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  "body": "\n## Example\n\nLet $\\omega_0 \u003e 0$, $\\zeta \\ge 0$, $F \u003e 0$ and $\\omega \u003e 0$ be constants. The forced oscillator is\n$$x'' + 2 \\zeta \\omega_0\\, x' + \\omega_0^2\\, x = F \\cos(\\omega t).$$\nIf $\\zeta \u003e 0$, it has the particular solution $x_p(t) = a(\\omega) \\cos(\\omega t - \\delta)$ for a phase $\\delta$, with amplitude\n$$a(\\omega) = \\frac{F}{\\sqrt{\\left(\\omega_0^2 - \\omega^2\\right)^2 + \\left(2 \\zeta \\omega_0 \\omega\\right)^2}}.$$\nEach solution is $x_p$ plus a solution of the [[damped equation|0RNXVVTJTJ]], which tends to $0$. So $x_p$ is the steady state.\nIf $\\zeta = 0$ and $\\omega = \\omega_0$, a particular solution is $x_p(t) = \\frac{F}{2 \\omega_0}\\, t \\sin(\\omega_0 t)$, which is unbounded. This is resonance.\n\n## Remarks\n\nFor $0 \u003c \\zeta \u003c 1/\\sqrt{2}$, the amplitude $a(\\omega)$ is largest at $\\omega = \\omega_0 \\sqrt{1 - 2 \\zeta^2}$. For small $\\zeta$, the maximum is close to $F / (2 \\zeta \\omega_0^2)$, which is large.\nFor $\\zeta = 0$ and $\\omega \\ne \\omega_0$, all solutions are bounded: $x_p = F \\cos(\\omega t) / (\\omega_0^2 - \\omega^2)$.\nThe factor $t$ in the resonant solution comes from the rule of the [[method of undetermined coefficients|0XRB469Z32]]: $i \\omega_0$ is a root of the characteristic polynomial.\n\n## Explanations\n\n### L1 human\n\nIf you push a swing at the right rhythm, each push adds to the motion, and the swing goes higher and higher.\nThis is resonance: the driving frequency matches the natural frequency $\\omega_0$.\nFriction limits the growth, but the response is still very large near $\\omega_0$.\n",
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        "locator": "The driven, damped harmonic oscillator",
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