How can a phase reversal occur?

Recall that the formulation for the standing wave equation in temporal frequency space is

$ (-\omega^2+\omega_o^2)F(\omega) = Forcing(\omega) $

Note that the forcing has a sign change about the resonant condition $\omega_0$. So what happens if temporarily a forcing is applied that is near the resonance condition but with a frequency on the side of the peak that has the opposite sign of the prevailing standing wave phase? I will assert that this may be enough to force the output to change sign, and that this most likely occur at a zero crossing where the impact would be strongest.

I can easily test this out but may have to add a stronger dampening term to make sure that the disturbance can die out.

Recall that the formulation for the standing wave equation in temporal frequency space is

$ (-\omega^2+\omega_o^2)F(\omega) = Forcing(\omega) $

Note that the forcing has a sign change about the resonant condition $\omega_0$. So what happens if temporarily a forcing is applied that is near the resonance condition but with a frequency on the side of the peak that has the opposite sign of the prevailing standing wave phase? I will assert that this may be enough to force the output to change sign, and that this most likely occur at a zero crossing where the impact would be strongest.

I can easily test this out but may have to add a stronger dampening term to make sure that the disturbance can die out.