Decrement Factor from Damping Depth

The decrement factor is the fraction of the outdoor temperature swing that is still present at depth $x$, and it is the whole of the damping story in one exponential: $f = e^{-x/d}$. Thickness is measured in damping depths, and each one costs a factor of $e$. One damping depth leaves 37%, two leave 13.5%, three leave 5%. Because the decay is exponential rather than linear, the first few centimetres do most of the work and the last few do almost none — halving $f$ always costs another $0.693d$ of material, and each halving removes a swing that was already smaller than the one before.

A name collision worth stating plainly, because both terms live on this site. The vibration pages carry the logarithmic decrement, which is the natural logarithm of the ratio of two successive peaks of a freely decaying oscillation, and which is used to recover a damping ratio from a ring-down trace. The decrement factor here is an entirely different quantity: a dimensionless amplitude ratio for a steady periodic wave that never decays in time at all — it decays in space, with depth into the material. One is about a system losing energy as the seconds pass; the other is about a wave shrinking as the millimetres pass. They share a word and nothing else, they are never substituted for one another, and if you arrived here from a vibration problem you want the other page.

Now the honesty, because this equation is quoted about buildings far more often than it is derived. It is the semi-infinite solution. It assumes the wall goes on forever behind the point you are asking about, so nothing reflects. A real 400 mm earth wall has an inner face, an air film, and a room; part of the wave turns around there, and measured decrement factors on real assemblies come out several times larger than this predicts. The standard that handles the finite layered wall properly is ISO 13786, which represents each layer as a complex transfer matrix at the driving frequency and multiplies the chain together to get periodic thermal transmittance, internal areal heat capacity, and its own decrement factor. That method is iterative, it is standardised, and it is copyrighted. This is not it. Use this page to understand why mass works and roughly how much thickness it takes; use ISO 13786 when a number has to survive a code review.

Two further limits. The driving swing is assumed sinusoidal, and real weather is not — what is solved here is the first harmonic of the outdoor cycle. That is more forgiving than it sounds, because the higher harmonics have shorter periods, therefore shallower damping depths, therefore die faster than the fundamental; the deeper you go, the better the sine assumption gets. And surface films are ignored entirely, so the temperatures in this equation are surface temperatures, not air temperatures.

The last trap is conceptual and it costs real money: thermal mass is not insulation. The decrement factor says how much of the swing gets through. It says nothing whatever about where the average sits, and the average is set by the R-value and the heat balance of the building. A heavy wall with no insulation gives you a room that is uniformly cold instead of variably cold, and there are climates where that is a good trade and climates where it is not.