---
title: "When Two Waves Meet Interactive Lab | Sandbox Physics"
description: "Why do passing waves leave a stable fringe pattern? Because the waves add as signed amplitudes first, and only then is the intensity taken."
canonical_url: "https://sandboxphysics.com/labs/wave-interference/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/wave-interference/"
last_modified: "2026-10-27"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

Read the wave field

## Where do the waves agree?

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 →](https://sandboxphysics.com/labs/diffraction/) [MIT: interference and diffraction](https://ocw.mit.edu/courses/6-007-electromagnetic-energy-from-motors-to-lasers-spring-2011/resources/mit6_007s11_lec27/)

Physics tutorial

## Wave interference: how phase becomes fringes

**Background** When linear waves meet, their amplitudes add. The scalar detector here records the time average of the squared sum, so a difference in path length shows up as a difference in signal.

**Why it matters** Interference can measure tiny changes in distance, refractive index and phase, and it builds the intuition you need for quantum amplitudes.

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 misconception** At a dark fringe, the two waves destroy each other and the energy is gone.

**Better mental model** Fields 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 misconception** Any two equal-frequency sources make a stable pattern.

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

### Run the experiment

1. 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.

2. 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.

3. 03

   #### Separate the sources

   Hold $\lambda$ fixed and record a larger $d$.

   **What to observe:** The screen fringes become denser.

Physical conclusion

An interference pattern is a phase map that encodes path difference as brightness.

## Related experiments and articles

Continue exploring

### From sources to phasors

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

1. [You are here **Coherent Wave Studio** Scan two spherical sources and vary their relative phase.](https://sandboxphysics.com/labs/wave-interference/)
2. [Lab **Coherent Wave Composer** Combine plane waves, then test beating and exposure averaging.](https://sandboxphysics.com/labs/coherent-wave-composer/)

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)