Wave Interference Calculator

Wave interference arises from the superposition principle: when two or more waves overlap in the same medium, the net displacement at every point is the algebraic sum of the individual displacements. For two coherent sinusoidal waves of the same frequency, the outcome depends entirely on their phase relationship. When the waves are perfectly in phase (phase difference δ = 0, 2π, 4π, …), their crests and troughs align and the amplitudes add constructively. When they are perfectly out of phase (δ = π, 3π, …), crests meet troughs and the amplitudes subtract destructively. All intermediate cases produce partial interference.

The resultant amplitude is given by the vector addition formula: A_R = √(A₁² + A₂² + 2A₁A₂ cos δ), where the phase difference δ = 2π Δx / λ converts path difference Δx and wavelength λ into the angular mismatch between the two waves. Intensity is proportional to amplitude squared, so the relative intensity I_R / I_max = A_R² / (A₁ + A₂)², which equals 1 for full constructive interference and 0 for full destructive interference when the amplitudes are equal. This calculator outputs both the resultant amplitude and the relative intensity so users can quickly characterise the interference condition.

Practical applications span a wide range of fields. In physical optics, thin-film interference and anti-reflection coatings exploit path differences of a fraction of a wavelength to eliminate unwanted reflections. In acoustics, engineers designing concert halls and noise-cancellation headphones must account for interference between direct and reflected sound paths. In radio engineering, antenna arrays are spaced and phased to steer and shape radiation patterns through constructive interference in desired directions and destructive interference elsewhere. In quantum mechanics, the same mathematics describes probability-amplitude interference in electron diffraction and neutron scattering experiments.

This calculator assumes the two waves are perfectly coherent (same frequency, stable phase relationship) and have the same polarisation direction. Real sources — including thermal light and most acoustic sources — are only partially coherent, so the observed fringe visibility will be lower than predicted. The formula also treats waves as one-dimensional plane waves; in three dimensions, amplitude falls with distance (1/r for spherical waves), which is not accounted for here. Path difference is assumed to be the only source of phase offset; any intrinsic initial phase difference between sources must be added to δ manually. Finally, the formula is linear and does not account for nonlinear medium effects that arise at very large amplitudes, nor for the vector nature of electromagnetic waves when polarisations differ.