Handbook for Sound Engineers

Sound System Design 1253

where,

x is any distance,

other terms are as before.

Example:
Let
Do = 128 ft,
EAD = 4 ft,
Q = 5,
S = 28,000 ft^2 ,
a = 0.35.

then,

and,

These two equations are mathematically equivalent to
Eqs. 34-21 and 34-22 and will produce the same answers
given the same data. Eq. 34-21 and 34-22 may be easier
to understand and to insert in a computer program.
In addition, it should be noted that some users prefer
to place the NOM (number of open microphones) and
6 dB feedback stability margin (FSM) terms in the NAG
equation rather than in the PAG equation. This author
believes that they belong in the PAG equation since
including them produces a value of PAG more nearly
equal to that which will be measured in the installed
system. While PAG and NAG values will differ with
placement of the two terms, the PAG – NAG value
(which is the most important result) will be the same
regardless of the placement of the two terms.
Also, as before, if PAG is greater than or equal to
NAG, it’s reasonable to assume that the system will be
stable and not feed back.
In Eq. 34-21, the terms Ds, D 1 , D 2 , Do, and NOM are
as explained in the simplified (outdoor) system. In Eq.
34-22, use the simplified system estimate for EAD Eq.
34-7, ignoring the effects of the reverberant field. This
puts the estimate on the safe side for the NAG calcula-
tion. The equations for PAG and NAG are similar to the
equations given for the simplified system, Eqs. 34-5 and
34-8, except for the 10log[ ] terms that can be interpreted
as modifications caused by the room reverberant field.

34.2.3.4.6 The Effect of Directional Microphones and

Loudspeakers

In a reverberant room, the effect of directional micro-

phones is less significant than in the outdoor case. The

reason is that the amount of reverberant sound energy

picked up by the microphone depends very little on the

microphone’s pickup pattern (a cardioid microphone

picks up more reverberant sound from the front, which

compensates for its reduced rear pickup). A directional

microphone will, however, exhibit higher gain in the

direction of the talker which has the same effect on

feedback as a reduction in Ds. This improvement may

be as much as 2 or 3 dB and it is not included in any of

the PAG or NAG equations.

Directional loudspeakers may reduce the amount of

direct sound energy reaching the microphone but do not

substantially reduce the amount of reverberant sound

reaching the microphone, since this is usually domi-

nated by the nondirectional low-frequency loud-

speakers. Note that the indoor PAG equation Eq. 34-21

already includes the effect of directional loudspeakers

on the reverberant field. Thus, it is best to assume that

no additional gain before feedback will be provided by

directional loudspeakers.

34.2.3.5 Validity of the Model in a Geometrically

Complex Room

In effect, the equations just presented form a mathemati-

cal model of the interactions between a room and a

sound system. The question arises, “Just how valid is

this model?” The answer is to remember that the model

assumes a well-developed, statistically random reverber-

ant field in a room with simple geometry. Thus the

model can be very accurate in a room like a high-school

gymnasium or a rectangular church. Add balconies,

transepts, or other complexities, and the equations, while

still useful, cannot adequately describe the entire room.

One way to deal with more complex rooms is to treat

them as two or more acoustically separate spaces. A

large stage with hardwood floors and reflecting walls

and ceiling, for example, may be coupled to an audience

seating area with padded seats, carpeting, and draped

walls. A reverberant cathedral may have an

under-balcony area that is very different acoustically

from the main room. System design and the use of the

equations will be improved by treating these different

spaces as entirely different rooms that just happen to

share a common boundary (an imaginary wall).

An equation for the overall combined reverberation

time in such a dual-space room is

'x 10 Q

4 Sx^2

------------^4

Sâ

–= log©¹§·----- -+

'Do=33.6

'EAD 16=

NAG 33.6+= 16.0

=17.7 dB