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Introduction

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 [1] 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 latest piece of technology, or thousands of heavy metal fans gather to see Metallica [2] 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.

First, the examination of the elementary piston 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 [3]
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

It is desirable to keep the Helmholtz number to a minimum, as this results in a uniform sound radiation. A large Helmholtz number results in a very narrow 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. This can be modelled by treating each loudspeaker 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.

Dealing with 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 is known as acoustical feedback[4]. This phenomenom 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 clearly 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 the amplitude of the signal outputted to the listener, $Y(\omega )$.

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

As one observes the closed loop transfer function defined by $O(\omega )$, the poles of the function for which the denominator vanishes are of immediate interest. Thus, if the value of 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 horrific results. Similarly, if the open loop gain is much smaller than unity, the value of the closed loop transfer function becomes increasingly large as the value of $KG(\omega )$ tends to unity. 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. Thus, 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:

$20log({\frac {K}{K_{o}}})<-12dB$, where K_o is the critical amplifier gain value for which $O(\omega )$ explodes 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 incredibly weak. A wiser strategy in reducing the effects of acoustical feedback involves in attempting to reduce G(&omega), while increasing G'(&omega). Although this may prove to be a tricky feat, 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 [5] are commonly used as they effectively reject sounds propagating from other directions other than those originating from the source (i.e from the speaker.)

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, one must ensure that an adequate speech intelligibility, which is measured in accordance with the speech transmission index [6] is achieved.

Loudspeaker location must be chosen in such a way that the audience is supplied as large as a uniform direct sound 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.

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 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 radiating the 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 is the reverberation time

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, yields:

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

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 ${\frac {log_{e}({\frac {10^{\mathrm {-6} }(1+m)}{m}})}{13.8}})$

Therefore, an increase in sound intensity by 1 dB will increase the reverberation time by 88%, ensuring that further increases in sound intensity will not have much of an effect on the reverberation time.

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/R². 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 m3
T is the reverberation time in seconds 

At the microphone, the magnitude of the sound intensity, I_o becomes ${\frac {AI_{o}}{R_{o}^{2}}}$, so that A = R_o². 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

To cap off this section, 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 [[7]] 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 [8] 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:

$\phi (t)=ksin(w_{m}t)$

makes the system non linear, making the analysis more complex.

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.