{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "3ZEW7MC05Z"
    }
  ],
  "body": "\n## Summary\n\nThe method of multiple scales builds approximations that stay valid over long times for problems with a small parameter.\nIt treats a fast time $t$ and a slow time $T = \\varepsilon t$ as independent variables.\nThe slow dependence is chosen to remove secular terms, which would otherwise make the error grow.\nIt handles damping, slow modulation of amplitude and phase, and other effects that the [[Poincaré–Lindstedt method|YHJHPRKQ6D]] cannot capture.\n\n## Explanations\n\n### L1 human\n\nA weakly damped oscillator swings quickly while its amplitude shrinks slowly.\nThe method describes the swing with a fast clock and the shrinking with a slow clock.\nFor $x'' + 2\\varepsilon x' + x = 0$ it gives $x \\approx e^{-\\varepsilon t} (a \\cos t + b \\sin t)$.\nA plain expansion in $\\varepsilon$ would instead give $a \\cos t$ plus a term growing like $\\varepsilon t$, which is wrong once $t$ is of order $1/\\varepsilon$.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "34E13",
      "34E10"
    ],
    "name": "Method of multiple scales",
    "same_as": {
      "wikidata": "Q6934874",
      "wikipedia": "Multiple-scale analysis"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Multiple-scale_analysis"
      }
    ],
    "tag": "2V5E1B1Q9H"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/2V5E1B1Q9H/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/2V5E1B1Q9H.md",
  "msc": [
    "34E13",
    "34E10"
  ],
  "name": "Method of multiple scales",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/2V5/2V5E1B1Q9H.md",
  "status": null,
  "tag": "2V5E1B1Q9H",
  "uri": "m2://2V5E1B1Q9H",
  "url": "https://math2.tamnd.com/wiki/method-of-multiple-scales/",
  "writing": {
    "tag": "2V5E1B1Q9H",
    "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
      }
    ]
  }
}