# Antenna Beam

Polar · r = f(θ, t)

`r = abs(∫₋₁¹ cos((4 + 3 · sin(t)) · u · sin(θ)) du)`

[Open in the app](https://www.wavelace.com/app#p=112) · [This page](https://www.wavelace.com/presets/antenna-beam)

### What it draws

`r` is the size of an integral, computed numerically at every angle: `r = |∫₋₁^1 cos(k·u·sin θ) du|`, with `k = 4 + 3·sin t`. Read `u` as position along a straight radiator of length 2, a line of antennas driven in step, and `θ` as the direction of a distant observer, measured from the perpendicular. A wave from the point `u` arrives with a phase lead of `k·u·sin θ`, `k = 2π/λ` being the wavenumber, and the integral sums the line. The integrand is even in `u`, so the sum has a closed form, `2·sin(k·sin θ)/(k·sin θ)`, a sinc. Its peak is 2 at `θ = 0` and `θ = π`, broadside, where the whole line arrives in step. The two main lobes lie along the `x` axis; the radiator runs along `z`, where the pattern is only `2·sin k / k`.

The clock breathes the wavenumber: `k` runs from 1 to 7 and back every `2π ≈ 6.28` seconds at `Speed` 1. In wavelengths the line is `k/π` long, from a third of a wave to just over two.

### Why the beam narrows

The sinc's first zero is at `k·sin θ = π`, so the pattern has a null where `sin θ = π/k`. At `k = 4` that is `θ ≈ 51.8°`; at `k = 7` it has moved in to `26.7°`, and a second null follows at `63.8°`. While `k < π` there is no null, `π/k` being above 1: the short line radiates a fat blob, at `k = 1` still `1.68` out of 2 along the line. A longer line in wavelengths makes a narrower beam, the rule that sizes every dish and array.

Beyond the first null the sinc rises into a side lobe, its peak `0.217` of the main lobe, about `13` dB down, the figure of every uniformly driven aperture. It shows in full once `k` passes `4.49`, and at `k = 7` it peaks at `θ ≈ 39.9°`. The absolute value hides its sign: the side lobe is negative, half a wave out of step with the main beam.

### Try

- Freeze the length, `abs(integral(cos(7·u·sin(th)), u, −1, 1))`: the narrowest beam, nulls at `26.7°` and `63.8°`, a side lobe between and a stub along the line.
- Double the line, `abs(integral(cos((4 + 3·sin(t))·u·sin(th)), u, −2, 2))`: every null halves in sine, so at `t = 0` the first moves in from `51.8°` to `23.1°`.
- Steer it, `abs(integral(cos((4 + 3·sin(t))·u·(sin(th) − 0.5)), u, −1, 1))`: a phase ramp along the line points the beam where `sin θ = 0.5`, at `30°` and `150°`, with nothing turning, a phased array.
- `Top`: the pattern in plan, beams along `x`, radiator along `z`, the nulls sliding in and out as `k` breathes.

### Read more

- [Radiation pattern](https://en.wikipedia.org/wiki/Radiation_pattern)
- [Sinc function](https://en.wikipedia.org/wiki/Sinc_function)
- [Phased array](https://en.wikipedia.org/wiki/Phased_array)
- [Sidelobes](https://en.wikipedia.org/wiki/Sidelobes)
- [Antenna (radio)](https://en.wikipedia.org/wiki/Antenna_(radio))