Phase matching for a displacement referenced driven oscillator ($(k-m{\omega }^2)\sin {\varphi }_0=m\gamma \omega \cos {\varphi }_0$)

For the forced harmonic oscillator $m\ddot{z}+m\gamma \dot{z}+kz=F_0\cos (\omega t+{\varphi }_0)$, a response $z=A\cos \omega t$ requires $A(k-m{\omega }^2)=F_0\cos {\varphi }_0$ and $Am\gamma \omega =F_0\sin {\varphi }_0$. Squaring and adding gives the amplitude $A=F_0/\sqrt{(k-m{\omega }^2{)}^2+(m\gamma \omega {)}^2}$ for positive $A,F_0$. The two coefficient equations determine the full quadrant of the forcing phase, while their ratio alone can lose that information. In particular the amplitude cannot generally be independent of damping: at $m=k=F_0=\omega =1$ and ${\varphi }_0=\pi /2$, it is $A=1/\gamma$.