---
tag: NT73Y95RY5
kind: concept
name: Shallow water equations
msc: [76B15, 35Q35, 35Q86]
specializes: [XN081NCPF5]
sources:
  - url: https://en.wikipedia.org/wiki/Shallow_water_equations
same_as:
  wikidata: Q4008384
  wikipedia: Shallow water equations
---

## Summary

The shallow water equations describe a layer of fluid whose depth is small compared with the horizontal length scale of its motion.
Their unknowns are the depth of the layer and the horizontal velocity, both depending on horizontal position and time.
They form a system of [[hyperbolic conservation laws|XN081NCPF5]] for mass and momentum.
They model tides, tsunamis, floods and large-scale flows in the atmosphere and the ocean.

## Explanations

### L1 human

When water is shallow compared with the size of its waves, the water in a vertical column moves together.
So one only needs to know how deep the water is and how fast it moves horizontally at each place.
Water flows from where the surface is high to where it is low, and the amount of water is conserved.
Small waves travel at speed $\sqrt{g h}$, where $h$ is the depth and $g$ the gravitational acceleration, so a tsunami in a deep ocean is very fast.
Like other conservation laws, the equations can develop jumps, which model hydraulic jumps and bores.