Value chain · CLM-1463
In plain terms
If each antenna in a row starts its wave a moment after its neighbour, the beam leans away from straight ahead. The size of the lean follows from three things: how big the time step is, how fast the waves travel, and how far apart the antennas are. A bigger step, or closer antennas, gives a bigger lean.
The claim as recorded
For a straight row of identical elements a distance d apart, each sending the same wave a fixed time dt later than its neighbour, the waves add in full in the direction theta from the row's normal where sin(theta) = c dt / d, with c the wave speed. Measured in wavelengths and wave periods this is sin(theta) = s / d, where s is the delay step in periods. For example, a quarter-period step on half-wavelength spacing gives sin(theta) = 0.5, so theta = 30 degrees. When c dt exceeds d this direction does not exist, but at a single frequency a delay of whole periods changes nothing, so the row may still send a full-strength beam in another direction (CLM-1465).
Evidence behind this: T0: derived from the sum of the waves from a straight row of identical elements, and checked against a second implementation that shares no code (CALC-142). No textbook is held to corroborate it.
Scope and limits
A law for a straight, even row in the far field. It gives the direction of the strongest beam, not its width or the size of any other lobe. Where each delay is set as a phase, the law holds at the frequency the phases were set for. Elements that do not send equally in every direction make the beam weaker as it is swung. Says nothing about how a real design sets its delays.
What it assumes
Far field; identical elements; a straight, evenly spaced row; a single frequency, so that a delay and a phase shift of the same fraction of a period are the same thing.
Where it comes from
This is the project's own working, shown in full:
CALC-142 step 1. In a far direction theta the path from element n is shorter by n d sin(theta) than from element 0, so its wave arrives n (d sin(theta) - c dt) ahead; all arrive together when d sin(theta) = c dt (plus whole wavelengths, which are the extra beams of CLM-1465). Over eight cases (spacings 0.5 and 0.85, steps 0.05 to 0.4 periods), the page model's steer law matches the same law solved by bisection to within 1e-14 degrees, and the strongest direction the model finds by summing the waves matches it to within 2e-7 degrees in seven. In the eighth (spacing 0.85, step 0.4) the sum returns an extra beam of equal strength at -44.90 degrees, as CLM-1465 predicts.
Basis: spacing and delay in any consistent units; angle measured from the line straight out of the row
Related records
- Answers question
- QST-013 not published in this release
Cite this
energyaperture.com/c/clm-1463
CLM-1463, Energy Aperture, as of 2026-10-17.