Math LabLab viii Why it sings

Lab viii

Resonance — pushing at the right time

Push a swing at just the right moment and it flies. Sing the exact note of a glass and it rings. That matching is resonance — the reason every instrument has a body.

works for:Ages 6–9Ages 10–13Ages 14+

Everything that can wobble has one favourite speed to wobble at. Push it at THAT speed and the wobbles pile up bigger and bigger — that's resonance. Push a swing at just the right moment and it soars; push at the wrong moment and you fight it. Slide the pushes below and watch the swing wake up.

its favourite speed is 1 push per second — find it

In tune! The swing is flying. ✦

the response curve — how big it wobbles

drive — how fast you push1.00 per sec

response — the wobble size

× 8.00

A = 1 ÷ √( (1 − (f/f0)²)² + (f/(Q·f0))² )  ·  Q = 8

At the favourite speed the wobble is exactly Q = 8× a lazy push. Off to the side it barely moves.

Everything has a favourite speed

A swing, a guitar string, a wine glass, a bridge — each one likes to wobble at one special speed, called its natural frequency. Give it a poke and that is the speed it hums back at you.

Push in time and it grows

Match your pushes to that favourite speed and every push adds to the last, so the wobble climbs and climbs. That's the tall spike on the curve. Push out of time and your pushes cancel out.

Why instruments have bodies

A guitar's hollow body and a drum's shell are resonators: the air inside has its own favourite speed, so it catches the string's wobble and rings along, making a tiny sound big enough to fill a room.

For the classroom

Learning goals

  • Every oscillator has a natural frequency f0; driving it near f0 makes the response amplitude grow — this is resonance.
  • The response-vs-drive curve peaks at f0; the quality factor Q sets both the peak height (≈Q) and its narrowness (width ≈ f0/Q).
  • Resonance is why instruments have bodies (a guitar shell, a drum, an air cavity) — the resonator amplifies a small vibration by ringing along at its own frequency.

Try this

  1. 1.Put the drive at 0.5, then 1.0, then 2.0 pushes per second and read the response number each time. Ask: which one made the swing fly, and why is 1.0 special?
  2. 2.Hold a ruler off the edge of a desk and twang it; slide it out to make it floppy and in to make it stiff. Each length has its own favourite wobble speed — that's f0 changing. Match a finger-tap rhythm to it and watch the twang grow.