Thermal Damping Depth

Every periodic heat problem has a natural ruler, and this is it. Drive the surface of a solid with a temperature swing that repeats with period $P$, and the swing does not travel inward undiminished; it decays exponentially, and $d = \sqrt{\alpha P/\pi}$ is the distance over which it falls by a factor of $e$ — down to 36.8% of what it was. At two damping depths 13.5% survives, at three 5.0%, at four 1.8%. A wall thin compared with $d$ barely delays anything; a wall three or four times thicker has swallowed the cycle entirely and further thickness is wasted material.

The solution is Fourier's, published in the Théorie analytique de la chaleur in 1822, but the person who made it a working tool was Anders Angstrom, who in 1861 measured the daily temperature wave in the ground at Uppsala and used the way it faded with depth to recover the diffusivity of the soil itself. That inversion is still the best way to get a real number for real ground, and it is on this page: bury two loggers, find the depth at which the swing has fallen to about 37% of the surface swing, and $\alpha = \pi d^2/P$ gives you the diffusivity of the material as it actually is — moisture, stones, roots and all — rather than as a table imagines it.

The square root on the period is where all the interesting behaviour comes from. The annual cycle is 365 times longer than the daily one, so its damping depth is only $\sqrt{365} = 19$ times deeper. In soil at 0.5 mm²/s the daily wave has a damping depth of 117 mm and the annual wave 2.24 m. That single pair of numbers explains a great deal: half a metre of soil shrugs off any hot afternoon, so a shallow buried pipe never notices the weather, while the seasons reach down several metres, which is why frost lines are where they are, why a root cellar has to be genuinely deep to hold a steady temperature, and why ground-source heat pump loops are buried below the annual wave rather than below the daily one.

Two traps. The period must be the period of the repeating cycle, not the length of the event you happen to be interested in — a three-day heat wave is not a three-day cycle, it is three cycles of the daily one riding on a shift in the mean. And the period must be entered in the right unit: typing 24 into a box set to seconds instead of hours shrinks the damping depth by a factor of 60, which is the single most common numerical error on these pages.

Finally, remember what solid the derivation describes. This is the semi-infinite half-space: material that goes on forever, so the wave has nothing to reflect off. Soil, rock, a cave roof and a wall much thicker than $d$ all qualify. A finite wall with a room on the other side does not, and the honest treatment of that case is ISO 13786's transfer-matrix method rather than anything on this page.