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Derivation

Power Absorption and the Lorentzian Lineshape

Statement

For a linear damped oscillator driven sinusoidally at angular frequency \( \omega \), the time-averaged power delivered by the driving force is, in the neighbourhood of resonance, a Lorentzian function of \( \omega \) centred on \( \omega_0 \) with full width at half maximum equal to the damping rate \( \gamma \): \[ \langle P \rangle(\omega) \;=\; \langle P\rangle_{\max}\,\frac{(\gamma/2)^2}{(\omega-\omega_0)^2+(\gamma/2)^2}, \qquad \langle P\rangle_{\max}=\frac{F_0^2}{2m\gamma}, \] so that the half-power bandwidth \( \Delta\omega_{\rm FWHM}=\gamma \) and the quality factor obey \( Q=\omega_0/\gamma=\omega_0/\Delta\omega_{\rm FWHM} \).

Why it matters

The Lorentzian is the universal lineshape of a weakly damped resonance responding linearly to a harmonic drive. It is what a spectrometer records when a laser sweeps across an atomic transition, what a network analyser traces across an RLC filter, and what sets the linewidth of a laser cavity or a mechanical Q-meter. Reading a width off the curve and dividing it into the centre frequency is the single most common way to measure \( Q \) experimentally.

It also makes the energy story concrete: the same \( \gamma \) that governs how fast free oscillations decay governs how sharply the driven system absorbs. Absorption bandwidth and ringdown time are two faces of one number, which is why a high-\( Q \) bell rings for a long time yet responds only to a very narrow band of frequencies.

Assumptions
Linear restoring force and linear (viscous) damping.If either is nonlinear the response is no longer a single-frequency sinusoid; harmonics appear, the peak bends over (Duffing backbone), and the lineshape ceases to be Lorentzian.
Steady state has been reached.During the transient, input and dissipated powers differ; the clean equality \( \langle P_{\rm in}\rangle=\langle P_{\rm diss}\rangle \) that lets us evaluate the average holds only after transients have decayed over a time \( \sim 1/\gamma \).
Sinusoidal drive of fixed amplitude \( F_0 \) and frequency \( \omega \).A drive with its own spectral width convolves into the response, broadening the observed line beyond the intrinsic \( \gamma \).
Weak damping / high \( Q \): \( \gamma \ll \omega_0 \).The Lorentzian form is an approximation about the peak. If \( \gamma \) is not small the exact curve is asymmetric (the \( \omega^2 \) numerator and the \( \omega_0+\omega \) factor no longer cancel), the peak of the amplitude shifts below \( \omega_0 \), and FWHM \( =\gamma \) fails at second order.
Time-averaging over a whole number of drive cycles.If averaged over a partial cycle or a window comparable to \( 2\pi/\omega \), cross terms \( \langle \sin\cos\rangle \) do not vanish and \( \langle P\rangle \) picks up a spurious oscillating part.
Derivation
1
\[ A(\omega)=\frac{F_0/m}{\sqrt{(\omega_0^2-\omega^2)^2+(\gamma\omega)^2}},\qquad x(t)=A\cos(\omega t-\delta) \]
Steady-state amplitude of \( m\ddot x+b\dot x+kx=F_0\cos\omega t \) with \( \gamma=b/m,\ \omega_0^2=k/m \), imported from the driven-oscillator resonance result. A
2
\[ \langle P\rangle=\langle P_{\rm diss}\rangle=\big\langle b\,\dot x^2\big\rangle,\qquad \dot x=-A\omega\sin(\omega t-\delta) \]
In steady state the cycle-averaged input power equals the cycle-averaged dissipated power \( b\dot x^2 \), since the stored energy returns to the same value each period. B
3
\[ \big\langle \sin^2(\omega t-\delta)\big\rangle=\tfrac12 \;\Rightarrow\; \langle P\rangle=\tfrac12\,b\,\omega^2 A^2=\tfrac12\,m\gamma\,\omega^2 A^2 \]
Average of \( \sin^2 \) over a full cycle is \( 1/2 \); substitute \( b=m\gamma \). Symbols only, no numbers yet. A
4
\[ \langle P\rangle(\omega)=\frac{F_0^2\,\gamma\,\omega^2}{2m\big[(\omega_0^2-\omega^2)^2+(\gamma\omega)^2\big]} \]
Insert \( A^2 \) from Step 1. This is the exact power-absorption curve, valid for any \( \gamma \). A
5
\[ \omega_0^2-\omega^2=(\omega_0-\omega)(\omega_0+\omega)\;\approx\;2\omega_0(\omega_0-\omega),\qquad \omega\approx\omega_0 \]
Near resonance, factor the difference of squares and keep the leading term; the slowly varying factor \( \omega_0+\omega\to2\omega_0 \) and \( \omega\to\omega_0 \) elsewhere. Legal because \( \gamma\ll\omega_0 \) confines all the action to \( |\omega-\omega_0|\lesssim\gamma \). B
6
\[ \langle P\rangle\approx\frac{F_0^2\,\gamma\,\omega_0^2}{2m\big[4\omega_0^2(\omega_0-\omega)^2+\gamma^2\omega_0^2\big]} =\frac{F_0^2}{2m}\,\frac{\gamma}{4(\omega-\omega_0)^2+\gamma^2} \]
Substitute the approximation of Step 5 into Step 4 and cancel the common \( \omega_0^2 \) from numerator and denominator. B
7
\[ \frac{\gamma}{4(\omega-\omega_0)^2+\gamma^2}=\frac{\gamma}{4}\,\frac{1}{(\omega-\omega_0)^2+(\gamma/2)^2} \;\Rightarrow\; \langle P\rangle\approx\frac{F_0^2}{2m\gamma}\,\frac{(\gamma/2)^2}{(\omega-\omega_0)^2+(\gamma/2)^2} \]
Divide top and bottom by 4 to expose the standard Lorentzian denominator with half-width \( \gamma/2 \); the prefactor is the peak value \( \langle P\rangle_{\max}=F_0^2/2m\gamma \) reached at \( \omega=\omega_0 \). A
Result
\[ \boxed{\;\langle P\rangle(\omega)=\frac{F_0^2}{2m\gamma}\,\frac{(\gamma/2)^2}{(\omega-\omega_0)^2+(\gamma/2)^2}\;,\qquad \Delta\omega_{\rm FWHM}=\gamma=\frac{\omega_0}{Q}\;} \]

Reading. The absorbed power traces a symmetric bell centred on \( \omega_0 \). Its peak \( F_0^2/2m\gamma \) grows as damping falls, while its width \( \gamma \) shrinks in step: a sharper resonance absorbs more strongly but over a narrower band. The half-power points sit at \( \omega=\omega_0\pm\gamma/2 \), so measuring the width and the centre delivers \( Q=\omega_0/\gamma \) directly.

Units check. \( \dfrac{F_0^2}{m\gamma} \) has units \( \dfrac{\mathrm{N}^2}{\mathrm{kg}\cdot\mathrm{s}^{-1}}=\dfrac{(\mathrm{kg\,m\,s^{-2}})^2\,\mathrm{s}}{\mathrm{kg}}=\mathrm{kg\,m^2\,s^{-3}}=\mathrm{N\,m\,s^{-1}}=\mathrm{W} \). The Lorentzian factor is dimensionless (both terms are \( (\mathrm{rad/s})^2 \)), so \( \langle P\rangle \) is a power.

Limiting cases
  • On resonance \( \omega=\omega_0 \): \( \langle P\rangle=F_0^2/2m\gamma \), the maximum; all drive work is dissipated at the largest possible rate.
  • Far off resonance \( |\omega-\omega_0|\gg\gamma \): \( \langle P\rangle\to \dfrac{F_0^2\gamma}{8m(\omega-\omega_0)^2}\to 0 \); the wings fall as \( 1/(\omega-\omega_0)^2 \).
  • High \( Q \) \( (\gamma\to0) \): a tall, needle-thin line; peak \( \propto1/\gamma \), width \( \propto\gamma \), so peak\(\times\)width \( =F_0^2/2m \) is fixed (integrated absorption is \( \gamma \)-independent).
  • Half-power detuning \( \omega-\omega_0=\pm\gamma/2 \): \( \langle P\rangle=\tfrac12\langle P\rangle_{\max} \), which defines the FWHM \( =\gamma \).
  • Static limit \( \omega\to0 \): the exact curve gives \( \langle P\rangle\to0 \) as \( \omega^2 \); a DC force does no time-averaged work.
Breaks when
  • Damping is not small. When \( \gamma\sim\omega_0 \) the difference-of-squares linearisation of Step 5 fails: the exact curve becomes asymmetric, its features shift, and the width is no longer exactly \( \gamma \). The Lorentzian is only the leading approximation about the peak.
  • Nonlinearity sets in. A large-amplitude drive on a real (anharmonic) spring or a nonlinear damping law produces a folded, hysteretic response (Duffing) with jumps — not a single-valued Lorentzian at all.
  • Inhomogeneous or Doppler broadening dominates. An ensemble of oscillators with a spread of \( \omega_0 \) gives a Gaussian (or Voigt) envelope; the intrinsic Lorentzian survives only as the homogeneous core.
  • Neighbouring modes overlap. If another resonance lies within \( \sim\gamma \), the individual Lorentzians interfere (Fano lineshapes, avoided crossings) and neither peaks nor widths read off cleanly.
Failure modes
  • Amplitude-peak vs power-peak confusion. The amplitude resonance sits slightly below \( \omega_0 \) at \( \sqrt{\omega_0^2-\gamma^2/2} \), but the power (and velocity) resonance is exactly at \( \omega_0 \). Students quote the amplitude shift for the absorption peak.
  • Half-width vs full-width slip. The Lorentzian denominator carries \( (\gamma/2)^2 \); the FWHM is \( \gamma \), not \( \gamma/2 \). Writing FWHM \( =\gamma/2 \) doubles the inferred \( Q \).
  • Forgetting the cycle average. Using instantaneous \( P=F\dot x \) instead of \( \langle P\rangle \) leaves an oscillating term and gives the wrong peak by a factor of 2.
  • Using \( x \) instead of \( \dot x \) for dissipation. Power dissipated is \( b\dot x^2 \), so it scales with \( \omega^2 A^2 \); those who write \( \propto A^2 \) misplace the peak toward low \( \omega \).
  • Dropping the \( \omega^2 \) numerator too early. Setting \( \omega\to\omega_0 \) in the numerator before checking the far wings hides the correct asymmetric \( 1/(\omega-\omega_0)^2 \) tail.
Discussion

The result welds two apparently separate measurements into one. The quality factor was first met as an energy quantity — \( Q=\omega_0/\gamma \) counts the radians of free oscillation over which the stored energy falls by \( 1/e \), and a lightly damped system rings for \( \sim Q/\pi \) periods. Here the very same \( \gamma \) reappears as the width in frequency over which the driven system absorbs. Ringdown time in the time domain and absorption width in the frequency domain are a Fourier pair; their product is fixed, which is the resonance version of a bandwidth–duration relation.

The integrated line strength is revealing. Integrating the near-resonance Lorentzian over all \( \omega \) gives \( \int\langle P\rangle\,d\omega=\pi F_0^2/4m \), independent of \( \gamma \). Damping redistributes a fixed total absorption between height and width but never changes the area. This is why oscillator strengths and integrated cross sections, not peak heights, are the robust spectroscopic observables: they survive any broadening that conserves the transition's total coupling.

The Lorentzian is not a coincidence of this problem but the signature of a single complex pole. Near resonance the susceptibility \( \chi(\omega)=1/(\omega_0^2-\omega^2-i\gamma\omega) \) has a pole at \( \omega\approx\omega_0-i\gamma/2 \) in the lower half-plane; the absorbed power is \( \propto\omega\,\mathrm{Im}\,\chi \), and the imaginary part of a simple pole is exactly a Lorentzian whose width is twice the pole's imaginary part. Causality (the pole lies in the lower half-plane) ties the absorptive \( \mathrm{Im}\,\chi \) to the dispersive \( \mathrm{Re}\,\chi \) through the Kramers–Kronig relations, so the very \( \gamma \) that broadens the absorption line also sets the anomalous dispersion across it. Every damped mode that couples linearly to a drive — an atomic dipole, an LC tank, a cavity field, a nuclear spin — inherits this one pole and hence this one lineshape.

Common misconceptions. The absorption peak is not where the amplitude is largest; power tracks velocity, which peaks exactly at \( \omega_0 \). A higher \( Q \) does not mean more total energy absorbed — it means the same integrated absorption squeezed into a narrower, taller line. And the width \( \gamma \) is an intrinsic (homogeneous) property of the oscillator; it is not made narrower by driving more gently, only masked by extra inhomogeneous broadening.

Worked examples
1
\[ \text{Mechanical Q-meter: } m=0.050\,\mathrm{kg},\ \omega_0=200\,\mathrm{rad/s},\ \gamma=4.0\,\mathrm{rad/s},\ F_0=0.10\,\mathrm{N}. \]
Given data; find the peak absorbed power, the FWHM, \( Q \), and the power at a detuning of one half-width. A
2
\[ \langle P\rangle_{\max}=\frac{F_0^2}{2m\gamma}=\frac{(0.10)^2}{2(0.050)(4.0)}=\frac{0.010}{0.40}=0.025\,\mathrm{W}=25\,\mathrm{mW} \]
Peak formula; symbols first, then numbers with units. A
3
\[ \Delta\omega_{\rm FWHM}=\gamma=4.0\,\mathrm{rad/s},\qquad Q=\frac{\omega_0}{\gamma}=\frac{200}{4.0}=50 \]
Width equals the damping rate; \( Q \) is centre over width. A
4
\[ \text{At }\omega-\omega_0=\gamma/2=2.0:\quad \frac{\langle P\rangle}{\langle P\rangle_{\max}}=\frac{(\gamma/2)^2}{(\gamma/2)^2+(\gamma/2)^2}=\tfrac12 \;\Rightarrow\; \langle P\rangle=12.5\,\mathrm{mW} \]
Half-power point, by construction of the FWHM. A
\[ \langle P\rangle_{\max}=25\,\mathrm{mW},\quad \Delta\omega_{\rm FWHM}=4.0\,\mathrm{rad/s},\quad Q=50 \]

Reading. A modestly damped mechanical resonator dumps 25 mW at line centre and half that a mere 2 rad/s off — a 1% detuning already halves the absorption, the hallmark of \( Q=50 \).

Units check. \( \mathrm{N}^2/(\mathrm{kg\cdot s^{-1}})=\mathrm{W} \); rad/s for the width; \( Q \) dimensionless.

1
\[ \text{Series RLC filter: } L=1.0\,\mathrm{mH},\ C=1.0\,\mathrm{nF},\ R=10\,\Omega,\ V_0=5.0\,\mathrm{V}. \]
The circuit analogue: \( L\ddot q+R\dot q+q/C=V_0\cos\omega t \), so \( \omega_0=1/\sqrt{LC} \) and \( \gamma=R/L \). The absorbed power \( \langle P\rangle=\tfrac12 V_0^2R/[R^2+(\omega L-1/\omega C)^2] \) is Lorentzian near \( \omega_0 \). B
2
\[ \omega_0=\frac{1}{\sqrt{LC}}=\frac{1}{\sqrt{(10^{-3})(10^{-9})}}=\frac{1}{\sqrt{10^{-12}}}=1.0\times10^{6}\,\mathrm{rad/s} \]
Resonant frequency from \( L \) and \( C \). A
3
\[ \gamma=\frac{R}{L}=\frac{10}{10^{-3}}=1.0\times10^{4}\,\mathrm{rad/s},\qquad Q=\frac{\omega_0}{\gamma}=\frac{10^{6}}{10^{4}}=100 \]
Damping rate and quality factor; \( \Delta\omega_{\rm FWHM}=\gamma=10^4\,\mathrm{rad/s} \). A
4
\[ \langle P\rangle_{\max}=\frac{V_0^2}{2R}=\frac{(5.0)^2}{2(10)}=\frac{25}{20}=1.25\,\mathrm{W},\qquad \Delta f_{\rm FWHM}=\frac{\gamma}{2\pi}\approx1.6\,\mathrm{kHz} \]
At resonance the reactances cancel and the source sees pure \( R \); convert the angular width to Hz. A
\[ \omega_0=1.0\times10^{6}\,\mathrm{rad/s},\quad Q=100,\quad \langle P\rangle_{\max}=1.25\,\mathrm{W},\quad \Delta f_{\rm FWHM}\approx1.6\,\mathrm{kHz} \]

Reading. The same Lorentzian machinery reads a bandpass filter: a \( Q=100 \) tank passes 1.25 W at \( \approx159\,\mathrm{kHz} \) over a 1.6 kHz window — the electrical twin of the mechanical resonator above.

Units check. \( V^2/\Omega=\mathrm{W} \); \( R/L=\Omega/\mathrm{H}=\mathrm{s^{-1}} \); \( 1/\sqrt{LC}=1/\sqrt{\mathrm{H\,F}}=\mathrm{s^{-1}} \).

Problems
  1. (A) Width to Q. A driven oscillator has \( f_0=440\,\mathrm{Hz} \) and its power-absorption line is measured to have FWHM \( 22\,\mathrm{Hz} \). Find \( Q \) and the damping rate \( \gamma \).
    Solution \( \Delta\omega_{\rm FWHM}=2\pi\times22=138\,\mathrm{rad/s}=\gamma \). \( Q=\omega_0/\gamma=f_0/\Delta f_{\rm FWHM}=440/22=20 \). So \( \gamma\approx1.4\times10^{2}\,\mathrm{rad/s} \), \( Q=20 \).
  2. (A) Peak power. For the oscillator \( m=0.20\,\mathrm{kg} \), \( \gamma=5.0\,\mathrm{rad/s} \) driven by \( F_0=0.40\,\mathrm{N} \) at resonance, compute \( \langle P\rangle_{\max} \).
    Solution \( \langle P\rangle_{\max}=F_0^2/2m\gamma=(0.40)^2/[2(0.20)(5.0)]=0.16/2.0=0.080\,\mathrm{W}=80\,\mathrm{mW} \).
  3. (B) Detuning. Using the Lorentzian, at what detuning \( |\omega-\omega_0| \) (in units of \( \gamma \)) does the absorbed power fall to one tenth of its peak?
    Solution Set \( (\gamma/2)^2/[(\omega-\omega_0)^2+(\gamma/2)^2]=0.10 \). Then \( (\omega-\omega_0)^2=9(\gamma/2)^2 \Rightarrow |\omega-\omega_0|=3(\gamma/2)=1.5\,\gamma \). The 10% points lie at \( \pm1.5\gamma \), i.e. \( \pm3 \) half-widths.
  4. (B) Integrated absorption. Show that the area under the near-resonance Lorentzian, \( \int_{-\infty}^{\infty}\langle P\rangle\,d\omega \), is independent of \( \gamma \), and evaluate it.
    Solution \( \int_{-\infty}^{\infty}\dfrac{(\gamma/2)^2}{(\omega-\omega_0)^2+(\gamma/2)^2}\,d\omega=(\gamma/2)^2\cdot\dfrac{\pi}{(\gamma/2)}=\dfrac{\pi\gamma}{2} \). Multiplying by \( \langle P\rangle_{\max}=F_0^2/2m\gamma \) gives \( \int\langle P\rangle\,d\omega=\dfrac{F_0^2}{2m\gamma}\cdot\dfrac{\pi\gamma}{2}=\dfrac{\pi F_0^2}{4m} \), which contains no \( \gamma \). Height and width trade off to keep the area fixed.
  5. (C) Peak-shift subtlety. Starting from the exact power \( \langle P\rangle(\omega)=\dfrac{F_0^2\gamma\omega^2}{2m[(\omega_0^2-\omega^2)^2+\gamma^2\omega^2]} \), prove that the maximum lies exactly at \( \omega=\omega_0 \) (not below it), in contrast to the amplitude resonance at \( \sqrt{\omega_0^2-\gamma^2/2} \).
    Solution Write \( \langle P\rangle\propto\dfrac{\omega^2}{(\omega_0^2-\omega^2)^2+\gamma^2\omega^2} \). Divide numerator and denominator by \( \omega^2 \): \( \langle P\rangle\propto\dfrac{1}{(\omega_0^2-\omega^2)^2/\omega^2+\gamma^2}=\dfrac{1}{(\omega_0^2/\omega-\omega)^2+\gamma^2} \). The bracket \( g(\omega)=\omega_0^2/\omega-\omega \) equals zero when \( \omega_0^2/\omega=\omega \Rightarrow\omega=\omega_0 \); there the denominator is smallest \( (=\gamma^2) \), so \( \langle P\rangle \) is maximal exactly at \( \omega_0 \), for any \( \gamma \). The amplitude, lacking the \( \omega^2 \) numerator, peaks at the lower \( \sqrt{\omega_0^2-\gamma^2/2} \).