The simplest example of Newton's Model of Cooling is given by the differential equation

where A is a constant ambient temperature. This equation is easily solved using separation of variables as shown below.

In many cases, however, the ambient temperature is not constant. For example, we might be interested in the fluctuations of the temperature of a lake as the air temperature varies over the course of several days or over the course of a year. Other related phenomena include the fluctuation of air temperature over the course of the day or over the course of a year. Notice that even though the sun is strongest close to noon, the day is hottest several hours later. Similarly, the hottest weather occurs a month or two after the summer solstice.

We investigate these phenomena by looking at the differential equation

obtained by replacing the constant ambient temperature A in the first model by a fluctuating ambient temperature.

This differential equation is easily solved using the standard recipe for solving first order linear differential equations as shown below.

Notice that solving this differential equation requires a fairly complicated integration by parts. Then putting the solution into a form that emphasizes amplitude and phase shift requires standard trigonometric trickery.

We are interested in the effects of two things.

• The size and insulation of the object being studied. This is represented mathematically by the constant k. If this constant is large than heat is transfered to and from the object very quickly. This is what happens if the object is small and poorly insulated. If this constant is small then the heat transfer is slow. This is what happens if the object is alrge and well insulated.

• The frequency with which the ambient temperature oscillates. This is represented by the greek letter omega .

We can also see these effects directly from the solution obtained above. Notice the following

• The term
  

  involving the constant C that comes from the initial condition goes to zero as t goes to infinity. Thus, over the long term the effect of the initial condition dies out and the function looks like

  

• If we fix the frequency and vary the constant k then as k goes to infinity the lag represented by the greek letter delta goes to zero and the amplitude goes to one. Thus, the temperature of the object is very close to the ambient temperature. Since large values of k represent small poorly insulated objects this is exactly what we would expect.

• If we fix the frequency and vary the constant k then as k goes to zero the lag represented by the greek letter delta goes to pi/2 and the amplitude goes to zero. Thus, the temperature of the object doesn't vary very much and it lags behind the variation in the ambient temperature. Since small values of k represent large well insulated objects this is exactly what we would expect.

• If we fix the constant k and increase the frequency represented by the greek letter omega then as omega goes to infinity the lag represented by the greek letter delta goes to pi/2 and the amplitude goes to zero. Thus, the temperature of the object doesn't vary very much and it lags behind the variation in the ambient temperature. This is what you would expect when the ambient temperature fluctuates so rapidly that the object doesn't have enough time to respond.