---
tag: 21DD2334NR
kind: concept
name: Tricomi equation
msc: [35M10, 76H05]
specializes: [GC5BDZ6QBC]
sources:
  - url: https://en.wikipedia.org/wiki/Euler–Tricomi_equation
same_as:
  wikidata: Q5409013
  wikipedia: Euler–Tricomi equation
---

## Summary

The Tricomi equation, or Euler–Tricomi equation, is the linear second-order equation $y u_{xx} + u_{yy} = 0$ for a function $u(x, y)$ in the plane.
It is the model [[equation of mixed type|7WFKPAMQFG]]: it is [[elliptic|B4TTNHSTRX]] in the upper half-plane, [[hyperbolic|92C3S03AQZ]] in the lower half-plane, and degenerate on the $x$-axis.
It arises in the study of transonic flow, where the flow changes from subsonic to supersonic.

## Explanations

### L1 human

Some equations behave like the Laplace equation in one region and like the wave equation in another.
In the Tricomi equation the coefficient $y$ switches sign across the $x$-axis, and the behaviour switches with it.
Above the axis solutions are smooth and spread information in all directions, as for a steady temperature.
Below the axis information travels along curves, as for waves.
This models air flowing around a wing, where the flow is slower than sound in one region and faster in another.