Mean Absorption and Equivalent Absorption Coefficient of a Band Spectrum

The note deals with the absorption of a band in which the lines have the dispersion form; the effect of the overlapping of neighboring lines upon the absorption is taken into account. The analytical problem can be solved for a periodic pattern of lines with equal intensity $S$, width $2\alpha$ and distance $d$ from each other. Simple expressions are derived for the limiting cases of small $\frac{\alpha }{d}$ (formula (8)) and large $\frac{\alpha }{d}$. In the latter case the transmission by a layer of thickness $x$ is given by $I=I_0{e}^{-\left(\frac{\mathrm{Sx}}{d}\right)tanh\left(\frac{2\pi \alpha }{d}\right)}.$ It appears that this formula gives a fairly good mean approximation even for small values of $\frac{\alpha }{d}$, while the true transmission in this case is not exponential. It is shown that in problems of radiative transfer and radiative equilibria this approximate formula for the transmission can be used; i.e. the factor of $x$ in the above exponent may be substituted in the equations of transfer as equivalent of a mean absorption coefficient.