# Engineering Acoustics/Electro Acoustic Installations in Room

As the development of new and sophisticated technology is advancing at
an exponential rate, it has become quite evident that the study of room
acoustics would be rather incomplete without the analysis of the effect
of electroacoustic devices. Whether the Canadian Prime Minister is
addressing the members of parliament, a university professor is giving a
lecture in a large auditorium, the CEO of one of the largest tech
companies holds a press conference to unveil their company's technology,
or thousands of heavy metal fans gather to see Metallica perform live,
loudspeakers and microphones are frequently used as a means of speech
amplification. This addition to the Engineering Acoustics Wikibook will
aim to aid the reader in strategic placement of electroacoustic devices
in any room. It will also serve as an extension to the previous Wikibook
entry, entitled Room Acoustics and Concert Halls.

# Loudspeaker Directivity

Although, a common sound amplification setup consists of a microphone,
amplifier and a loudspeaker, the loudspeaker is perhaps the crucial
component of the arrangement. Loudspeakers must be designed in order to
withstand high power while not causing any distortions in the radiated
sound.\[Kuttruff\]

First, a description of the radiation characteristics of a loudspeaker
is of interest. !Motion of piston creating soundwaves at an angle theta
from the
horizontal
When waves having high frequencies begin propagating due to the motion
of the piston, the radius of the piston can no longer be considered
small relative to the wavelength of the radiated sound. Thus, a
significant phase difference occurs between the radiated wave motion and
the motion of the piston. This leads to various degrees of interference
between both elements, which may lead to total cancellation of all
sound. Thus, it is useful to define a directivity function as follows:

$\delta(\theta)=\frac{2J_1(kasin(\theta))}{kasin(\theta)}$

`J1 is the first order Bessel Function`\
`θ is the angle of radiation`\
`k is the wave number, which is the ratio of angular frequency to the speed of sound `\
`a is the radius of the piston`\
`ka is referred to as the Helmholtz number`

A low Helmholtz number to a minimum, as this leads to a uniform sound
directivity. A large Helmholtz number results in a very directive sound
radiation as shown in the figures below.

Next, let us consider the horn loudspeaker, which provides a more
practical model of the loudspeaker. The main advantage of the horn model
is that it provides a broader directivity than the narrow scope provided
by the piston loudspeaker. It is often desirable to combine several horn
loudspeakers. The horn loudspeaker increases radiation resistance. Since
the directional characteristics of horn loudspeakers depend on both the
size and shape of the opening, as well as on the entire shape of the
horn, an expression for the directivity becomes complex.

Multiple loudspeakers will often be found in a room. Each loudspeaker
may be modeled as a point source. A directivity function for the
ensemble may be defined similarly as for a single piston loudspeaker.

$\delta_a (\theta)=\frac{sin(0.5*Nkdsin(\theta))}{Nsin(0.5kdsin(\theta))}$

Here, the angle of radiation is taken along the normal direction of the
array containing N point sources, spaced along a straight line at an
equal distance, d, from each other.

# Acoustical Feedback

If both the loudspeakers and microphone of an electroacoustic setup are
in the same room, the microphone will pick up sound propagating from the
loudspeakers as well as from the original source. This occurrence leads
to a phenomenon known as acoustical feedback
1. This phenomenon often
leads to a disruption in the entire electroacoustic setup and one may
hear loud howling or whistling sounds. As already discussed, loudspeaker
positioning plays a vital role in the minimization of acoustic feedback.
In order to effectively analyze the effects of acoustical feedback in a
room, it is wise to represent the propagation of sound waves from a
source such as a loudspeaker by means of a block diagram. !Block
Diagram Representation of Acoustical feedback in a room of sound
radiating from a
source

$\frac{Y(\omega)}{S(\omega)}=O(\omega)=\frac{KG'(\omega)}{1-G(\omega)K(\omega)}$

$Y(\omega)$` is the complex amplitude spectrum of the signal at the listener's seat`\
$S(\omega)$` is the sound input (into a microphone)`\
$K$` is the amplifier gain`\
$G'(\omega)$` is a complex transfer function representing the path by which the sound will reach the listener `\
$G(\omega)$` is the transmission function representing the path taken by sound radiating from the loudspeaker back into the microphone`

The closed loop transfer function at right shows how acoustical feedback
is caused by the original source signal passing continuously through the
loop, ending up back at the microphone. The term in the denominator of
complex transfer function, $KG(\omega)$, represents the open-loop gain
of the system. The magnitude of this quantity greatly impacts the
amplitude of the signal outputted to the listener, $Y(\omega)$. This
outputted variable represents the sound waves that actually propagate
toward the listener while the rest enter the feedback loop where they
make there way back to the microphone.

!Open Loop Block Diagram of Sound travelling through a room. This
diagram idealizes the actual closed loop system by neglecting the
presence of any acoustical
feedback

By observing the closed loop transfer function defined by $O(\omega)$,
one can see that the poles of the function for which the denominator
vanishes are of immediate interest. If the value of the open loop gain
equals unity, the entire system becomes unstable. If the open loop gain
equals unity, the closed loop transfer function tends to infinity,
producing a large scale instability. Similarly, if the open loop gain is
than unity, the value of the closed loop transfer function becomes
increasingly large as the value of $KG(\omega)$ tends to one. At such
large values of the open loop transfer function, the sound heard by the
listener will be distorted. This effect is better explained when an
impulsive signal is applied at the source, in which case ringing effects
are heard as the sound waves reach the listener. It is of the highest
importance to allow the system to operate within an appropriate range of
amplifier gain values. Through experimentation (See reference 1,) the
amplifier gain must be set in such a way that the following equations is
satisfied:

$20log(\frac{K}{K_o})<-12 dB$, where K_o is the critical amplifier gain
value for which $O(\omega)$ grows without bound.

Naturally, it is not in one\'s best interest to reduce the amplifier
gain to infinitessimily small values. This would simply render the
amplifier useless and the radiated sound signal would be weak. A wiser
strategy in reducing the effects of acoustical feedback involves in
attempting to reduce G(ω), while increasing G\'(ω). Although this may
prove to be difficult, the meticulous selection of the loudspeaker
directivity, with the main lobe pointing towards the listeners, while
the microphone is positioned away from the main lobes, in a position of
weak radiation, will greatly reduce undesirable feedback effects. Such
unidirectional microphones as Cardioid microphones are commonly used as
they effectively reject sounds propagating from other directions other
than those originating from the source (i.e from the speaker.) The
stability of closed loop feedback systems may also be analysed via such
tools as root locus plots or by using the Routh-Hurwitz criterion. In
traditional control theory, lead or lag compensators are frequently used
to improve the stability of the system
2.

# Loudspeaker Positioning

When deciding on an appropriate location for each loudspeaker in a room,
it is necessary to take into account certain factors. As seen in the
previous section, loudspeaker directivity can be manipulated depending
on the amount of loudspeakers present and if they are arranged linearly
in a group. Regardless, loudspeakers should be placed in such a manner
that a uniform sound energy is supplied to all listeners in the room.
Also, it is necessary to establish an adequate speech intelligibility,
which is measured in accordance with the speech intelligibility index
3.

Loudspeaker location must be chosen in such a way that the audience is
supplied direct sound that is as uniform direct as possible. One way of
achieving this is by mounting the loudspeaker at a higher altitude than
the sound source (i.e. microphone), which also limits feedback. This
also ensures that the direct sound will also arrive to the listener with
a relatively uniform directivity in the horizontal plane. Usually, the
human ear is not sensitive to sound variations in the vertical plane,
thereby making the elevation of loudspeakers a wise chose.

It is also of interest to derive a quantity known as the radius of
reverberation, R~o~. Consider sound leaving the loudspeaker with
intensity, I~o~. This intensity then makes it way through the closed
loop system, arriving back at the microphone input. It is then
re-amplified by a gain A to a value of AI~o~ as it passes through the
amplifier and makes its way back to the source. The intensity near the
loudspeaker will then decay as a function of the radial distance,R, from
the loudspeaker according to 1/R^2^. The intensity will decay until a
distance of R~O~ is reached. This radius of reverberation is then
defined as:

$R_O=0.057sqrt{\frac{V}{T}}$ where

`V is the volume of the room in m`^`3`^\
`T is the reverberation time in seconds`

## Loudspeaker Positioning for the Improvement of the reverberation of a Room using Direct Feedback

Since modern amphitheatres and stadiums serve as multipurpose venues,
controlling the reverberation time of the room to suit a specific event
is of importance. For example, Montreal, Quebec\'s Bell Centre serves as
the home of the Montreal Canadiens hockey team while also hosting a
multitude of live concerts and shows every year. The reverberation time
of such multipurpose venues can be modified with the aide of rotating or
removable walls or ceilings or even thick curtains. Such movable items
can play a role in altering the room\'s absorbtion and reflection
properties. However, the implementation of such devices may prove to be
quite costly. Thus, one\'s attention should be turned to the strategic
placement of loudspeakers in order to increase the reverberation time of
a room. A popular method has been discussed in a 1971 paper by Guelke
and Broadhurst.

In order to increase the reverberation time in a room by means of direct
feedback, it should be noted that the total sound intensity in a room is
the sum of the intensity of the source along with contributions from the
pressure from each of the room modes. Both contributions will decay
exponentially according to the relation:

$I_\mathrm{Tot}=I_\mathrm{Room}e^\mathrm{-kt}+I_\mathrm{sys}e^\mathrm{-Lt}$

$I_\mathrm{Tot}$` is the total sound intensity radiated in the room`\
$I_\mathrm{sys}$` is the sound intensity prom the acoustical system comprising of a microphone, amplifier and loudspeaker`\
$I_\mathrm{Room}$` is the sound intensity due to the pressure modes of the room`\
`k is the room damping coefficient`\
`L is the system damping coefficient`\
$T_\mathrm{60}$` is the reverberation time or more specifically, `\
`the time it takes for the energy density or sound pressure level in the room to decay by 60 dB, once the source is abruptly shut off.`

Furthermore, the system damping coefficient as well as the system sound
intensity may be related to the room\'s damping coefficient and radiated
sound intensity by constants n and m respectively, such that:

$L=nK$

$I_\mathrm{sys}=mI_\mathrm{Room}$

Under initial conditions at time, t=0, the total sound intensity in the
room is:

$I_\mathrm{Tot}=I_\mathrm{room}+mI_\mathrm{room}$

Next, the reverberation time is defined to be the point where the sound
intensity reaches 10^−6^ of its original value. Thus, the above equation
becomes:

$10^\mathrm{-6}(1+m)=e^\mathrm{-kt}+me^\mathrm{-nkt}$

Solving for the reverberation time,$T_\mathrm{60}$, yields:

$T_\mathrm{60}=\frac{-1}{nk}log_{e}(\frac{10^\mathrm{-6}(1+m)}{m})$

If the reverberation time of an isolated room is defined as:

$\frac{log_{e}10^6}{k}=\frac{13.8}{k}$

then the total reverberation time, t, may be written as follows:

$t=\frac{t_\mathrm{room}f(m)}{n}$ where f(m) is
$\frac{log_{e}(\frac{10^\mathrm{-6}(1+m)}{m})}{13.8})$

At the microphone, the magnitude of the sound intensity, I~o~ becomes
$\frac{AI_o}{R_o^2}$, so that A = R~o~^2^. Consequently, the larger the
the radial distance, the larger the average sound intensity in the room.
The directivity factor of the loudspeakers may help to create a more
diffusive sound field. Hence, the radial distance, R, is increased by an
increase in the directivity factor. Also it has been should that delay
times greater than 75 ms produce undesirable whining noises. It is
unwise to increase the reverberation time, T, of such a simple feedback
system beyond 1.5 s as the system very near the point of instability has
at least one frequency with a very long reverberation time.

By taking a value of 75 ms for $\tau$ in the following equation, the
reverberation time, t, in open space is defined by the relation:

$t=\frac{60\tau}{20log_{10}(A)}$, where $\tau = \frac{d}{c}$ c is the
speed of in air, 344 m/s and d is the distance between microphone and
loudspeaker

It is noteworthy to mention the works of M.R. Schroeder presented in his
1964 paper on \"Acoustic-Feedback Stability by Frequency Shifting\" In
essence,in order to avoid any ringing or howling noises of any kind in
the feedback loop, Schroeder suggested a frequency shift of 4 Hz into
the feedback loop. This was shown to greatly improve the stability of
the system. However,the method produces undesirable sidebands Because of
this phenomenon, the method is undesirable for concert halls. To improve
on Schroeder\'s method, Guelke and Broadhurst suggest the introduction
of phase modulation into the system in order to achieve longer
reverberation times and increased system stability.

By definition, if a linear, time invariant (LTI) system, such as the one
described by the closed feedback loop of a microphone, amplifier and
loudspeaker, is unstable, radiated sound will increase exponentially and
without bound at certain frequencies. By the implementation of a phase
reversal switch, sounds at the initially unstable frequency will
stabilize. However, a new set of frequencies that were once stable will
begin to grow without bound. Nonetheless, such a consequence of the
utilization of phase reversal is minor as it may be accounted for by
ensuring that the switch is activated a couple of times a second. This
method allows for the reverberation time to be held constant while
assuring that all strange ringing or howling noises vanish.

Such rapid triggering of the switch does induce another concern in the
continued presence of transient effects. Luckily, such effects become
negligible if the phase is varied sinusoidally.

Ultimately, the modulating frequency must be chosen such that:

`1. The change in phase should be enough to stabilize the system `\
`2. The sidebands produced are so small that they be neglected`

The first criteria may be satisfied if the modulation frequency is set
to a value less than the reciprocal of the delay time, τ. The second
criterion may be satisfied by ensuring the the modulation frequency is
below one hertz. A modulation frequency of 1 Hz is so negligible that it
will not be picked up by the human ear. Briefly, based on the definition
of τ increase the distance between microphone and loudspeaker to achieve
a minimal modulation frequency, which will stabilize the system and be
practically non-existent to the human ear.

For further reading on ensuring that the phase modulation is sinusoidal,
it is advised to consult reference \[2\] or a textbook on control
systems, if desired. Introducing a sinusoidal phase modulator of the
form:

$\phi(t)=ksin(w_mt)$

Introducing a modulator that is purely sinusoidal in nature greatly
simplifies the usually non-linear nature of a real modulator, which
greatly simplifies the problem at hand.

# References

\[1\] Kuttruff, Heinrich; *Room Acoustics, Fourth Edition*, Spon Press
Publishing, 342 pgs

\[2\] Guelke, R. W. and A. D. Broadhurst (1971). \"Reverberation Time
Control by Direct Feedback.\" Acta Acustica united with Acustica 24(1):
33-41.

\[3\] Schroeder, M. R. (1964). \"Improvement of Acoustic‐Feedback
Stability by Frequency Shifting.\" The Journal of the Acoustical Society
of America 36(9): 1718-1724.

