And temperature

In this appendix, we give an elementary mathematical treatment of the relationship between heating and temperature. We will explain two things: why the temperature range is smaller if a body has a larger heat capacity and why there is a lag between heating and temperatures. We model the system with the simple equation

Here, S is the heating source, T is the temperature, and t, the time. The parameter C is the heat capacity of the system, and X is a constant that determines how fast the body cools when it is hot. The equation says that the heat capacity times the rate of the temperature increase (the left-hand side) is equal to the heating (S) minus the cooling (AT). Let us further suppose that the heating is cyclic, with S = S0 cos ~t, where ~ is the frequency of the heating and S0 is its amplitude.

To solve the equation, we write S = Re S0 exp(ict) and seek solutions of the form T = Re T0 exp(ict) where Re means "take the real part" and T0 is a constant to be determined. Substituting into equation 5.3, we have

CT0irneim S0eim AT0eim, (5.4)

where only the real part of the equation is relevant. From this equation, we straightforwardly obtain

T Re

What does this equation tell us? If the heat capacity is negligible, or if the frequency ~ is very small (i.e., very slow variations in forcing), then s

The temperature is in phase with the heating, and the amplitude of the cycle is S0/A. If the heat capacity is large or the frequency is high, then

The amplitude of the cycle is S„/C®, which is small because C is large. That is, because C~ > X in this case, the variations in temperature are smaller than they are in the case with slow forcing variations given by equation 5.6. Because high frequencies are damped more than low frequencies, we say that the variations are reddened. Note too that the temperature variations are now out of phase with the heating. Thus, the maximum temperature occurs when the heating is least. It is this effect that accounts for the delay in the maximum temperature in maritime climates, with the hottest part of the year occurring in late summer or even the beginning of autumn.

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