This is just a solution to a second-order differential wave equation. What is being modified is the characteristic frequency of the wave equation. Here is a simpler one:
This is cast as a variant of a Hill or Mathieu differential equation where the modulation of the characteristic frequency $\omega^2$ is shown in the lower panel. Of course this is a contrived example but it illustrates how a perturbation will speed up or slow down the frequency of the oscillation transiently.
To me, this is a no-brainer to do this kind of analysis. If you ever did audio electronics this is just a LC resonant circuit with a variable capacitor which will create a variable frequency tone.
I didn't mean to make it so complicated, but I have to make it quantitative -- otherwise it is just hand-waving.