5.20 Reverberation
165
5.20 Reverberation
‘Reverberation’ is the collection of residual vibrations within a bound system due to
wave pulse reflections which continue for some time even after a localized source
of power has stopped. Enclosures, such as bathrooms, orchestra halls, and voids
in the gut, reverberate from sound reflected back from the enclosing boundaries.
The amplitude of the wall vibrations due to impinging sound waves depends on
the stiffness of the wall, its inertia, and internal damping. In response, the vibrating
surfaces produce a scattered wave, with a fraction of the sound energy lost by being
conducted into the wall or converted to heat.
The time for the reverberation level to drop by 60 dB is called the ‘reverberation
time’. This figure for drop is used because the crescendo of a typical orchestra is
about 100 dB, while the background hall noise is about 40 dB. At 500 Hz, the Boston
Symphony Hall has a reverberation time of 1.8 s; the Vienna Musikvereinsaal, 2.1 s;
the Royal Albert Hall, 2.8 s.
In 1895, Wallace Clement Sabine, the ‘father’ of modern architectural acoustics,
showed 26 that the reverberation time for an enclosed room is given by
T R = (0.161 s/m)
V
α i A i
,
(5.51)
where V is the volume of the room, α i is the sound absorptivity of a surface in the
room and A i is its area. The constant 0.161 comes from 60/(10 log 10 {e})(4/v) with
v being the speed of sound.
The argument goes like this: Suppose a room is sufficiently large that standing
waves do not develop in the room before the sound from a localized source
diminishes by absorptive loss. Assume a pulse of sound with energy E o is generated
and then left to bounce around in a room. If the walls have sound absorptivity α, at
each encounter with a wall a fraction 1 − α of the sound energy E is returned to
the pulse. After N such encounters, the energy has dropped to E o (1 − α) N . The
time between pulse hits will be L/v, where L is the average distance between
the walls and v is the speed of the sound pulse, so in a time t there will be
N = vt/L such encounters. After a time t, the pulse has sound energy E =
E o (1 − α) vt/L = E o exp {(vt/L) ln e (1 − α)}. One can show that the geometric
average distance L between the walls of a box of volume V and total wall surface
A is 4V /A. For α << 1, ln e (1 − α) ≈ −α, giving E = E o exp {−(vαA/(4V ))t}.
With a variety of absorbing surfaces, we define an average sound absorptivity by
¯
α =
α i A i /A.
The decrease in the sound level in a time t is then
β = 10 log 10 (e
−
v ¯
αA
4V t ) = 10
−v ¯
αA
4V
t
log 10 (e)
26 The Collected Papers on Acoustics by Wallace Clement Sabine, [Harvard University Press]
(1922).
165
5.20 Reverberation
‘Reverberation’ is the collection of residual vibrations within a bound system due to
wave pulse reflections which continue for some time even after a localized source
of power has stopped. Enclosures, such as bathrooms, orchestra halls, and voids
in the gut, reverberate from sound reflected back from the enclosing boundaries.
The amplitude of the wall vibrations due to impinging sound waves depends on
the stiffness of the wall, its inertia, and internal damping. In response, the vibrating
surfaces produce a scattered wave, with a fraction of the sound energy lost by being
conducted into the wall or converted to heat.
The time for the reverberation level to drop by 60 dB is called the ‘reverberation
time’. This figure for drop is used because the crescendo of a typical orchestra is
about 100 dB, while the background hall noise is about 40 dB. At 500 Hz, the Boston
Symphony Hall has a reverberation time of 1.8 s; the Vienna Musikvereinsaal, 2.1 s;
the Royal Albert Hall, 2.8 s.
In 1895, Wallace Clement Sabine, the ‘father’ of modern architectural acoustics,
showed 26 that the reverberation time for an enclosed room is given by
T R = (0.161 s/m)
V
α i A i
,
(5.51)
where V is the volume of the room, α i is the sound absorptivity of a surface in the
room and A i is its area. The constant 0.161 comes from 60/(10 log 10 {e})(4/v) with
v being the speed of sound.
The argument goes like this: Suppose a room is sufficiently large that standing
waves do not develop in the room before the sound from a localized source
diminishes by absorptive loss. Assume a pulse of sound with energy E o is generated
and then left to bounce around in a room. If the walls have sound absorptivity α, at
each encounter with a wall a fraction 1 − α of the sound energy E is returned to
the pulse. After N such encounters, the energy has dropped to E o (1 − α) N . The
time between pulse hits will be L/v, where L is the average distance between
the walls and v is the speed of the sound pulse, so in a time t there will be
N = vt/L such encounters. After a time t, the pulse has sound energy E =
E o (1 − α) vt/L = E o exp {(vt/L) ln e (1 − α)}. One can show that the geometric
average distance L between the walls of a box of volume V and total wall surface
A is 4V /A. For α << 1, ln e (1 − α) ≈ −α, giving E = E o exp {−(vαA/(4V ))t}.
With a variety of absorbing surfaces, we define an average sound absorptivity by
¯
α =
α i A i /A.
The decrease in the sound level in a time t is then
β = 10 log 10 (e
−
v ¯
αA
4V t ) = 10
−v ¯
αA
4V
t
log 10 (e)
26 The Collected Papers on Acoustics by Wallace Clement Sabine, [Harvard University Press]
(1922).