Resonance

You are pushing a child on a swing. You are not strong - each shove is gentle. Yet after a dozen well-timed pushes the swing is soaring. Somewhere else a singer holds a note and a wine glass across the room shatters. A radio picks one station out of the hundreds crossing the room at once. These all run on the same trick, and it is not about force. It is about timing.

Every springy, swinging, or bouncing thing from the last few lessons has a rhythm it naturally wants to move at - its natural frequency, the ${\omega }_0=\sqrt{k/m}$ we found for simple harmonic motion. This lesson asks what happens when you push it, over and over, at some rhythm of your choosing.

First, a guess

Sweep the drive frequency

On the left is a mass on a spring being pushed rhythmically. The push never changes size. All you change is how fast you push it - the drive frequency. The curve on the right records how big the steady swing gets at each drive rate. Sweep the drive slider and find the spike.

The swing stays small at most drive rates, then blows up in a narrow band right around the spring's natural frequency. Then turn up the damping and watch the spike shrink and spread. Those two behaviors are the entire lesson.

Why the peak is there

Drive an oscillator at some frequency and, after it settles, it moves at the frequency you drive it at, with a steady swing size. How big that swing is depends on how close your driving rhythm is to the oscillator's own. When the two match, every push lands in step with the motion, feeding energy in on every single cycle. The energy accumulates and the amplitude climbs. That match is resonance.

Push faster or slower than the natural rhythm and your pushes drift in and out of step: sometimes you help the motion, sometimes you fight it, and they largely cancel. So the response is small everywhere except in a narrow window around ${\omega }_0$, where it towers. The steady swing size as a function of drive frequency $\omega$ is

You do not need to derive this by hand - take it as stated - but you can read it. When $\omega$ nears ${\omega }_0$, the first term under the square root collapses toward zero, so the whole fraction blows up. That is the peak. The only thing holding it back is the second term, which comes entirely from damping.

Damping: the height of the peak

Real oscillators lose energy - to friction, to air, to heat. damping, written $\gamma$ (gamma), is that drain. It sets a ceiling on how high resonance can climb: energy fed in by the pushes has to balance energy bled off by damping, and where they balance is where the amplitude stops growing. Little damping means a tall, razor-sharp peak - a wine glass has almost none, so a singer on exactly its pitch can drive it past breaking. Heavy damping means a low, broad hump - your car's shock absorbers are heavily damped on purpose, so bumps do not set the whole car ringing.

The same trick, five places

A swing: you push once per period, at the natural rhythm, and small pushes build a big arc. A microwave oven: an electric field flips back and forth billions of times a second and shoves water molecules along with it, and the friction of all that jostling is the heat. A wine glass: sound at exactly the glass's natural pitch drives its walls until they flex past their limit. A radio: the dial tunes a circuit's natural frequency onto one station. And a bridge or building: engineers work hard to keep its natural frequencies away from the rhythms of wind and footfall, because a structure driven at resonance can swing itself apart. Same physics, from a toy to a skyscraper.

Check yourself