Every real oscillator stops eventually. The swing slows, the guitar note fades, the shaken car settles. Something is quietly stealing the energy — friction, air resistance, oil in a piston. That thief is damping, and how greedy it is decides everything: whether the system wobbles to a halt over a minute, glides to rest in an instant, or creeps back so slowly you lose patience.
📘 What you need to know
Damping is the reduction in energy and amplitude of oscillations caused by resistive forces
Resistive forces act opposite to the motion of the oscillator
With light damping the amplitude decays exponentially, and the time period does not change
Critical damping returns the system to equilibrium in the shortest possible time without oscillating
Heavy damping also avoids oscillating, but takes a long time to return
On a resonance curve, more damping makes the peak lower, broader, and shifts it slightly left
Damping does not change the natural frequencyf0
What damping is
Set a mass on a spring going and it does not swing forever. A resistive force — air resistance, friction in the spring, oil in a dashpot — always points against the motion. It does negative work, so mechanical energy leaks away, and since energy depends on amplitude, the amplitude shrinks.
Damping, definedthe reduction in energy and amplitude of oscillationsdue to resistive forces acting on the system
Damping keeps going until the oscillator comes to rest at the equilibrium position. Not somewhere random — at equilibrium, because that is where the restoring force is zero.
Two forces, two jobs, and students mix them up every year. The restoring force pulls the oscillator back towards equilibrium — it is what makes it oscillate at all. The resistive force opposes whichever way it happens to be moving — it is what makes it stop. One creates the oscillation; the other kills it.
Light damping
This is the gentle case: a pendulum swinging in air, a guitar string ringing. The oscillator completes many cycles, each a little smaller than the last.
The amplitude does not fall linearly — it decays exponentially with time
The peaks sit under a smooth exponential envelope
Crucially, the frequency and time period stay constant. The peaks and troughs remain equally spaced
The amplitude halves, halves again, and again — but the clock keeps perfect time. A damped pendulum is a worse ornament and an equally good timekeeper.
Critical and heavy damping
Turn the resistive force up far enough and the oscillator never makes it past equilibrium even once. It simply returns and stops.
Critical damping — returns to equilibrium in the shortest possible time, without oscillating. Car suspension is designed this way, so you take one bump and not a dozen
Heavy damping — also no oscillation, but the system crawls back. A door closer, so the door never slams
Both avoid overshooting. The difference is how long you wait. Critical damping is the sweet spot: any less and it wobbles, any more and it dawdles.
Damping
Does it oscillate?
Time to reach equilibrium
Everyday example
Light
Yes — many times, amplitude decaying exponentially
Long
Swinging pendulum
Critical
No
The shortest possible
Car suspension
Heavy
No
Long, slower than critical
Door closer
Watch the wording: critical damping is not “the fastest to stop moving” in general — it is the fastest to reach equilibrium without oscillating. Slightly less damping gets you across the line quicker, but you overshoot and wobble, so it doesn’t count.
What damping does to resonance
Now bring back the resonance curve from last page. Damping drains energy away as fast as the driver pours it in, so the amplitude can never run away.
The dashed line is the natural frequency, and it stays exactly where it is. What moves is the peak of the response — and only noticeably once the damping is heavy.
As damping increases:
The amplitude at resonance decreases, so the peak lowers
The peak broadens — the system responds over a wider range of driving frequencies
The peak moves slightly to the left of f0 when damping is heavy
Overall, damping reduces the sharpness of the resonance
And the sentence examiners hide a mark behind: the resonant frequency (where the peak actually sits) can shift when you add damping, but the natural frequencyf0 — the frequency the thing oscillates at when nothing is damping it — is a property of the system and does not change. Damping never touches f0.
More damping
drains energy
Lower, broader resonance peak
but
f0 unchanged
🛠 Choosing the right damping
Is oscillating acceptable? If yes, light damping is fine (a pendulum, a guitar).
Must it not overshoot? Then critical or heavy.
Does it need to settle quickly too? That pins it down to critical.
Does slowness actually help? Then heavy (a door closer, so it never slams).
WE 1
State, with a reason, which degree of damping is best for each: (a) a car’s suspension after driving over a speed bump (b) a heavy fire door fitted with a closing mechanism (c) the pendulum of a grandfather clock
(a) car suspensionCritical. Passengers must not bounce, so no oscillation is allowed, and the car must settle quickly ready for the next bump.
(b) fire door closerHeavy. It must not swing past shut, and here the slow return is the whole point — the door closes gently instead of slamming.
(c) grandfather clock pendulumLight. It must keep oscillating for a long time. Light damping barely reduces the amplitude each swing, and the period is unaffected anyway.
Critical and heavy both forbid oscillation. The tiebreaker is always how fast must it settle.
WE 2
A lightly damped pendulum has a time period of 0.50 s. Its amplitude falls from 12.0 cm to 6.0 cm after 10 complete oscillations. Determine the time this takes, the amplitude after a further 10 oscillations, and state the frequency at that moment.
Step 1 — time for 10 oscillationst = 10T = 10 × 0.50t = 5.0 sStep 2 — exponential decay means equal times give equal fractions
Every 10 oscillations the amplitude halves.
12.0 → 6.0 → 3.0 cmamplitude = 3.0 cmStep 3 — the frequency
Light damping does not change the period, so f = 1/T = 1/0.50
f = 2.0 HzExponential decay is why it halves again rather than dropping another 6.0 cm to zero. Equal intervals, equal ratios — not equal subtractions.
WE 3
A driven system has a natural frequency of 50 Hz. Lightly damped, its resonance peak is 8.0 cm tall and sits at 50 Hz. Heavily damped, the peak is 3.0 cm tall and sits at 48 Hz. Calculate the percentage reduction in peak amplitude, and state the natural frequency of the heavily damped system.
Step 1 — percentage reduction in the peak(8.0 − 3.0) / 8.0 × 10062.5% reductionStep 2 — what moved and what didn’t
The peak has moved to 48 Hz, so the resonant frequency has shifted left.
Step 3 — the natural frequency
Damping does not change f₀.
f₀ = 50 HzClassic trap. The question hands you 48 Hz hoping you’ll call it the new natural frequency. It’s the new resonant frequency — a different thing.
💡 Top tips
Define damping using resistive forces reducing energy and amplitude — mention both.
Under light damping the amplitude falls exponentially and the period is unchanged. Say both.
Critical = shortest time to equilibrium without oscillating. The last four words earn the mark.
More damping → peak lower, broader, slightly left. Three separate marks.
Resistive force opposes the motion. Restoring force pulls back to equilibrium. Never swap them.
⚠ Common mistakes
Saying damping changes the natural frequency — it doesn’t. Only the resonant peak shifts
Drawing a lightly damped graph with the peaks getting closer together. The period is constant
Claiming the amplitude decays linearly under light damping — it decays exponentially
Saying a heavily damped system returns to equilibrium fastest — that is critical damping
Drawing critically or heavily damped curves that dip below the axis. They never oscillate
Confusing the resistive force with the restoring force
Quick recap:Damping is the loss of energy and amplitude to resistive forces. Light damping gives many oscillations under an exponential envelope with an unchanged period. Critical damping returns the system to equilibrium in the shortest time without oscillating; heavy damping does so slowly. On a resonance curve, more damping means a lower, broader peak that shifts slightly left — while the natural frequencyf0 stays exactly where it was.
And that closes Standing Waves & Resonance. Look at what you can now do: build a standing wave out of two travelling waves, name its nodes and antinodes, let the ends decide which harmonics survive, drive the thing at one of those frequencies to get resonance — and then damp it to keep the amplitude sensible. Every piece connects to the one before it. Go back and re-read the first page; it should look completely different now.
Damping dampening your confidence?
Book a free meeting and we’ll go through light, critical and heavy damping, resonance curves and past-paper questions together.
