The Physic Concepts in the Designing of Concert Halls

A very important but little known acoustical phenomena is the Inverse Square Law. As a sound wave propagates spherically, the sound energy is distributed over the ever-increasing surface diameter of the wave front surface. So, when the distance from the source is doubled, the energy carried by the sound is spread over double the distance in all directions, or four times the original area. This means that the intensity will be reduced to a quarter of the original distance. For example, at three times the original distance from a particular source, the intensity will be reduced to one-ninth of that at the starting point. This is refered to as an inverse square law. The sound intensity from a point source of sound will obey the inverse square law if there are no reflections or reverberation.

Reverberation is the collection of reflected sounds from the surfaces in an enclosure like an auditorium. It is a desirable property of auditoriums to the extent that it helps to overcome the inverse square law dropoff of sound intensity in the enclosure. However, too much reverberation makes the sounds run together with loss of articulation - the sound becomes muddy and garbled. A measure that is used to characterize the reverberation in a room is the reverberation time. Technically speaking, the reverb time is the amount of time it takes for sound pressure level or intensity to decay to 1/1,000,000th (60 dB) of it's original value (or 1/1000th of it's original amplitude.) Longer reverberation times mean that the sound energy stays in the room longer before being absorbed. Reverberation time is associated with what we sometimes call the 'size' of the room. Concert halls have reverberation times of about 1.5 to 2 seconds. The reverberation time is controlled primarily by two factors - the surfaces in the room, and the size of the room. The surfaces of the room determine how much energy is lost in each reflection. Highly reflective materials, such as a concrete or tile floor, brick walls, and windows, will increase the reverb time as they are very rigid. Absorptive materials, such as curtains, heavy carpet, and people, reduce the reverberation time (and the absorptivity of most materials usually varies with frequency). Bigger rooms tend to have longer reverberation times since, on the average, the sound waves travel a longer distance between reflections. The air in the room itself will also weaken the sound waves, reducing the reverberation time. This weakening varies with the humidity and temperature, and high frequencies are affected most. The optimum reverberation time for an auditorium or room of course depends upon its intended use. Around 2 seconds is desirable for a medium-sized, general purpose auditorium that is to be used for both speech and music. A classroom should be much shorter, less than a second, so that the students can hear the teacher or lecturer clearly. A recording studio should minimize reverberation time in most cases for clarity of recording. Some examples of concert halls with suitable reverbaration time are Vienna, Musikvereinsaal (2.05 seconds), Boston, Symphony Hall (1.8 seconds) and New York, Carnegie Hall (1.7 seconds)

The reverberation time is strongly influenced by the absorption coefficients of the surfaces, but it also depends upon the volume of the room as shown in the Sabine formula. You won't get a long reverberation time with a small room.
(a + x)T = kV where V volume of the room
T time of reverberation a initial absorbtion of the room and x the amount added by cushions and other absorbents

When a sound wave in a room strikes a surface, a certain fraction of it is absorbed, and a certain amount is transmitted into the surface. Both of these amounts are lost from the room, and the fractional loss is characterized by an absorption coefficient a which can take values between 0 and 1, 1 being a perfect absorber.The effective absorbing area is a factor in determining the reverberation time of an auditorium. The absorption coefficient of a surface typically changes with frequency, so the reverberation time is likewise frequency dependent.The higher the absorption coefficient, the longer the reverberation time. However, the Sabine formula neglects air absorption, which can be significant for large auditoriums.