Damped Harmonic Motion
Suppose a particle of mass, m, moves in one dimension, subject to both a restoring force, =  kx (Hooke's Law)
and also a damping force which is proportional to the velocity of the particle, =  bv =  b dx/dt. The equation
of motion is therefore
The solution to this equation is
(
)
where ω, the oscillation angular frequency, is
√
(
)
√ being the undamped angular frequency.
We note that in the limit of a vanishingly small resistive force, i.e., as b→0, the solution becomes that of
simple harmonic motion. An example of damped harmonic motion is plotted below. The amplitude of the
oscillation is
. This function, plotted by itself, represents an envelope which limits the maxima and
minima in the curve.