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Re: Harmonic Motion and Simple Harmonic Motion



This is in response to Paul Johnson's and John Denker's discussion about the
definition of harmonic motion.

For plain old "harmonic motion" (not simple harmonic motion), I think it is
sufficient to say the motion is periodic; it is not necessary to say it is a
sum of sine functions.

According to Fourier theory/analysis, all periodic motion, no matter how
complex, can be expressed as a sum of sine functions [x(t) = B(0) +
B(k)sin(kt+phase)], or a sum of sine and cosine functions [x(t) = A(0) +
A(k)cos(kt) + B(k)sin(kt)].

Therefore, saying harmonic motion is "periodic" and saying harmonic motion
can be described as "a sum of sine functions" is redundant. Given redundant
definitions, I would opt for the simpler description... harmonic motion is
periodic... or more complete: harmonic motion is motion for which the
equation of motion (position as a function of time) is a periodic function.

* * * * Below is a slight change of subject. * * * *

I have had some discussion with our math professors about periodic functions
and harmonic functions. There might be a slight difference in the way they
use the words and the way physicists use the words. As a physicist I often
view periodic functions and harmonic functions as equivalent. But my math
colleagues have a tendency to call a function periodic even when the
independent variable is not time, and a tendency to call it both periodic
and harmonic if the independent variable is time. I usually think of period
as a time interval. They think of period as any interval of space or time
or whatever in which the function completes "one cycle."

This cropped up as a result of an exam question for which I drew a graph of
a sound wave by drawing air pressure as a function of position. I indicated
(with dimension lines) a particular distance and asked the students to give
a name for that distance. I was expecting them to say "wavelength," and
that's what most students said. But some students called it the "period"
and I counted it wrong. The students who called it "period" complained.
They said "period" would be correct on a math exam. I consulted the math
professors and they agreed they would indeed call it the period if such a
graph appeared on a math test.

The dictionaries are on my side, they specifically say the root of period is
periodus and this specifically refers to a time interval. But I am
completely outnumbered by current-day math professors who are teaching scads
of students that period does not need to be a time interval. Nonetheless, I
still expect students to work problems in context. If I ask students for
the wavelength of a sound wave I expect them to report a length, and if I
ask for the period of a sound wave I expect them to report a time. However,
it is certainly an eye-opener to talk to math professors and realize how
much of the terminology used in math classes these days is stuff I never
heard of, or stuff that is different than the way I learned it and teach it.

Michael D. Edmiston, Ph.D. Phone/voice-mail: 419-358-3270
Professor of Chemistry & Physics FAX: 419-358-3323
Chairman, Science Department E-Mail edmiston@bluffton.edu
Bluffton College
280 West College Avenue
Bluffton, OH 45817