See a string vibrate — in 3D.

A picture on paper can only show a string swinging one way: up and down. A real string is not stuck on paper. It can swing up-and-down and side-to-side at the same time — so instead of a flat wave it draws a whirling shape in the air, like a skipping rope. Here it is, for real.

Physics: wavesTrigonometryFractions

the real thing — a string swinging in three dimensions

y = A · sin(nπx / L) · cos(2πft)

harmonic n:

Drag the picture to walk around the string. We run it at a few swings a second so you can watch; the real string does this 82 times a second.

how hard you plucked75%

A in the equation. Louder, not higher.

flat → whirl35%

The sideways swing. At 0 it’s a flat drawing; at 100% it skips like a rope.

how fast we draw it0.50 × / sec

Only the picture slows down. The note doesn’t.

bellies (antinodes)

The rust rings. Each one is the ellipse that bit of string traces — a flat line when the spin is off, a circle when it’s full.

still points (nodes)

At every k/3 along the string. Always one more than the bellies, because the two tied-down ends count.

it sounds at

3 × 82.4 =

247.2Hz

Spin changes nothing here. The pitch is set by the shape along the string, not the direction it swings.

the only two words you need

node — the still point

A node is a spot on the string that never moves. Not a little bit. Not at all. The two ends are nodes because somebody tied them down — but the surprising ones are in the middle, where the string holds itself still. Those are the green dots, and the number under each one tells you how far along the string it sits: 1/2 is halfway, 1/3 is a third of the way.

antinode — the belly

An antinode is the opposite: the spot that swings the hardest, exactly halfway between two still points. They are the rust-coloured rings. Turn the spin up and you can see what the ring really is — the loop that bit of string travels round, over and over, thousands of times a second.

Now count them. Harmonic 1 has 1 belly and 2 still points. Harmonic 2 has 2 bellies and 3 still points. Harmonic 5 has 5 and 6. There is always exactly one more still point than belly, and it is not a coincidence: the bellies sit between the still points, like the gaps between fence posts. Six posts, five gaps. Every time.

And here is the part worth remembering: spinning the string does not change its note. Slide the spin from flat to full whirl and the pitch never budges. The note comes from how the string is divided up along its length — how many bellies — not from which direction it happens to be swinging.

For the classroom

Learning goals

Define a node and an antinode physically, and state the rule that harmonic n has n antinodes and n + 1 nodes — counting the two fixed ends.
Locate the nodes as exact fractions of the string, x = k·L/n, and read them straight off the model (1/2, 1/3, 2/3, 1/4 …).
Separate the two properties of a vibration that students routinely confuse: the shape along the string sets the pitch, while the direction and size of the swing do not.

Try this

1.Set the spin to zero and count the bellies and still points for harmonics 1 to 6, writing the pair of numbers each time. Predict harmonic 7 before you are told, then say the general rule in your own words.
2.Choose harmonic 4 and slide the spin from 0 to 100% while a partner holds a finger over the still point at 1/4 and listens. Does the note change? Now change the harmonic instead. Write one sentence explaining which control changed the pitch and why.

Want the flat version, with a finger you can rest anywhere on the string? The natural harmonics lab → Or go back to where it all started — one string and a movable bridge →