This example illustrates periodic variation of amplitude which occurs when you have a forced,

undamped vibration. Suppose a vibration is modelled by y′′+y′=cos(ωt),y(0)=0,y′(0)=0.

Then the solution will be y=(21−ω2sin(2(1−ω)t))sin(2(1+ω)t).

If ω is close to 1, then 1−ω is small, and hence the factor sin(2(1+ω)t)

is oscillating much faster than 21−ω2sin(2(1−ω)t).

So, we think of | | 21−ω2sin(2(1−ω)t)| | (shown in red) as periodically varying the amplitude of the sine curve.
Here are plots for various values of ω:

ω=0.7
ω=0.9
ω=0.95
Here's an animation showing the solution for many values of ω: