---
tag: M37M411KP7
kind: concept
name: Traveling wave
msc: [35C07, 35K57]
sources:
  - url: https://en.wikipedia.org/wiki/Wave
---

## Summary

A traveling wave is a solution of an evolution equation that moves at constant speed without changing its shape.
In one space dimension it has the form $u(x, t) = U(x - ct)$, with a profile $U$ and a speed $c$.
Substituting this form turns the partial differential equation into an ordinary differential equation for $U$.
Traveling waves describe fronts in reaction–diffusion equations, viscous shock profiles and [[solitons|63D4AXYCT4]].
See the [[definition|HAZ0NKCAXS]].

## Explanations

### L1 human

Think of a surfer riding a wave: in the frame that moves with the surfer, the water surface does not change.
A traveling wave is a solution that looks frozen when one moves along with it at the right speed.
The simplest case is the [[transport equation|PVD7VZJZ9B]] $u_t + a u_x = 0$, whose solutions $u = g(x - at)$ all travel at speed $a$.
In nonlinear equations only special shapes travel unchanged, and finding them is a question about an ordinary differential equation.