Line: closed-form array factor (uniform taper) or numeric sum (tapers). Dots: independent brute-force sum of every emitter at a distant point. Rings every 10 dB down to −40 dB.
The science
Each emitter is a line source radiating a cylindrical wave of amplitude an/√r. With wavenumber k = 2π/λ the field is the real part of the phasor sum
S(x,z) = Σ a_n r_n^(-1/2) e^( j (k r_n + φ_n) )
φ_n = k x_n sinθ0 (progressive phase)
Giving each emitter a phase step k d sinθ0 makes all waves arrive in step in direction θ0 - no moving parts. Far away the sum becomes the array factor
AF(θ) = sin(Nψ/2) / (N sin(ψ/2)), ψ = k d (sinθ - sinθ0)
- First null:
sinθ = sinθ0 ± λ/(N d). - Half-power beamwidth ≈
0.886 λ / (N d cosθ0): a longer array is sharper, and the beam widens as it is steered off broadside. - Grating lobes appear at
sinθ = sinθ0 + mλ/d when the spacing is too large - the same effect as extra orders of a diffraction grating. - Uniform weights give a first sidelobe near −13.3 dB; tapers (Hamming, Hann) trade a wider main lobe for much lower sidelobes.
Simplifications: 2-D scalar waves on a half-plane (baffled line sources, isotropic element pattern, no damping, no mutual coupling). The same maths describes radar and antenna arrays, sonar and ultrasound probes, but real elements have their own directivity. The pattern is a far-field result; the receiver buoy sits in the near field of a large aperture (Fraunhofer distance 2D²/λ), so its level only approximately follows the polar plot. Scene units are arbitrary and the wave speed is a fixed visual value.
The in-page checks sum every emitter with exact distances (large radius) and compare with the closed-form array factor; the null, beamwidth, sidelobe and grating-lobe values are textbook formulas.
Independent educational model. Wave surface, polar plot and numbers are computed live in your browser.