Resonance Lineshape and Bandwidth
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You should be comfortable with the equation of motion for a damped, driven harmonic oscillator and the definition of the quality factor \( Q \) as a measure of energy storage relative to dissipation.
Why this matters
The shape of a resonance peak tells you how sharply a system selects its natural frequency.
In spectroscopy, radio engineering, and mechanical design, the width of a resonance line is often as important as its centre frequency. A narrow line implies high selectivity and long-lived oscillations, while a broad line indicates strong damping and rapid energy loss. Understanding the Lorentzian lineshape allows you to predict how much power a system absorbs at a given driving frequency and to quantify the bandwidth over which the system remains effective. This concept is fundamental to interpreting experimental data in physics and engineering, where the "sharpness" of a resonance directly impacts performance and measurement precision.
The Shape of Resonance
Imagine pushing a child on a swing. If you push at the right rhythm, the swing goes very high. If you push slightly off-rhythm, the swing still moves, but not as high. The "height" of the swing corresponds to the power absorbed by the system. The "rhythm" is the driving frequency. The relationship between how close you are to the perfect rhythm and how high the swing goes is not linear; it follows a specific, symmetric curve known as a Lorentzian. This curve has a sharp peak at the natural frequency and falls off smoothly on either side. The width of this peak at half its maximum height is a direct measure of how "damped" the system is. A very damped system (like a swing with heavy friction) has a broad, low peak, meaning it absorbs energy over a wide range of frequencies but never reaches a high amplitude. A lightly damped system (like a well-oiled pendulum) has a narrow, tall peak, meaning it is very selective about the driving frequency but can store a lot of energy when driven at resonance.
The Lorentzian Function
The time-averaged power absorbed by a linear damped oscillator driven sinusoidally is, near resonance and for light damping, very well approximated by a Lorentzian function of the driving angular frequency \( \omega \). This function is centred on the natural angular frequency \( \omega_0 \) and has a characteristic width determined by the damping rate \( \gamma \). The mathematical form is:
Here, \( \langle P\rangle_{\max} \) is the maximum power absorbed, which occurs exactly at \( \omega = \omega_0 \). The term \( \gamma \) represents the damping rate, which is inversely proportional to the quality factor \( Q \). The Lorentzian shape is symmetric about \( \omega_0 \) and decays as \( 1/(\omega - \omega_0)^2 \) far from resonance. This functional form is universal for linear systems with exponential decay, making it a cornerstone of wave physics and quantum mechanics.
Bandwidth and Quality Factor
The full width at half maximum (FWHM) of the Lorentzian peak is a key parameter. It is defined as the difference between the two frequencies where the power is half of its maximum value. For the Lorentzian function above, this width is exactly equal to the damping rate \( \gamma \):
The quality factor \( Q \) is defined as the ratio of the resonant frequency to the bandwidth: \( Q = \omega_0 / \Delta\omega_{\rm FWHM} \). A high \( Q \) value indicates a narrow bandwidth and low damping, meaning the system oscillates for many cycles before its energy decays significantly. Conversely, a low \( Q \) value indicates a broad bandwidth and high damping. In practical terms, \( Q \) quantifies the efficiency of energy storage relative to energy loss per cycle. For example, a tuning fork has a high \( Q \) (narrow bandwidth), while a doorbell has a low \( Q \) (broad bandwidth).
Maximum Power Absorption
The maximum power absorbed at resonance depends on the amplitude of the driving force \( F_0 \), the mass \( m \), and the damping rate \( \gamma \). The expression for the peak power is:
This result shows that for a fixed driving force, the power absorbed at resonance is inversely proportional to the damping. Lightly damped systems (small \( \gamma \)) absorb much more power at resonance than heavily damped systems. This is why resonance can be dangerous in structures: if the damping is low, even a small periodic force can lead to large energy absorption and potentially destructive amplitudes. The relationship also highlights the trade-off between selectivity (narrow bandwidth) and power handling (high peak power) in resonant systems.
The derivation
Power Absorption and the Lorentzian Lineshape
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 \( Q=\omega_0/\gamma=\omega_0/\Delta\omega_{\rm FWHM} \).
A common misconception is that the resonant frequency shifts significantly with damping. In reality, for a linear damped oscillator, the frequency of maximum power absorption is exactly \( \omega_0 \), the natural frequency of the undamped system. The damping affects the width and height of the peak, but not its centre position. This is a subtle point that distinguishes the power resonance from the amplitude resonance, which can shift slightly for very high damping, but in the standard linear theory used here, the centre remains fixed at \( \omega_0 \). C
Worked example
Problem. A damped harmonic oscillator has a mass \( m = 0.5\ \mathrm{kg} \), a natural angular frequency \( \omega_0 = 100\ \mathrm{rad/s} \), and is driven by a sinusoidal force with amplitude \( F_0 = 10\ \mathrm{N} \). The quality factor is \( Q = 50 \). Calculate the full width at half maximum (FWHM) of the resonance peak and the maximum power absorbed.
- Determine the damping rate from the definition of \( Q \): \( \gamma = \omega_0 / Q = 100/50 = 2.0\ \mathrm{s^{-1}} \).
- The FWHM of the power resonance equals \( \gamma \): \( \Delta\omega_{\rm FWHM} = 2.0\ \mathrm{rad/s} \).
- The maximum mean power absorbed (at \( \omega = \omega_0 \)) is \( \langle P\rangle_{\max} = F_0^2 / (2m\gamma) = 10^2/(2 \times 0.5 \times 2.0) = 50\ \mathrm{W} \).
Units check. \( \gamma \) has units of \( \mathrm{rad/s} \) (equivalent to \( \mathrm{s^{-1}} \)). \( F_0^2 \) is \( \mathrm{N^2} = \mathrm{kg^2 m^2 s^{-4}} \). \( m\gamma \) is \( \mathrm{kg \cdot s^{-1}} \). The ratio \( F_0^2 / (m\gamma) \) is \( \mathrm{kg m^2 s^{-3}} = \mathrm{W} \). The units are consistent.
Three quick tests
- If the damping rate \( \gamma \) is doubled, what happens to the FWHM?
Answer
The FWHM doubles, as \( \Delta\omega_{\rm FWHM} = \gamma \). - If the quality factor \( Q \) is increased by a factor of 10, what happens to the maximum power absorbed (assuming \( F_0 \) and \( m \) are constant)?
Answer
The maximum power absorbed increases by a factor of 10, since \( \langle P\rangle_{\max} \propto 1/\gamma \propto Q \). - Is the Lorentzian lineshape symmetric about \( \omega_0 \)?
Answer
Yes, the function \( \frac{(\gamma/2)^2}{(\omega-\omega_0)^2+(\gamma/2)^2} \) is symmetric with respect to \( \omega = \omega_0 \).
Problem set for PU-103 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes drafted with a local model and checked line by line by the editor; every number in the worked example was recomputed.