---
tag: 2V5E1B1Q9H
kind: concept
name: Method of multiple scales
msc: ["34E13", "34E10"]
sources:
  - url: https://en.wikipedia.org/wiki/Multiple-scale_analysis
same_as:
  wikidata: Q6934874
  wikipedia: Multiple-scale analysis
---

## Summary

The method of multiple scales builds approximations that stay valid over long times for problems with a small parameter.
It treats a fast time $t$ and a slow time $T = \varepsilon t$ as independent variables.
The slow dependence is chosen to remove secular terms, which would otherwise make the error grow.
It handles damping, slow modulation of amplitude and phase, and other effects that the [[Poincaré–Lindstedt method|YHJHPRKQ6D]] cannot capture.

## Explanations

### L1 human

A weakly damped oscillator swings quickly while its amplitude shrinks slowly.
The method describes the swing with a fast clock and the shrinking with a slow clock.
For $x'' + 2\varepsilon x' + x = 0$ it gives $x \approx e^{-\varepsilon t} (a \cos t + b \sin t)$.
A 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 $1/\varepsilon$.