 Resonant forcing

Resonant forcing couples an input to a natural homogeneous mode. For the undamped harmonic oscillator equation $y^{\prime \prime }+{\omega }_0^{2}y=F\cos ({\omega }_{0}t)$, ${\omega }_0>0$, a particular response isIts linearly growing envelope follows by direct substitution and reflects coherent energy input each cycle. Initial conditions add bounded homogeneous sine and cosine terms, which cannot remove that secular growth.

Positive damping changes the conclusion. For $y^{\prime \prime }+2\gamma y^{\prime }+{\omega }_0^{2}y=F\cos (\Omega t)$ with $\gamma >0$, the steady harmonic amplitude isIt is finite at $\Omega ={\omega }_0$, where it equals $|F|/(2\gamma {\omega }_0)$. A finite resonance peak in a damped system is therefore distinct from the unbounded undamped resonant solution. The damping and forcing conventions must be specified before asserting linear amplitude growth.