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:
Here's an animation showing the solution for many values of ω: