# Membrane Modes

Surface · z = f(x, y, t)

`z = Σ_(k=1)⁶ sin(k · x) · sin(k · y) · cos(k · t)/k²`

[Open in the app](https://www.wavelace.com/app#p=123) · [This page](https://www.wavelace.com/presets/membrane-modes)

### What it draws

Six standing waves added together. The `k`-th term is `sin(kx) · sin(ky)`, a checkerboard of humps and hollows whose squares are `π/k` across, and `cos(kt)` rocks the whole pattern up and down in place. Nothing travels: a product of a shape in space and a shape in time has fixed nodes, and every term here shares them.

The weights are `1/k²`, so the first mode arrives at full strength, the second at `0.25`, and the sixth at `0.028`. The coarse pattern therefore sets the shape and the fine ones only roughen it, which is why the surface looks like one lumpy sheet rather than six grids laid over each other.

### Why it comes back

Each mode is finer and faster in the same proportion: term `k` has wavelength `2π/k` and period `2π/k`. Frequency rises in step with wavenumber, which is what a medium without dispersion does, and it means every period divides the first one. The whole sum therefore returns to exactly where it started every `2π ≈ 6.28` seconds at `Speed` 1, however many modes are in it.

The shared nodes come from the same place. Every term vanishes wherever `x` or `y` is a multiple of `π ≈ 3.14`, so those lines are still on the surface no matter what the modes are doing.

### Try

- Keep one mode with `sin(x) · sin(y) · cos(t)`. A plain grid of humps, and the baseline the other five roughen.
- Go to twelve modes by raising the 6. The surface gets no taller, because the terms added are worth `1/49` and less.
- Drop the weighting to `1/k` instead of `1/k²`. The fine modes now count for much more and the sheet turns spiky.
- Press `Top`. Looking straight down, the grid of nodal lines at multiples of `π` is easiest to pick out.

### Read more

- [Vibrations of a circular membrane](https://en.wikipedia.org/wiki/Vibrations_of_a_circular_membrane)
- [Standing wave](https://en.wikipedia.org/wiki/Standing_wave)
- [Normal mode](https://en.wikipedia.org/wiki/Normal_mode)
- [Dispersion relation](https://en.wikipedia.org/wiki/Dispersion_relation)