Thermal Time Lag

This is the equation behind every thick-walled building on earth, and it deserves to be famous. $\varphi = (x/2)\sqrt{P/(\pi\alpha)}$ is how many hours late the outdoor peak arrives at depth $x$. Work a real wall: 400 mm of rammed earth at $\alpha$ = 0.5 mm²/s under a 24-hour cycle gives $\varphi$ = 13.0 hours. The two o'clock sun lands on the outside face and the heat reaches the room at three in the morning — by which time the outdoors has gone cold and pulls it straight back out again. Pair that with a decrement factor of 0.033 and you have the entire physics of an adobe, cob, rammed-earth or stone building in two numbers: the peak is delayed until it is no longer a problem, and most of it is gone anyway. That is why the house is cool at noon and warm at midnight, and it is arithmetic rather than folklore.

Notice how the lag is built. It is proportional to thickness, so doubling the wall doubles the delay — unlike the decrement factor, which falls exponentially and saturates. And it goes as $1/\sqrt{\alpha}$, so the slower the material diffuses, the longer the delay. There is no separate term for mass, density or heat capacity: the material enters only through $\alpha$, and $\rho c$ sits in that quantity's denominator.

Which produces a result most people find wrong at first sight. Put 400 mm of solid softwood next to that earth wall and it lags 23.8 hours — nearly twice as long — because its diffusivity is 0.15 against earth's 0.5 and the lag ratio is $\sqrt{0.5/0.15} = 1.83$. Wood beats earth per unit thickness, on both lag and damping. So why is the effect associated with masonry rather than timber? Because nobody builds 400 mm of solid wood except in a log or cross-laminated wall, and those genuinely do behave this way. A conventional stud wall is 12 mm of sheathing, 140 mm of cavity that conducts almost nothing, and 12 mm of gypsum: there is no 400 mm of material for a wave to crawl through, and no equation on this page applies to it. Mass is not what delays heat. Low diffusivity is, and continuous thickness of real material is how you get to use it.

The $\sqrt{P}$ again. The annual wave lags about nineteen times longer than the daily one through the same material, which is why deep soil is at its coldest in spring and its warmest in autumn, several months out of step with the air above it. Buried pipe, cave temperatures, the frost phase lag that puts the deepest freezing weeks after the coldest weather — all the same equation with $P$ set to a year.

Design with it in the inverse direction: name the delay you want and it returns the thickness. Ten to twelve hours puts the afternoon's heat into the evening, which is what a hot dry climate wants. Half a period — twelve hours on a daily cycle — puts the inside peak exactly at the outdoor trough, which is the theoretical ideal. But check the decrement factor before pouring anything, because thickness bought for lag also flattens the swing, and past about three damping depths there is very little swing left to delay.

Two honest limits. This is the pure conduction lag between two surfaces: surface films, solar gain heating the outside face well above air temperature, and ventilation indoors all shift what you would actually measure. And it is again the semi-infinite solution, so for a finite wall treat the lag as indicative and the damping as optimistic.