Wave Interference

The field equation

The total displacement at point r from N coherent sources is the real-valued superposition of circular waves:

$$ \begin{aligned} u(r, t) = \sum_i r_{i}^{-\delta} \sin(k r_{i} - \omega t + \phi_i) \\ k = \frac{2\pi}{\lambda}, \quad \omega = 2\pi f \end{aligned} $$

The decay exponent δ controls fall-off with distance: δ = 0 gives a plane-wave approximation, δ = 0.5 is the physical cylindrical-wave law (intensity ∝ 1/r). Colour encodes signed amplitude — cyan is positive (crest), orange-red is negative (trough), near-black is zero.

What to watch for

With two in-phase sources, the bright central maximum and flanking fringes are the same pattern as Young's double-slit experiment — the path difference |r₁ − r₂| = mλ locus is a set of hyperbolas. Move the sources closer together and the fringes widen; increase λ and they widen further — the spacing is inversely proportional to source separation and directly proportional to wavelength.

The Anti-phase preset shifts one source by π radians (half a wavelength). The central axis becomes a nodal line instead of a maximum — the pattern is the same geometry rotated by one fringe. The Quad array scene shows a 2×2 phased array with alternating signs: the principal lobes collapse and the field takes on a more complex, higher-order topology — the same principle exploited in phased-array radar and radio telescopes.

Interaction

Drag any numbered source to an arbitrary position. The interference pattern updates in real time at half resolution (bilinearly upscaled) to maintain 60 fps across all device speeds. Up to six sources are supported simultaneously.

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