Fourier Analysis: Independent Study for Senior in HS

[cscott]

I have a chance to do an Independent Study in place of a regular class (senior, high school), and with my physics teacher as an advisor I've thought of the possibility of doing some work with Fourier Analysis. My problem is I don't know at what level this is normally taught at and what math/physics is required for a good understanding. Any other ideas would be appreciated!

 

[Facial]

I'm not a math major, but Fourier analysis is something taught in upper-division math courses at my university.

 

[Jonny_trigonometry]

ya, if you want to take it in college, you must have already taken calculus 1,2,3, and 4. I'm sure you can find some of its main ideas online somewhere. I would just plug away and learn all you can about it, even if you don't really get to the level where you understand the process. It requires knowledge of limits, sequences, and series (convergence and divergence), and of course derivitives and integrals. Check out taylor series, that's a good thing to understand, and maybe even some things in complex analysis like laurent series because that ties into a small extent with Fourier analysis. It sounds like an interesting subject, and as far as i understand, it's all about approximation. Apparently, you can approximate any function which meets certain criteria with a Fourier transform.

 

[Integral]

All the math that is required to understand Fourier analysis is trigonometry, integration and infinite series. There are algorithms and/or software packages available which will do the work without your having to understand the underlying math.

There are many potentially very interesting facets of Fourier analysis which should be accessible to an advanced HS student.

As a hands on Physics exercise,
You could capture different sound patterns and do F.A. to observe the structure of the waveform in what is known as the Frequency domain.

As a Computer exercise you could compute the Fourier Transform of various functions and examine how the resulting infinite series converges to the starting function. The F.T. of a square wave is quite interesting and can display features which impact the function of a computer.

 

[cscott]

Thanks for this information.

Do you think it's possible to do anything worthwhile in nanotechnology or quantum computer at my level? My guess is no...

 

[Doc Al]

Understanding Fourier analysis

All the math that is required to understand Fourier analysis is trigonometry, integration and infinite series. There are algorithms and/or software packages available which will do the work without your having to understand the underlying math.

As Integral says, algorithms and software packages exist to do the dirty work for you. But if you want to understand the idea of Fourier analysis, check out this book: https://www.amazon.com/gp/product/0964350408/?tag=pfamazon01-20. It's written in a somewhat goofy manner supposedly by a bunch of kids, but if you read through it (and it's not hard) you will understand Fourier better than some grad students I have known. (And I'm not kidding. )

 

[cscott]

I'll add that book to my ever growing list.

I'll definitely check out the packages, etc. but there's no fun if you don't do a bit of dirty work!

 

[berkeman]

I have a chance to do an Independent Study in place of a regular class (senior, high school), and with my physics teacher as an advisor I've thought of the possibility of doing some work with Fourier Analysis. My problem is I don't know at what level this is normally taught at and what math/physics is required for a good understanding. Any other ideas would be appreciated!

To help us help you better, what level of math and physics have you made it to so far? Do you have intro to differential and integral calculus yet? Have you worked much with complex numbers (real and imaginary components) yet? There are lots of fun projects at the senior HS level that will teach you a lot. My extra credit project as a HS senior was to build a laser from scratch. It didn't work in the end, but now I know why. It was a real education researching it and building it though.

 

[cscott]

To help us help you better, what level of math and physics have you made it to so far? Do you have intro to differential and integral calculus yet? Have you worked much with complex numbers (real and imaginary components) yet? There are lots of fun projects at the senior HS level that will teach you a lot. My extra credit project as a HS senior was to build a laser from scratch. It didn't work in the end, but now I know why. It was a real education researching it and building it though.

I'm taking Trig & 3-Space right now and we're actually just starting complex numbers (supplementary unit for top my level class). My first calc class is next semester but I've studied on my own so I know some differential and integral calc (integral to a lesser extent). In physics the last two units we did were rotational motion and oscillations, and presently we're starting electricity.

I have to have enough to do to make it a full semester's worth of work.

 

[cscott]

I decided to put this in the same thread because it's the same problem, just with a different choice of topic for my study. I've now settled on differential equations instead of Fourier analysis. My advisor asked me to compile a list of things that I expect to cover, but my problem is I don't know all the necessary theory around this topic. I plan on covering first/second-order ODE's. I have some calc background but what else is needed? Also, doing some programming would be nice and any ideas on that would be appreciated (euler's method maybe?)

 

[FredGarvin]

Personally, I'd think you'd be better off with Fourier, but that's just my opinion.

 

[cscott]

Personally, I'd think you'd be better off with Fourier, but that's just my opinion.

Could you elaborate as to why? Difficulty?

My only gripe with Fourier is that the only other student that did this with him was last year and chose that topic.

 

[FredGarvin]

Fourier I think is a bit easier at your level because it can be expressed in as a series with integrations. You can do some hand waving to avoid a lot of the diff eqn's stuff which I think is tougher. If you feel comfortable going into the diff eq. part then go for it. You will definitely be the better for doing I can tell you that!

I can especially see your reasoning because someone just did it the year before. I would want to distibguish myself as much as possible too.

Like was already mentioned by Integral, Fourier has some really cool applications in the physical world, especially in one of my pet areas, vibrations. So, I guess I am a bit biased. There's tons of good stuff for diff eq's though. Everything seems to stem from them in one way or another. Mechanics, fluids, etc...

 

[cscott]

Since the education board wants quite a detailed overview of my 'course' I've been looking for a textbook as model--one that I'll probably end up buying. I found Ordinary Differential Equations by Tanenbaum/Pollard, which seems to be a nice book, but I was wondering if there are better 'standard' textbooks for this subject.