散乱媒質中の輻射の減衰(積雪の輻射に対する性質の研究1)

抄録

In a medium which consists of numerous small particles of transparent substance, like snow, fog and cloud, the intensity of radiation, passing through it, decreases. However, there is a question, if the rate of decrease obeys an exponential law, as was expected by many investigators. In this regard, Dietzius, in 1922, obtained equations: solving Schuster's equations and tried to discuss the decrease of brightness in fog. In the above equation, A₀ means intensity of incident radiation at x=0 and A intensity of radiation advancing to x direction at the point x and B that of radiation returning backwards by the effect of diffuse reflexion of the part of the medium ahead. And, 2_??_ means a coefficient of diffuse reflexion of the medium, h its thickness, and μ the albedo at the base (x=h).
The present writer investigated the value of γ and β, when the medium is not homogeneous, that is,
(1) when
(2) when the medium consists of two layers, whose coefficients are _??_ and _??_ respectively.
In the case (1), the decrease of the radiation is no more linear, but parabolic.
In the case (2), we can show that the relations
hold, where A', B' are the proceeding and backing radiation in the second medium, whose coefficient is _??_; A'_h'_??_ advancing radiation at the boundary of two layers; h' the thickness of the first medium; and the other notations are same as before. The above equations contain only _??_, and independent of _??_, so they fit _??_or computation of _??_ from experimental data. After we have got _??_ by use of _??_e above relation, then we can use the following relations for the first medium, where μ^* means
The relations (4) may enable us to calculate the value of _??_ This method will be also applied to the medium which consists of more than two layers and will be extended further to the medium whose coefficient changes continuously, if we divide the whole layer into many thin strata and carry out the above procedure for each stratum.
Returning to the case of the uniform medium, the radiation, which reaches the base, is obtained, put ng x=hin the equation (1), viz.
This means, _??__h depends on μ extraordinarily, and in the extreme case, μ=1, _??__h becomes 1. In the other words, if we place a mirror at the base of a fog stratum, the incident ray at the top, after passing through it, reaches the mirror without any drop in its strength. This is far beyond our imagination based upon experience. Such a misleading conclusions may come from the unnatural assumption of no absorptive power in the medium.
Let us take absorption into account, and put From this equation we get, under the same boundary condition for the uniform medium, where At x=0, we have and at x=h, we have
When _??_ is large enough, we have for smaller value of x, that is, at the portion near the surface; for the value of x not considerably different from h that is, at the portion near the base plane.
These equations (11), (12) teach us the-following facts in the case of fog, for example. If the fog layer is sufficiently thick, the decrease at the top portion obeys exponential law, while at the base (that is at the ground), the intensity changes linearly.
From the equation (11), we get
We must take special care in treating with the surface condition, as will be shown in another article. However, we may think that β₀ is an approximate value of albedo of the fog (and also, of the cloud). As we see, β₀ and accordingly the albedo does not depend on absolute values of k and _??_, but on, their ratio k/_??_ Also, we have to remark, that β₀ has no dependence upon μ the albedo of the base, against Dietzius' solution (1).
It is also interesting to consider about the condition at the base.