Fourier Analysis (and guitar jammin') - Sixty Symbols

that to me this is one of the nicest signs of the public and then we leave it about right we're going to do sigma mathematicians i guess make fear that we're in trojan on their territory but let's just define what what Sigma means it means a summation so for example if we wanted to write down 1 plus 2 Plus 3 plus 4 plus 5 then what we'd have would be something like this it connects into very very many areas of physics including quantum mechanics but what I really want to focus on is something called Fourier analysis this was a guy called jean-baptiste Fourier who developed an entire theory with all those but largely he put the key ideas an entire theory of the analysis of waves quantum mechanics fundamentally relies on an understanding and usage of waves but that's not the only area of course of physics we can think of music we can think of sound engineer and we can think of engineering problems all of these things involved waves the general picture of a wave is something that looks like this that's a standard picture where something that's very simple very regular moves back and forth in a very simple what we call this mathematically a sinusoidal or a cool sinusoidal pattern but you can take any complicated wave and the beauty of Fourier analysis as you can see well that just turns prises a set of simple sinusoidal and cosinusoidal or harmonic components or harmonic bits let me just get a really really boring wave up so so what we have is probably what you can do most people will consider what sound waves look like a very simple sine wave but the key thing is is that down here we've got the wave in time and down here we've got the wave in frequency this is a rubber wire which I'm oscillating at a particular frequency so well this is an incredibly incredibly per Les Paul copy that I got for 100 quid awhile ago and you can see that there is a displacement at the end in here so I damn Pitt down and it will build up again and no displacement there or in the middle these are called the nodes but the only time I used to play a lot of guitar but the only time I get to play guitar now is actually when I bring it into two lectures and this is a standing wave on this string just like on a guitar string and I've irritated quite a few generations of undergraduates bringing in guitars and tell them about Fourier analysis but now you can see that there is a wave that comes up and goes down and comes up and it's oscillating with time if I were to change the frequency it will go disappear and then you'll if I'm clever enough I'll get next harmonic up and it will resonate and this will have a higher frequency but now I've got a node here and a node here and of course at the end we'll have to turn things off a little bit but what I want to get across first of all is this idea that you think of waves or generally we think of waves as that very simple signal we saw very simple sine wave them and then you can have other mode that will disappear and this will make a bit of a noise don't worry it's not going to fall the building in but that's the frequency of this and there's a node node note and we're going through and building up different ways so what you can see is instead of it being a very simple regular one component as we said one a sinusoidal component you can see that it's it's a much more complex signal so if we look at that on a spectrum analyzer then what we see is that that particular guitar tone is not just made up of one frequency it's made up of a number of different frequencies and this is key it's that it's that combination of frequencies combination of harmonics that allows us to distinguish a near note on guitar from an e note on piano or near note on a trumpet or near note on of whatever so if I were to turn this off and displace the wave by plucking it at a certain point I can think of this as being the fundamental which is the one that goes up sinusoidal II like that plus a little bit of the next harmonic plus a bit of the next harmonic when it's going up and down like that when you play the string we have this is the ends one end of the guitar on it so this is here this is the knot and here's the bridge and so that the fundamental mode or the fundamental frequency looks like that the next one and we get all these difference what are term standing waves but you can see that the nice sign with components and what you do is you add those up and that and the mixture of those and the strengths of those gives you the sound of the NOTAM and it's the sum of all these different wave-like motions that makes up the guitar string as I pluck it in a particular place and as I pluck it in different places you'll get a different shape the key thing is that you're building up these notes you're taking many different ingredients making a mix making a stew making a kick whatever you want to call it but by mixing those ingredients in different ways you get different tones you get different flavors the idea of Fourier analysis is to build up any shape like this however you pluck it if you plug it in two places or whatever as are some of these different standing waves but the neatest way of accentuate noise frequencies and playing around with those frequencies is to use something called a while pedal and this idea can be used throughout physics if I were to make this thing longer and longer and longer I would change the frequencies that I get many more frequencies coming out this idea is due to Monsieur Fourier a French mathematician if you make this longer we call that a Fourier transform which is what professional physicists use but first of all you have to learn about the Fourier series well let's just focus on one note so when you play that and if I keep the wire pedal fully up so I'm accentuating those low notes and then if I push it forward you see that I'm now accentuating the high notes and I've pulled down the low notes at the expense of the high notes and then when you sweep those two and they will still be the ones up to the point where you can't hear which in your case is going to be much higher frequencies than an old man like me who's lost half his hearing at high frequency range it used to be an x-ray machine in here and and it wheezed and when I was 2530 I could hear the whizzing noise and the students can hear it and it would drive you to distraction when I go up to 40 I couldn't hear it at all the students complained to me and I could remember that it made this horrible noise but you lose that higher frequency range

Channel: Sixty Symbols

Views: 203,952

Rating: 4.9240599 out of 5

Keywords: fourier, analysis, waves

Id: u1Lz8pm2npQ

Length: 7min 25sec (445 seconds)

Published: Sun May 24 2009