Inside Swing Amplitude

The decrement factor is a ratio, and nobody feels a ratio. $A_i = f A_o$ turns it into degrees. Take a desert day swinging 15 C° either side of its mean — a 30 C° range from dawn to mid-afternoon — and pass it through a 400 mm earth wall with $f$ = 0.033: the inner face moves 0.50 C° either side of its mean, a range of one degree. That is a room that feels as though it has no weather at all, and it is achieved with no insulation, no glazing strategy and no machinery.

Get the amplitude convention right or every answer is out by two. Amplitude here means the half-swing: the peak deviation from the mean, which is the convention the sine solution is written in. If you are holding a daily high and low from a weather record, the amplitude is half the difference between them. The ratio itself is indifferent — feed it two peak-to-peak figures and it hands back a peak-to-peak figure — but the moment you mix a half-swing with a full range you are out by a factor of two, and the error is invisible because both numbers look plausible.

The equation says nothing about where the swing sits, and this is the single most important thing to understand about thermal mass. It moves the swing; it does not move the average. The mean indoor temperature is decided by the heat balance of the whole building — the R-value of the envelope, the solar gain, the ventilation rate, and whatever heating or cooling is running. A heavy uninsulated wall in a cold climate gives you a building that is steadily cold rather than variably cold, which is not usually what the owner had in mind. Mass and insulation solve different problems, they are not substitutes, and a claim that a wall "does not need insulation because it has thermal mass" is a claim about the swing being offered as an answer about the average.

Where mass genuinely does the whole job is where the outdoor mean over the cycle is already comfortable and only the swing is the problem. That is the classic hot-dry climate: a mean around 22 °C with a 30 C° daily range. Flatten the swing and you have a comfortable house for nothing. In a maritime climate with a small daily range and a cold mean, the same wall delivers almost no benefit and a great deal of weight.

Run the equation backwards and it becomes a diagnostic. Log the swing indoors and out over several settled days, divide, and you have the decrement factor of a real building, which is worth more than any formula because it includes the surface films, the reflected wave at the inner face, and every gap and bridge in the construction. Expect it to be several times larger than the semi-infinite formula predicts; that is normal. If it comes out above 1, the room is swinging more than the outdoors, which passive conduction cannot do — look for solar gain through glazing, an intermittently cycling heater, or night ventilation.