{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "G82RA93DWJ"
    }
  ],
  "body": "\n## Summary\n\nThe Klein–Gordon equation $u_{tt} - \\Delta u + m^2 u = 0$ is the [[wave equation|5EHD2XBQDG]] with a mass term.\nIt is the relativistic wave equation of a free scalar particle of mass $m$, written in units with $c = \\hbar = 1$.\nIt is hyperbolic, so signals travel no faster than light, and for $m \u003e 0$ it is also dispersive: waves of different frequencies travel at different speeds.\nIts nonlinear versions, such as $u_{tt} - \\Delta u + m^2 u = \\pm \\abs{u}^{p-1}u$, are model problems in nonlinear wave theory.\n\n## Explanations\n\n### L1 human\n\nTake a string that vibrates and is also held to its rest position by springs along its length.\nThe springs add the term $m^2 u$, which pulls the string back toward zero.\nA plane wave $e^{i(kx - \\omega t)}$ solves the equation exactly when $\\omega^2 = k^2 + m^2$.\nSo short waves move almost at speed $1$, while long waves move more slowly, and a bump spreads out as it travels.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "35Q40",
      "35L05",
      "35L71"
    ],
    "name": "Klein–Gordon equation",
    "same_as": {
      "wikidata": "Q868967",
      "wikipedia": "Klein–Gordon equation"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Klein%E2%80%93Gordon_equation"
      }
    ],
    "specializes": [
      "92C3S03AQZ"
    ],
    "tag": "AYCCDQWCJF"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/AYCCDQWCJF/",
  "links": [
    {
      "field": "specializes",
      "target": "92C3S03AQZ"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/AYCCDQWCJF.md",
  "msc": [
    "35Q40",
    "35L05",
    "35L71"
  ],
  "name": "Klein–Gordon equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/AYC/AYCCDQWCJF.md",
  "status": null,
  "tag": "AYCCDQWCJF",
  "uri": "m2://AYCCDQWCJF",
  "url": "https://tamnd.github.io/math2/wiki/klein-gordon-equation/",
  "writing": {
    "tag": "AYCCDQWCJF",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}