Resonance

Resonance is the large response produced when periodic forcing aligns with a natural mode.

A resonator stores oscillatory energy in one or more normal modes. Driving near a mode's frequency can produce resonance. Losses limit the response and determine its quality factor. After the drive is removed, an isolated weakly damped mode decays approximately exponentially; an observed room response may combine many modes.

A frequency at which a specified driven response is resonant. The value can depend on the observable and damping: the electric current amplitude of a driven series RLC circuit peaks at $f_0=1/(2\pi \sqrt{LC})$, while a different circuit readout need not peak at the same frequency.

For a resonator, the quality factor is $2\pi$ times the stored energy divided by the energy dissipated per cycle, using a specified steady-state stored-energy convention. A series RLC circuit at resonance has $Q=L{\omega }_0/R$, because its maximum magnetic energy is $L{I}_M^2/2$ and one cycle of Joule heating dissipates $\pi R{I}_M^2/{\omega }_0$.

For a linear oscillator with viscous drag, this parameter puts the underdamped/critical boundary at $Q=1/2$. The energy-per-cycle interpretation and narrow displacement linewidth are most useful at high quality factor; critical and overdamped free motion do not complete oscillation cycles. Amplitude and averaged-energy ringdown times differ by a factor of two. The damping classification is not the same as the condition for a nonzero-frequency displacement maximum.

Minimizing $({\omega }_0^2-{\omega }^2{)}^2+4{\beta }^2{\omega }^2$ gives a displacement maximum at $\sqrt{{\omega }_0^2-2{\beta }^2}$ only when $\beta <{\omega }_0/\sqrt{2}$. An underdamped oscillator with $1/2<Q\le 1/\sqrt{2}$ consequently has no nonzero-frequency displacement peak. The absorbed-power response instead peaks at ${\omega }_0$, with exact half-power bandwidth $2\beta$ for this model. Comparing bandwidth definitions requires naming the measured response.

For a constant-amplitude force on $m\ddot{x}+b\dot{x}+kx=F_0\cos \Omega t$, the cycle-averaged absorbed power is $b{F}_0^2{\Omega }^2/[2((k-m{\Omega }^2{)}^2+b^2{\Omega }^2)]$. It peaks at ${\omega }_0=\sqrt{k/m}$. Its half-power frequencies are ${\Omega }_{\pm }=(\sqrt{b^2+4mk}\pm b)/(2m)$, hence their separation is exactly $b/m$ and $Q={\omega }_0/\Delta \Omega$. A displacement-amplitude resonance has different half-power frequencies away from weak damping; the measured observable must be specified.

Converting the two half-power frequencies using $\lambda =2\pi c/\omega$ gives $\Delta \lambda =2\pi c(1/{\omega }_{-}-1/{\omega }_{+})$. Since ${\omega }_{-}{\omega }_{+}={\omega }_0^2$, this reduces exactly to the displayed expression. The wavelength endpoints are not symmetric around $2\pi c/{\omega }_0$; a symmetric narrow-line sketch is an approximation. This conversion concerns the same measured response versus a reparametrized axis, not a transformation between spectral density conventions.

For an isolated weakly damped resonator mode, $Q=\omega E/P_{\mathrm{loss}}$. If $E(t)=E(0)e^{-t/{\tau }_E}$, differentiating gives $P_{\mathrm{loss}}=E/{\tau }_E$, so $Q=\omega {\tau }_E$. Since energy is quadratic in wave amplitude, the wave amplitude exponential relaxation time is ${\tau }_A=2{\tau }_E$. A decay by an energy factor $D$ in a measured interval $t_d$ gives $Q=\omega t_d/\ln D$. For a conventional reverberation time, $Q=2\pi f{T}_{60}/[6\ln (10)]$ under this exponential single-rate approximation.

For two weakly damped normal modes with initial intensity ratio $R_0>1$, independent exponential energy decay gives $I_1(t)/I_2(t)=R_0{e}^{-t({\omega }_1/Q_1-{\omega }_2/Q_2)}$. An observed reversal to a ratio below one implies the displayed upper bound on $Q_1$, provided the denominator is positive. A strict intensity comparison does not determine a unique quality factor; equality fixes only the crossing threshold.