In this laboratory exercise we will investigate some of the acoustic behavior of rooms. We will calculate, observe, and map the lowest few standing wave modes in the laboratory classroom. We will also calculate and measure the reverberation time in McKinnon Theater (a small auditorium in the GMI Academic Building). In addition to these measurements we will observe some of the good and bad properties of an acoustic environment
The indoors acoustic environment introduces boundaries which reflect sound. If the boundaries completely reflect all incident sound without any absorption then the resulting sound field is termed diffuse or reverberant. In a diffuse sound field the time average of the mean square sound pressure is the same everywhere through the enclosure. The flow of energy is equally probable in all directions. If the boundaries absorb some of the incident sound and reflect the rest, then the sound field is called semireverberant. Energy flows in more than one direction. Much of the energy is truly diffuse, though there are regions of the sound field that have a definable direction of propagation from the noise source. Semireverberant fields are the most widely encountered in the majority of architectural acoustic environments.
When a sound source is enclosed the radiated sound energy is retained within the enclosure. If the boundaries are perfectly reflective then the sound energy inside the enclosure could theoretically grow until a pressure is reached that would be explosive. Fortunately, most realistic boundaries are at least partly absorbing (air also absorbs sound) and the kinds of sound sources usually encountered in a room (eg., human speech) are not extremely powerful. For example the sound power produced by human speech is very small. In his book Speech and Hearing in Communication Harvey Fletcher states that it would take " . . . 500 people talking continuously for one year to produce enough energy to heat a cup of tea." Typical male and female speakers generate 34 µW and18 µW, respectively, at a distance of 3.28 ft. So, common sound sources are not excessively powerful, the sound energy in the enclosure travels about the enclosure and slowly decays as it is absorbed by the boundaries and the medium.
What are the essential parameters of a typical room necessary to determine its acoustical behavior? First, an enclosed space has an internal volume V. Second, it has a total boundary surface area S. Third, each of the individual surface areas has an absorption coefficient $\alpha$. The average absorption coefficient, $\bar{\alpha}$, for all surfaces together is given by
$$ \bar{\alpha} = {s_1 \alpha_1 + s_2 \alpha_2 + \cdots + s_n \alpha_n \over S} \EQN alpha $$
where $s_{1,2,\dots,n}$ are the individual surface areas, $\alpha$1,2, . . . ,n are the individual absorption coefficients of the individual surface areas, and S is the total boundary surface area of the enclosure. In Lab Exercise #4 you became familiar with one method of determine the absorption coefficient of a material. The table at the end of this write-up lists absorption coefficients for some common building materials (see also Table 2.5.1 on page 68 of Wilson {\sl Noise Control}). Finally, the room will possess a reverberation time RT60. This is the time in seconds required for the steady state sound level to drop 60 dB after the sound source has been turned off. Since the boundaries of the room reflect incident sound energy, the sound signal received by a listener at some location in the room will consist of sound which arrives directly from the source, sound which arrives after reflecting from one surface, and sound which has undergone several reflections. The average distance between reflections in such a space is called the mean free path and is related to the dimensions of the room by
$$ \hbox{MFP} = 4 {V \over S}$$
Each of the signals arriving at the listener's ear will have experience more attenuation at some frequencies than at others due to spreading, absorption, reflection, refraction, and diffraction. The resulting sound signal is very different than the pure direct sound signal which one would hear in a free field.
\FFig{soundpaths}(a) shows the paths of the direct sound and several reflections in a typical room. Plotting the amplitude of a short-duration signal as a versus time yields a chart like that shown in \Fig{soundpaths}(c). This chart clearly shows that the direct sound arrives at the listener's ears first. After a time gap, reflections from the walls, ceiling, stage enclosure, and other surfaces arrive in rapid succession. The initial time delay between the direct sound and the first reflected sound, and the initial reverberation consisting of the first few reflections are very important in the perception of the acoustical quality of a room. In fact, the relation of the direct sound to early reflections is more important to the perceived quality of a room than is the reverberant sound field level.
\FFig{largeroom}(a) is an idealized acoustic response chart for a large room, showing the direct sound, a time gap, the arrival of early reflections, and the ensuing reverberant sound field. The graphs in \Fig{largeroom}(b) and (c) show measurements representing an excellent listening environment.
When a source of sound is started in a live room, the reflections from the walls produce a sound energy distribution that becomes increasingly uniform with time. Eventually, except close to the source or to the absorbing surfaces, the distribution of energy may be assumed to be completely uniform, and the direction of energy flow at a specific location in the room may be considered to be random.
Let Pr be the spatially averaged effective pressure amplitude of the reverberant sound field. The acoustic energy density $\xi$ in the room may then be expressed as (math involves integrating over a hemisphere)
$$ \xi = {P_r^2 \over \rho c^2} . \EQN energydensity $$
The fundamental differential equation governing the growth of sound in a room is
$$ \Pi = V{d\xi \over dt} + \xi {c\,\bar{\alpha}S \over 4} .\EQN growth $$
This equation states that the rate at which energy is absorbed by the surfaces ($\xi {\bar{\alpha}S c \over 4}$) plus the rate ($V{d\xi \over dt}$) at which it increases throughout the room must equal the rate 
at which it is being produced ( is the power produced by the source). Assuming that the source starts at time t=0, the solution of this differential equation may be expressed, in terms of the average pressure, as
$$ P_r^2 = {4 \Pi \rho c \over \bar{\alpha} S}[1 - e^{-{c\,\bar{\alpha}S \over 4V}\,t}] \EQN growth-Pr $$
This growth has the form shown in \Fig{growth-decay}(a). For a small room the time required to achieve a diffuse sound field may be on the order of 50 ms, while for a large auditorium it may approach 1 s.
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The differential equation governing the decay of uniformly diffuse sound in a reverberant room is obtained by setting $\Pi \!=\!0$ in \Eq{growth}. If the sound source is turned off at time $t\!=\!0$ the pressure amplitude at any later time is
$$ P_r^2 = P_r^2(0) e^{-{c \,\bar{\alpha}S \over 4V}\,t} . \EQN decay-Pr $$
\FFig{growth-decay}(b) shows the log of the average pressure as the sound energy in the room decays. The decrease in the sound pressure level is
$$\eqalign{ \triangle \hbox{SPL} &= 10 \log({P_r^2 \over P_r^2(0)}) = 10 \log({e^{-c\, \bar{\alpha}S \over 4V}t}) = 10 ({-c \,\bar{\alpha}S \over 4V}t) \log (e) \cr &= - 4.35 ({c \bar{\alpha}S \over 4V})\ t \cr} \EQN SPL-1 $$
The reverberation time, RT60, is defined as the time required for the sound pressure level to drop by 60 dB. From \Eq{SPL-1} this becomes
$$ RT_{60} = 13.8 ({4V\over c\, \bar{\alpha}S }) = {55.2 V \over c\,\bar{\alpha}S} . \EQN RT-60 $$
This expression for the reverberation time was derived around 1895 by the father of modern architectural acoustics, Clement Sabine. Since the absorption coefficients of absorbing materials vary with frequency, the reverberation time in a room will also vary with frequency. Usually, the reverberation time is measured in octave bands centered at 125, 250, 500, 1000, 2000, and 4000 Hz. If no frequency is specified with a reverberation time measurements then it is usually understood to refer to 500 Hz (Sabine's original work was restricted to a single frequency of 512 Hz). One possible correction to \Eq{RT-60} accounts for the additional absorption provided by the air in the room. This air absorption may become significant at higher frequencies in large rooms, and may account for the majority of the absorption in extremely reverberant rooms. Sabine's reverberation time may be corrected as
$$ RT_{60} = {55.2 V \over c\, \bar{\alpha} S + 4 m V}. \EQN RT-60cor $$
For relative humidities h (in percent) between 20 and 70%, and for frequencies f between 1.5 and 10 kHz, the constant m may be approximated by
$$ m = 5.5 \times 10^{-4} \, (50/h)(f/1000)^{1.7}. $$
There are two basic methods for measuring the reverberation time of a large room. One method uses an impulsive noise source (pistol, canon, balloon). The other method uses a continuous noise source which is turned off at some specified time. The sound level in the room is measured as a function on a frequency analyzer or on a chart recorder. The slope of the sound level as a function of time may then be used to determine the reverberation time from
$$ RT_{60} = 60 dB (\triangle \hbox{time} \over \triangle \hbox{SPL}) .$$
Manfred Schroeder defined the large-room frequency (FL) to the the frequency above which a large number of room modes will be excited to vibrate at the source frequency:
$$ F_L = 2000 \sqrt{RT_{60} \over V} .\EQN FL $$
An alternate calculation, with almost identical results, is
$$ F_L = {3 (\hbox{velocity of sound}) \over \hbox{Room's smallest dimension}}. $$
\FFig{FL} shows the transition from small room to large room (fc in the figure is the same as FL). The large-room frequency marks the transition between large and small rooms. For physically small rooms FL is in the neighborhood of 250-500 Hz. For installation of sound systems, the large-room frequency has a lower limit of 80 Hz for speech and 30 Hz for music. A reverberation time of 1.6 sec is approximately the decay time for a minimum density sound field. Using \Eq{FL} then, a large-room volume is approximately 1000 m3 for speech and 7100 m3 for music.
The primary difference between small rooms and large rooms is that the sound field in a large room is dominated by the statistical reverberant field whereas the sound field in a small room is dominated by resonant room modes. In fact, a reverberant sound field does not exist in a small room because the sound energy is not able to develop a diffuse, uniformly random distribution. \FFig{small-big} shows clearly the difference in the decay of a sound field in a small and large room. Since there is no reverberant field in a small room, random placement of absorbing material will not improve the acoustic quality of the room. Instead, placement must coincide with resonant room modes, and must consider reflected paths. Helmholtz resonators are also very effective at knocking out specific room modes.
The natural frequencies of the standing waves in a rectangular room with dimensions $L_x \times L_y \times L_a$ are given by
$$ f_{n_x,n_y,n_z} = {c \over 2} \sqrt{(n_x \over L_x)^2 + (n_y \over L_y)^2 + (n_z \over L_z)^2} , \EQN room-mode-freq $$
where nx,ny,nz = 0,1,2,3, . . . , and c is the speed of sound in the room. There are three designations or classifications of room modes.
There are attached figures at the end of this lab write-up which show some representations of these mode shapes.
