Waves · phase · superposition

Coherent Wave Studio

Give two spherical waves a phase and amplitude. Walk a probe along the scan line, watch their field add, and compare a saved setup with the next.

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 / ONE FIELD

Make a quiet place between waves

Phase-averaged referenceRemove the cross term to compare with uncorrelated sources.Paths to the probeCompare the two distances behind

\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

Ideal coherent scalar spherical waves in free space. The surface shows a horizontal field slice, not water. No walls, reflections or source coupling. Signal is normalized mean-square field, not directional acoustic power. Animation is slowed; physical frequency uses a fixed wave speed. Sliders apply when recorded.

A three-dimensional field slice and a separate linear detector scan share the same complex amplitudes. Metre coordinates; ideal scalar free-space waves.

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

CONTINUE EXPLORING

From sources to phasors

Which changes move the fringes, and which reduce their contrast?

Start with the essentials

Focus question: If both sources emit waves, why do some locations remain dark?
One-sentence intuition: Energies do not add first; signed field amplitudes add first, and the result is then squared.

Core mathematical model

Frequency and slow motion

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

Wave speed is a fixed teaching reference. The displayed phase advances by 1.5 radians per second and is not the physical frequency.

Field and intensity

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

In-phase amplitudes reinforce and opposite amplitudes cancel, while intensity remains nonnegative.

Far-field fringe spacing

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

Larger wavelength $\lambda$ spreads fringes; larger source separation $d$ compresses them.

Common difficulties

Cancellation does not destroy energy

Typical misconceptionThe two waves erase energy at a dark fringe.

Better mental modelLocal field cancellation does not destroy energy. This Lab calculates mean-square scalar field, not the full acoustic energy flux or source power.

Stable phase is required

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

Better mental modelRelative phase must remain stable during observation or 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: Path difference is zero and the central region reinforces.
02
Lengthen the wavelength
Keep the current scan, then increase $\lambda$ and record again.
What to observe: Wavefront and bright-fringe spacing 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?