 Frequency bandwidth of viscous oscillator power absorption

For a forced harmonic oscillator with constant viscous damping coefficient $m\gamma >0$, the average power divided by its maximum is ${\gamma }^2{\omega }^2/[({\omega }_0^2-{\omega }^2{)}^2+{\gamma }^2{\omega }^2]$. It peaks at ${\omega }_0=\sqrt{k/m}$. Its two positive half-power angular frequencies are ${\omega }_{1,2}=(\sqrt{4{\omega }_0^2+{\gamma }^2}\mp \gamma )/2$. Hence their separation is exactly $\gamma$, and their product is ${\omega }_0^2$. The narrow-resonance approximation ${\omega }_{1,2}\simeq {\omega }_0\mp \gamma /2$ is needed for symmetry about the peak, not for the exact bandwidth in angular-frequency units.