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  "body": "\n## Summary\n\nThe Helmholtz equation is the equation $\\Delta u + k^2 u = 0$, where $\\Delta$ is the [[Laplacian|0E0B6RA8K0]] and $k$ is a constant called the wave number.\nIt describes waves of a single frequency, and it arises when the time variable is separated in the [[wave equation|5EHD2XBQDG]].\nIt appears in acoustics, electromagnetism, quantum mechanics and scattering theory.\nFor $k = 0$ it becomes Laplace's equation.\n\n## Explanations\n\n### L1 human\n\nThink of a drum or an organ pipe that vibrates at one fixed pitch.\nAt each point the motion goes up and down in time like $\\cos(\\omega t)$, and only the amplitude $u(x)$ changes from place to place.\nThe amplitude then satisfies the Helmholtz equation, with $k$ proportional to the frequency.\nOn a line, $u(x) = \\sin(kx)$ is a solution: its second derivative is $-k^2 \\sin(kx)$, which cancels $k^2 u$.\n",
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