◉ Ripple Tank: interference from multiple sources

Water waves from two or more sources: maxima, nodal lines, intensity, and a calculator for two coherent sources.

Wavelength λ25 px

Phase of the second source0°

Animation speed×1,0

View:

Sources:

Highlight nodal lines

Pattern parameters

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Sources N

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Wavelength λ, px

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Source spacing d (for two)

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Maximum lines (for two)

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Nodal lines (for two)

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Angle of the first maximum sin θ = λ / d

Field: y(x, y, t) = (1/N) · Σ sin(2π r_i / λ − ωt) / (1 + 0.005 r_i) (ω = 9 rad/s, as in the original)
Maximum: r₂ − r₁ = mλ Minimum: r₂ − r₁ = (m + ½)λ (for two sources in phase)
Number of lines: maxima 2⌊d/λ⌋ + 1, minima 2⌊d/λ + ½⌋

Each white circle is a source emitting circular waves of the same frequency. Brightness shows the displacement of the water surface, and the sum of the waves from all sources is divided by N, as in the original. The amplitude decreases slightly with distance.
Sources can be dragged with the mouse (a double click removes a source). On touch screens, use the layout buttons and “Add source” so the page can still scroll. Up to 8 sources in total.
Between two sources the waves travel toward each other and form a standing pattern with stationary nodes. Farther out, the nodal lines spread apart as hyperbolas. The larger the distance d compared with λ, the more lines there are and the narrower the angles between them.
The “Intensity” view averages the pattern over time and keeps only the amplitude: bright regions are maxima, dark lines are nodes. The phase slider of the second source shifts the whole pattern; in antiphase, nodes and maxima swap places.

Two coherent point sources at distance d, placed symmetrically about the origin (along the x axis). Enter the frequency, the wave speed and the observation point: you get the path difference, the phase difference, and whether that point is a maximum or a minimum.

Source spacing d, m

Frequency f, Hz

Wave speed v, m/s

Observation point x, m

Observation point y, m

Phase shift of the second source, °

Examples:

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Wavelength λ = v / f, m

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Path difference Δ / λ (r₂ − r₁)

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Total phase difference at the point, °

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Relative amplitude |cos(Φ/2)|

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What is at this point

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Maxima / minima (lines)

λ = v / f, Δ = r₂ − r₁, Φ = 2π Δ / λ + Δφ_src
Relative amplitude: |cos(Φ / 2)| (1 — maximum, 0 — minimum)

Number of lines versus d / λ (two sources)

d / λ	In phase: maxima	In phase: minima	In antiphase: maxima	In antiphase: minima	Angle of 1st maximum
0,75	1	2	2	1	none
1,50	3	4	4	3	41,8°
2,50	5	6	6	5	23,6°
3,50	7	8	8	7	16,6°
5,50	11	12	12	11	10,5°
10,50	21	22	22	21	5,5°

Ripple Tank Online: Wave Interference from Multiple Sources

A free ripple tank simulation in which several point sources excite circular waves of the same frequency. Add up to eight sources, drag them with the mouse, change the wavelength and phase shift, and watch the maximum lines and nodal lines form. The “Intensity” view shows the time-averaged pattern, and the calculator computes the path difference, phase difference and interference type at any point for two coherent sources.

Key concepts

Path difference

The difference between the distances from two sources to a point. It determines the phase in which the waves arrive and whether they reinforce each other.

Δ = r₂ − r₁

Nodal lines

Lines where the waves from two sources are always in antiphase. The water surface there stays still. They are hyperbolas with the sources as foci.

Δ = (m + ½) λ

Coherence

The sources have the same frequency and a constant phase difference. Otherwise the pattern gets smeared out over time and no interference is visible.

f₁ = f₂, Δφ = const

Where it is used

School and university labs on wave optics and acoustics
Loudspeaker placement: where sound is reinforced and where there are “dead zones”
Phased antenna arrays and radar
Explaining Young’s experiment and diffraction gratings through interference
Calculating the interference of light from several sources
Noise cancellation: choosing the positions and phase of the sources

How to use

Watch two sources: a standing pattern is visible on the line between them, and hyperbolas spread out beyond
Change the wavelength: the shorter it is, the more maximum lines there are
Drag the sources to change the distance between them
Switch to “Intensity” and turn on the nodal line highlighting
Add a third source or choose the circular layout
In the “Calculator” tab, enter real values of d, f, v and the observation point

Frequently asked questions

What is a ripple tank?

It is a shallow tank of water in which vibrating sources create waves on the surface. Light is projected through it onto a screen, and you can see the waves, their reflection, diffraction and interference. The simulation reproduces such a pattern for several point sources.

Why do stationary points form between two sources?

The waves from two sources travel toward each other and overlap, forming a standing pattern. At the nodes, the crest of one wave always meets the trough of the other, so the surface does not move. The nodes are spaced half a wavelength apart.

Why are the nodal lines hyperbolas?

A nodal line is the set of points with a constant difference of distances to the two sources: r₂ − r₁ = (m + ½)λ. The locus of points with a constant difference of distances to two points is a hyperbola whose foci are at the sources.

What changes if the sources work in antiphase?

All the conditions are reversed: where there were maxima, minima appear, and the nodal lines become maximum lines. The central line, equidistant from the sources, is a nodal line in antiphase rather than a maximum. Try the 180° phase slider.

What does the number of maximum lines depend on?

On the ratio of the source spacing to the wavelength: for two sources in phase the number of maxima is 2⌊d/λ⌋ + 1. The larger d or the smaller λ, the more lines there are and the smaller the angle between them (sin θ = λ/d for the first maximum in the far field).