like the symmetry argument we just made There39s no way to get any of that particular frequency Phys 1240 Fa 05 SJP 85 DRAFT Modulation When radio stations send music through the air they are NOT sending pressure waves They are sending electromagnetic waves You cannot hear or detect those with any of your senses the frequency is in the MegaHz MHz range and no organs in our bodies detect this form of electromagnetic wave You CAN detect much higher frequencies that s what your eyeballs do That39s up in the 10A15 Hz range though Your ears do NOT detect elelectromagnetic waves at all they can only detect air pressure variations The radio waves jiggle electrons they make electrical currents so your radio or TV if you still have rabbit ear antennae picks up the electromagnetic signals which makes electrical currents that wiggle back and forth If you amplify this signal and send it through wires to speakers you can then easily build a mechanical device which will physically wiggle in response to the electric currents This physical wiggle can couple to say a cardboard speaker cone which wiggles the air which makes a sound The problem is that if you did this directly you d be converting MHz electrical vibrations into MHz physical vibrations and that is WAY above anything your ear can detect even now that it39s been quotconvertedquot into air motion So instead there is a trick you MOD ULATE the very high frequency MHz signal with a much LOWER frequency signal It s this lower one that is the music you want to hear in the 20 20000 Hz range There are many ways to quotmodulatequot but the two simplest are AM and FM AM Amplitude modulation quot The dashed line is a low frequency signal The solid line is the high frequency carrier not to scale So the idea is that the high frequency signal in quot quot 7 quot I the MHz range is relatively SLOWLY varying in amplitude Hence the name AMPLITUDE modulation What you see in the picture above is a sinusoidal variation in the amplitude If you build an electronic device which filters away the high frequency signal then you are left with JUST the quotenvelopequot the lower frequency signal which might be directly converted into speaker motion and thus sound Here you would hear a quotpure sin tonequot corresponding to the frequency of that dashed grey envelope But any sound is just some variation of this some repeating waveform and so if that waveform is the envelope it39s what you39ll end up hearing FM Frequency modulation Now the amplitude of the high frequency quotcarrier wavequot signal is constant and the frequency is nearly constant at the carrier frequency but with some variation that encodes the signal not to scale So you re SLIGHTLY varying the frequency of the signal a little high a little low a little high a little low This variation the quotmodulation of the frequency becomes your desired signal The circuitry required here is a little fancier to convert that variation into a physical motion of the cone One advantage of this technique is that you don39t CARE about the strength of the signal as much the amplitude is irrelevant only the frequency matters So if weather or obstructions or shadows accidentally absorb some signal strength that won t change what you hear So FM tends to sound quotcleanerquot less static and noise than AM signals


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