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