t = 0.164 V ÷ a (1)
For good acoustical conditions, that is, for a short time of reverberation, the volume V should be small and the absorbing materials, represented by a, large. This is the case in a small room with plenty of curtains and rugs and furniture. If, however, the volume of the room is great, as in the case of an auditorium, and the amount of absorbing materials small, a troublesome reverberation will result.
Professor Sabine determined the absorbing powers of a number of different materials. Calling an open window a perfect absorber of sound, the results obtained may be written approximately as follows:
| One square meter of open window space | 1.000 |
| One square meter of glass, plaster, or brick | .025 |
| One square meter of heavy rugs, curtains, etc. | .25 |
| One square meter of hair felt, 1 inch thick | .75 |
| One square meter of audience | .96 |
These values, together with the formula, allow a calculation to be made in advance of construction for the time of reverberation. This pioneer work cleared the subject of architectural acoustics from the fog of mystery that hung over it and allowed the essential principles to be seen in the light of scientific experiment.
In a later investigation[6] Sabine showed that the reverberation depended also on the pitch of sound. As a concrete example, the high notes of a violin might be less reverberant with a large audience than the lower tones of the bass viol, although both might have the same reverberation in the room with no audience. Again, the voice of a man with notes of low pitch might give satisfactory results in an auditorium while the voice of a woman with higher pitched notes would be unsatisfactory.
These considerations show that the acoustics in an auditorium vary with other factors than the volume of the room and the amount of absorbing material present. The audience may be large or small, the speaker’s voice high or low, the entertainment a musical number or an address. The best arrangement for good acoustics is then a compromise where the average conditions are satisfied. The solution offered by Professor Sabine is such an average one, and has proved satisfactory in practice.
The problem of architectural acoustics has been attacked experimentally by other workers. Stewart[7] proposed a cure for the poor acoustical conditions in the Sibley Auditorium at Cornell University. His experiments confirmed the work of Sabine. Marage[8], after investigating the properties of six halls in Paris, approved Sabine’s results and advocated a time of reverberation of from ½ to 1 second for the case of speech.
Formulae for Reverberation of Sound in a Room.—On the theoretical side, Sabine’s formula has been developed by Franklin,[9] who obtained the relation t = 0.1625 V ÷ a, an interesting confirmation, since Sabine’s experimental value for the constant was 0.164.
A later development has been given by Jäger,[10] who assumes for a room whose dimensions are not greater than about 60 feet, that the sound, after filling the room, passes equally in all directions through any point, and that the average energy is the same in different parts of the room. By using the theory of probability and considering that a beam of sound in any direction may be likened to a particle with a definite velocity, he was able to deduce Sabine’s formula and write down the factors that enter into the constants. Applying his results to the case of reflection of sound from a wall, he showed that sound would be reflected in greater volume when the mass of the wall was increased and the pitch of the sound made higher. He showed also that when sound impinges on a porous wall, more energy is absorbed when the pitch of the sound is high than when it is low, since the vibrations of the air are more frequent, and more friction is introduced in the interstices of the material.