diminished, so that at a depth of only 4 in. it was already reduced
to about 6° F., and to less than 2° at 10 in. At the same time,
the epoch of maximum or minimum was retarded, about 4 hours
at 4 in., and nearly 12 hours at 10 in., where the maximum
temperature was reached between 1 and 2 a.m. The form of
the wave was also changed. At 4 in. the rise was steeper than
the fall, at 10 in. 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.
Fig. 4.
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—
| θ−θ0 = A sin 2π𝑛𝑡 = A sin 2π𝑡/T, | (5) |
where θ0 is the mean temperature of the surface, A the amplitude of the cycle, 𝑛 the frequency, and T the period. In this simple case the temperature cycle at a depth 𝑥 is a precisely similar curve of the same period, but with the amplitude reduced in the proportion 𝑒𝑚𝑥, and the phase retarded by the fraction 𝑚𝑥/2π of a cycle. The index-coefficient 𝑚 is √(π𝑛𝑐/𝑘). The wave at a depth 𝑥 is represented analytically by the equation
| θ−θ0 = A𝑒−𝑚𝑥 sin (2π𝑛𝑡−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. 5 are drawn for a particular 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 θ =±𝑒−𝑚𝑥, 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 in., at which depth the amplitude of the variation is reduced to less than one five-hundredth part (𝑒−2π) of that at the surface, so that for all practical purposes the oscillation may be neglected beyond one wave-length At half a wave-length the amplitude is only 123rd of that at the surface. The wave-length in any case is 2π/𝑚.
The diffusivity can be deduced from observations at different depths 𝑥′ and 𝑥″, by observing the ratio of the amplitudes, which is 𝑒𝑚(𝑥′−𝑥″) for a simple-harmonic wave. The values obtained in this way for waves having a period of one second and a wave-length of half an inch agreed very well with those obtained in the same cast-iron by Ångström’s method (see below), with waves having a period of 1 hour and a length of 30 in. 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.
Speed, 42 revolutions per minute; range, 20° at surface.
Fig. 5.
15. Annual Variation.—A similar method has frequently been applied to the study of variations of soil-temperatures 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 the actual degree of wetness of each may vary considerably. The following are a few typical values for sand or gravel deduced from the annual wave in different localities:—
Table II.—Diffusivity of Sandy Soils. C.G.S. Units.
| Observer. | Soil. | Locality. | Thermo- meter. | Diffusivity. | ||
| Kelvin, 1860 | Garden sand | Edinburgh | Mercury | ·0087 | ||
| Neumann, 1863 | Sandy loam | .. | ” | ·0136 | ||
| Everett, 1860. | Gravel | Greenwich | ” | ·0125 | ||
| Ångström, 1861 | Sandy clay | Upsala | ” | ·0057 | ||
| ” | ” | ” | ” | ·0045 | ||
| Ångström | Coarse sand | ” | ” | ·0094 | ||
| Rudberg | The same soil, place and instruments | ·0061 | ||||
| Quetelet | reduced for different years | ·0074 | ||||
| Callendar, 1895 | Garden sand | Montreal | Platinum | ·0036 | ||
| Rambaut, 1900 | Gravel | Oxford | ” | ·0074 | ||
The low value at Montreal is chiefly due to the absence of percolation during the winter. A. A. Rambaut’s results were obtained with similar instruments similarly located, but he did not investigate the seasonal variations of diffusivity, or the effect of percolation. It is probable that the coarser soils, permitting more rapid percolation, would generally give higher results. In any case, it is evident that the transmission of heat by percolation would be much greater in porous soils and in the upper layers of the earth’s crust than in the lower strata or in solid rocks. It is probable for this reason that the average conductivity of the earth’s crust, as deduced from surface observations, is too large; and that estimates of the age of the earth based on such measurements are too low, and require to be raised; they would thereby be brought into better agreement with the conclusions of geologists derived from other lines of argument.
16. Ångström’s Method consists in observing the propagation of heat waves in a bar, and is probably the most accurate method for
