Physics 136 / Music 126 Duke University Fall 2012 Handout 18

Some Formulae Related to Room Acoustics

Derivation of the Sabine Equation

Given: A = room total absorption (sq ft), S = room total surface area (sq ft), V = room volume (cu ft), c = velocity of sound in air = 1130 ft/sec, and approximate mean free path between reflections in a compact room m = 4V/S (ft)

[Aside: this approximation for m was made by Sabine. In 1947, calculations by A. E. Bate et al. confirmed its accuracy for a variety of compact shapes, as reported in Proc. Phy. Soc. 59, 535-541. This relationship means that the mean free path will equal 2/3 of an edge for a cubic room, 4/3 the radius for a spherical space, and vary from 2/3 to twice the smallest dimension of a rectangular room.]

Quantities to be derived: a = average absorption coefficient, d = average decay per reflection (dB/reflection), R = average reflection rate (reflections/sec), r = average decay rate for sound (dB/sec), and T = reverberation time for a 60 dB decay (sec).

Overall approach: From total absorption and total room surface area determine the average absorption coefficient and the average dB decrease (SPL decay) per reflection. Multiplying the latter by the average reflection rate will yield the average decay rate. The time required for any specified range of decay (e.g. 60 dB) can then be determined.

average absorption coefficient:

a = A/S

average decay per reflection: On average at each reflection: the fraction of energy lost is a and the fraction of energy reflected is (1 - a). So the ratio of reflected sound pressure to incident sound pressure is the square root of (1 - a). And the average change in SPL is

d = 10 log(1/(1 - a)) dB/reflection

average reflection rate: R = (distance traveled in 1 sec, i.e. velocity of sound)/(average distance between reflections, i.e. mean free path)

R = c/m reflections/sec

average decay rate: r = (average reflection rate)(average decay per reflection)

r = Rd = (c/m) 10 log (1/(1 - a)) = (cS/4V) 10 log (1/(1 - a)) dB/sec

now, adopting foot units for length, we plug in a value for c and note that, for small a, -2.3 log(1 - a) = a:

r = 1230 (S/V) (-2.3 log (1 - a)) = 1230 Sa/V = 1230 A/V dB/sec

reverberation time: for a 60 dB decay, the time required will be

T = 60/r = .049 V/A sec.

Normal Modes of a Rectalinear Room

The standing wave modes in a pipe closed at both ends have frequencies given by

where n = 1,2,3,. . . is an integer index, v the speed of sound in air, and L the length of the pipe. For these normal modes in one dimension, the index n is simply the harmonic number. The sound pressure amplitude at a distance x from the end of the pipe is proportional to

Now, since

every normal mode of the pipe has pressure antinodes at x = 0 and x = L and, for the nth normal mode, the number of pressure nodes within the pipe is equal to n.

Now consider a room whose surfaces are all rectangles that intersect at right angles. The normal modes of this three-dimensional room are a straightforward generalization of the one-dimensional pipe, with three indices -- n_x, n_y, and n_z, one for each spatial dimension -- instead of a single harmonic number. All three indices must be specified in order to identify a particular normal mode. These indices may take the value zero or any positive integer, but at least one of the three must be non-zero in order to define a normal mode. The length, width, and height of the room replace the single L of the one-dimensional pipe.

We can note immediately that the lowest frequency mode will have a one as its only non-zero index, the index corresponding to the longest dimension of the room. The sound pressure amplitude for any normal mode at any point (x,y,z) in the room is proportional to

Refer to the diagram above -- a room that is L_x wide,L_y deep, and L_z tall. The symmetry of such a rectalinear room makes it easy to predict the form and locations of the pressure antinodes for many of its normal modes. The rules and examples that follow can be verified with the above equation for P.

Normal modes for which all three indices are non-zero have pressure antinodes only at isolated points. If any one of the indices is zero, then the pressure antinodes occur along lines within the room. If two of the three indices are zero, then the pressure antinodes of the associated mode form planes. The diagrams below show examples of the latter two cases. Each diagram is a view perpendicular to the x-y plane -- think of them as a views looking down toward the floor of the room. The indices n_x,n_y,n_z are indicated below each diagram in that order. On each diagram, vertical pressure antinode planes are shown as solid red lines, vertical pressure antinode lines as red dots, and vertical pressure node planes as dashed green lines. Since the n_z index is zero in each of these examples, there are no pressure variations along the vertical direction.

For the 2,0,0 mode, then, the pressure antinodes form three vertcial y-z planes at the surface of each side wall and midway between them. For the 1,1,0 mode the pressure antinodes are four vertical lines at the corners of the floor and ceiling, and for the 2,1,0 mode there are two additional such lines midway the long walls. Note that, in analogy with the one-dimensional pipe case, the value of the index for each of the spatial dimensions corresponds to the number of pressure nodes along that direction (shown here in green).

It is instructive as well to notice how the pressure antinodes of many different normal modes tend to be concentrated at certain locations within such a room. At any of the eight corners -- for instance at the point labeled with a red dot and the number 1 in our first diagram, with coordinates (x,y,z) = (0,0,0) -- the value of P will be 1 for all combinations of the indices. This means that every normal mode of the room has a pressure antinode in each corner of the room. This means, for instance, that if one wants to put sound energy into the available normal modes of a room with maximum efficiency, a corner is the place to do it. Many years ago Paul Klipsch developed a series of loudspeaker designs to exploit this fact [they also exploited the walls near the corner as a final flare, giving the designs very impressive effective horn sizes]. A corner is also an excellent microphone location if your primary goal is to accurately sample all of the room's normal modes.

Verify for yourself that, similarly, the middle point of any edge of the room is a pressure antinode for half of all the room's normal modes. (The light blue point 2 in our diagram, for example, at coordinates (0,0,L_z/2). In this case P is 0 if n_z is odd and 1 if n_z is even.) By the same arguments, 1/4 of all normal modes have pressure antinodes at the center of the floor, ceiling, or any wall (for example, the green point 3 at (L_x/2, L_y/2,0)), and 1/8 of all normal modes have pressure antinodes at the center of the room (our purple point 4 at (L_x/2,L_y/2,L_z/2). The latter, of course, is a favorite location for public address loudspeakers in many rectalinear high school basketball venues, where the room's normal modes are the last place you want much of the sound energy to go!

Reverberation Time in a Rectalinear Room:
The Fitzroy Equation

The elegant simplicity of the well known Sabine equation for reverberation time,

is based on the assumption that acoustic conditions are very homogenous throughout the room. V is the volume of air in the room, and A_tot the total sound absorption of the room's surfaces. c is a constant that is 0.049 if foot units are used for V and A, and 0.161 if meter units are used. The total absorption typically is summed over the individual surface areas and their coefficients of absorption:

Note that the absorption coefficients, and the total absorption and reverberation time calculated from them, are frequency dependent.

Various departures from the Sabine equation's underlying assumptions require various modifications to it. One common example is a rectalinear room in which the absorption is disproportionate for sound moving along certain directions. A prime example would be a classroom with a relatively absorptive "acoustical" tile ceiling and carpet on the floor but highly reflective plaster walls. In that example most of the absorption would be concentrated on sound moving vertically and the Sabine equation would significantly underestimate the reverberation in the horizontal plane. A relatively simple extension to the Sabine equation, called the Fitzroy equation, deals with this complication:

Here the surfaces perpendicular to the three axes of the room are analyzed separately and their effects combined in a way that takes the room's geometry into account. Specifically,

and similarly for absorbing surfaces perpendicular to the y and z axes.

Note that if all of the surfaces share the same average absorption coefficient then the Fitzroy equation reduces to the Sabine equation.

Reflecting Planes and Image Sources

The red dot in each of the four diagrams above is a source radiating sound equally in all directions. If the source is located in front of a flat reflecting wall, as shown from the side in the second diagram, the specular reflections will appear to come from an image of the source (shown as a blue dot) an equal distance behind the wall. If there are two, perpendicular flat reflective surfaces nearby (a floor and a wall, as shown in the third diagram, or two intersecting walls) there will be three images. (If you look into two mirrored walls at right angles you'll see three reflections of yourself; one behind each wall and one behind the crease between them.) If, as in our fourth diagram, the sound source is located near a corner, i.e. the intersection of three mutually perpendicular reflecting planes, the sound will seem to come from seven images as well as the source itself. (Continuing the mirrored wall -- and ceiling! -- analogy, there'd be images behind each of the three flat surfaces, images behind each of the three intersecting creases, and a final image beyond the corner itself.)

If the source is close to the wall(s) in each case and we listen from a relatively great distance, the sound that is radiated evenly in all directions in the first case, will be directed evenly to one half of all directions in the second case, one fourth of all directions in the third, and only one eighth of all directions in the final case. The dashed purple lines in the diagrams indicate a "wavefront" for the sound at a radius r from the source, and take the form of a sphere in the first case, and then 1/2, 1/4, and 1/8 of a sphere in the other examples. The following table summarizes for each case the number of apparent sources (including images), the sound pressure at radius r, the sound pressure level (and sound intensity level), the intensity, the surface over which the sound is distributed at radius r, the sound power, and rhe power level. All these quantities are described relative to the first case.

Sources + Images	1	2	4	8
Sound Pressure at r	P	2P	4P	8P
Sound Level	0 dB	+6 dB	+12 dB	+18 dB
Intensity at r	I	4I	16I	64I
Surface at r	A	A/2	A/4	A/8
Sound Power	W = IA	2W = 4IA/2	4W = 16IA/4	8W = 64IA/8
Power Level	0 dB	+3 dB	+6 dB	+9 dB

Spending some time with this table is a good way to develop and/or check your understanding of a variety of topics covered in this course.