Resonance

When you pull a mass on a spring aside and release it, or push a child on a swing once and step back, the system oscillates on its own. This is a free oscillation: there is no external driving force keeping it going, and (in the ideal case) no energy is lost, so it oscillates with constant amplitude at a single, characteristic frequency.

That characteristic frequency is the natural frequency, $f_0$ — the frequency at which the system oscillates when displaced and released. It is a property of the system itself, fixed by the physical quantities that control the restoring force and the inertia:

• Mass–spring system: $f_0=\dfrac{1}{2\pi}\sqrt{\dfrac{k}{m}}$ — a stiffer spring (larger $k$) or a smaller mass gives a higher $f_0$.
• Simple pendulum: $f_0=\dfrac{1}{2\pi}\sqrt{\dfrac{g}{L}}$ — a shorter pendulum gives a higher $f_0$. Notice it depends only on length and $g$, not on the mass of the bob or (for small swings) the amplitude.

These come straight from Simple Harmonic Motion: for SHM the angular frequency is $\omega=\sqrt{k/m}$ (spring) or $\omega=\sqrt{g/L}$ (pendulum), and since $\omega=2\pi f_0$, dividing by $2\pi$ gives the natural frequency in hertz. Keep $\omega$ (in $\text{rad s}^{-1}$) and $f_0$ (in $\text{Hz}$) distinct — the missing $2\pi$ is a very common slip.

The key idea to lock in now, before we add a driver, is that $f_0$ belongs to the system. Later, when an external force drives the system, that force has its own frequency (the driving frequency). Whether the response is dramatic or feeble depends entirely on how the driving frequency compares with this natural frequency.

Real oscillators do not swing forever. Resistive forces — air resistance, friction, viscous drag in a fluid — do negative work against the motion, transferring the oscillator's mechanical energy to the surroundings (as heat). This gradual loss of energy is damping, and because the energy of an oscillator is proportional to the square of its amplitude ($E\propto A^2$), losing energy means the amplitude falls.

The amount of damping decides the character of the motion. There are three standard cases:

• Light damping. Only a little energy is lost per cycle, so the system keeps oscillating, but the amplitude decays exponentially — each successive peak is a fixed fraction of the one before, so the oscillations sit inside a shrinking 'envelope'. The period is almost unchanged. (Think of a pendulum swinging in air, gradually dying away.)
• Critical damping. The system returns to equilibrium in the shortest possible time without oscillating at all — it does not overshoot past the equilibrium position. This is the ideal for anything that must settle quickly.
• Heavy (over) damping. The resistive force is so large that the system also does not oscillate, but it creeps back to equilibrium slowly — more slowly than critical damping.

Why the distinction matters. Critical damping is engineered into systems that must settle quickly and cleanly:

• Car suspension / shock absorbers — the car body should return to level after a bump quickly, without bouncing up and down.
• Analogue meter needles — the pointer should move straight to the reading without wobbling either side of it.
• Self-closing doors — the door should close briskly but not slam or swing back.

In every case light damping would leave an unwanted oscillation, and heavy damping would make the response sluggish; critical damping is the sweet spot.

Now attach a driver — a continuous external periodic force, such as a vibration generator, someone pushing a swing repeatedly, or people walking in step across a bridge. The system is now undergoing a forced oscillation, and the crucial fact is that it oscillates at the driving frequency (the frequency of the external force), not at its own natural frequency.

How big the resulting oscillation is depends on how the driving frequency compares with the natural frequency $f_0$:

• Drive well below $f_0$ → small amplitude; the system just about keeps up with the driver.
• Drive close to $f_0$ → the amplitude grows rapidly.
• Drive exactly at $f_0$ → the amplitude reaches a maximum. This is resonance.
• Drive well above $f_0$ → the amplitude falls again.

Resonance is therefore the condition in which the driving frequency equals the natural frequency, $f_{\text{driving}}=f_0$. At this point the driver is 'in step' with the system's own rhythm, so it feeds in energy at exactly the right moments each cycle. The system absorbs the maximum energy from the driver, and — because energy is stored as amplitude — the amplitude is a maximum.

A neat demonstration is Barton's pendulums: several pendulums of different lengths hang from one string, and a single 'driver' pendulum is set swinging. All the others are forced to oscillate through the shared string, but only the one whose length (and hence natural frequency) matches the driver swings with a large amplitude — it is being driven at resonance. This makes the resonance condition visible: matching frequencies, maximum response.

Because resonance is a matching condition, the amplitude falls whether you move the driving frequency above or below $f_0$. It is not a 'high frequency' effect — a slow-swinging swing resonates at a low frequency just as readily.

Plotting the amplitude of a forced oscillation against the driving frequency gives the resonance curve. It rises to a peak at the natural frequency $f_0$ and falls away on either side. The shape of this peak is controlled by the amount of damping.

Reading the curve, increasing the damping does three things — you should be able to state and explain all three:

1. The peak gets lower. With more damping, more energy is dissipated each cycle, so the system cannot build up as large an oscillation — the maximum amplitude at resonance falls.
2. The peak gets broader (less sharp). A heavily damped system responds appreciably over a wider range of driving frequencies, so the curve is flatter and wider around the peak. A lightly damped system responds strongly only very close to $f_0$ (a sharp peak).
3. The peak shifts to a slightly lower frequency. For light damping the maximum sits essentially at $f_0$; heavier damping moves the maximum slightly below $f_0$.

Exam discipline. A resonance-curve question almost always wants both 'lower' and 'broader'. Writing only that the peak gets lower is the single most common way to drop a mark here. Say: "the peak becomes lower and broader (and shifts to a slightly lower frequency)."

The underlying cause in every case is the same: greater damping means larger resistive forces removing more energy per cycle, so the oscillator stores less energy and responds over a wider spread of frequencies.

Resonance is a double-edged phenomenon: engineers deliberately exploit it in some devices and fight to suppress it in others. The physics is identical — a driving force at the system's natural frequency producing a maximum-amplitude response — but the consequences differ.

Useful resonance

• Radio and TV tuning. A tuning circuit has a natural (resonant) frequency you adjust with a dial. When it matches the frequency of one broadcast, that signal produces a large response while all others are barely picked up — this is how a single station is selected.
• MRI (magnetic resonance imaging). Hydrogen nuclei in the body resonate with radio-frequency pulses at their natural (Larmor) frequency, allowing detailed images of soft tissue.
• Musical instruments. The air column in a wind instrument, or a string on a guitar, resonates at particular natural frequencies to produce loud, sustained notes.
• Microwave ovens. Microwaves drive water molecules near a resonant frequency, transferring energy efficiently to heat food.

Problematic resonance

• Bridges. A footbridge has natural frequencies of sway. If people walk in step at a matching pace (or the wind provides a periodic push), the bridge resonates and the sway amplitude can build to dangerous levels — famously the Tacoma Narrows Bridge (driven by wind) and the London Millennium Bridge (driven by pedestrians in step).
• Buildings in earthquakes. Ground shaking at a building's natural frequency drives it into large-amplitude resonant oscillation, which can cause structural failure. Tall buildings are especially at risk.
• Machinery. Rotating parts can drive a mounting or panel at its natural frequency, causing loud, damaging vibrations.

The cure is damping. To reduce unwanted resonance, engineers add damping — devices that introduce resistive forces to dissipate the oscillation energy (as heat) before it can build up. Adding damping lowers and broadens the resonance peak, so the structure no longer responds violently to one precise driving frequency. Examples include shock-absorber-style dampers on bridges, tuned mass dampers (a large mass that oscillates out of phase to soak up energy, as in Taipei 101), and rubber mountings on machinery. The alternative approach is to design the natural frequency away from any likely driving frequency in the first place.