# How does the period of a temperature fluctuation affect the depth to which the ground freezes?

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
Hi Bhargav. To help us organize the site better, please tag your questions with relevant tags and give descriptive titles that communicate some of the question's content. I've done this for you here. Welcome to phys.SE – Mark Eichenlaub Aug 25 '11 at 11:48

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.