Sabine's FormulaProf. Wallace C. Sabine (1868 - 1919) of Harvard University investigated architectural acoustics scientifically, particularly with reference to reverberation time. He deduced experimentally, that the reverberation time is:
$$T α \frac{V}{Σ aA}$$
Where, V is the volume of the hall, a is the absorption coefficient of an area A. If the volume is measured in cubic feet and area in square feet, then the experimentally obtained value of the constant of proportionality, according to Sabine is 0.05. Then,
$$T = \frac{0.05V}{Σ aA}$$
If there are different absorbing surfaces of area $A_1, A_2, A_3, A_4$ etc., having absorption coefficients $a_1, a_2, a_3, a_4$ etc., then,
$$T = \frac{0.05V}{a_1A_1 + a_2A_2 + a_3A_3 + a_4A_4 + ..}$$
If the area is measured in square meters and the volume in cubic meters, then Sabine's formula can be written as:
$$T = \frac{0.16V}{ΣaA}$$
Increasing the effective area of complete absorption like, changing the wall materials or adding more furniture may decrease an excessive reverberation time for a hall. But this also decreases the intensity of a steady tone. Also, too much absorption will make the reverberation time too short and cause the room to sound acoustically 'dead'. Hence, the optimum reverberation time is a compromise between clarity of sound and its intensity.
The absorption coefficient determines how far into a material light of a particular wavelength can penetrate before it is absorbed. In a material with a low absorption coefficient, light is only poorly absorbed, and if the material is thin enough, it will appear transparent to that wavelength. The absorption coefficient depends on the material and also on the wavelength of light which is being absorbed. Semiconductor materials have a sharp edge in their absorption coefficient, since light which has energy below the band gap does not have sufficient energy to excite an electron into the conduction band from the valence band. Consequently this light is not absorbed. The absorption coefficient for several semiconductor materials is shown below.
The above graph shows that even for those photons which have an energy above the band gap, the absorption coefficient is not constant, but still depends strongly on wavelength. The probability of absorbing a photon depends on the likelihood of having a photon and an electron interact in such a way as to move from one energy band to another. For photons which have an energy very close to that of the band gap, the absorption is relatively low since only those electrons directly at the valence band edge can interact with the photon to cause absorption. As the photon energy increases, not just the electrons already having energy close to that of the band gap can interact with the photon. Therefore, a larger number of electrons can interact with the photon and result in the photon being absorbed.
The absorption coefficient, α, is related to the extinction coefficient, k, by the following formula:
$$α = \frac{4πk}{λ}$$
Where λ is the wavelength. If λ is in nm, multiply by $10^7$ to get the absorption coefficient in the units of $cm^{-1}$.
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