Left to itself, every real oscillator eventually stops. Friction and air resistance quietly drain its energy, cycle after cycle — that draining is damping. The exam skill here is reading and drawing graphs: shrinking oscillations, curves that don’t oscillate at all, and resonance peaks that sag when damping grows.
📘 What you need to know
Damping is the reduction in energy and amplitude of oscillations caused by resistive forces (like friction and air resistance) acting on the oscillating system
Resistive forces always act in the opposite direction to the motion, and damping continues until the oscillator rests at equilibrium
Key fact: as the amplitude shrinks, the frequency (and period) of the oscillations does not change
Light damping: the system still oscillates, and the amplitude decays exponentially — not linearly
Critical damping: the system returns to equilibrium in the shortest possible time without oscillating (e.g. car suspension)
Heavy damping: the system returns slowly, again without oscillating (e.g. a soft-close door damper)
On a resonance curve, more damping makes the peak lower and broader, and shifts it slightly left of f0 — but the natural frequency itself doesn’t change
What Is Damping?
In practice, every oscillator loses energy. Resistive forces — friction at a pivot, air resistance on a swinging bob — push against the motion at every instant, doing negative work and bleeding energy out of the system. The result is damping: the amplitude gets smaller and smaller until the oscillator settles at its equilibrium position.
Be careful with a favourite examiner trap — there are two different forces in this story, and they do different jobs:
the restoring force pulls the oscillator back towards equilibrium — it’s what makes it oscillate at all
the resistive force opposes the motion itself — it’s what causes the damping
resistive force opposes motion
removes energy every cycle
amplitude shrinks
but frequency stays the same
rest at equilibrium
That third box hides the fact examiners love: the frequency of damped oscillations doesn’t change as the amplitude decreases. A child’s swing dying down still takes the same time per swing — the swings just get smaller.
Light Damping
With light damping, the system keeps oscillating but each swing is a little smaller than the last — a pendulum gradually settling is the classic picture. The amplitude does not decrease linearly; it decays exponentially with time, hugging a smooth “envelope” curve.
Light damping: the oscillation continues, but each peak is smaller than the last, tracing an exponential envelope (dashed). Notice the peaks stay equally spaced — the period never changes.
Key features to mention when describing (or drawing) this graph:
A sine or cosine shape with gradually decreasing amplitude, on both the positive and negative sides
The decay is exponential — steep at first, gentler later
The time period is constant, so peaks and troughs stay equally spaced
Critical and Heavy Damping
Turn the damping up far enough and the system stops oscillating altogether — displaced and released, it just slides back to equilibrium without ever crossing it. There are two flavours:
Critical damping: the oscillator returns to rest at equilibrium in the shortest possible time, without oscillating. That’s exactly what car suspension does — after a bump, the car settles instantly instead of bouncing down the road
Heavy damping: the oscillator also never oscillates, but takes a long time to creep back to equilibrium — think of a soft-close door damper easing a door shut
Both systems start displaced at x₀ and return to equilibrium without oscillating. Critical damping (teal) gets there in the shortest possible time — its gradient falls fast and the curve then sits flat on the axis. Heavy damping (blue) drifts back over a long period of time.
What Damping Does to Resonance
Last page’s resonance curve assumed one amount of damping. Change the damping and the curve changes shape — this is a very common exam graph:
The peak of the curve lowers — damping reduces the amplitude of resonance vibrations
The peak broadens — the response is less sharply focused around one frequency
With heavy damping, the peak sits slightly to the left of the natural frequency
The natural frequencyf0itself does not change — damping never alters it
The more damping, the lower and broader the resonance peak — and under heavy damping the peak sits slightly left of f₀. The natural frequency itself stays exactly where it was.
🎨 Reading and drawing damping graphs
Does it oscillate? If the curve crosses the axis repeatedly → light damping; if it never crosses → critical or heavy
Fast or slow return? Shortest possible time to equilibrium → critical; a long, lazy drift → heavy
Sketching light damping: draw the shrinking wave inside a smooth exponential envelope, and keep the peaks equally spaced — the period never changes
Resonance curves: more damping = lower, wider peak, nudged slightly left — and always the same f0
Quick recap: damping = resistive forces draining energy, so amplitude falls (exponentially, for light damping) while frequency stays the same. Critical damping reaches equilibrium fastest without oscillating; heavy damping is slower. More damping = lower, broader resonance peak.
WE 1
A laboratory balance has a pointer that swings back and forth around the correct reading before settling, which makes it slow and awkward to read. Suggest, with a reason, whether light, critical or heavy damping should be applied to the pointer.
What do we want?
The pointer should settle at the correct (equilibrium) reading without swinging — so light damping is out
Critical or heavy?
The balance is read as soon as something is placed on it, so the pointer should settle as quickly as possible — heavy damping would make the user wait
critical dampingCritical damping brings the pointer to rest at the reading in the shortest possible time without oscillating.
WE 2
A lightly damped pendulum starts with an amplitude of 12 cm. After each complete oscillation, its amplitude falls to 75% of its previous value.
(a) Calculate the amplitude after three complete oscillations.
(b) State what happens to the period of the pendulum as the amplitude decreases.
Part (a)
Multiply by 0.75 once per oscillation
A = 12 × 0.75 × 0.75 × 0.75 = 12 × 0.75³A ≈ 5.1 cmPart (b)
Nothing — the period stays constant
T is unchangedSame fraction lost every cycle = exponential decay. And notice the peaks would still be equally spaced on the graph.
💡 Top tips
Keep the two forces straight: resistive force opposes the motion and causes damping; restoring force points to equilibrium and causes the oscillation
In sketches of light damping, shrink the amplitude but keep the peaks evenly spaced — changing the period is a classic mark-loser
“Critical” doesn’t mean “the most damping” — it means the fastest return to equilibrium without oscillation; heavy damping is more damping but slower
On the resonance graph, damping changes the curve, never the natural frequency — say both in written answers
⚠ Common mistakes
Drawing the amplitude decreasing in a straight line — the decay is exponential, not linear
Letting the wavelength of the sketch shrink along with the amplitude — the period must stay the same
Saying a critically damped system “doesn’t move” — it moves back to equilibrium, it just never oscillates
Claiming damping lowers the natural frequency — f0 is a property of the system and doesn’t change (only the resonant peak of the curve can shift)
And that’s a wrap on Standing Waves & Resonance — superposition, nodes and antinodes, boundary conditions, harmonics, resonance and now damping. Together they explain everything from a guitar note to a car’s suspension; next we carry these ideas onward through the rest of the waves topic.
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