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Acoustic Study of Hill Auditorium

Submitted by: Joe Arifin Atin Tandon Ross Penniman Thomas Rainwater

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Contents

Introduction ........................................................................................ 3 Design Criteria for Auditoria .............................................................. 4 Mathematical Model ........................................................................ 8 Behavior of Sound in a Room ............................................................ 12 Experiments ................................................................................. 15 Analysis ..................................................................................... 17 Conclusion .................................................................................. 20 Appendix A ........................................................................................................... 21 Appendix B ................................................................................. 22 Appendix C ................................................................................. 23 Appendix D .................................................................................. 29 Appendix E ................................................................................. 32 Bibliography ................................................................................ 33

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Introduction

For our Mechanical Engineering 424 term project, our group is looking to find technical explanations and quantifications of subjective characteristics of a concert hall. These characteristics include things such as reverberation, clarity, and intimacy. We are specifically investigating the acoustics of Hill Auditorium. We carried out this project in three stages; the first was information gathering, the second was experimentation, and the third was analysis. Information gathering on architectural acoustics was needed to find out how to do a theoretical analysis of the acoustics of a concert hall and what experiments could be done in the concert hall to find certain acoustic parameters. The experiments were done to obtain results for analysis and also to compare to the results obtained by theoretical calculations. These three stages ultimately served to help us fulfill our motivation and purpose for this project. Hill Auditorium's Acoustics Before Renovation Hill Auditorium first opened in 1913 after it finished being built at a cost of $282,000. It has a built-in area of approximately 124,000 square feet and before renovation, it had a seating capacity of 4,200 seats, making it the largest concert hall on the University of Michigan campus. It was designed by architect Albert Khan and his associate Ernest Wilby and its acoustics were excellent as many hailed it as a "monument to perfect acoustics". This was partly in due to architect Albert Khan's collaboration with renowned acoustical engineer Hugh Tallant. For an audience member, it was known if he sat anywhere in the auditorium, even the topmost corner seat in the second balcony, he would not only have a full and clear view of the stage, but he would also hear clearly every sound being played on the stage. To a performer, the tremendous focus of the ceiling onto the audience plane allows it to act as an architectural amplifier that returns the applause of 12,000 people for an audience of only 3,500. Hill Auditorium's Renovations and Acoustical Improvements Although Hill Auditorium had excellent acoustics within the main concert hall itself, its main problem lay in the interfering noises that intruded into the main hall. Many musicians complained about the difficulty of hearing on stage and this was partly in fact due to the interfering noise of the mechanical systems within the auditorium building. Toilets just outside the main concert hall, lighting dimmers that caused filament hum, and sounds from the lobby area also contributed to the interfering noise making its way into the main hall. Thus the main objective of the renovation was to maintain the already excellent architectural acoustics of the main hall but minimize interfering noise. Many improvements were made to minimize the interfering noise coming into the main concert hall. The aging mechanical heating ventilation system that emitted many sounds was changed to a more quiet and modern electrical one which also provided air conditioning. The lighting systems were improved to reduce the hum and the mechanical systems were moved into a separate structure outside the envelope of the original building. The toilets near the main concert hall were redesigned to reduce the audibility of their operations and the doors to the main hall were doubled and improved to minimize noise coming in from the lobby area. Even the seating selection was also worked on to maintain or reduce sound absorption.

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Design Criteria for Auditoria

Construction and Design In the construction of an auditorium, acousticians take into consideration a variety of acoustic phenomena that will have the greatest impact on the sound in that enclosure. The reverberation time, early decay time, strength factor, initial time-delay, and bass ratio of the room are just some of the measurable parameters which can be determined experimentally in order to optimize the auditorium for its intended purpose. For example, an auditorium that will be used primarily for speech would require a shorter reverberation time than a hall designed for music in order for speech to be understood clearly. Auditoriums built for the main purpose of musical performances have to take into account the acoustical characteristics of different musical instruments and have to accommodate for solo instrument performances and performances by groups of instruments such as small ensembles or entire orchestras. With this in mind, there are therefore, also a number of subjective factors such as warmth, intimacy, and the clarity of the sound in the enclosure, which are not as clearly defined, but are equally as important as the measurable parameters mentioned previously, when designing a quality auditorium as a concert venue. Definitions The acoustic phenomena of a concert hall depend primarily on six parameters. These are the early decay time (EDT), apparent source width (ASW), strength factor (Gmid), sound diffusion index (SDI), initial time delay gap (ITDG), and bass ratio (BR). These values, except the sound diffusion index, can all be easily measured experimentally. The sound diffusion index can be estimated subjectively from a visual inspection of the irregularities in a hall. Early Decay Time Early decay time (EDT) is the time associated with the early part of reverberation time. Reverberation time (RT), as first defined by W.C. Sabine, is the time taken for sound energy to fall 60 dB below its original level, and it is directly related to the surface area, volume and total absorption of the enclosure. An ideal reverberation time for a reverberant hall is between 1.8 and 2.0 seconds. EDT is defined as the length of time, multiplied by a factor of six that it takes for a sound to decay 10 dB after the instant it is turned off. The factor of six is used to make comparison between the EDT and RT easier. The EDT can be determined experimentally as the average of sound measurements over a range of frequencies from a variety of locations around the hall. The EDT is typically ten percent higher than RT. Apparent Source Width The apparent source width is a measure of the feeling of being immersed in sound due to lateral reflections in the hall. These lateral reflections, or reflections from the side walls, increase the apparent source width by giving the listener the feeling that the sound source (band, orchestra, choir, etc) is larger than it actually is. This gives the audience the feeling of being immersed in or surrounded by the sound emanating from the source. The early lateral reflections are reflections that reach the ear within the first 80 milliseconds from the side walls and are the most important. The lateral direction is shown in Figure 1.

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Figure 1: Apparent Source Width

Strength Factor The strength factor (Gmid) is the measure of the sound intensity or loudness. Loudness is dependent on the frequency at which it is measured, so Gmid is calculated as an average of the values of loudness at frequencies 500 Hz and 1000 Hz. The strength factor is dependent on both the direct and reflected sounds and thus is proportional to the EDT and inversely proportional to the volume of the hall. Sound Diffusion Index The sound diffusion index is the qualitative measure of how well the sound is diffused in the hall. The diffusion of sound adds to the listener's feeling of being enveloped by the sound, so that the greater the diffusion of sound, the greater the listener's envelopment. Sound is diffused off irregular surfaces and ideally reaches every listener in equal strength from all directions. Initial Time Delay Gap The initial time delay gap (ITDG) is the measure of the intimacy an audience perceives of the hall, where intimacy is a subjective measure of the size of a hall. The initial time delay gap is defined as the interval between when the direct sound reaches a location and when the first reflected sound reaches that same location. A good concert hall typically has an ITDG of approximately 16 to 28 milliseconds. Bass Ratio The bass ratio (BR) is an average of reverberation times for an occupied hall, and is a measurement of the warmth of the hall. It is calculated by taking the average of the RTs at 125 Hz and 250 Hz and dividing them by the average of the RTs at 500 Hz and 1000 Hz. The recommended BR value for halls with a RT greater than 1.8 seconds is between 1.1 and 1.25. Importance of Reverberation Time in the Design of Rooms and Auditoria In a room with highly reflecting surfaces, such as a bathroom, the reverberation time is relatively long, while in an anechoic chamber where all the walls, the ceiling and the floor are covered by a highly absorbent material, the reverberation time is nearly zero. The absorption of different materials varies widely with the frequency of the incident sound and the angle of incidence. It also follows that the reverberation time is liable to vary with frequency. Generally, the reverberation time is longer at lower frequencies because these are usually less effectively absorbed than higher frequencies. It is important that the reverberation time suits the intended use of the room. Too long a reverberation time renders speech less intelligible and music more cacophonous and produces higher background noise levels. A short reverberation time deadens background noise, but muffles speech and makes music sound "thin" and staccato.

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Volume, Floor Area, and Seating In designing concert halls, acousticians want to design the space-volume in order to optimize certain characteristics that will contribute to the quality of sound. The volume of a concert hall, and its floor area, are directly related to the seating capacity. However, starting a design based on the number of seats is not the best method. Instead, by specifying a desirable value for the strength factor (Gmid) and the reverberation time, RTmid (RTmid is average of RT at 500 and 1000 Hz), acousticians can calculate the proposed volume, V, (in m3) of the hall using the equation: Gmid = 10log[ RTmid / V ] + 44.4 For optimum performance, Gmid should be between 4.0 and 5.5 dB (as seen in Appendix A: Figure 1) and RTmid should be between 1.8 and 2.0 sec. Once the desired volume has been calculated, the number of seats, N, can be derived from the equation: N = [ 0.2V / RTmid ] The total floor area of the hall, ST, is then calculated from the equation: ST = 0.69N The values of area ST, volume V, and number of seats N are then known to approximate ranges which allow for variations that arise out of the design process. Shape of the Hall Once the approximate volume of the hall is determined, the next step is the layout or shaping of the hall. It is desirable to maximize the seating capacity without sacrificing acoustic quality. Tall, narrow rectangular or horseshoe halls provide early lateral reflections, a desirable characteristic, and minimize reflections from overhead. Another popular design is a fan shaped auditorium, which has the advantage of seating more listeners closer to the stage, but typically causes undesirable acoustic effects due to an increased ITDG and decreased lateral reflections. If balconies and boxes are incorporated into the design, then the audience can be brought closer to the performers without losing desired acoustic qualities. The surfaces of the balconies also help to increase the diffusion of sound in the hall. In designing the balconies, it is desirable to have the depth of the balcony overhang as small as possible, and be no greater than the opening height.

Figure 2: Balcony Height vs. Depth

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Material Properties of the Hall The best-rated music halls typically have walls of plaster on masonry. These walls have low absorption coefficients, providing excellent reverberation from the side. Sound absorbing panels or wood diffusers can be hung from the walls and ceilings to modify the sound pattern in the hall. The reverberation time is inversely related to the total room absorption in the following equation where is the sum of the absorption coefficient for all surface areas: RT = [ 0.16V/ ] Thus, if a reverberation time of between 1.8 and 2.0 seconds is desired, then the materials of the ceiling, floor, seats, and acoustic panels can be decided. In order to have an adequate amount of diffusion it is also important to have surfaces that diffuse sound well, such as moldings, statues, and surface irregularities. Diffuse sound in the hall ensures a smooth decay of sound, eliminating annoying effects such as echo. Isolation of Sound and Elimination of Background Noise In order to keep a large concert hall's audience comfortable, heating and cooling systems are essential and thus provide very important design considerations. The mechanical noise of the systems, the background noise due to fans and the air flow through the ducts can be disruptive and can mask quieter musical performances. These systems should thus be located on a site that minimizes the noise intrusion. The duct system can be designed using filters to minimize the ambient noise they create and it is also important to minimize the noise from outside sources such as traffic, railways, or airports. The consideration of these factors in the initial design can ultimately help reduce later high-cost renovation. As in the case of Hill Auditorium, it underwent a $38.6 million restoration/renovation compared to its initial building cost of $282,000. Although the renovation was completed in 2004 as compared to it being built in 1913, the difference in costs is still staggering.

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Mathematical Model

A Simple Mathematical Model for the Growth of Sound in a Room If a source of sound is operated continuously in an enclosure, absorption in air and the surrounding surfaces prevents the acoustic pressure amplitude from becoming infinitely large. In smaller enclosures, absorption in air is negligible so that both the rate at which amplitude increases and its ultimate value are controlled by surface absorption. If the total sound absorption is large, the pressure amplitude quickly reaches an ultimate value only slightly in excess of that produced by the direct wave alone. Rooms with very high absorption are called dead, or anechoic. By contrast, if the total absorption of the room is small, considerable time will elapse before higher amplitude is reached. Rooms of this type are considered live, reverberant or echoic. When a source of sound is started in a live room, reflections at the walls produce a sound energy distribution that becomes more and more uniform with increasing time. Ultimately, except close to the source or absorbing surfaces, this energy distribution may be assumed to be completely diffuse. This lends itself to ray acoustic descriptions. Let S be an element of a boundary and dV an element of volume in the air at a distance r from S, where r makes an angle with the normal to S. Let the acoustic energy density E be uniform throughout the region so that the acoustic energy present in dV is EdV. The amount of V this energy that will strike S by direct transmission is E multiplied by the projection of 4r 2 S on the sphere of radius r centered on dV.

E

dV S cos 4r 2

(1)

Now, let dV be part of a hemispherical shell of thickness r and radius r centered on S. The acoustic energy E contributed to S by this entire shell is obtained by assuming that energy arrives from any direction with equal probability. Integrating over the hemisphere with V = 2r sin r r d yields ESr E= 2

/2

sin cos d

0

(2)

This energy arrives during a time interval t = as

r , and equation (2) can thus be rewritten c

E EcS dE = . Therefore, the rate at which energy falls on a unit area of wall is t 4 dt dE Ec = dt 4

(3)

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Assume at any point within the room: 1. energy is arriving and departing along individual ray paths 2. the rays have random phases The energy density E at the point is then the sum of the energy densities Ej of each of the rays. If the jth ray has effective pressure amplitude Pej, we have E = Ej = (P2ej/p0 c2) and thus E = P2r/p0 c2

(4)

where Pr= (P2ej)1/2 is the effective pressure amplitude of the reverberant sound field. If the total sound absorption by the surfaces of and within the room is A, then from equation (3), the rate at which energy is being absorbed is AEc/4. The sound absorption A has the dimensions of area and is expressed as either metric Sabine (m2) or English Sabine (ft2). This rate of absorption of energy by the surfaces plus the rate VdE/dt at which it increases in the volume V of the room must equal the input power . Therefore, the differential equation governing growth of sound energy in a live room is

V

dE Ac + E= dt 4

(5)

If the sound source is started at t = 0, the solution is given by E= (4 /Ac) (1-e-t/E)

(6)

where E = 4V/Ac is the time constant. It is clear that 1/ E = 2I, with being the temporal absorption coefficient. If the space has a large volume and small total absorption, then E is large and a relatively long time will be required for the energy density to approach its limiting value. The final equilibrium energy density is E () =P2r()/poc2=4 /Ac For the same input power, the smaller A is, the larger E () will become. Since these results are based on diffused field, there are limitations. For instance equation (5) may not be used until the energy traveling along each ray path has had enough time to accumulate several reflections at the boundaries. This time can range from 50ms for a small room to above 1s for an auditorium. Equation (7) indicates that the final energy density and effective pressure amplitudes are independent of the volume and the shape of the room, are the same at all points in the room, and depend only on the source strength and the total absorption. This is not true for a room that has sound focusing surfaces or deep recesses, or that is coupled to another space by an opening. These equations may also be invalid if some large surfaces of the

(7)

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room are abnormally absorptive, since the energy density near such surfaces may be lower than elsewhere.

Sabine's Formula for Reverberation Time The sound pressure level reduces with time as

SPL = 4.34t

E

The reverberation time RT is defined as the time required for the level of sound to drop by 60 dB, R T = 13.82 E = 55.3V / Ac Expressing V in cubic meters, A in metric sabin and c = 343m/sec, we obtain the metric form of Sabine Reverberation Formula, RT = and in English units, RT = 0.049V A 0.161V A

where V is in cubic feet, A in English sabin and c = 1125ft/s. The total surface area of the room is S, the average Sabine absorptivity a is defined by

a=

Hence reverberation time can be defined as: RT =

A S

0.161V Sa

A depends on the area and absorptive properties of all the diverse materials within the room, the form of this dependence is subject to a variety of simplifying assumptions. Sabine assumed that the total sound absorption is the sum of the sound absorptions Ai of the individual surfaces, A = Ai = S i a i

i i

where ai is the Sabine absorptivity for the ith surface of area Si. With this assumption, the average sound absorptivity a is the area-weighted average of the individual absorptivity ai.

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Each a for a surface or object is evaluated from standardized measurements on a sample of the material in a reverberation chamber. In the development of reverberation time, acoustic losses in the volume of the air are neglected. The importance of absorption in the air is determined by the ratio: 4mV Sa Since m increases with f whereas a tends to decrease above 1 kHz, absorption in the air can be significant at higher frequencies in large volumes. Also, in highly reverberant spaces the major portion of the sound absorption may occur in air rather than at the surfaces. For relative humidity h in percent between 20 and 70 and frequencies in the range 1.5 to 10 kHz, a sufficiently accurate approximation for most architectural applications is: m = 5.5 × 10 -4 (50 / h)( f / 1000)1.7

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Behavior of Sound in a Room

A knowledge of the behavior of sound in a room is necessary if we wish to adapt the room for speech or music and if we want to attenuate external noise. Consider the effect of placing a sound source in a room, when sound energy (Ei) from the source strikes a room boundary, the reflected sound energy (Er) contributes to the sound-field in the room, the absorbed sound (Ea) dissipates as heat and the transmitted sound energy (Et) propagates away through the boundary layer.

Reflection of Sound If the wavelength of an incident sound-wave is much smaller than the dimensions of the reflecting surface, then the angle of reflection of the sound-wave equals the angle of incidence. We can use this geometrical behavior to predict the pattern of sound rays in a room, a limitation being that only the primary and possibly the secondary reflections can be studied before the reverberant field begins to mask the ray paths. In larger rooms such as concert halls, 'ray tracing' can identify problematic echoes, where an echo is defined as a reflection which arrives more than 50 ms after the direct sound. An echo can also be thought of as a reflected ray with a path-length that is at least 17 m longer than that of the direct ray. Echo problems in large enclosures are solved by reducing the path length of the reflected ray. This can be done either by lowering the ceiling or by suspending reflectors from the ceiling. By observing the behavior of the reflections in a room, we can control subjective properties such as intimacy, the quality of which depends on early arrival of reflections after the direct sound, and diffusion which is the evenness of the reverberant field.

Figure 3: Reflection of Sound

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Absorption of Sound We can understand the effect of absorption by measuring, at a given position in a room, the sound pressure level caused by a steady sound power source. Instead of rising indefinitely as an increasing number of reflections arrive at the measuring position, the sound pressure level soon stabilizes. This must mean that the rate of energy input is exactly compensated by the rate at which the energy is absorbed by the different surfaces of the room. If more absorption material is put in the room, the sound pressure level is less because the energy in the reflections is reduced.

Typical absorbing surfaces in a room include carpets and curtains. These are simple porous absorbers which absorb sound energy by restricting the movement of air particles, creating frictional forces that cause the dissipation of energy as heat. Porous absorbers are most effective when placed at a point on the sound-wave which has maximum particle velocity. This position is a quarter wavelengths away from a reflecting surface (when a wave is incident at right-angles) and is therefore frequency dependent. A carpet is an example of a porous absorber close to a reflective boundary. It absorbs best at high frequencies because the dimensions of the quarter wavelengths are then comparable with the thickness of carpet. Other surfaces in the room absorb different frequencies to different extents, and by controlling the proportions of these absorbers it is possible to adjust the warmth of a room for music, or its clarity for speech.

Figure 4: Absorption of Sound

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Build-up and Decay of Sound in a Room If we position a microphone in a room and then switch on a steady sound-source, we notice that the sound pressure level does not immediately reach a steady level. This is because the first reflection and subsequent reflections take a finite time to reach the microphone. In the resulting equilibrium state, interference between the sound-waves causes a spatial distribution of pressure maxima and minima which can be detected by moving the microphone around the room. These natural resonances or normal room modes are associated with the geometry of the room and the wavelengths emitted by the sound source. Interesting consequences of these modes are that pressure doubling occurs at reflective boundaries, and that since all the room modes have antinodes at the corners of the room, they can all be "driven" by a sound-source placed there.

If the sound-source is now switched off, the collection of decaying room modes is called the reverberant sound-field. The rate of decay depends on the amount and positioning of absorption in the room.

Figure 5: Build-up and Decay of Sound

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Experiments

Process On the evening of Saturday, February 21, Ross Penniman, Joe Arifin, and Professor Jason Corey (School of Music) met at Hill Auditorium to do some tests on the hall. The recording equipment used was a pair of DPA 4006 microphones fed into a Mackie VLZ series mixer and then into an Alesis Masterlink digital audio recorder. The microphones had an omni-directional pickup pattern and had a nearly flat frequency response from 20 Hz to 20 kHz. In all experimental test cases the microphones were positioned 13 inches apart and were mounted on a stand about 5 feet off the floor. A loudspeaker (Genelec 1029A) was used to produce tones. The ambient temperature in the room was 66 degrees Fahrenheit, which meant that the speed of sound was approximately 343 meters per second (1125 feet per second).

To test the properties of the auditorium, we produced sounds on stage and then recorded these sounds from various points in the room. By examining how these sounds were changed by the acoustic environment, we could then learn about the acoustic properties of the auditorium. Our first test involved producing sustained sounds using a loudspeaker. We produced several sine tones at several different frequencies that were abruptly shut off. This allowed us to examine how the reverberation of the room changed for different frequencies. We also produced broadband sounds such as white noise to get a more general picture of the frequency response of the room and of the reverberation. The second test involved producing very short sounds which was done by popping balloons. Due to the fact that the sound was very short in duration, it allowed us to examine how it was reflected in the auditorium. We recorded these balloon pop sounds from three different locations in the auditorium and for each location we popped a balloon on the left, center, and right side of the stage (refer to Appendix C: Figure 1). Using the experimental recordings obtained to generate the experimental results needed for analysis of the hall proved to be a challenge. To obtain the 60 dB decay times, we loaded the digitally recorded audio files into MATLAB and ran them through a code we wrote (refer to Appendix B) that would produce an amplitude (dB) versus time (s) plot of the recording. From the waveforms of the recordings and the plots of amplitude versus time, we noticed an amplitude discrepancy between the two microphones (left and right channel). This was likely due to different phase cancellations occurring at each of the microphones. To compensate for this, when we obtained the 60 dB decay times for the different sine tone frequencies from their respective plots, we took the average of the two values (left and right channel) obtained. To obtain the values of the initial time delay gap, we plotted the absolute values of the waveform versus time. The initial time delay gap is the duration of time between when the sound reaches the microphone directly from the source and when the first reflected sound arrives at the microphone. This could then be discerned approximately from the plots. Again, due to waveform discrepancies between the two microphones (left and right channel), we took the average of the two values (left and right channel) of the initial time delay gap obtained. For the

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most part, the recordings could only generate plots that showed us the general distribution and strength of reflections but were not clear enough for us to do a mathematical analysis. A summary of the experimental results obtained such as the 60 dB decay time and initial time delay gap for different sine tone frequencies and different microphone and source positions respectively is tabulated below:

Frequency of Sine Tone 60 dB Decay Time (Hertz) (seconds) 125 2.15 250 1.43 500 1.79 1000 1.94 2000 1.38 4000 0.79 Table 1: Reverberation Time

Microphone Position

Front Edge of Stage Center of Main Floor

Center of Balcony

Initial Time Delay Gap (milliseconds) Center stage 24 Center stage 46 Stage left 24.5 Stage right 26.5 Center stage 36.5 Stage left 23 Stage right 23 Table 2: Initial Time Delay Gap

Source Position

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Analysis

Experimental Analysis For the most part, the reverberation times agree well with the expected values, however there are two exceptions (refer to Table 1). At 250 Hz the reverberation time is significantly shorter than expected (trend expects about 1.9 sec). At 1000 Hz, the reverberation time is significantly longer than expected (trend expects about 1.6 sec). The possible causes for this include reflections and room modes. If a strong reflection arrives at the microphone out of phase with the direct sound, it could cause cancellation of sound. Room modes are standing waves which occur between two or more surfaces in a room. If the microphones happened to be positioned at a node or anti-node of one of these modes, it could make the reverberation time significantly shorter or longer. Decay envelope plots (refer to sample in Appendix C: Figure 2) obtained using the MATLAB code were used to find the reverberation times at the different sine tone frequencies. The onset envelope plots (refer to sample in Appendix C: Figure 3) for the sine tone frequencies of 500Hz and 1000Hz were used to find the Gmid value of 4.3 dB by estimating how much in dB the auditorium amplified those frequencies and finding the average of the estimated values.

It can be seen that the initial time delay gap (ITDG) varies both with the sound source location and also with the pickup location (refer to Table 2). When the source is in the center of the stage, the ITDG is considerably longer than when the source is on either side of the stage. This is likely due to the geometry involved. On the side of the stage the source is much closer to the walls, which are strong reflecting surfaces. It can also be seen that the ITDG is generally shorter when the listener is sitting in the balcony than when the listener is sitting on the main floor. The most noticeable difference is for the center source position; however it is slightly shorter for the side source positions as well. The phenomenon of the ITDG being shorter for the balcony than the main floor is a common feature among concert halls. For instance at Symphony Hall in Boston, the ITDG is 15 ms for the main floor and only 7 ms for the balcony. At Orchestra Hall in Chicago, the ITDG is 40 ms for the main floor and only 24 ms for the balcony. Number of Seats 3540* 2630 ITDG (Floor) 46 ms 15 ITDG (Balcony) 36.5 ms 7 RT 60 dB 125 Hz 2.15 s 2.2

Hill Auditorium Symphony Hall, Boston Orchestra Hall, 2580 40 24 Chicago Carnegie Hall, 2760 23 16 1.8 New York Opera House, 3250 51 30 San Francisco (ITDG times for Hill Auditorium are for center source) * number of seats after renovation

500 Hz 1.79 s 1.8 1.3 1.8 1.7

2000 Hz 1.38 s 1.7 1.6 -

Table 3: Concert Hall Comparison

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Compared to some of the more revered concert halls of the world (refer to Table 3), Hill Auditorium actually compares rather poorly in respect to the ITDG times. This may be excusable since Hill is such a large auditorium. Having a seating capacity of 3540, it is larger than all of the best concert halls. Another feature that may inhibit the ITDG times is that the shape of the ceiling is designed to project the sound from the stage out into the audience with maximum efficiency. The motivation is more for everybody to be able to hear rather than to create an intimate sonic experience. In terms of sound diffusion, looking at the plots of the balloon pops helps to show how sound is reflected in the auditorium (refer to Appendix C: Figure 4 - 7). For the microphone being on the front edge of the stage, the direct sound is very strong in comparison to the reflected sound. In addition there is a "hump" of reflected sound that occurs quite late, probably being reflected off of the back wall. This is something performers frequently encounter on stage and is often referred to as "slap back". As the microphones are moved back from the stage, the sound field becomes more diffuse because more of the sound energy a listener receives is coming from all directions as opposed to directly from the source. It is also interesting to examine the frequency response of the hall. Two plots are shown (Appendix C: Figure 8 - 9). The first is the spectrum of white noise as analyzed by an FFT operation. The second plot is the response of the hall to white noise. The sound was produced on stage using a loudspeaker and recorded from the center of the main floor. It can be clearly seen that the high frequencies are significantly decreased in comparison to the low frequencies. Some of this is due to the high frequency absorption of the air, but most is likely due to the reflective properties of the materials in the hall. In addition, some frequencies are significantly more attenuated than others. This is likely due to phase cancellations at the location of the microphone and the presence of room modes that create spots in the hall where particular frequencies will be particularly large or small in amplitude. Finally, a spectrogram (Appendix C: Figure 10) shows how the sound decays when the white noise is shut off. The horizontal axis is time, the vertical axis is frequency and the color represents amplitude. The spectrogram gives a nice visual representation of how the high frequencies decay much faster than the low frequencies.

Theoretical Analysis Atin Tandon and Ross Penniman took approximate measurements of the Hill Auditorium's hall dimensions and features. Atin Tandon and Thomas Rainwater then divided the hall into blocked sections and simplified curved shapes into polygons (refer to block drawings in Appendix D: Figure 1 - 4). From the approximated dimensions, the areas and volume of the auditorium were then calculated. The materials in the Hill Auditorium hall were then approximated and their acoustic properties estimated, using standard material data tables. The theoretical equations were then employed to calculate various acoustical parameters of the auditorium as described in the design criteria section of the paper. The results are tabulated (refer to Appendix D: Table 1). Practical Analysis Examining the background information, test data and observations of Hill Auditorium, it seems clear that it was designed with very specific goals in mind. While it does serve as an excellent

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venue for many types of events, it excels most highly at being economically successful. Its size and shape allows for seating many people and it is larger than most other concert halls. In addition, its hemispherical shaped ceiling is highly efficient at reflecting the sound on stage into the audience. This means that even small and quiet ensembles can be heard. In addition, the loudness of sound is based on using the initial sound energy very efficiently, rather than building up sound with many reflections. This way, the reverberation time is kept fairly short as to preserve the clarity of music or spoken words. This design does make some sacrifices in the process, though. As mentioned before, the ITDG times are not optimal, leading to the sense of intimacy being sacrificed, especially for the seats on the main floor. Performers are also presented with a challenge as the shape of the hall is so effective in reflecting sound away from the stage that it is often difficult for members of an ensemble to hear each other.

Evaluation Plotting the reverberation times calculated with the Sabine equation and the reverberation times experimentally measured (refer to Appendix F: Figure 1), a reasonably good agreement can be seen. The two outlying values (500 Hz and 1000 Hz) in the experimental measurements (as previously discussed) distort the shape of the experimental curve and show less agreement with their corresponding calculated values. However, both ends of the curves are in very good agreement. The experimental measurements are also 0.3 - 0.4 sec longer than the calculated values.

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Conclusion

The theoretical and experimental analysis of Hill Auditorium provided us with insight into the unique acoustic qualities which have the greatest impact on the listening experience. Our findings on reverberation time explain why the hall brings a sense of liveliness to a musical performance. The Gmid value obtained (4.3 dB) also helps to explain how it is possible to hear a pin drop on stage while sitting in the very back of the hall. The only area where Hill Auditorium's acoustics were not as fine was in its quality of intimacy between audience and performer as the ITDG values obtained show a reasonable but less than optimal range. However, considering that Hill Auditorium is a large capacity hall, it is undoubtedly a marvel of architectural acoustics. For a further or more complete theoretical analysis, we would recommend obtaining detailed architectural drawings in order to find precise dimensions of the hall as well as employing the actual properties of the actual materials used in the auditorium. The experimental data could be made to agree more closely with concert conditions by performing tests while the hall was occupied rather than empty. In addition, the effect of room modes could be countered by measuring the reverberation times using many different source and microphone positions.

Some websites for additional information on Hill Auditorium, especially its recent renovation: UM Plant Extension Project Website for the Renovation of Hill Auditorium: http://www.plantext.bf.umich.edu/plantext/projects/Hill/index.html Hill Auditorium Reopening Website: http://www.umich.edu/%7Eurel/hill/index.html

20

Appendix A

Figure 1: Measured Gmid Values as a Function of EDT/V

21

Appendix B

MATLAB code % This code finds the loudness in dB of the signal passed into the function and plots a % graph of loudness (dB) vs. time (s). % Code for left and right channel are the same except for commented lines. function PlotLeftdB(sound); % function PlotRightdB(sound) for right channel S=sound(:,1); % S=sound(:,2) for right channel S=S.'; L=length(S); for i=401:400:L Smax(i-400:i)=max(S((i-400):i)); end S_dB=20*log10(Smax); SdBlength = length(S_dB); t=0:(1/44100):((SdBlength - 1)/44100); plot(t, S_dB); xlabel('Time (s)') ylabel('Loudness (dB)') title('Left Channel Decay') % title(`Right Channel Decay') for right channel

22

Appendix C

Figure 1: Microphone and Source Positions

23

Figure 2: Sample Decay Envelope

Figure 3: Sample Onset Envelope

24

Figure 4: Center Source, Microphone on Balcony

Figure 5: Source on Stage Left, Microphone on Balcony

25

Figure 6: Center Source, Microphone on Front Edge of Stage

Figure 7: Center Source, Microphone on Main Floor

26

Figure 8: Spectrum of White Noise

Figure 9: Spectrum of Steady State Response to White Noise

27

Figure 10: Spectrogram Showing Decay of White Noise

28

Appendix D

Figure 1: Main Floor

Figure 2: Mezzanine

29

Figure 3: Balcony

Figure 4: Hall Structure

30

Table 1: Theoretical Analysis of Hall

Material Main Floor Stage Seating Back Wall Mezzanine Underside Mezzanine Seating Back Wall Balcony Underside Balcony Seating Back Wall Ceiling Shell Floor Area ft^2 6840 10968 1170 3360 125 Hz 0.15 1026 0.49 5374.3 0.013 15.21 0.013 43.68 0.49 5062.7 0.013 16.38 0.013 41.86 0.49 9387.4 0.013 10.764 0.18 559.26 0.013 401.73 0.02 24 0.36 3888 250 Hz 0.11 752.4 0.66 7238.9 0.015 17.55 0.015 50.4 0.66 6819.1 0.015 18.9 0.015 48.3 0.66 12644 0.015 12.42 0.06 186.42 0.015 463.53 0.06 72 0.44 4752 500 Hz 0.1 684 0.8 8774.4 0.02 23.4 0.02 67.2 0.8 8265.6 0.02 25.2 0.02 64.4 0.8 15326 0.02 16.56 0.04 124.28 0.02 618.04 0.14 168 0.31 3348 1000 Hz 0.07 478.8 0.88 9651.8 0.03 35.1 0.03 100.8 0.88 9092.2 0.03 37.8 0.03 96.6 0.88 16859 0.03 24.84 0.03 93.21 0.03 927.06 0.37 444 0.29 3132 12566 1047198 25851 1.9849 2.1834 1.310242 1.441266 0.852297 1.34947 33076 1.5513 1.7065 Seats 37505 1.3681 1.505 4555.7 40973 1.2523 1.3776 52776 0.9723 1.0695 74172 0.6918 0.761 2000 Hz 0.06 410.4 0.82 8993.8 0.04 46.8 0.04 134.4 0.82 8472.2 0.04 50.4 0.04 128.8 0.82 15710 0.04 33.12 0.02 62.14 0.04 1236.1 0.6 720 0.39 4212 39794 4000 Hz 0.07 478.8 0.7 7677.6 0.05 58.5 0.05 168 0.7 7232.4 0.05 63 0.05 161 0.7 13411 0.05 41.4 0.02 62.14 0.05 1545.1 0.65 780 0.25 2700

Wood Upholstered Plaster Plaster

Upholstered Plaster Plaster

10332 1260 3220

Upholstered Plaster Glass Plaster Carpet Concrete

19158 828 3107 30902 1200 10800

Air Absorption Volume Total Absorption RT EDT RT_mid EDT_mid G_mid BR

31

Appendix E

Figure 1: Comparison of Calculated and Measured Reverberation Times

32

Bibliography

Ballou, G. and Howard W. Sams & Co. (1987). Handbook for sound engineers : the new audio cyclopedia. Indianapolis, Ind., H. W. Sams. Egan, M. D. (1988). Architectural acoustics. New York, McGraw-Hill. Kinsler, L. E. (2000). Fundamentals of acoustics. New York, Wiley. Mankovskii, V. S. (1971). Acoustics of studios and auditoria. New York, Communication Arts Books. Rossing, Thomas F., Richard Moore and Paul Wheeler (2002). The science of sound. San Francisco, Addison Wesley. Mehta, M., J. Johnson, et al. (1999). Architectural acoustics : principles and design. Upper Saddle River, N.J., Prentice Hall. Pfeiffer, Scott. Head Acoustician for Hill Auditorium Renovations. Kirkergaard and Assosciates. Email Interview. 09 April 2004 Rettinger, M. (1977). Acoustic design and noise control. New York, Chemical Pub. Co.

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