Periodic Temperature Profile

This is the complete solution, and both earlier pages are sitting inside it. $T(x,t) = T_m + A e^{-x/d}\sin(2\pi t/P - x/d)$: the exponential is the decrement factor, the $x/d$ subtracted inside the sine is the phase lag in radians, and the same damping depth does both jobs at once. Fourier gave the method in 1822, Angstrom put it to work on soil in 1861, and Carslaw and Jaeger's Conduction of Heat in Solids is where the modern statement is usually found. Use this page when you want a temperature at a stated depth and hour rather than a ratio: soil temperature for an agronomy problem, the inner face of a wall at four in the afternoon, the wall of a wine cave in August.

The time origin is the trap, and it catches everybody once. $t$ is measured from the instant the surface crosses its mean on the way up — not from midnight, not from January the first, and not from the surface peak. For a daily ground-temperature cycle peaking around 14:00, the surface crosses its mean going up about six hours earlier, near 08:00, so $t = 0$ is 08:00 and clock time is $t$ plus eight hours. For an annual cycle in the northern hemisphere the mean upward crossing is somewhere in April. Getting this wrong shifts every answer by a fixed number of hours or weeks, and because the answers still look like plausible temperatures nothing warns you.

Solving for the time is where the equation stops being invertible, and that is not a defect — it is what periodic means. Give the page a temperature and ask when it occurs, and there is no single answer: the wave passes every value it reaches twice in each cycle, once climbing and once falling, and then does the whole thing again next cycle and the cycle after. The arcsine returns one branch, the principal one, and this page normalises it into the first period so the answer is a time of day rather than a negative number. The companion time in the same cycle is $P/2 + xP/(2\pi d) - t$, and both are equally correct. The answer note prints both, because a reader asking "when is the cellar at 12 °C" almost always wants to know that it is twice a day and not once.

The same three limits apply as everywhere in this shard, and they are worth restating because a full temperature profile looks more authoritative than a ratio does. It is the semi-infinite solid, so it describes soil, rock and a cave roof very well and a finite wall with a room behind it only approximately — for that case ISO 13786's transfer-matrix method is the authority and this is not it. The driving swing is a pure sine, so what is being solved is the first harmonic of real weather; the higher harmonics damp faster and matter less with depth, which is why the assumption improves as you go deeper. And surface films are absent, so $T_m$ and $A$ are surface values, not air values. If all you have is air temperature, the real surface swing is smaller and slightly later than the air swing, and the film is doing part of the job you are about to credit to the wall.