---
title: "Excitable waves and spirals Interactive Lab | Sandbox Physics"
description: "Why does a wave curl into a spiral, and why do two waves disappear when they meet?"
canonical_url: "https://sandboxphysics.com/labs/excitable-waves/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/excitable-waves/"
last_modified: "2026-10-28"
---

FR21 · Excitation and recovery

# Excitable waves and spirals

**Why does a wave curl into a spiral, and why do two waves disappear when they meet?** A broken wave front can curl around its recovering wake and keep exciting fresh material. Colliding fronts meet regions that are still recovering, so they usually die out.

Touch a recovering sheet to launch a wave. Follow a spiral, collide two fronts and watch the material become excitable again.

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

Model assumptions

The dimensionless Barkley model represents excitation and recovery on a square with reflecting boundaries. It has no calibrated cell chemistry, membrane voltage or three-dimensional tissue geometry. Colors show computed fields.

Cyan: excitation Amber: recovery

Time $6.00$

Excited area $0.100$

Probe excitation $0.500$

Probe recovery $0.300$

### A point becomes excitable again

Both curves come from the white probe on the sheet. Cyan rises quickly; amber takes longer to fall.

### Watch the fronts meet

Every row stores the actual excitation along the center line. Position increases rightward, time downward. The latest row is at the bottom.

Physics tutorial

## Excitable waves and spirals

**Background** A small stimulus can launch a large wave if each newly excited patch triggers its neighbor. Behind the front, the material needs time to recover.

**Why it matters** A spiral can keep turning because its tip always finds fresh material ahead. Recovery decides which paths remain available.

Start with the essentials

- **Focus question**: Why does a wave curl into a spiral, and why do two waves disappear when they meet?
- **One-sentence intuition**: An excitation wave carries a temporary change of state. The recovering material behind each front explains both spiral rotation and annihilation on collision.

### Core mathematical model

#### Excitation spreads and recovery follows

$$
\partial_t u=D\nabla^2u+\varepsilon^{-1}u(1-u)\left(u-\frac{v+b}{a}\right),\qquad\partial_t v=u-v
$$

Both fields and time are dimensionless. Excitation is the fast field; the recovery field changes locally and raises the threshold. The boundary reflects the diffusing field.

#### The resting threshold

$$
u_{\mathrm{th}}=\frac{v+b}{a}
$$

With recovery held fixed, an excitation above the threshold initially grows. Recovery rises after excitation, so a second stimulus can fail.

### Common difficulties

#### A wave does not carry the material across the sheet

**Typical misconception** The bright patch is a lump of substance moving through the medium.

**Better mental model** The color indicates a local state. Each patch rises and falls as its neighbors trigger it. Only the fast field has a diffusive flux in this model.

#### Spirals and stationary patterns need different tests

**Typical misconception** Any reaction–diffusion pattern is a stationary Turing instability.

**Better mental model** Here a finite stimulus leaves a recovering trail in an otherwise stable medium. Follow the wave and recovery time, rather than a growing infinitesimal stationary mode.

### Run the experiment

1. 01

   #### Predict the collision

   Select Two fronts collide. Follow the slanted bands in the saved center-line record.

   **What to observe:** The incoming fronts meet and end. Each would need to enter material that the other front has just left recovering.

2. 02

   #### Stimulate a recovering region

   Select A half-recovered sheet. Drag the same short stroke above and below the midline.

   **What to observe:** The fresh region supports a growing front. A sufficiently recovered region can be excited again later.

Physical conclusion

The dimensionless Barkley model represents excitation and recovery on a square with reflecting boundaries. It has no calibrated cell chemistry, membrane voltage or three-dimensional tissue geometry. Colors show computed fields.

## Related experiments and articles

Morphogenesis and reaction–diffusion · Reading path

### Reaction, diffusion, growth and active stress

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How a local stimulus becomes a rotating spiral**](https://sandboxphysics.com/stories/morphogenesis/excitation-and-recovery/)

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)