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  "body": "\n## Summary\n\nThe zeros of a [[Bessel function of the first kind|CZCMRDWDYW]] $J_\\nu$ with $\\nu \u003e -1$ are real.\nIts positive zeros form an increasing sequence $j_{\\nu,1} \u003c j_{\\nu,2} \u003c \\cdots$ tending to infinity.\nThese numbers fix the frequencies of a vibrating circular drum and the modes of heat flow in a cylinder.\nThey obey interlacing and spacing rules, and they are the subject of [[Bourget's hypothesis|38D9RPAXQ8]].\n\n## Explanations\n\n### L1 human\n\nThe graph of $J_0(x)$ for $x \u003e 0$ looks like a damped cosine, so it crosses the axis again and again.\nThe crossing points are the zeros $j_{0,1} \\approx 2.405$, $j_{0,2} \\approx 5.520$, and so on.\nFar out they are spaced by almost exactly $\\pi$, like the zeros of a cosine.\nA drum of radius $1$ fixed at its rim vibrates in its lowest radially symmetric mode with frequency proportional to $j_{0,1}$.\n",
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