---
title: "Shallow-Water Waves & Dam Break Interactive Lab | Sandbox Physics"
description: "Does a dam-break flood front move at the speed of small water waves? No."
canonical_url: "https://sandboxphysics.com/labs/shallow-water-dam-break/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/shallow-water-dam-break/"
last_modified: "2026-10-27"
---

M100 · Fluid mechanics

# Shallow-Water Waves & Dam Break

**Does a dam-break flood front move at the speed of small water waves?** No. Small waves travel at a speed set by the local depth: the deeper the water, the faster they go. A dam-break front is a large, nonlinear wave, and its speed comes from the full solution. When the flood runs onto dry ground, the front moves at twice the small-wave speed of the original reservoir.

Open a gate, launch a pulse, hold a still lake over a ridge or look at a hydraulic jump. Scrub through the finite-volume solution, compare it with the exact dam-break reference and refine the mesh. The water in the flume plus what has flowed out through the ends should stay constant.

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

The flow is one-dimensional, depth-averaged and inviscid. The scheme uses hydrostatic reconstruction with a diffusive first-order flux. The 3D flume shows cell averages and does not resolve vertical turbulence or the rollers of breaking waves.

Physics tutorial

## Waves, wetting fronts and conservative numerics

**Background** This experiment tells you something only when the flume geometry, the conserved quantities and the recorded data agree.

**Why it matters** You connect a flume to quantitative checks, such as the exact dam-break solution and the volume balance, and then test where the model holds and where it fails.

Start with the essentials

- **Focus question**: Does the flood on screen obey the conservation laws?
- **One-sentence intuition**: Compare the full physical model, its approximation and the separately recorded data.

### Core mathematical model

#### Conservation of depth and momentum

$$
\begin{aligned}h_t+(hu)_x&=0\\(hu)_t+\left(hu^2+\frac{gh^2}{2}\right)_x&=-ghb_x\end{aligned}
$$

The flow is hydrostatic and depth-averaged in a channel of constant width. The bed enters as a source term in the momentum equation.

#### Local speeds of small waves

$$
\lambda_\pm=u\pm\sqrt{gh}
$$

A flood shock does not in general travel like a small long wave. The front over a dry bed moves at yet another speed.

#### Froude number

$$
\mathrm{Fr}=\frac{|u|}{\sqrt{gh}}
$$

In subcritical flow, signals can travel both upstream and downstream. Supercritical flow carries both families of signals downstream. A dry cell has no Froude number.

#### Volume conservation with flow through the ends

$$
\begin{aligned}V&=W\sum_i h_i\Delta x\\V(t)+\int_0^t W(q_R-q_L)\,ds&=V(0)\end{aligned}
$$

Water leaving through the ends can change the volume in the flume without breaking conservation. Scanning the raw cell depths gives a separate estimate of the integral.

#### Adaptive step and grid error

$$
\begin{aligned}\mathrm{CFL}&=\frac{\Delta t}{\Delta x}\max_i\left(|u_i|+\sqrt{gh_i}\right)\leq0.4\\E_h&=\frac1N\sum_i|h_i-h_{\mathrm{exact}}(x_i,t)|\end{aligned}
$$

The first-order scheme smears shocks out. The mean depth error is available only for a flat-bed dam break, before the waves reach the ends.

#### Stationary hydraulic jump

$$
\begin{aligned}q_L&=q_R=q\\\frac{q^2}{h_L}+\frac{gh_L^2}{2}&=\frac{q^2}{h_R}+\frac{gh_R^2}{2}\end{aligned}
$$

The preset balances the mass and momentum fluxes across a depth increase that satisfies the entropy condition. The numerical solution smears the jump over several cells.

### Common difficulties

#### A reference curve is not a measurement

**Typical misconception** The reference curve on screen is a measured result.

**Better mental model** Only the sensor channels you record count as observations, and changing the setup clears them. The perturbations are deterministic, so they give no statistical certification.

### Run the experiment

1. 01

   #### Release the gate

   Compare the wet-bed and dry-bed dam presets, and move the time slider.

   **What to observe:** The rarefaction, the shock and the dry front all come from the same balance laws.

2. 02

   #### Refine and compare

   Double the cell count, keeping the same initial state and time.

   **What to observe:** The reference curve stays the same, and the numerical smearing shrinks.

3. 03

   #### Test a still lake

   Select Lake over a ridge, then change the ridge height.

   **What to observe:** The well-balanced hydrostatic reconstruction keeps this lake at rest, with no false currents.

4. 04

   #### Check the volume balance

   Compare reflecting and transmissive ends, and acquire every cell.

   **What to observe:** The volume in the flume plus the net outflow stays constant. The raw heights integrate into a separate volume estimate that carries its own sensor perturbations.

Physical conclusion

Neither an approximation nor a polished scene can stand in for the full model, an independent reference check and raw measurements.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

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