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  "body": "\n## Summary\n\nA [[damped harmonic oscillator|0RNXVVTJTJ]] $x'' + 2 \\zeta \\omega_0 x' + \\omega_0^2 x = 0$ with $\\omega_0 \u003e 0$ is underdamped when its damping ratio satisfies $0 \u003c \\zeta \u003c 1$.\nThen the characteristic roots are $-\\zeta \\omega_0 \\pm i \\omega_1$ with $\\omega_1 = \\omega_0 \\sqrt{1 - \\zeta^2}$.\nEach nonzero solution has the form $x(t) = A e^{-\\zeta \\omega_0 t} \\cos(\\omega_1 t + \\varphi)$ with $A \u003e 0$: an oscillation whose amplitude decays exponentially.\n\n## Remarks\n\nThe other cases are critical damping, $\\zeta = 1$, and overdamping, $\\zeta \u003e 1$. In these cases the solution changes sign at most one time.\nThe damped frequency $\\omega_1$ is smaller than the undamped frequency $\\omega_0$.\nThe zeros of an underdamped solution are spaced $\\pi / \\omega_1$ apart.\n\n## Explanations\n\n### L1 human\n\nAn underdamped system overshoots its rest position again and again.\nEach swing is smaller than the previous one, by the same factor.\nA guitar string or a car on soft springs behaves like this.\n",
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