Decrement Factor from Diffusivity and Period

The same amplitude ratio, written straight from the material and the cycle: $f = \exp(-x\sqrt{\pi/(\alpha P)})$. The damping depth is still in there — it is the reciprocal of that square root — but not having to compute it first makes this the convenient form for comparing things, because you can change one input and read the consequence directly.

Try the comparison the equation is best at. Take 400 mm of rammed earth at $\alpha$ = 0.5 mm²/s against a 24-hour cycle: $f$ = 0.033, so three per cent of the swing survives. Now change nothing but the period, and ask about the annual cycle instead: $f$ = 0.84, so 84% survives. The same wall that erases the day barely touches the season. This is not a subtlety; it is the reason a thick-walled house still needs heating in January, and the reason people who have read one article about thermal mass are sometimes disappointed by it.

Change the material instead and something more surprising happens. Solid softwood at $\alpha$ = 0.15 mm²/s, at the same 400 mm and the same daily cycle, gives $f$ = 0.002 — it damps better than earth, by a factor of seventeen. Diffusivity is what governs, and softwood's is lower because its conductivity is far lower even though its $\rho c$ is smaller too. What earth and stone have that timber construction usually has not is that all 400 mm is continuous material. A framed wall is two thin skins with a cavity between them, and the periodic conduction solution does not describe it at all — the transport across that cavity is convection and radiation, not conduction, and the mass is not there to absorb anything.

The inverse forms are the field method. Angstrom's 1861 experiment had two halves: an amplitude ratio and a phase lag, each of which independently yields the diffusivity. Solving this page for $\alpha$ is the amplitude half. Solving the time-lag page for $\alpha$ is the phase half. Run both on the same data and compare. If they agree, the material is behaving as a uniform conductor and you can trust the number. If the lag-derived diffusivity comes out noticeably larger than the amplitude-derived one, heat is arriving by some route conduction does not model — usually moisture migrating as vapour and condensing, sometimes air leaking through a crack — and the disagreement is telling you something more useful than either number would have.

Two input errors account for most wrong answers here, and both concern $P$. One is the unit: a period entered in seconds where hours were meant is out by a factor of 3600, and since the answer depends on $\sqrt{P}$ the damping depth is out by 60. The other is conceptual: $P$ is the period of the repeating cycle, always 24 hours or a year for weather, never the duration of the spell of weather you are thinking about.