Wave Interference Simulator

What is wave interference?

Interference is what happens when two or more waves meet in the same space. Each wave carries energy and oscillates the medium — or field — it travels through. When waves overlap, their amplitudes add together at every point. Where two crests coincide the combined amplitude is doubled, producing a bright region. Where a crest meets a trough the waves cancel, producing a dark or silent region.

This is not a special edge case. It is how all waves behave, from water ripples in a pond to sound in a concert hall to light passing through a narrow slit. The two-source case shown in this simulator is the cleanest possible demonstration: two identical point sources emit circular waves with the same frequency and phase, and the pattern they create is called a two-source interference pattern.

Constructive and destructive interference

Constructive interference occurs wherever the two waves arrive in phase. That means the path lengths from the two sources to that point differ by an exact whole number of wavelengths: 0, λ, 2λ, 3λ, and so on. At those points the crests always coincide with crests and the amplitude is at its maximum. These are the bright bands you see on the right side of the canvas.

Destructive interference occurs wherever the path difference is an odd half-wavelength: λ/2, 3λ/2, 5λ/2. At those points a crest from one source always arrives at the same time as a trough from the other. The dashed lines in the simulator trace the nodal lines — loci where the path difference is always exactly half a wavelength — from the midpoint between the sources to the screen edge.

Understanding the formula d · sin(θ) = m · λ

The formula comes from geometry. Pick any bright fringe at angle θ from the central axis. Draw lines from each source to that fringe. If the sources are close compared to the distance to the screen, those lines are nearly parallel. The extra distance one wave travels compared to the other — the path difference — is approximately d · sin(θ), where d is the source separation.

For constructive interference, that path difference must equal a whole number of wavelengths: d · sin(θ) = m · λ. Rearranging gives sin(θ) = mλ/d, so the fringe angles are set entirely by the ratio of wavelength to source separation. A larger wavelength or smaller separation pushes the fringes to wider angles, making them appear farther from the centre on the screen.

How source separation affects fringe spacing

Increasing the separation d between the two sources compresses the fringe pattern. The formula shows why: sin(θ) = mλ/d, so a larger d gives a smaller angle for each fringe order. In practice, the bands pack together and more of them become visible across the same screen area.

Decreasing the separation has the opposite effect. As d shrinks toward the wavelength, the first-order fringe angle grows toward 90 degrees and eventually the fringe moves off screen. At very small separations only the central maximum — the straight-ahead bright band at m = 0 — remains visible. Try dragging the source separation slider from 40 down to 5 and watch the bands spread apart.

How frequency affects the pattern

Frequency and wavelength are linked by the wave speed: λ = v / f. A higher frequency means a shorter wavelength. Substituting back into the fringe formula, shorter λ gives a smaller sin(θ) for every fringe order, so the fringes move closer to the centre and more bands fit within the visible area.

Lowering the frequency stretches the wavelength and spreads the fringes further apart. At the minimum frequency setting of 0.5 Hz the wavelength is large relative to the source separation, so only one or two bright bands are visible. At 3 Hz the shorter wavelength packs several bands clearly into view. This is the direct wave equivalent of changing the colour of light in a real double-slit experiment: blue light has higher frequency and narrower fringe spacing than red light.

What the amplitude slider controls

Amplitude sets how strongly each source emits. In this simulator it controls the peak oscillation height of each circular wave. At full amplitude the constructive bright bands reach their maximum brightness and the contrast between bright and dark regions is sharpest.

Lowering the amplitude dims both the bright and dark regions proportionally, because the wave heights shrink. The pattern itself — where the fringes fall — does not change. Amplitude has no effect on fringe position, only on how vivid the contrast appears. This mirrors real physics: increasing the power of two loudspeakers makes the sound louder everywhere but does not shift where the quiet nodal lines are in the room.

One source vs two sources

Toggle the source count to see what interference actually adds. With one source you get a single set of concentric circular ripples expanding outward. There is no fixed fringe pattern, just the natural spreading of a wave from a point — amplitude falls with distance as the energy spreads around a growing circle.

Switching to two sources immediately creates a stable fringe pattern. The nodal lines and bright bands are fixed in space even as the wave crests travel outward. That stability is the signature of interference: it only appears when two coherent sources combine. In real experiments coherence — keeping the same phase relationship — is the difficult part; in this simulator both sources are always perfectly coherent.

Real-world applications

Young's double-slit experiment, performed in 1801, was the first clear demonstration that light behaves as a wave. Two closely spaced slits act as two coherent sources and the resulting fringe pattern on a screen proved light must have a wavelength. The same geometry described by d · sin(θ) = m · λ applies whether the waves are water, sound, or light.

Engineers exploit interference in noise-cancelling headphones, which generate a second sound wave with inverted phase to produce destructive interference at the ear. Thin-film coatings on camera lenses use the same principle with light: a thin layer is chosen so reflected waves interfere destructively, reducing glare. Antenna arrays in radio and radar use constructive interference to focus a beam in a chosen direction. Understanding the fringe formula is the starting point for all of these technologies.

Practical examples

FAQ