Much like a physical prism, which displays the frequency components of a light wave, Fourier analysis can be thought of as a mathematical prism that can tell us what harmonics or frequency components are contained in a recording of a sound wave. We wrote the MacScope II program1 so that the user could not only see a plot of the harmonic amplitudes but also see the harmonic waves and how these waves add together to recreate the original sound curve.
REFERENCES
1.
MacScope II is now a shareware program that can be downloaded from http://www.physics2000.com and used anywhere. The program has been tested to run on Windows 98 & XP and on Mac OS8,9 & OSX. The version on the web is the latest version, and may be considered an upgrade for those who own the $10 Physics2000 CD. Any questions concerning the use of MacScope should be addressed to [email protected].
2.
We developed the noncalculus derivation of the Fourier coefficients for our daughter's high school physics class. The result was that the formula for the nth harmonic cn is where f(t) is the function being analyzed and φn is a phase factor. For the single slit we are discussing, f(t) is equal to one inside the slit and zero outside. The phase factor φn moves the cosine wave left or right so that it has a maximum within the slit. The derivation can be found on page 16–28 of the Physics2000 introductory physics text and in the “Fourier Analysis” paper in the MacScope Instruction Manual.
3.
Theo Lasser, Optical Design II, http://lob.epfl.ch/webdav/site/lob/shared/Teaching/Optical%20Microscopy%20and%20Metrology/Fourier%20optics.
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© 2007 American Association of Physics Teachers.
2007
American Association of Physics Teachers
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