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The thermal wave equation

The thermal wave equation - surface a little way below...

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The thermal wave equation If the heat flow through a volume changes with time the rate of change of temperature is given by the thermal wave equation: where K is the coefficient of thermal conductivity. If we look at the penetration into a planetary surface of temperature variations due to an external heat source, the we can reduce this to one dimension: We shall assume the forcing function is periodic. It could be the diurnal variability of solar input, or the annual or seasonal variation of heating. We can also assume that the temperature varies with depth as f(z) (where z is depth). Then we can substitute for T(z,t): This has a general solution: The first term is physically meaningless as it represents a temperature exponentially growing with depth. Thus the equation for T(z,t) can be written: This is a sinusoidally travelling wave moving into the material from the surface, with an amplitude that drops exponentially with depth. Thus if we apply a sinusoidal variation at the

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Unformatted text preview: surface, a little way below there will be a sinusoidal variation with a smaller amplitude and phase shifted from the input. The further in the larger the phase shift and the smaller the amplitude. The temperature at a great depth approaches the average value of the input variation. This equation can be used to explain things like the existence of permafrost. In the tundra regions, although the surface temperature may go well above zero in the summer months, a few meters below the surface it oscillates around the yearly average of the surface, and if that is below zero, the ice below the surface will never melt. Notice also that the peak temperature will be phase lagged compared to the surface - there will be a layer below the surface where the maximum temperature occurs at the point where the surface is at its coldest....
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