🚀 go-pugleaf

About Fourier Transforms:
The Fourier Transform is a very complicated mathematical fuction, and it
is very difficult to gain a conceptual idea of what it means. If I had
known what it meant before I took quantum mechanics, my life would have
been much easier in college.
I treat the Fourier transform on a purely conceptual level in my class.
Simple trigonometry is used, but no calculus, although I do walk through
the calclus with some of the advanced students who are a bit ahead of the
others in math.(I have a few students who took Calc in 11th grade).
The two media I use to teach FT are sound and light.
I have not yet seen FT mentioned in the area of sound in any low level
physics texts, however any good music theory book, or electronic music
book will have a chapter on the subject, and this is what I work from.
I introduce the idea of sound as a wave, using a function generator, a
speaker, and an oscillascope.  THe students can simulatneously hear and
see what you sound wave looks like. Things then get interesting when you
combine sounds.
I then, using a synthesizer, show them what different colors of sound
waves look like. Sawtooth, triangle waves, sqaure waves etc.
I then introduce the idea, using graphing calculators, that each of these
waves can be comprised up of several SIN waves.
Any music theory book will have a nice picture or graph of the harmonics
used to make a square wave. This graph of the harmonics, is the Fourier
Transform of the actual wave.

Next year, I do plan to purchase the ULI cards and use the Vernier Sound
programs to analyze the FT of other sounds.

When we get into optics, things get more confusing.
I won't attempt a detailed explanation here, but suffice to say that I
introduce the concept of diffraction as a Fourier Transform of an image.
This is especially important when teaching about holography, where a
hologram is a diffraction grating, and the image produced is the FT of
the diffraction grating.  The theory that deals with this is called the
Abbes Theory of imaging.
For those of you looking for background on the subject, check out
"Optics" by Eugene Hecht and Alfred Zajac.  Addison-Wesley. 2nd ed.
        Chap. 14. p 559-600
The treatment of the matter is not as simple as I would like, and is
easily confusing, but it is the best that I know of.  If anyone out there
knows of a more basic treatment, by all means, tell me.
At some point in the next couple years, I will be releasing a simple
written treatment of this topic, but until then, enjoy.



For those interested, I will be giving a workshop at the NorthEast
regional conference, along with a doctoral student from Bryn Mawr
College. The workshop will be on how to build a laser and how to
integrate it into the curriculum.  Participants get to build and take
home a He-Ne laser.  During the second half of the workshop we will
discuss how to integrate Lasers into the curriculum, and at that point I
will mention more about Fourier Transforms, but I will save giving a
paper on the topic for another time.


Cheers,

Gary Garber
Germantown Friends School
31 W. Coulter St.
Philly, The City of Brotherly Love  :)
215-951-2300
"If you can't laugh, you are better off dead!"  Roger Rabbit.


>Gary:
>Sounds like a very interesting program.  I suppose I am rather
>traditional,still spending one semester on mechanics.  Interestingly,
>we used to use Estes rockets, but our school has gotten so sensitive
>on lawesuits, that we no longer do so.  I would like to know how you
>introduce Fourier transforms ( what math background do the kids
>have?). Are you planning to talk about the program at the NE Regional
>meeting in Philadelphia in  October? I think it would be a great paper
>Harold Lefcourt