Two things happen to the wave on the way down
Heat moves through ground by conduction, and a temperature that oscillates at the surface sends a wave down into it. The wave loses amplitude exponentially with depth and it is delayed linearly with depth, and both effects are controlled by one length: the damping depth, which is the square root of the diffusivity times the period divided by pi.
At a diffusivity of 0.55 square feet a day over a 365 day cycle, the damping depth is 7.99 feet. That is the depth at which the swing is down to 37 percent of the surface swing, which is one over e. Every further 7.99 feet takes another factor of e off it. So a 22 degree surface swing is 8.09 degrees at 8 feet, 2.97 at 16 feet, 1.09 at 24 feet, and 0.40 at 32.
The lag over the same distance is depth times the damping depth divided by twice the diffusivity, which for this soil is 7.27 days per foot. At six feet that is 43.6 days, so the ground down there reaches its annual low around six weeks after the surface does. At 25.1 feet the lag is 182.5 days, exactly half the cycle, which is why the page prints that depth as the one where the ground is warmest at the moment the surface is coldest.
What that means for a horizontal loop
A trench loop at six feet in this soil sits in ground that swings between 41.6 and 62.4 degrees over the year. With the surface peaking on day 205, the ground at that depth peaks on day 249 and bottoms out on day 66 — which is early March, not January. That matters, because the coldest ground and the coldest weather are not on the same day, and the loop is doing its hardest work at the point where the ground has been giving up heat all winter and has not yet started recovering.
It also explains why going deeper with a horizontal loop pays and then stops paying. The annual minimum at three feet is 36.9 degrees, at six feet 41.6, at nine feet 44.9 and at twelve feet 47.1. The first three feet of extra depth buy 4.7 degrees, the next three buy 3.3, the next three 2.2. The excavation cost meanwhile climbs in a straight line, and past about twelve feet the swing left to flatten is under five degrees either way. Where that crossover sits is different in every soil, which is why the diffusivity is a field on this page rather than a constant.
Why a vertical bore is a different animal
Below about thirty feet in most ground the annual wave has died to under a degree, and the temperature there is effectively the mean annual value all year round. A vertical borehole spends nearly all of its length in that region, which is the real reason vertical loops perform more consistently than horizontal ones: they are not sitting in the part of the ground the seasons reach.
What they do instead is change the ground themselves. A bore field pulls heat from the rock around it over a winter and that rock cools, so entering water temperature falls through the season and, if the annual balance is one-sided, over the years. That is a different mechanism from the surface wave and this page does not model it. The two get confused often enough to be worth naming.
The daily wave, and a check on the model
Put 1 in the period box and the same arithmetic describes the day and night cycle. The damping depth collapses to 0.42 feet, so a 20 degree daily swing at the surface is down to 7.4 degrees at five inches and under two degrees at a foot. Anyone who has put a hand on a paving slab at midday and then dug under it has met this result directly.
That is a useful sanity check on the whole model, because it is a result everyone already knows from experience. If the annual numbers look surprising, run the daily case first and see whether the shape matches what you have felt.
Where the inputs come from, and where they do not
The mean annual ground temperature is a measurement, taken deep enough that the wave has died. A well log gives it, and so does a temperature probe at the bottom of a bore. Regional maps give a regional figure that can be a couple of degrees away from a specific garden, and a couple of degrees is a large fraction of the swing at depth.
The surface amplitude is half the range of monthly mean ground surface temperatures, and it is not the air temperature range. Ground surface runs warmer than air in summer under sun and colder in clear winter nights without snow, and snow cover breaks the relationship entirely for the part of the winter it lasts.
Diffusivity is the input with the widest honest range. It is conductivity divided by volumetric heat capacity, both of which move with moisture content, and moisture content moves through the year. Saturated clay and dry sand are not close to each other. If the diffusivity is a guess, run the calculation at the top and bottom of the range you think it might be, and use the spread as the real answer.
Questions people ask
How deep do I have to dig for the ground to stay the same temperature?
It depends on the soil, which is why it is a field rather than a constant. The annual swing falls by a factor of e for every damping depth, and the damping depth is the square root of the diffusivity times 365 divided by pi. At 0.55 square feet a day that is 7.99 feet, so the swing is down to a tenth of the surface swing at 18.4 feet and a twentieth at 23.9 feet. In slower ground both figures get shallower and in faster ground deeper.
Why is the ground coldest in March rather than January?
Because the temperature wave takes time to travel down. The lag is depth times the damping depth divided by twice the diffusivity, which at these defaults is 7.27 days per foot. Six feet down that is about 44 days behind the surface, and the surface itself already lags the solstice. Deeper still and the lag keeps growing until, at pi times the damping depth — 25.1 feet here — the ground is exactly half a cycle behind and is warmest when the surface is coldest.
Is this the frost line?
No, and it should not be used as one. Frost depth for footings, foundations, water services and anything else structural is set by the code the local authority has adopted, and that figure carries margin for the things a smooth conduction model cannot see: bare ground without snow cover, a hard winter well outside the average, ground that has been cleared or paved, and soil that behaves differently when it is wet. Take frost depth from the authority with jurisdiction, not from a thermal wave.
What is a thermal diffusivity and where do I get one?
It is thermal conductivity divided by volumetric heat capacity, and it says how fast a temperature change travels through a material rather than how much heat it carries. For soils it is usually quoted in square feet per day or square metres per second. Get it from the soil or formation data for the site, ideally from the same investigation that produced the conductivity. It moves substantially with moisture content, which moves through the year, so treating it as one fixed number is already an approximation.
Does this tell me what a ground loop will run at?
It tells you the undisturbed ground temperature the loop starts from, which is not the same thing. Once a loop is running it changes the ground around it, pulling the local temperature down through a heating season and pushing it up through a cooling one, and that effect is what the loop design has to model. The undisturbed profile on this page is the baseline that design works from, and the difference between it and the fluid temperature is the design temperature difference on the loop length page.