Wave Interference Patterns - 2D Ripple Field and Detector Screen Intensity

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Live readouts

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Numerical checks

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How to use

The red dots are coherent sources; the right-hand screen plots the time-averaged intensity. Try: (1) Two sources and watch the dark nodal lines between bright fans; (2) Out of phase swaps maxima and minima on the centre line; (3) Diffraction grating sharpens the maxima to N^2 tall, narrow peaks; (4) Close pair: when d < lambda/2 there are no nodal lines at all; (5) Steered beam: a phase step between sources tilts the main beam (phased array). Space pauses, R restarts.

The science

Superposition. Linear waves add. Each source i radiates a circular wave with amplitude falling as 1/sqrt(r) in 2D:

psi(x,t) = sum_i cos(k r_i - w t - phi_i) / sqrt(r_i), k = 2 pi / lambda, w = k c

Conditions. With equal phases, a point is bright when the path difference is a whole number of wavelengths and dark when it is a half-integer number:

r2 - r1 = m lambda (bright), r2 - r1 = (m + 1/2) lambda (dark)

Far from the pair, d sin(theta) = m lambda, so the fringe spacing on a screen at distance L is about lambda L / d. For two sources there are 2 floor(d/lambda + 1/2) nodal lines. For N sources the far-field intensity is

I(theta) = I1 [ sin(N delta/2) / sin(delta/2) ]^2, delta = 2 pi d sin(theta)/lambda - dphi

The peak is N^2 times a single source (the energy is concentrated, not created); the phase step dphi steers the beam to sin(theta) = dphi lambda/(2 pi d).

Checks. (1) the first-order intensity maximum on a screen 120 cm away sits at the position where the exact path difference equals lambda; (2) the central intensity equals N^2 times one source; (3) the nodal-line count along a far arc matches 2 floor(d/lambda + 1/2).

Simplifications. Scalar waves, identical point sources, no reflections or damping, 2D cylindrical spreading; the intensity trace on the screen is the exact sum, the dashed curve the far-field formula (they differ at small L).