Waves · phase · superposition

Coherent Wave Studio

Why do passing waves leave a stable fringe pattern? Because the waves add as signed amplitudes first, and only then is the intensity taken. Where the path difference keeps them out of step, they cancel; where it keeps them in step, they reinforce. Those places stay fixed, so the fringes do too.

Set the phase and amplitude of two spherical waves. Move a probe along the scan line to see how their fields add, and save a setup to compare with the next one.

Probe position $\text{—}$

Mean-square signal $\text{—}$

Path difference $\text{—}$

Paraxial fringe spacing $\text{—}$

Local phase-scan visibility $\text{—}$

Frequency at the fixed wave speed $\text{—}$

Two sources add up to one field

Make the two waves cancel at one spot

Phase-averaged referenceDrops the cross term to show what uncorrelated sources would give.Paths to the probeShows the two distances that make up

\Delta r=r_B-r_A

.Wavelength

\lambda

0.6\,\mathrm m

Source separation

d

1.2\,\mathrm m

Scan distance

L

3\,\mathrm m

Source B phase

\phi_B

0^\circ

Source B / A amplitude

q

1

The sources emit ideal, coherent scalar spherical waves into free space. The surface is a horizontal slice through the field, not a water surface. There are no walls or reflections, and the sources do not couple to each other. The signal is the normalized mean-square field, not a directional acoustic power. The animation is slowed down; the physical frequency is set by a fixed wave speed. Slider changes take effect when you record.

The three-dimensional field slice and the separate detector scan along a line are computed from the same complex amplitudes. Coordinates are in metres, and the waves are ideal scalar waves in free space.

Move the probe

0.00\,\mathrm m

Mint: current record. Coral: saved record. Gold: phase-averaged reference with no cross term. The cursor selects an acquired sample; all numbers and CSV values stay linear. Screen brightness uses a square-root display mapping, clipped at one, to reveal faint fringes.

S=\frac14\left|\frac L{r_A}e^{ikr_A}+q\frac L{r_B}e^{i(kr_B+\phi_B)}\right|^2

The field excludes a radius of 0.18 m around either ideal point source; no detector samples enter that region. The equal-phase on-axis reference for coincident unit sources is one. Each scan distance uses its own reference scale, so this is not an absolute loss measurement. The phase-averaged curve is a statistical reference; the moving field remains coherent.

Continue to diffraction →MIT: interference and diffraction

Start with the essentials

Focus question: If both sources emit waves, why do some locations remain dark?
One-sentence intuition: You do not add the energies. You add the signed field amplitudes first, then square the result.

Core mathematical model

Frequency and slow motion

f=\frac{c}{\lambda},\qquad c=343\,\mathrm{m\,s^{-1}}

The wave speed is fixed at a teaching reference value. The displayed phase advances only 1.5 radians per second; this is slowed down and is not the physical frequency.

Field and intensity

u=u_1+u_2,\qquad I\propto\langle u^2\rangle_t

Amplitudes in phase reinforce and amplitudes in opposite phase cancel; the intensity is never negative.

Far-field fringe spacing

\Delta y\approx\frac{\lambda L}{d}

A longer wavelength $\lambda$ spreads the fringes apart; a larger source separation $d$ squeezes them together.

Common difficulties

Cancellation does not destroy energy

Typical misconceptionAt a dark fringe, the two waves destroy each other and the energy is gone.

Better mental modelFields that cancel at one place do not destroy energy. This Lab calculates only the mean-square scalar field, not the full acoustic energy flux or the source power.

Stable phase is required

Typical misconceptionAny two equal-frequency sources make a stable pattern.

Better mental modelThe relative phase has to stay steady during the observation; otherwise time averaging washes the pattern out.

Run the experiment

01
Find zero path difference
Choose In phase and move the probe slider to the centre.
What to observe: At the centre the path difference is zero, so the two waves always reinforce there.
02
Lengthen the wavelength
Keep the current scan, then increase $\lambda$ and record again.
What to observe: The spacing of the wavefronts and of the bright fringes both grow.
03
Separate the sources
Hold $\lambda$ fixed and record a larger $d$ .
What to observe: The screen fringes become denser.

Coherent Wave Studio

Where do the waves agree?

Wave interference: how phase becomes fringes

Core mathematical model

Frequency and slow motion

Field and intensity

Far-field fringe spacing

Common difficulties

Cancellation does not destroy energy

Stable phase is required

Run the experiment

Find zero path difference

Lengthen the wavelength

Separate the sources

Related experiments and articles

From sources to phasors