Thermal Time Lag from Damping Depth

The same delay, written through the damping depth: $\varphi = xP/(2\pi d)$. Nothing new is being claimed — substitute $d = \sqrt{\alpha P/\pi}$ and you get the previous page back — but the structure of the answer becomes visible, and the structure is worth having in your head.

Read $x/d$ as a count of damping depths. The formula then says the delay is that count, divided by $2\pi$, times the period. In other words one damping depth costs exactly one radian of phase. And since the same damping depth costs exactly one factor of $e$ in amplitude, the two halves of the periodic solution are locked together at a fixed exchange rate. One radian of a 24-hour day is 3.82 hours, so:

One damping depth: 3.8 hours of delay, 37% of the swing left. Two: 7.6 hours, 13.5%. Three: 11.5 hours, 5.0%. Four: 15.3 hours, 1.8%. Those four lines are most of what a designer needs, and they explain the shape of the trade-off. The lag keeps growing in a straight line forever, while the swing being delayed disappears exponentially — so past three damping depths you are building a heavier wall to delay something that has already stopped mattering. That is the sense in which very thick walls stop paying for themselves, and it is why traditional earth walls converge on 400 to 600 mm rather than a metre.

A pleasant special case falls out exactly: a delay of half a period requires $x/d = \pi$, so $x = \pi d$. That is 3.14 damping depths, the peak lands precisely at the outdoor trough, and about 4.3% of the swing survives to do it. For soil at 0.5 mm²/s and a daily cycle that is 368 mm — which is, to within a builder's tolerance, the thickness that hot dry climates arrived at by trial and error long before anyone wrote the equation down.

The inverse is the field measurement that matters most, because a lag is easy to measure honestly. Two loggers, a fortnight of settled weather, find the offset between the two daily peaks, and $d = xP/(2\pi\varphi)$ gives the damping depth of the wall in front of you. From there the damping-depth page gives $\alpha$, and the diffusivity page gives whichever of $k$, $\rho$ or $c$ you did not know. That chain — lag, depth, diffusivity, property — is Angstrom's whole 1861 experiment, and it still works. Use clear settled weather: a passing front puts a step into the record, and the sine assumption underneath the entire method quietly fails when it does.