Wave Interference Simulator

Explore how waves from two point sources create interference patterns, how thin films produce iridescent colors, and how slightly mismatched frequencies produce beats. All calculations run in your browser.

Controls

Mode

Presets

Wavelength (\u03bb)40 px

Source Separation (d)120 px

Amplitude1.00

Results

\Delta d = d_2 - d_1 = n\lambda

Wavelength (\u03bb)

40px

Separation (d)

120px

d / \u03bb ratio

3.00

Bright fringe orders

±3

Centre intensity

1.0004I\u2080

Sample \u0394d

0.0px

Constructive orders: -3, -2, -1, 0, 1, 2, 3

Reference Guide

Wave Superposition

When two or more waves occupy the same region, their displacements add at every point. This is the principle of superposition and it is the foundation of all interference phenomena.

y_{\text{total}}(x, t) = y_1(x, t) + y_2(x, t)

For two identical sources separated by distance d, the superposed field forms a stationary pattern of bright and dark regions called the interference pattern.

The intensity at any point depends on the phase relationship between the two arriving waves, which in turn depends on the path difference.

Constructive and Destructive Interference

Constructive interference occurs when the path difference is an integer multiple of the wavelength. The waves arrive in phase and their amplitudes add:

\Delta d = n\lambda, \quad n = 0, \pm 1, \pm 2, \ldots

Destructive interference occurs when the path difference is a half-integer multiple of the wavelength. The waves arrive exactly out of phase and cancel:

\Delta d = \left(n + \tfrac{1}{2}\right)\lambda, \quad n = 0, \pm 1, \pm 2, \ldots

The resulting intensity follows $I = 4I_0 \cos^2\!\left(\frac{\pi \Delta d}{\lambda}\right)$ , so it peaks at $4I_0$ for constructive and reaches zero for destructive.

Path Difference and Fringe Location

For two coherent point sources separated by d, a distant screen at distance L shows bright fringes at positions $y_m$ where:

y_m = \frac{m \lambda L}{d}, \quad m = 0, \pm 1, \pm 2, \ldots

The path difference from any point P to the two sources is:

\Delta d = d_2 - d_1 = \sqrt{(x - x_2)^2 + (y - y_2)^2} - \sqrt{(x - x_1)^2 + (y - y_1)^2}

The coloured map in the simulator shades each pixel by the intensity computed from this exact path difference, revealing hyperbolic nodal lines and bright antinodal lines radiating from the midpoint.

Beats and Thin-Film Interference

Beats arise when two waves with slightly different frequencies overlap. The combined amplitude swells and fades at the beat frequency:

f_{\text{beat}} = |f_1 - f_2|

Thin-film interference occurs because light partially reflects at the top and bottom surfaces of a thin transparent layer. The two reflected rays interfere. For a film in air (one phase inversion), constructive interference requires:

2nt = \left(m - \tfrac{1}{2}\right)\lambda, \quad m = 1, 2, 3, \ldots

Here $n$ is the refractive index and $t$ is the film thickness. Different wavelengths satisfy the condition at different thicknesses, producing the rainbow colors seen in soap bubbles and oil slicks.