{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "GAJDYXNHN2"
    },
    {
      "field": "uses",
      "target": "RN8Y1823FX"
    },
    {
      "field": "defines",
      "target": "HAZ0NKCAXS"
    }
  ],
  "body": "\n## Summary\n\nA traveling wave is a solution of an evolution equation that moves at constant speed without changing its shape.\nIn one space dimension it has the form $u(x, t) = U(x - ct)$, with a profile $U$ and a speed $c$.\nSubstituting this form turns the partial differential equation into an ordinary differential equation for $U$.\nTraveling waves describe fronts in reaction–diffusion equations, viscous shock profiles and [[solitons|63D4AXYCT4]].\nSee the [[definition|HAZ0NKCAXS]].\n\n## Explanations\n\n### L1 human\n\nThink of a surfer riding a wave: in the frame that moves with the surfer, the water surface does not change.\nA traveling wave is a solution that looks frozen when one moves along with it at the right speed.\nThe simplest case is the [[transport equation|PVD7VZJZ9B]] $u_t + a u_x = 0$, whose solutions $u = g(x - at)$ all travel at speed $a$.\nIn nonlinear equations only special shapes travel unchanged, and finding them is a question about an ordinary differential equation.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "35C07",
      "35K57"
    ],
    "name": "Traveling wave",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Wave"
      }
    ],
    "tag": "M37M411KP7"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/M37M411KP7/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/M37M411KP7.md",
  "msc": [
    "35C07",
    "35K57"
  ],
  "name": "Traveling wave",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/M37/M37M411KP7.md",
  "status": null,
  "tag": "M37M411KP7",
  "uri": "m2://M37M411KP7",
  "url": "https://tamnd.github.io/math2/wiki/traveling-wave/",
  "writing": {
    "tag": "M37M411KP7",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}