Audible effect of comb filters shown in Figure 1-5
Subjectively, the effect of such a comb filter is not particularly noticeable on normal program material as long as several peaks and dips occur within each one-third octave band. See Figure 1-6. Actually, the controlling factor is the critical bandwidth. In general, amplitude variations that occur within a critical band will not be noticed as such. Rather, the ear will respond to the signal power contained within that band. For practical work in sound system design and architectural acoustics, we can assume that the critical bandwidth of the human ear is very nearly one-third octave wide. In houses of worship, the system should be suspended high overhead and centered. In spaces which do not have considerable height, there is a strong temptation to use two loudspeakers, one on either side of the platform, feeding both the same program. We do not recommend this.

Diffraction of Sound

Diffraction refers to the bending of sound waves as they move around obstacles. When sound strikes a hard, non-porous obstacle, it may be reflected or
diffracted, depending on the size of the obstacle relative to the wavelength. If the obstacle is large compared to the wavelength, it acts as an effective barrier, reflecting most of the sound and casting a substantial shadow behind the object. On the other hand, if it is small compared with the wavelength, sound simply bends around it as if it were not there. This is shown in Figure 1-7. An interesting example of sound diffraction occurs when hard, perforated material is placed in the path of sound waves. So far as sound is concerned, such material does not consist of a solid barrier interrupted by perforations, but rather as an open area obstructed by a number of small individual objects. At frequencies whose wavelengths are small compared with the spacing between perforations, most of the sound is reflected. At these frequencies, the percentage of sound traveling through the openings is essentially proportional to the ratio between open and closed areas. At lower frequencies (those whose wavelengths are large compared with the spacing between perforations), most of the sound passes through the openings, even though they may account only for 20 or 30 percent of the total area.
Figure 1-7. Diffraction of sound around obstacles
Effects of Temperature Gradients on Sound Propagation
If sound is propagated over large distances out of doors, its behavior may seem erratic. Differences (gradients) in temperature above ground level will affect propagation as shown in Figure 1-8. Refraction of sound refers to its changing direction as its velocity increases slightly with elevated temperatures. At Figure 1-8A, we observe a situation which often occurs at nightfall, when the ground is still warm. The case shown at B may occur in the morning, and its skipping characteristic may give rise to hot spots and dead spots in the listening area.

Effects of Wind Velocity and Gradients on Sound Propagation
Figure 1-9 shows the effect wind velocity gradients on sound propagation. The actual velocity of sound in this case is the velocity of sound in still air plus the velocity of the wind itself. Figure 1-10 shows the effect of a cross breeze on the apparent direction of a sound source. The effects shown in these two figures may be evident at large rock concerts, where the distances covered may be in the 200 - 300 m (600 - 900 ft) range.
Figure 1-8. Effects of temperature gradients on sound propagation
Figure 1-9. Effect of wind velocity gradients on sound propagation
Effects of Humidity on Sound Propagation
Contrary to what most people believe, there is more sound attenuation in dry air than in damp air. The effect is a complex one, and it is shown in Figure 1-11. Note that the effect is significant only at frequencies above 2 kHz. This means that high frequencies will be attenuated more with distance than low frequencies will be, and that the attenuation will be greatest when the relative humidity is 20 percent or less.
Figure 1-10. Effect of cross breeze on apparent direction of sound
Figure 1-1 1. Absorption of sound in air vs. relative humidity

Chapter 2: The Decibel

Introduction
In all phases of audio technology the decibel is used to express signal levels and level differences in sound pressure, power, voltage, and current. The reason the decibel is such a useful measure is that it enables us to use a comparatively small range of numbers to express large and often unwieldy quantities. The decibel also makes sense from a psychoacoustical point of view in that it relates directly to the effect of most sensory stimuli. signal. The convenience of using decibels is apparent; each of these power ratios can be expressed by the same level, 10 dB. Any 10 dB level difference, regardless of the actual powers involved, will represent a 2-to-1 difference in subjective loudness. We will now expand our power decibel table: P1 (watts) 1.25 1.60 2.5 3.15 6.Level in dB 8 10

Power Relationships

Fundamentally, the bel is defined as the common logarithm of a power ratio: bel = log (P1/P0) For convenience, we use the decibel, which is simply one-tenth bel. Thus:
Level in decibels (dB) = 10 log (P1/P0)
The following tabulation illustrates the usefulness of the concept. Letting P0 = 1 watt: P1 (watts) 10,000 20,000 Level in dB 40 43
This table is worth memorizing. Knowing it, you can almost immediately do mental calculations, arriving at power levels in dB above, or below, one watt. Here are some examples: 1. What power level is represented by 80 watts? First, locate 8 watts in the left column and note that the corresponding level is 9 dB. Then, note that 80 is 10 times 8, giving another 10 dB. Thus: 9 + 10 = 19 dB 2. What power level is represented by 1 milliwatt? 0.1 watt represents a level of minus 10 dB, and 0.01 represents a level 10 dB lower. Finally, 0.001 represents an additional level decrease of 10 dB. Thus:

E E dB level = 10 log 1 = 20 log 1 , and E0 E0 I I dB level = 10 log 1 = 20 log 1 . I0 I0
Sound pressure is analogous to voltage, and levels are given by the equation:
P dB level = 20 log 1 . P0
Figure 2-1. Nomograph for determining power ratios directly in dB
The normal reference level for voltage, E0, is one volt. For sound pressure, the reference is the extremely low value of 20 x 10-6 newtons/m2. This reference pressure corresponds roughly to the minimum audible sound pressure for persons with normal hearing. More commonly, we state pressure in pascals (Pa), where 1 Pa = 1 newton/m2. As a convenient point of reference, note that an rms pressure of 1 pascal corresponds to a sound pressure level of 94 dB. We now present a table useful for determining levels in dB for ratios given in voltage, current, or sound pressure: Voltage, Current or Pressure Ratios 1 1.25 1.2.5 3.5 6.10
If we simply compare input and output voltages, we still get 0 dB as our answer. The voltage gain is in fact unity, or one. Recalling that decibels refer primarily to power ratios, we must take the differing input and output impedances into account and actually compute the input and output powers.
Input power = E= watt Z 600 E= Z 15

Output power =

Level in dB 20
600 Thus, 10 log = 10 log 40 = 16 dB 15
Fortunately, such calculations as the above are not often made. In audio transmission, we keep track of operating levels primarily through voltage level calculations in which the voltage reference value of 0.775 volts has an assigned level of 0 dBu. The value of 0.775 volts is that which is applied to a 600ohm load to produce a power of 1 milliwatt (mW). A power level of 0 dBm corresponds to 1 mW. Stated somewhat differently, level values in dBu and dBm will have the same numerical value only when the load impedance under consideration is 600 ohms. The level difference in dB can be converted back to a voltage, current, or pressure ratio by means of the following equation: Ratio = 10dB/20 For example, find the voltage ratio corresponding to a level difference of 66 dB: voltage ratio = 1066/20 = 103.3 = 2000.
This table may be used exactly the same way as the previous one. Remember, however, that the reference impedance, whether electrical or acoustical, must remain fixed when using these ratios to determine level differences in dB. A few examples are given: 1. Find the level difference in dB between 2 volts and 10 volts. Directly from the table we observe = 14 dB. 2. Find the level difference between 1 volt and 100 volts. A 10-to-1 ratio corresponds to a level difference of 20 dB. Since 1-to-100 represents the product of two such ratios (1-to-10 and 10-to-100), the answer is 20 + 20 = 40 dB. 3. The signal input to an amplifier is 1 volt, and the input impedance is 600 ohms. The output is also 1 volt, and the load impedance is 15 ohms. What is the gain of the amplifier in dB? Watch this one carefully!

6 dB beamwidth limits and when there is minimal radiation outside rated beamwidth will the correlation be good. For many types of radiators, especially those operating at wavelengths large compared with their physical dimensions, Molloys equation will not hold.
A Comparison of Polar Plots, Beamwidth Plots, Directivity Plots, and Isobars
There is no one method of presenting directional data on radiators which is complete in all regards. Polar plots (Figure 3-4A) are normally presented in only the horizontal and vertical planes. A single polar plot covers only a single frequency, or frequency band, and a complete set of polar plots takes up considerable space. Polars are, however, the only method of presentation giving a clear picture of a radiators response outside its normal operating beamwidth. Beamwidth plots of the 6 dB down coverage angles (Figure 3-4B) are very common because considerable information is contained in a single plot. By itself, a plot of Dl or Q conveys information only about the on-axis performance of a radiator (Figure 3-4C). Taken together, horizontal and vertical beamwidth plots and Dl or Q plots convey sufficient information for most sound reinforcement design requirements.
Figure 3-4. Methods of presenting directional information
Isobars have become popular in recent years. They give the angular contours in spherical coordinates about the principal axis along which the response is -3, -6, and -9 dB, relative to the on-axis maximum. It is relatively easy to interpolate visually between adjacent isobars to arrive at a reasonable estimate of relative response over the useful frontal solid radiation angle of the horn. Isobars are useful in advanced computer layout techniques for determining sound coverage over entire seating areas. The normal method of isobar presentation is shown in Figure 3-4D. Still another way to show the directional characteristics of radiators is by means of a family of off-axis frequency response curves, as shown in Figure 3-5. At A, note that the off-axis response curves of the JBL model 2360 Bi-Radial horn run almost parallel to the on-axis response curve. What this means is that a listener seated off the main axis will perceive smooth response when a Bi-Radial constant coverage horn is used. Contrast this with the off-axis response curves of the older (and obsolete) JBL model 2350 radial horn shown at B. If this device is equalized for flat on-axis response, then listeners off-axis will perceive rolled-off HF response.

Directivity of Circular Radiators
Any radiator has little directional control for frequencies whose wavelengths are large compared with the radiating area. Even when the radiating area is large compared to the wavelength, constant pattern control will not result unless the device has been specifically designed to maintain a constant pattern. Nothing demonstrates this better than a simple radiating piston. Figure 3-6 shows the sharpening of on-axis response of a piston mounted in a flat baffle. The wavelength varies over a 24-to-1 range. If the piston were, say a 300 mm (12) loudspeaker, then the wavelength illustrated in the figure would correspond to frequencies spanning the range from about 350 Hz to 8 kHz. Among other things, this illustration points out why full range, single-cone loudspeakers are of little use in sound reinforcement engineering. While the on-axis response can be maintained through equalization, off-axis response falls off drastically above the frequency whose wavelength is about equal to the diameter of the piston. Note that when the diameter equals the wavelength, the radiation pattern is approximately a 90 cone with - 6 dB response at 45.
Figure 3-5. Families of off-axis frequency response curves
The values of DI and Q given in Figure 3-6 are the on-axis values, that is, along the axis of maximum loudspeaker sensitivity. This is almost always the case for published values of Dl and Q. However, values of Dl and Q exist along any axis of the radiator, and they can be determined by inspection of the polar plot. For example, in Figure 3-6, examine the polar plot corresponding to Diameter = l. Here, the on-axis Dl is 10 dB. If we simply move off-axis to a point where the response has dropped 10 dB, then the Dl along that direction will be 10 - 10, or 0 dB, and the Q will be unity. The off-axis angle where the response is 10 dB down is marked on the plot and is at about 55. Normally, we will not be concerned with values of Dl and Q along axes other than the principal one; however, there are certain calculations involving interaction of microphones and loudspeakers where a knowledge of off-axis directivity is essential.

Omnidirectional microphones with circular diaphragms respond to on- and off-axis signals in a manner similar to the data shown in Figure 3-6. Let us assume that a given microphone has a diaphragm about 25 mm (1) in diameter. The frequency corresponding to l/4 is about 3500 Hz, and the response will be quite smooth both on and off axis. However, by the time we reach 13 or 14 kHz, the diameter of the diaphragm is about equal to l, and the Dl of the microphone is about 10 dB. That is, it will be 10 dB more sensitive to sounds arriving on axis than to sounds which are randomly incident to the microphone. Of course, a piston is a very simple radiator or receiver. Horns such as JBLs Bi-Radial series are complex by comparison, and they have been designed to maintain constant HF coverage through attention to wave-guide principles in their design. One thing is certain: no radiator can exhibit much pattern control at frequencies whose wavelengths are much larger than the circumference of the radiating surface.
Figure 3-6. Directional characteristics of a circular-piston source mounted in an infinite baffle as a function of diameter and
The Importance of Flat Power Response
If a radiator exhibits flat power response, then the power it radiates, integrated over all directions, will be constant with frequency. Typical compression drivers inherently have a rolled-off response when measured on a plane wave tube (PWT), as shown in Figure 3-7A. When such a driver is mounted on a typical radial horn such as the JBL model 2350, the on-axis response of the combination will be the sum of the PWT response and the Dl of the horn. Observe at B that the combination is fairly flat on axis and does not need additional equalization. Off-axis response falls off, both vertically and horizontally, and the total power response of the combination will be the same as observed on the PWT; that is, it rolls off above about 3 kHz.
Now, let us mount the same driver on a BiRadial uniform coverage horn, as shown at C. Note that both on-and off-axis response curves are rolled off but run parallel with each other. Since the Dl of the horn is essentially flat, the on-axis response will be virtually the same as the PWT response. At D, we have inserted a HF boost to compensate for the drivers rolled off power response, and the result is now flat response both on and off axis. Listeners anywhere in the area covered by the horn will appreciate the smooth and extended response of the system. Flat power response makes sense only with components exhibiting constant angular coverage. If we had equalized the 2350 horn for flat power response, then the on-axis response would have been too bright and edgy sounding.

Figure 3-7. Power response of HF systems
The rising DI of most typical radial horns is accomplished through a narrowing of the vertical pattern with rising frequency, while the horizontal pattern remains fairly constant, as shown in Figure 3-8A. Such a horn can give excellent horizontal coverage, and since it is self equalizing through its rising DI, there may be no need at all for external equalization. The smooth-running horizontal and vertical coverage angles of a Bi-Radial, as shown at Figure 3-8B, will always require power response HF boosting.
Measurement of Directional Characteristics
Polar plots and isobar plots require that the radiator under test be rotated about several of its axes and the response recorded. Beamwidth plots may be taken directly from this data. DI and Q can be calculated from polar data by integration using the following equation:
2 DI = 10 log (P )2 sind o
PQ is taken as unity, and q is taken in 10 increments. The integral is solved for a value of DI in the horizontal plane and a value in the vertical plane. The resulting DI and Q for the radiator are given as:

DI = DIh DIv + 2 2

Q = Qn Q v
(Note: There are slight variations of this method, and of course all commonly use methods are only approximations in that they make use of limited polar data.)
Figure 3-8. Increasing DI through narrowing verticalbeamwidth
Using Directivity Information
A knowledge of the coverage angles of an HF horn is essential if the device is to be oriented properly with respect to an audience area. If polar plots or isobars are available, then the sound contractor can make calculations such as those indicated in Figure 3-9. The horn used in this example is the JBL 2360 Bi-Radial. We note from the isobars for this horn that the -3 dB angle off the vertical is 14. The -6 dB and -9 dB angles are 23 and 30 respectively. This data is for the octave band centered at 2 kHz. The horn is aimed so that its major axis is pointed at the farthest seats. This will ensure maximum reach, or throw, to those seats. We now look at the -3 dB angle of the horn and compare the reduction in the horns output along that angle with the inverse square advantage at the closer-in seats covered along that axis. Ideally, we would like for the inverse square advantage to exactly match the horns off-axis fall-off, but this is not always possible. We similarly look at the response along the -6 and -9 dB axes of the horn,
comparing them with the inverse square advantages afforded by the closer-in seats. When the designer has flexibility in choosing the horns location, a good compromise, such as that shown in this figure, will be possible. Beyond the -9 dB angle, the horns output falls off so rapidly that additional devices, driven at much lower levels, would be needed to cover the front seats (often called front fill loudspeakers). Aiming a horn as shown here may result in a good bit of power being radiated toward the back wall. Ideally, that surface should be fairly absorptive so that reflections from it do not become a problem.

Figure 4-1.

A simple outdoor reinforcement system
Figure 4-2. Electrical response of a sound system 3 dB below sustained acoustical feedback
The Concept of Acoustical Gain
Boner (4) quantified the concept of acoustical gain, and we will now present its simple but elegant derivation. Acoustical gain is defined as the increase in level that a given listener in the audience perceives with the system turned on, as compared to the level the listener hears directly from the talker when the system is off. Referring to Figure 4-3, let us assume that both the loudspeaker and microphone are omnidirectional; that is, DI = 0 dB and Q = 1. Then by inverse square loss, the level at the listener will be: 70 dB - 20 log (7/1) = 70 - 17 = 53 dB
If the loudspeaker produces a level of 70 dB at the microphone, it will produce a level at the listener of: 70 - 20 log (6/4) = 70 - 3.5 = 66.5 dB With no safety margin, the maximum gain this system can produce is: 66.5 - 53 = 13.5 dB Rewriting our equations: Maximum gain = 70 - 20 log (D2/D1) - 70 - 20 log (D0/Ds) This simplifies to:
Now, we turn the system on and advance the gain until we are just at the onset of feedback. This will occur when the loudspeaker, along the D1 path, produces a level at the microphone equal to that of the talker, 70 dB.
Maximum gain = 20 log D0 - 20 log Ds + 20 log D1 - 20 log D2
Figure 4-3. System gain calculations, loudspeaker and microphone both omnidirectional
Adding a 6 dB safety factor gives us the usual form of the equation: Maximum gain = 20 log D0 - 20 log Ds + 20 log D1 - 20 log D2 - 6 In this form, the gain equation tells us several things, some of them intuitively obvious: 1. That gain is independent of the level of the talker 2. That decreasing Ds will increase gain 3. That increasing D1 will increase gain.
The Influence of Directional Microphones and Loudspeakers on System Maximum Gain
Let us rework the example of Figure 4-3, this time making use of a directional loudspeaker whose midband polar characteristics are as shown in Figure 4-4A. It is obvious from looking at Figure 4-4A that sound arriving at the microphone along the D1 direction will be reduced 6 dB relative to the omnidirectional loudspeaker. This 6 dB results directly in added gain potential for the system.
The same holds for directional microphones, as shown in Figure 4-5A. In Figure 4-5B, we show a system using an omnidirectional loudspeaker and a cardioid microphone with its -6 dB axis facing toward the loudspeaker. This system is equivalent to the one shown in Figure 4-4B; both exhibit a 6 dB increase in maximum gain over the earlier case where both microphone and loudspeaker were omnidirectional. Finally, we can use both directional loudspeakers and microphones to pick up additional gain. We simply calculate the maximum gain using omnidirectional elements, and then add to that value the off-axis pattern advantage in dB for both loudspeaker and microphone. As a practical matter, however, it is not wise to rely too heavily on directional microphones and loudspeakers to make a significant increase in system gain. Most designers are content to realize no more than 4-to-6 dB overall added gain from the use of directional elements. The reason for this is that microphone and loudspeaker directional patterns are not constant with frequency. Most directional loudspeakers will, at low frequencies, appear to be nearly omnidirectional. If more gain is called for, the most straightforward way to get it is to reduce Ds or increase D1.

Figure 4-4. System gain calculations, directionalloudspeaker
Figure 4-5. System gain calculations, directionalmicrophone

How Much Gain is Needed?

The parameters of a given sound reinforcement system may be such that we have more gain than we need. When this is the case, we simply turn things down to a comfortable point, and everyone is happy. But things often do not work out so well. What is needed is some way of determining beforehand how much gain we will need so that we can avoid specifying a system which will not work. One way of doing this is by specifying the equivalent, or effective, acoustical distance (EAD), as shown in Figure 4-6. Sound reinforcement systems may be thought of as effectively moving the talker closer to the listener. In a quiet environment, we may not want to bring the talker any closer than, say, 3 meters from the listener. What this means, roughly, is that the loudness produced by the reinforcement system should approximate, for a listener at D0, the loudness level of an actual talker at a distance of 3 meters. The gain necessary to do this is calculated from the inverse square relation between D0 and EAD: Necessary gain = 20 log D0 - 20 log EAD In our earlier example, D0 = 7 meters. Setting EAD = 3 meters, then: Necessary gain = 20 log (7) - 20 log (3) = 17 - 9.5 = 7.5 dB Assuming that both loudspeaker and microphone are omnidirectional, the maximum gain we can expect is: Maximum gain = 20 log (7) - 20 log (1) + 20 log (4) - 20 log (6) - 6 Maximum gain = 17 - 0 + 12 - 15.5 - 6 Maximum gain = 7.5 dB
As we can see, the necessary gain and the maximum gain are both 7.5 dB, so the system will be workable. If, for example, we were specifying a system for a noisier environment requiring a shorter EAD, then the system would not have sufficient gain. For example, a new EAD of 1.5 meters would require 6 dB more acoustical gain. As we have discussed, using a directional microphone and a directional loudspeaker would just about give us the needed 6 dB. A simpler, and better, solution would be to reduce Ds to 0.5 meter in order to get the added 6 dB of gain. In general, in an outdoor system, satisfactory articulation will result when speech peaks are about 25 dB higher than the A-weighted ambient noise level. Typical conversation takes place at levels of 60 to 65 dB at a distance of one meter. Thus, in an ambient noise field of 50 dB, we would require speech peaks of 75 to 80 dB for comfortable listening, and this would require an EAD as close as 0.25 meter, calculated as follows: Speech level at 1 meter = 65 dB Speech level at 0.5 meter = 71 dB Speech level at 0.25 meter = 77 dB Let us see what we must do to our outdoor system to make it work under these demanding conditions. First, we calculate the necessary acoustical gain: Necessary gain = 20 log D0 - 20 log EAD Necessary gain = 20 log (7) - 20 log (.25) Necessary gain = 17+ 12 = 29 dB

Figure 5-1. Sound impinging on a large boundary surface
When dealing with the behavior of sound in an enclosed space, we must be able to estimate how much sound energy will be lost each time a sound wave strikes one of the boundary surfaces or one of the objects inside the room. Tables of absorption coefficients for common building materials as well as special acoustical materials can be found in any architectural acoustics textbook or in data sheets supplied by manufacturers of construction materiaIs. Unless otherwise specified, published sound absorption coefficients represent average absorption over all possible angles of incidence. This is desirable from a practical standpoint since the random incidence coefficient fits the situation that exists in a typical enclosed space where sound waves rebound many times from each boundary surface in virtually all possible directions. Absorption ratings normally are given for a number of different frequency bands. Typically, each band of frequencies is one octave wide, and standard center frequencies of 125 Hz, 250 Hz, 500 Hz, 1 kHz, etc., are used. In sound system design, it usually is sufficient to know absorption characteristics of materials in three or four frequency ranges. In this handbook, we make use of absorption ratings in the bands centered at 125 Hz, 1 kHz and 4 kHz. The effects of mounting geometry are included in standardized absorption ratings by specifying the types of mounting according to an accepted numbering system. In our work, familiarity with at least three of these standard mountings is important.
Acoustical tile or other interior material cemented directly to a solid, non-absorptive surface is called No. 1 mounting (see Figure 5-2). To obtain greater absorption, especially at lower frequencies, the material may be spaced out on nominal two-inch thick furring strips and the cavity behind loosely filled with fiberglass blanket. This type of mounting is called out as No. 2. No. 7 mounting is the familiary suspended T-bar ceiling system. Here the material is spaced at least 0.6 meter (2) away from a solid structural boundary. Absorption coefficients fall within a scale from zero to one following the concept established by Sabine, the pioneer of modern architectural acoustics. Sabine suggested that an open window be considered a perfect absorber (since no sound is reflected) and that its sound absorption coefficient must therefore be 100 percent, or unity. At the other end of the scale, a material which reflects all sound and absorbs none has an absorption coefficient of zero. In older charts and textbooks, the total absorption in a room may be given in sabins. The sabin is a unit of absorption named after Sabine and is the equivalent of one square foot of open window. For example, suppose a given material has an absorption coefficient of 0.1 at 1 kHz. One hundred square feet of this material in a room has a total absorption of 10 sabins. (Note: When using SI units, the metric sabin is equal to one square meter of totally absorptive surface.)

Figure 5-2.

ASTM types of mounting (used in conducting sound absorption tests)
More recent publications usually express the absorption in an enclosed space in terms of the average absorption coefficient. For example, if a room has a total surface area of 1000 square meters consisting of 200 square meters of material with an absorption coefficient of.8 and 800 square meters of material with an absorption coefficient of.1, the average absorption coefficient for the entire internal surface area of the room is said to be.24: Area: 1000 a Coefficient: x 0.8 x 0.1 Sabins: = 160 = 80 240

= 240 = 0.24 1000

Although we commonly use published absorption coefficients without questioning their accuracy and perform simple arithmetic averaging to compute the average absorption coefficient of a room, the numbers themselves and the procedures we use are only approximations. While this does not upset the reliability of our calculations to a large degree, it is important to realize that the limit of confidence when working with published absorption coefficients is probably somewhere in the neighborhood of 10%. How does the absorption coefficient of the material relate to the intensity of the reflected sound wave? An absorption coefficient of 0.2 at some specified frequency and angle of incidence means that 20% of the sound energy will be absorbed and the remaining 80% reflected. The conversion to decibels is a simple 10 log function: 10 log10 0.8 = -0.97 dB In the example given, the ratio of reflected to direct sound energy is about -1 dB. In other words, the reflected wave is 1 dB weaker than it would have been if the surface were 100% reflective. See the table in Figure 5-3. Thinking in terms of decibels can be of real help in a practical situation. Suppose we want to improve the acoustics of a small auditorium which has a pronounced slap off the rear wall. To reduce the intensity of the slap by only 3 dB, the wall must be surfaced with some material having an absorption coefficient of 0.5! To make the slap half as loud (a reduction of 10 dB) requires acoustical treatment of the rear wall to increase its absorption coefficient to 0.9. The difficulty is heightened by the fact that most materials absorb substantially less sound energy from a wave striking head-on than their random incidence coefficients would indicate. Most acoustic materials are porous. They belong to the class which acousticians elegantly label fuzz. Sound is absorbed by offering resistance to the flow of air through the material and thereby changing some of the energy to heat. But when porous material is affixed directly to solid concrete or some other rigid non-absorptive surface, it is obvious that there can be no air motion and therefore no absorption at the boundary of the two materials.

The Growth and Decay of a Sound Field in a Room
At this point we should have sufficient understanding of the behavior of sound in free space and the effects of large boundary surfaces to understand what happens when sound is confined in an enclosure. The equations used to describe the behavior of sound systems in rooms all involve considerable averaging out of complicated phenomena. Our calculations, therefore, are made on the basis of what is typical or normal; they do not give precise answers for particular cases. In most situations, we can estimate with a considerable degree of confidence, but if we merely plug numbers into equations without understanding the underlying physical processes, we may find ourselves making laborious calculations on the basis of pure guesswork without realizing it. Suppose we have an omnidirectional sound source located somewhere near the center of a room. The source is turned on and from that instant sound radiates outward in all directions at 344 meters per second (1130 feet per second) until it strikes the boundaries of the room. When sound strikes a boundary surface, some of the energy is absorbed, some is transmitted through the boundary and the remainder is reflected back into the room where it travels on a different course until another reflection occurs. After a certain length of time, so many reflections have taken place that the sound field is now a random jumble of waves traveling in all directions throughout the enclosed space. If the source remains on and continues to emit sound at a steady rate, the energy inside the room builds up until a state of equilibrium is reached in which the sound energy being pumped into the room from the source exactly balances the sound energy dissipated through absorption and transmission through the boundaries. Statistically, all of the individual sound packets of varying intensities and varying directions can be averaged out, and at all points in the room not too close to the source or any of the boundary surfaces, we can say that a uniform diffuse sound field exists. The geometrical approach to architectural acoustics thus makes use of a sort of soup analogy. As long as a sufficient number of reflections have taken place, and as long as we can disregard such anomalies as strong focused reflections, prominent resonant frequencies, the direct field near the source, and the strong possibility that all room surfaces do not have the same absorption characteristics, this statistical model may be used to describe the sound field in an actual room. In practice, the approach works remarkably well. If one is careful to allow for some of the factors mentioned, 5-5

The ratio of direct to reverberant sound can be calculated from the simple equation shown below the chart, or estimated directly from the chart itself. For example, at four times DC the direct sound field is 12 dB less than the reverberant sound field. At one-half DC, the direct sound field is 6 dB greater than the reverberant sound field. Remember that, although critical distance depends on the directivity of the source and the absorption characteristics of the room, the relationships expressed in Figure 5-19 remain unchanged. Once DC is known, all other factors can be calculated without regard to room characteristics. With a directional sound source, however, a given set of calculations can be used only along a specified axis. On any other axis the critical distance will change and must be recalculated. Let us investigate these two factors in some detail: first the room constant R, and then the directivity factor Q. We have already mentioned that the room constant is related to the total absorption of an enclosed space, but that it is different from total absorption represented by Sa. One way to understand the room constant is first to consider that the total average energy density in a room is directly proportional to the power of the sound source and inversely proportional to the total absorption of the boundary surfaces. This
Figure 5-20. Relative SPL vs. distance from source in relation to critical distance
relationship is often indicated by the simple expression: 4W/cSa. W represents the output of the sound source, and the familiar expression Sa indicates the total absorption of the boundary surfaces. Remembering our statistical room model, we know that sound travels outward from a point source, following the inverse square law for a distance equal to the mean free path, whereupon it encounters a boundary surface having an absorption coefficient a. This direct sound has no part in establishing the reverberant sound field. The reverberant field proceeds to build up only after the first reflection. But the first reflection absorbs part of the total energy. For example, if a is 0.2, only 80% of the original energy is available to establish the reverberant field. In other words, to separate out the direct sound energy and perform calculations having to do with the reverberant field alone, we must multiply W by the factor (1 - a). This results in the equation: