# Bead String

Bodies · mechanical systems

```
springs
m=10^6  x=-3
m=1  x=-2  vy=0.4
m=1  x=-1
m=1  x=0
m=1  x=1
m=1  x=2
m=10^6  x=3
k=κ  from=1  to=2  rest=0.5
k=κ  from=2  to=3  rest=0.5
k=κ  from=3  to=4  rest=0.5
k=κ  from=4  to=5  rest=0.5
k=κ  from=5  to=6  rest=0.5
k=κ  from=6  to=7  rest=0.5
```

[Open in the app](https://www.wavelace.com/app#p=153) · [This page](https://www.wavelace.com/presets/bead-string)

### The setup

Five unit masses sit a unit apart on a line between two walls of a million, and six springs of stiffness `κ` join them in a chain. Every spring has natural length 0.5 and is stretched to 1, so the chain is under a tension of `κ/2` before anything moves. The first bead is plucked sideways, given a speed of 0.4 along y, and a pulse sets off down the chain.

### The wave

A string of beads is the oldest model of a wave on a string. A bead pulled sideways is pulled back by the two springs beside it, and pulls its neighbours sideways in turn, so the displacement travels. For a continuous string the speed is `√(T/μ)`, tension over mass per length; here that is `√(κ/2)`, 0.71 at `κ = 1`. The pulse is first felt at the far bead after 4.5 s at `κ = 1` and 2.5 s at `κ = 4`, a ratio of 1.8 against the 2 the square root predicts. Five beads are a coarse string, and a short pulse on a chain runs a little ahead of the continuous speed.

At the far wall the pulse turns back the way it came, and the chain settles into its standing modes. The slowest of them, every bead swinging together, has period `2π / (2√(κ/2) · sin(π/12))`, 17.2 s at `κ = 1`, the formula Lagrange found for the loaded string in 1759.

### Try

- Drag `κ` to 4: the pulse arrives in 2.5 s and the whole motion runs twice as fast. At 0.25 it crawls, arriving after 8 s.
- Slacken the string, `rest=0.9` on every spring line. The tension drops to a tenth, the pulse takes 7.4 s, and the plucked bead swings out to 0.69 before the chain pulls it back.
- Take the tension away, `rest=1` everywhere. The springs sit at their natural length and there is nothing to carry a sideways pulse: the plucked bead wanders out to 0.84 and the far bead barely hears of it before 9 s.
- `Top` in the deck shows the chain flat, the pulse a bump running along it.

### Read more

- [String vibration](https://en.wikipedia.org/wiki/String_vibration)
- [Normal mode](https://en.wikipedia.org/wiki/Normal_mode)
- [Hooke's law](https://en.wikipedia.org/wiki/Hooke%27s_law)
- [Wave equation](https://en.wikipedia.org/wiki/Wave_equation)