Antenna Beam
r = abs(∫₋₁¹ cos((4 + 3 · sin(t)) · u · sin(θ)) du)
Open in the app The dials and keys named below are the app's.
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.