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CONDUCTIO N

ing of a surface layer of ice and snow, a large quantity of cold water, percolating rapidly, gave for a short time values of the diffusivity as high as '0300. Excluding these exceptional cases, however, the variations of the diffusivity appeared to follow the variations of the seasons with considerable regularity in successive years. The presence of water in the soil always increased the value of &/c, and as it necessarily increased c, the increase of k must have been greater than that of k/c. 13. Periodic Plow of Heat.—The above method is perfectly general, and can be applied in any case in which the requisite observations can be taken. A case of special interest and importance is that in which the flow is periodic. The general characteristics of such a flow are illustrated in Fig. 3, showing the propagation

Fig. 3. of temperature waves due to diurnal variations in the temperature of the surface. The daily range of temperature of the air and of the surface of the soil was about 20° F. On a sunny day, the temperature reached a maximum about 2 p.m., and a minimum about 5 a.m. As the waves were propagated downwards through the soil the amplitude rapidly diminished, so that at a depth of only 4 inches it was already reduced to about 6° F., and to less than 2° at 10 inches. At the same time, the epoch of maximum or minimum was retarded, about 4 hours at 4 inches, and nearly 12 hours at 10 inches, where the maximum temperature was reached between 1 and 2 a.m. The form of the wave was also changed. At 4 inches the rise was steeper than the fall, at 10 inches the reverse was the case. This is due to the fact that the components of shorter period are more rapidly propagated. For instance, the velocity of propagation of a wave having a period of a day is nearly twenty times as great as that of a wave with a period of one year, but on the other hand the penetration of the diurnal wave is nearly twenty times less, and the shorter waves die out more rapidly. 14. A Simple-Harmonic or Sine Wave is the only kind which is propagated without change of form. In treating mathematically the propagation of other kinds of waves, it is necessary to analyse them into their simple-harmonic components, which may be treated as being propagated independently. To illustrate the main features of the calculation, we may suppose that the surface is subject to a simple-harmonic cycle of temperature variation, so that the temperature at any time t is given by an equation of the form— 0-da—A sin 2irnt=A sin 2TrtjT, (5) where da is the mean temperature of the surface, A the amplitude of the cycle, n the frequency, and T the period. In this simple

OF

HEAT

case the temperature cycle at a depth a; is a precisely similar curve of the same period, but with the amplitude reduced in the proportion e, and the phase retarded by the fraction mx/2ir of a cycle. The index-coefficient m is (irnc/fc)K The wave at a depth x is represented analytically by the equation— O-0o=Ae ^ sin (2irnt + mx). (6) A strictly periodic oscillation of this kind occurs in the working of a steam-engine, in which the walls of the cylinder are exposed to regular fluctuations of temperature with the admission and release of steam. The curves in Fig. 4 are drawn for a particular

Fig. 4. case, but they apply equally to the propagation of a simple-harmonic wave of any period in any substance changing only the scale on which they are drawn. The dotted boundary curves have the equation 0= +e > and show the rate of diminution of the amplitude of the temperature oscillation with depth in the metal. The wave-length in Fig. 4 is 0'60 inch, at which depth the amplitude of the variation is reduced to less than one-five hundredth part (e ~7r) of that at the surface, so that for all practical purposes the oscillation may be neglected beyond one wavelength. At half a wave-length the amplitude is only l/23rd of that at the surface. The wave-length in any case is 27r/m. The diffusivity can be deduced from observations at different depths x' and x", by observing the ratio of the amplitudes, which is e for a simple-harmonic wave. The values obtained in this way for waves having a period of one second and a wavelength of half an inch agreed very well with those obtained in the same cast-iron by Angstrom’s method (see below), with waves having a period of 1 hour and a length of 30 inches. This agreement was a very satisfactory test of the accuracy of the fundamental law of conduction, as the gradients and periods varied so widely in the two cases. 15. Annual Variation.—A similar method has frequently been applied to the study of variations of soiltemperatures by harmonic analysis of the annual waves. But the theory is not strictly applicable, as the phenomena are not accurately periodic, and the state of the soil is continually varying, and differs at different depths, particularly in regard to its degree of wetness. An additional difficulty arises in the case of observations made with long mercury thermometers buried in vertical holes, that the correction for the expansion of the liquid in the long stems is uncertain, and that the holes may serve as channels for percolation, and thus lead to exceptionally high values. The last error is best avoided by employing platinum thermometers buried horizontally. In any case results deduced from the annual wave must be expected to vary in different years according to the distribution of the rainfall, as the values represent averages depending chiefly on the diffusion of heat by percolating water. For this reason observations at different depths in the same locality often give very concordant results for the same period, as the total percolation and the average rate are necessarily nearly the same for the various strata, although