I think all the calculus-based
intro physics books on my shelf describe
_simple_ harmonic motion as systems
described by the differential equation:
d2x/dt2 + w2x = 0
X is position as a function of time.
w (omega) is angular frequency.
This differential equation results from
Newton's 2nd law by expressing a as the
second time derivative of position, and
expressing F as a linear restoring force
such as F equal minus kx. In this
particular case (a spring pendulum)we
have w2 = k/m
Therefore, the differential equation
that "defines" simple harmonic motion
results from a one-dimensional linear
restoring force.
It is also correct, as others have
mentioned, that an equivalent
form of the simple harmonic equation
would be
x = A cos(wt + phi)
x is position
A is amplitude
w is angular frequency
phi is phase constant
And, as mentioned, this is a single
or "pure" sinusoidal function.
On the other hand, motions that can be
described by a Fourier series such as
x = a1 cos (wt) + b1 sin (wt) +
a2 cos (2wt) + b2 sin (2wt) + ....
are harmonic, but are not simple harmonic.
Michael D. Edmiston, Ph.D.
Professor of Chemistry and Physics
Bluffton College
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu