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On account of the annual fluctuation of temperature the ground at the town of Ν freezes to a depth of 2 metres. To what depth would it freeze on account of a daily fluctuation of the same amplitude?

Found this on aops while gazing through some questions (So that people dont blame me for posting a phy q on math site). Any ideas?

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Probably $0$ metres. –  André Nicolas Aug 25 '11 at 7:52
    
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1 Answer 1

The characteristic length of the exponential decay in the ground is proportional to the square root of the period of the fluctuations, so the answer is roughly $2\text{m}/\sqrt{365.25}\approx10\text{cm}$. That's if we treat this as a linear isotropic heat diffusion problem and neglect non-linear effects from the heat of fusion, inhomogeneities in the ground, finite rate of heat transfer at the ground/air interface, and ground absorption of direct sunlight.

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