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03 

OU 158930 >m 



OSMANIA UNIVERSITY LIBRARY 

Call No. <T?>4 . %f//S ///? Accession No. 
Author 
Title 
This book shduld be returned on or before the date last marked below. 



ACOUSTICS AND ARCHITECTURE 



ACOUSTICS 

AND 

ARCHITECTURE 



BY 

PAUL E. SABINE, PH.D. 

Riverbank Laboratories 



FIRST EDITION 



McGRAW-HILL BOOK COMPANY, INC. 

NEW YORK AND L.ONDON 
1932 



COPYRIGHT, 1932, BY 
McGRAW-HiLL BOOK COMPANY, INC. 



PRINTED IN THE UNITED STATES OF AMERICA 

All rights reserved. This book, or 

parts thereof, may not be reproduced 

in any form without permission of 

the publishers. 



THE MAPLE PRESS COMPANY, YORK, PA. 



PREFACE 

The last fifteen years have sen a rapidly growing 
interest, both scientific and popular, in the subject of 
acoustics. The discovery of the thermionic effect and the 
resulting development of the vacuum tube have made 
possible the amplification and measurement of minute 
alternating currents, giving to physicists a powerful new 
d?vice for the quantitative study of acoustical phenomena. 
As a result, there have followed remarkable developments 
in the arts of communication and of the recording and 
reproduction of sound. These have led to a demand for 
increased knowledge of the principles underlying the 
control of sound, a demand which has been augmented 
by the necessity of minimizing the noise resulting from the 
ever increasing mechanization of all our activities. 

Thus it happens that acoustical problems have come to 
claim the attention of a large group of engineers and tech- 
nicians. Many of these have had to pick up most of their 
knowledge of acoustics as they went along. Even today, 
most colleges and technical schools give only scant instruc- 
tion in the subject. Further, the fundamental work of 
Professor Wallace Sabine has placed upon the architect 
the necessity of providing proper acoustic conditions In 
any auditorium which he may design. Some knowledge 
of the behavior of sound in rooms has thus become a neces- 
sary part of the architect's equipment. 

It is with the needs of this rather large group of possible 
readers in mind that the subject is here presented. No 
one can be more conscious than is the author of the lack of 
scientific elegance in this presentation. Thus, for example, 
the treatment of simple harmonic motion and the develop- 
ment of the wave equation in Chap. II would be much 



vi PREFACE 

more neatly handled for the mathematical reader by the 
use of the differential equation of the motion of a particle 
under the action of an elastic force. The only excuse for 
the treatment given is the hope that it may help the non- 
mathematical reader to visualize more clearly the dynamic 
properties of a wave and its propagation in a medium. 

In further extenuation of this fault, one may plead the 
inherent difficulties of a strictly logical approach* to tho 
problem of waves within a three-dimensional space whose 
dimensions are not great in comparison with the wave 
length. Thus, in Chap. Ill, conditions in the steady state 
are considered from the wave point of view; while in Chap. 
IV, we ignore the wave characteristics in order to handle 
the problem of the building up and decay of sound in room&. 
The theory of reverberation is based upon certain simplify- 
ing assumptions. An understanding of these assumptions 
and the degree to which they are realized in practical cases 
should lead to a more adequate appreciation of the precision 
of the solution reached. 

No attempt has been made to present a full account 
of all the researches that have been made in this field in 
very recent years. Valuable contributions to our knowl- 
edge of the subject are being made by physicists abroad, 
particularly in England and Germany. If undue promi- 
nence seems to be given to the results of work done in this 
country and particularly to that of the Riverbank Labora- 
tories, the author can only plead that this is the work 
about which he knows most. Perhaps no small part of 
his real motive in writing a book has been to give per- 
manent form to those portions of his researches which in 
his more confident moments he feels are worthy of thus 
preserving. 

Grateful recognition is made of the kindness of numerous 
authors in supplying reprints of their papers. It is also a 
pleasure to acknowledge the painstaking assistance of 
Miss Cora Jensen and Mr. C. A. Anderson of the staff of 
the Riverbank Laboratories in the preparation of the 
manuscript and drawings for the text. 



PREFACE vii 

In conclusion, the author would state that whatever 
is worth while in the following pages is dedicated to his 
friend Colonel George Fabyan, whose generous support 
and unfailing interest in the solution of acoustical problems 
have made the writing of those pages possible. 

P. E. S. 

RIVEBBANK LABORATORIES, 
GENEVA, ILLINOIS, 
July, 1932. 



CONTENTS 

PAGE 

PREFACE v 

CHAPTER I 
INTRODUCTION 1 

CHAPTER II 
NATURE AND PROPERTIES OF SOUND 12 

CHAPTER III 
SUSTAINED SOUND IN AN INCLOSURE 33 

CHAPTER IV v 
REVERBERATION THEORETICAL 47 

CHAPTER V 
REVERBERATION EXPERIMENTAL 66 

CHAPTER VI - 
MEASUREMENT OF ABSORPTION COEFFICIENTS 86 

CHAPTER VII - 
SOUND ABSORPTION COEFFICIENTS OF MATERIALS . ... 127 

CHAPTER VIII . 
REVERBERATION AND THE ACOUSTICS OF ROOMS 145 

CHAPTER IX 
ACOUSTICS IN AUDITORIUM DESIGN 171 

CHAPTER X 
MEASUREMENT AND CONTROL OF NOISE IN BUILDINGS . . 204 

CHAPTER XI 
THEORY AND MEASUREMENT OF SOUND TRANSMISSION. . . 232 

CHAPTER XII 
TRANSMISSION OF SOUND BY WALLS 253 

CHAPTER XIII 
MACHINE ISOLATION 282 

APPENDICES 307 

INDEX 323 

ix 



ACOUSTICS 

AND 

ARCHITECTURE 

CHAPTER I 
INTRODUCTION 

Historical. 

Next to mechanics, acoustics is the oldest branch of 
physics. 

Ideas of the nature of heat, light, and electricity have 
undergone profound changes in the course of the experi- 
mental and theoretical development of modern physics. 
Quite on the contrary, however, the true nature of sound 
as a wave motion, propagated in the air by virtue of its 
elastic properties, has been clearly discerned from the very 
beginning. Thus Galileo in speaking of the ratio of a 
musical interval says: "I assert that the ratio of a musical 
interval is determined by the ratio of their frequencies, 
that is, by the number of pulses of air waves which strike 
the tympanum of the ear causing it also to vibrate with the 
same frequency." In the "Principia," Newton states: 
" When pulses are propagated through a fluid, every particle 
oscillates with a very small motion and is accelerated and 
retarded by the same law as an oscillating pendulum." 
Thus we have a mental picture of a sound wave traveling 
through the air, each particle performing a to-and-fro 
motion, this motion being transmitted from particle to 
particle as the wave advances. On the theoretical side, 
the study of sound considered as the physical cause of the 
sensation of hearing is thus a branch of the much larger 
study of the mechanics of solids and fluids. 

1 



2 ACOUSTICS AND ARCHITECTURE 

Branches of Acoustics. 

On the physical side, acoustics naturally divides itself 
into three parts: (1) the study of vibrating bodies including 
solids and partially inclosed fluids; (2) the propagation of 
vibratory energy through elastic fluids; and (3) the study 
of the mechanism of the organ of perception bj; means of 
which the vibratory motion of the fluid medium is aftle to 
induce nerve stimuli. There is still another branch of 
acoustics, which involves not only the purely physical 
properties of sound but also the physiological and psycho- 
logical aspects of the subject as well as the study of sound 
in its relation to music and speech. 

Of the three divisions of purely physical acoustics, the 
study of the laws of vibrating bodies has, up until the last 
twenty-five years, received by far the greatest attention of 
physicists. The problems of vibrating strings, of thin 
membranes, of plates, and of air columns have all claimed 
the attention of the best mathematical minds. A list of the 
outstanding names in the field would include those of 
Huygens, Newton, Fourier, Poisson, Laplace, Lagrange, 
Kirchhoff, Helmholtz, and Rayleigh, on the mathematical 
side of the subject. Galileo, Chladni, Savart, Lissajous, 
Melde, Kundt, Tyndall, and Koenig are some who have 
made notable experimental contributions to the study of 
the vibrations of bodies. The problems of the vibrations 
of strings, bars, thin membranes, plates, and air columns 
have all been solved theoretically with more or less com- 
pleteness, and the theoretical solutions, in part, experiment- 
ally verified. It should be pointed out that in acoustics, 
the agreement between the theory and experiment is 
less exact than in any other branch of physics. This is due 
partly to the fact that in many cases it is impossible to set 
up experimental conditions in keeping with the assumptions 
made in deriving the theoretical solution. Moreover, the 
theoretical solution of a general problem may be obtained 
in mathematical expressions whose numerical values can 
be arrived at only approximately. 



INTRODUCTION 3 

Velocity of Sound. 

Turning from the question of the motion of the vibrating 
body at which sound originates, it is essential to know the 
changes taking place in the medium through which this 
energy is propagated. The first problem is to determine the 
velocity with which sound travels. The theoretical solu- 
tion of the problem was given by Newton in 1687. Starting 
with the assumption that the motion of the individual 
particle of air is one of pure vibration and that this motion 
is transmitted with a definite velocity from particle to 
particle, he deduced the law that the speed of travel of a 
disturbance through a solid, liquid, or gaseous medium is 
numerically equal to the square root of the ratio of the 
volume elasticity to the density of the medium. The 
volume elasticity of a substance is a measure of the resist- 
ance which the substance offers to compression or dilata- 
tion. Suppose, for example, that we have a given volume 
V of air under a given pressure and that a small change of 
pressure BP is produced. A small change of volume 
dV will result. The ratio of the change of pressure to the 
change of volume per unit volume gives us the measure of 
the elasticity, the so-called " coefficient of elasticity " of the 
air 



Boyle's law states a common property of all gases, 
namely, that if the temperature of a fixed mass of gas 
remains constant, the volume will be inversely proportional 
to the pressure. This is the law of the isothermal expansion 
and contraction of gases. It is easy to show that under the 
isothermal condition, the elasticity of a gas at any pressure 
is numerically equal to that pressure; so that Newton's 
law for the velocity c of propagation of sound in air becomes 

/pressure _ JP 
> density > p 

The pressure and density must of course be expressed in 
absolute units. The density of air at C. and a pressure 



4 ACOUSTICS AND ARCHITECTURE 

76 cm. of mercury is 0.001293 g. per cubic centimeter. A 
pressure of one atmosphere equals 76 X 13.6 X 980 = 
1,012,930 dynes per square centimeter; and by the Newton 
formula the value of c should be 



c = Vqipni'ooQ = 27,990 cm./sec. = 918.0 ft./sec. 



The experimentally determined value of c is about 18 
per cent greater than this theoretical value given by the 
Newton formula. This disagreement between theoTy and 
experiment was explained in 1816, by Laplace, who pointed 
out that the condition of constant temperature under which 
Boyle's law holds is not that which exists in the rapidly 
alternating compressions and rarefactions of the medium 
that are set up by the vibrations of sound. It is a matter 
of common experience that if a volume of gas be suddenly 
compressed, its temperature rises. This rise of temperature 
makes necessary a greater pressure to produce a given 
volume reduction than is necessary if the compression takes 
place slowly, allowing time for the heat of compression to 
be conducted away by the walls of the containing vessel or 
to other parts of the gas. In other words, the elasticity of 
air for the rapid variations of pressure in a sound wave is 
greater than for the slow isothermal changes assumed in 
Boyle's law. Laplace showed that the elasticity for the 
rapid changes with no heat transfer (adiabatic compression 
and rarefaction) is 7 times the isothermal elasticity where 
7 is the ratio of the specific heat of the medium at constant 
pressure to the specific heat at constant volume. The 
experimentally determined value of this quantity for air is 
1.40, so that the Laplace correction of the Newton formula 
gives at 76 cm. pressure and C. 



yP 1. 40 X 1,012,930 

C = _ = - - 



= 1,086.2 ft./sec. (1) 

Table I gives the results of some of the better known 
measurements of the velocity of sound. 



INTRODUCTION 



Other determinations have been made, all in close 
agreement with the values shown in Table I, so that it may 
be said that the velocity of sound in free atmosphere is 
known with a fairly high degree of accuracy. The weight 
of all the experimental evidence is to the effect that this 
velocity is independent of the pitch, quality, and intensity of 
the sound over a wide range of variation in these properties. 
TABLE I. SPEED OF SOUND IN OPEN Am AT C. 



Velocity 






Authority 






m./sec. 


ft./soc. 




331 20 


1,086 7 


Bureau des Longitudes, 1822 


331 40 


1,087.3 


Regnault, 1864 


331.57 


1,087 5 


Szathmari, 1877 


330 13 


1,082 8 


Blaikley, 1884 


331 67 


1,087.8 


Biaikley, 1884 


331 36 


1,087 1 


Violle, 1900 


331 30 


1,087 


Hebb, 1904 



Effect of Temperature. 

Equation (1) shows that the velocity of sound depends 
only upon the ratio of the elasticity and density of the trans- 
mitting medium. This implies that the velocity in free air 
is independent of the pressure, since a change in pressure 
produces a proportional change in density, leaving the ratio 
of pressure to density unchanged. On the other hand, since 
the density of air is inversely proportional to the tempera- 
ture measured on an absolute scale, it follows that the velocity 
will increase with rising temperature. The velocity of sound 
c t at the centigrade temperature t is given by the formula 



c t = 331.2^: 



1 + 



273 



(2a) 



or, if temperature is expressed on the Fahrenheit scale, 



c t = 331.2^ 



1 + 



t - _32 
491 



(26) 



A simpler though only approximate formula for the velocity 
of sound between and 20 C. is 



6 ACOUSTICS AND ARCHITECTURE 

c t = 331.2 + 0.60* 

Velocity of Sound in Water. 

As an illustration of the application of the fundamental 
equation for the velocity of sound in a liquid medium, we 
may compute the velocity of sound in fresh water. The 
compressibility of water is defined as the change of volume 
per unit volume for a unit change of pressure. For water at 
pressures less than 25 atmospheres, the compressibility as 
defined above is approximately 5 X 10~ u c.c. per c.c. per 
dyne per sq. cm. The coefficient of elasticity as defined 
above is the reciprocal of this quantity or 2 XlO 10 . The 
density of water is approximately unity, so that the velocity 
of sound in water is 



c w 



fe /2 X 10 10 1>M Anr , , 
\/~ = \/ -- i = 141,400 cm./sec. 
* * 1 



Colladon and Sturm, in 1826, found experimentally a 
velocity of 1,435 m. per second in the fresh water of Lake 
Geneva at a temperature of 8 C. Recent work by the 
U. S. Coast and Geodetic Survey gives values of sound in 
sea water ranging from 1,445 to more than 1,500 m. per 
second at temperatures ranging from to 22 C. for depths 
as great as 100 m. Here, as in the case of air, the difference 
between the isothermal and adiabatic compressibility 
tends to make the computed less than the measured 
theoretical value of the velocity. 1 

The velocity of sound in water is thus approximately four 
times as great as the velocity in air, although water has a 
density almost eighty times that of air. This is due to the 
much greater elasticity of water. 

Propagation of Sound in Open Air. 

Although the theory of propagation of sound in a homo- 
geneous medium is simple, yet the application of this theory 
to numerous phenomena of the transmission of sound in the 

1 It is important to have a clear idea of the meaning of the term "elastic- 
ity" as denned above. In popular thinking, there is frequently encoun- 
tered a confusion between the terms "elasticity" and "compressibility." 



INTRODUCTION 1 

free atmosphere has proved extremely difficult. For 
example, if we assume a source of sound of small area set up 
in the open air away from all reflecting surfaces, we should 
expect the energy to spread in spherical waves with the 
source of sound as the center. At a distance r from the 
source, the total energy from the source passes through 
the surface of a sphere of radius r, a total area of 4?rr 2 . If E 
is the energy generated per second at the source, then the 
energy passing through a unit surface of the sphere would 
be E/^Trr* ; that is, the intensity, defined as the energy per 
second through a unit area of the wave front, decreases as 



7, 
6 

| 

Jo 



5 



/dhtp velocity* 



\Wtnct31o4MRl 



10 15 20 25 30 35 
Distance in Thousands of Feet 



40 45 



FIG. 1. Variation of amplitude of sound in open air with distance from sou roe. 
(AfterL. V. King.) 



the square of the distance r increases. This is the well- 
known inverse-square law of variation of intensity with 
distance from the source, stated in all the elementary text- 
books on the subject. As a matter of fact, search of the 
literature fails to reveal any experimental verification of 
this law so frequently invoked in acoustical measurements. 
The difficulty comes in realizing experimentally the con- 
ditions of "no reflection" and a " homogeneous medium." 
Out of doors, reflection from the ground disturbs the ideal 

Elasticity is a measure of the ability of a substance to resist compression. 
In this sense, solids and liquids are more elastic because less compressible 
than gases. 



8 



ACOUSTICS AND ARCHITECTURE 



condition, and moving air currents and temperature 
variations nullify our assumption of homogeneity of the 
medium. Indoors, reflections from walls, floor, and ceiling 
of the room result in a distribution of intensity in which 
usually there is little or no correlation between the intensity 
and the distance from the source. 

Figure 1 is taken from a report of an investigation on the 
propagation of sound in free atmosphere made by Professor 
Louis V. King at Father Point, Quebec, in 1913! l A 
Webster phonometer was used to measure the intensity 



t 

i t'O 

I 1 c 



1 

j 
i 

.1 






T 
















\ 


^ 


^-"' 


r _. 




-.,/ 




^...., 


^ 20 40 6 


o a 

Dis-t 
00 feet 


100 120 140 160 
omce 


r 







FIG. 2. Variation of amplitude of sound in an enclosure with distance from 

source. 

of sound at varying distances from a diaphone fog signal. 
The solid lines indicate what the phonometer readings 
should be, assuming the inverse-square law to hold. 
The observed readings are shown by the lighter curves. 
Clearly, the law does not hold under the conditions pre- 
vailing at the time of these measurements. 

Indoors, the departure from the law is quite as marked. 
Figure 2 gives the results of measurements made in a large 
armory with the Webster phonometer using an organ 
pipe blown at constant pressure as the source. Here 
the heavy curve gives what the phonometer readings would 
have been on the assumption of an intensity decreasing as 
the square of the distance increases. The measured 
values are shown on the broken curve. There is obviously 
little correlation between the two. The actual intensity 
does not fall off with increasing distance from the source 

1 Phil. Tram. Roy. $oc. London, Ser. A, vol. 218, pp. 211-293. 



INTRODUCTION 9 

nearly so rapidly as would be the case if the intensity were 
simply that of a train of spherical waves proceeding from a 
source ; and we note that the intensity may actually increase 
as we go away from the source. 

Acoustic Properties of Inclosures. 

The measurements presented in Fig. 2 indicate that the 
behavior of sound within an inclosure cannot, in general, 
be profitably dealt with from the standpoint of progressive 
waves in a medium. The study of this behavior constitutes 
the subject matter of the first part of " Architectural 
Acoustics/' namely, the acoustic properties of audience 
rooms. One may draw the obvious inference from Fig. 
2 that, within an inclosed space bounded by sound-reflect- 
ing surfaces, the intensity at any point is the sum of two 
distinct components: (1) that due to the sound coming 
directly from the source, which may be considered to 
decrease with increasing distance from the source according 
to the inverse-square law; and (2) that which results from 
sound that has been reflected from the boundaries of the 
inclosure. From the practical point of view, the problem 
of auditorium acoustics is to provide conditions such that 
sound originating in one portion of the room shall be easily 
and naturally heard throughout the room. It follows then 
that the study of the subject of the acoustic properties of 
rooms involves an analysis of the effects that may be pro- 
duced by the reflected portion of the total sound intensity 
upon the audibility and intelligibility of the direct portion. 

Search of the literature reveals that practically no 
systematic scientific study of this problem was made prior 
to the year 1900. In Winkelmann's Handbuch der Physik, 
one entire volume of which is devoted to acoustics, only 
three pages are given to the acoustics of buildings, with 
only six references to scientific papers on the subject. 
On the architectural side, we find numerous references 
to the subject, beginning with the classic work on archi- 
tecture by Vitruvius ("De Architectural- In these 



10 ACOUSTICS AND ARCHITECTURE 

references, we find, for the most part, only opinion, based 
on more or less superficial observation. Nowhere is there 
evidence either of a thoroughgoing analysis of the problem 
or of any attempt at its scientific solution. 

In 1900, there appeared in the American Architect a 
series of articles by Wallace C. Sabine at that time an 
instructor in physics at Harvard University giving an 
analysis of the conditions necessary to secure good hearing 
in an auditorium. This was the first study ever made 
of the problem by scientific methods. The state of 
knowledge on the subject at that time can best be shown 
by quoting an introductory paragraph from the first of 
these papers. 1 

No one can appreciate the condition of architectural acoustics 
the science of sound as applied to buildings who has not with a pressing 
case in hand sought through the scattered literature for some safe 
guidance. Responsibility in a large and irretrievable expenditure of 
money compels a careful consideration and emphasizes the meagerness 
and inconsistency of the current suggestions. Thus the most definite 
and often-repeated statements are such as the following : that the dimen- 
sions of a room should be in the ratio 2:3:5 or, according to some 
writers, 1:1:2, and others, 2:3:4. It is probable that the basis of these 
suggestions is the ratios of the harmonic intervals in music, but the 
connection is untraced and remote. Moreover, such advice is rather 
difficult to apply; should one measure the length to the back or to the 
front of the galleries, to the back or front of the stage recess? Few 
rooms have a flat roof where should the height be measured? One 
writer, who had seen the Mormon Temple, recommended that all 
auditoriums be elliptical. Sanders Theater is by far the best auditorium 
in Cambridge and is semicircular in general shape but with a recess 
that makes it almost anything; and, on the other hand, the lecture 
room in the Fogg Art Museum is also semicircular, indeed was modeled 
after Sanders Theater, and it was the worst. But Sanders Theater 
is in wood and the Fogg lecture room is plaster on tile; one seizes on this 
only to be immediately reminded that Sayles Hall in Providence is 
largely lined with wood and is bad. Curiously enough, each suggestion 
is advanced as if it alone were sufficient. As examples of remedies may 
be cited the placing of vases about the room for the sake of resonance, 
wrongly supposed to have been the object of the vases in Greek theaters, 
and the stretching of wires, even now a frequent though useless device. 

1 SABINE, WALLACE C., "Collected Papers on Acoustics," Harvard 
University Press, p. 1. Cambridge 1922. 



INTRODUCTION 11 

In a succeeding paragraph, Sabine states very succinctly 
the necessary and sufficient conditions for securing good 
hearing conditions in any room. He says: 

In order that hearing may be good in any auditorium, it is necessary 
that the sound should be sufficiently loud; that the simultaneous 
components of a complex sound should maintain their proper relative 
intensities; and that the successive sounds in rapidly moving articula- 
tion, either of speech or music, should be clear and distinct, free from 
each other and from extraneous noises. These three are the necessary, 
as they are the entirely sufficient, conditions fof~good hearing.^ The 
architectural problem is, correspondingly, threefold, and in this intro- 
ductory paper an attempt will be made to sketch and define briefly the 
subject on this basis of classification. Within the three fields thus 
defined is comprised without exception the whole of acoustics. 

Very clearly, Sabine puts the problem of securing good 
acoustics largely as a matter of eliminating causes of 
acoustical difficulties rather than as one of improving hear- 
ing conditions by positive devices. Increasing knowledge 
gained by quantitative observations and experiment during 
the thirty years since the above paragraph was written 
confirms the correctness of this point of view. It is to be 
said that during the twenty-five years which Sabine himself 
devoted to this subject, his investigations were directed 
along the lines here suggested. Most of the work of others 
since his time has been guided by his pioneer work in this 
field. In the succeeding chapters, we shall undertake to 
present the subject of sound in buildings from his point 
of view, including only so much of acoustics in general 
as is necessary to a clear understanding of the problems 
in the special field. 



CHAPTER II 



NATURE AND PROPERTIES OF SOUND 

We may define sound either as the sensation produced 
by the stimulation of the auditory nerve, or we may define 
it as the physical cause of that stimulus. For the present 
purpose, we shall adopt the latter and define sound as the 
undulatory movement of the air or of any other elastic 
medium, a movement which, acting upon the auditory 
mechanism, is capable of producing the sensation of hearing. 
An undulatory or wave motion of a medium consists of the 
rapid to-and-fro movement of the individual particles, this 
motion being transmitted at a definite speed which is deter- 
mined by the ratio of the elasticity to the density of the 
medium. 

Simple Harmonic Motion. 

For a clear understanding of the origin and propagation 
of vibrational energy through a medium, let us consider 

in detail, though in an elemen- 
tary way, the ideal simple case of 
the transfer of a simple harmonic 
motion (S.H.M.) as a plane 
wave in a medium. Simple 
harmonic motion may be defined 
as the projection of uniform 
circular motion upon a diameter 
of the circular path. 

Thus if the particle P (Fig. 3) 
is moving with a constant speed 
upon the circumference of a 
circle of radius A, and P f is moving upon a horizontal 
diameter so that the vertical line through P always passes 
through P' , then the motion of P' is simple harmonic motion. 

12 




FIG. 3. Relation of simple 
harmonic motion to uniform 
circular motion. 



NATURE AND PROPERTIES OF SOUND 13 

If the angular velocity of P is co radians per second, and 
time is measured from the instant that P passes through 
N, and JP', moving to the right, passes through 0, then 
the displacement of P f is given by the equation 

= A sin <at (3) 

The position of P' as well as its direction of motion is given 
by the value of the angle coL This angle is called the 
phase angle and is a measure of the phase of the motion 
of P f . Thus when the phase angle is 90 deg., 7r/2 radians, 
P f is at the point of maximum excursion to the right. 
When the phase angle is 180 deg., TT radians, P f is in its 
undisplaced position, moving to the left. When the phase 
is 360 deg., or 2?r radians, P' is again in its undisplaced 
position moving to the right. The instantaneous velocity 
of P f is the component of the velocity of P parallel to the 
motion of P' or, as is easily seen from the figure, 

= AOJ cos co = ylco sin f coZ + ^) (4) 

In simple harmonic motion, the velocity is 90 deg. in 
phase in advance of the displacement. 

Now the acceleration of the particle P moving with a 
uniform speed s, on the circumference of a circle of radius 
A, is s 2 /A = (Aco) 2 /A = Aco 2 . Since the tangential speed 
is constant, this acceleration must be always toward the 
center of the circle. The acceleration of the particle P' 
is the horizontal component of the acceleration of JP, 
namely, Aco 2 sin co. Let m be the mass of the particle 
P' y its acceleration, and F x the force which produces its 
motion. Then by the second law of motion 



F x = m = mAu 2 sin ut = mAw 2 sin (o)t + TT) (5) 

Comparing Eqs. (3) and (5), it appears that the force 
is directly proportional to the displacement but of opposite 
sign. Thus if P' is displaced toward the right, it is acted 
upon by a force toward the left, which increases as the 
displacement increases. The force F x always acts to 



14 



ACOUSTICS AND ARCHITECTURE 



restore the particle to its undisplaced position, and its 
magnitude is directly proportional to the displacement. 

Now the restoring forces called into play when any 
elastic body is subjected to strain are of just this type. 
Therefore, it follows that a particle moving under the 
action of an elastic restoring force will perform a simple 
harmonic motion. Further, it can be shown that the 
free movement of any body under the action of elastic 
forces only can be expressed as the resultant of a series of 
simple harmonic motions. 




Displacement Velocity Acceleration 

PARTICLE 
FIG. 4. Simple harmonic motion projected on a uniformly moving film. 

Suppose now that the motion of P' is recorded on a 
film moving with uniform speed at right angles to the 
vibration. The trace of the motion will be a sine curve. 

For this reason simple harmonic motion is spoken of 
as sinusoidal motion. If at the same time we could devise 
mechanisms that would record particle velocity and particle 
acceleration, the traces of these magnitudes on the moving 
film would be shown as in Fig. 4. 

The maximum excursion on one side of the undisplaced 
position, the distance A, is the amplitude of the vibration. 
The number of complete to-and-fro excursions per second 
is the frequency/ of vibration. Since one complete to-and- 
fro movement of P 1 occurs for each complete rotation of P 9 



NATURE AND PROPERTIES OF SOUND 15 

corresponding to 2?r radians of angular motion, then 

co = 2irf 

Energy of Simple Harmonic Motion. 

It is evident that, in the ideal case we have assumed, 
the kinetic energy of the particle P r is a maximum at 0, 
the position of zero displacement and maximum velocity. 
Here the velocity is the same as that of P, namely coA, 
and the kinetic energy is 

t^znax. = Hrae^ 2 - 27T 2 W/ 2 A 2 (6) 

At the maximum excursion, when the particle is momen- 
tarily stationary, the kinetic energy is zero. The total 
energy here is potential. At intermediate points the sum 
of the potential and kinetic energies is constant and equal 
to i^racoM. 2 . The average kinetic as well as the average 
potential energy throughout the cycle is one-half the maxi- 
mum or j^wwM. 2 or Trra/M. 2 . 1 The total energy, kinetic 
and potential, of a vibrating particle equals J^mco 2 A 2 or 



Wave Motion. 

Having considered the motion of the individual particle, 
let us trace the propagation of this motion from particle 
to particle as a plane wave, that is, a wave in which all 
particles of the medium in any given plane at right angles 
to the direction of propagation have the same phase. Let 
the wave be generated by the rapid to-and-fro movement 
of the piston moving in parallel ways and driven with 
simple harmonic motion by the " disk-pin-and-slot " mech- 
anism indicated in Fig. 5a. The horizontal component 
of the uniform circular motion of the disk is transmitted 
to the piston by the pin, which slides up and down in the 
vertical slot in the piston head as the disk revolves. We 
may consider that the disturbance is kept from spreading 

1 The mathematical proof of this statement is not difficult. The interested 
reader with an elementary knowledge of the calculus may easily derive the 
proof for himself. 



16 



ACOUSTICS AND ARCHITECTURE 



laterally by having the propagating medium confined 
in a tube so long that we need not consider what happens at 
the open end. Represent the undisturbed condition of 
the air by 41 equally spaced particles. Let the distance 
from to 40 be the distance the disturbance travels during 
a complete vibration of a single particle. 1 This distance is 



> n *TL K 3 K 4 D D D 

X\t //'....> .?.; f h < 



R,o 
40 




FIG. 6. 

Fia. 5a. Compreseional plane wave moving to the right. 

Fia. 56. Compressional plane wave reflected to the left. 

FIG. 6. Stationary wave resultant of Figs. 5a and 56. 

called the wave length of the wave motion and is denoted 
by the Greek letter X. 

The line A shows the positions of each of the particles 
at the instant that P > the first particle, having made a 
complete vibration, is in its equilibrium position and is 

1 According to the conception of the kinetic theory of gases, the molecules 
of a gas are in a state of thermal agitation, and the pressure which the gas 
exerts is due to this random motion of its molecules. Here for the purpose 
of our illustration we shall consider stationary molecules held in place by 
elastic restraints. 



NATURE AND PROPERTIES OF SOUND 17 

moving to the right. The first member of the family 
of sine curves shown is the trace on a moving film of the 
motion of P . PI having performed only thirty-nine- 
fortieths of a vibration is, at this instant, displaced slightly 
to the left. Curve 1 represents its motion during the 
interval that P performs the motion shown by curve 0. 
The phase difference between the motions of two adjacent 
particles is 27T/40 radians, or 9 deg. P 20 is 180 deg. in 
phase behind P and is in its undisplaced position and 
moving to the left. P 40 is 2?r radians or 360 deg. behind 
P 0; and the motions of the two particles coincide, P having 
performed one more complete vibration than P 40 - 

Lines B, C, D, E (Fig. 5a) give the positions of the 41 
particles *, 3^, %> and 1 period respectively later than 
those shown at A. 

Types of Wave Motion. 

It will be noted that the motion of the particles is in 
the line of propagation of the disturbance. This type of 
motion is called a compressioiial wave, and, as appears, such 
a wave consists of alternate condensations and rarefactions 
of the medium. This is the only kind of wave that can 
be propagated through a gas. In solids, the particle motion 
may be at right angles to the direction of travel, and the 
wave would consist of alternate crests and troughs and is 
spoken of as a transverse wave. As a matter of fact, in 
general when any portion of a solid is disturbed, both 
compressional and transverse waves result and the motion 
becomes extremely complicated. A wave traveling through 
the body of a liquid is of the compressional type. At the 
free surface of a liquid, waves occur in which the particles 
move in closed loops under the combined action of gravity 
and surface tension. 

Equation of Wave Motion. 

Equation (3) gives the displacement of a vibrating par- 
ticle in terms of the time, measured from the instant the 
particle passed through its undisplaced position, and of the 



18 ACOUSTICS AND ARCHITECTURE 

amplitude and frequency of vibration. The equation 
of a wave gives the displacement of any particle in terms 
of the time and the coordinates that determine the undis- 
placed position of the particle. In the case of a plane wave, 
since all the particles in a given plane perpendicular to the 
direction of propagation have the same phase, the distance 
of this plane from the origin is sufficient to fix the phase 
of the particle's motion relative to that of a particle located 
at the origin. Call this distance x. 

In Fig. 5a, consider the motion of a particle at a distance 
x from the origin P . Let c be the velocity with which 
the disturbance travels, or the velocity of sound. Then the 
time required for the motion at P to be transmitted the 
distance x is x/c. The particle at x will repeat the motion 
of that at P , x/c sec. later. The equation therefore for 
the particle at x, referred to the time when P is in its neutral 
position, is 

= A sin Jt - ^ = A sin 2*f(t - ^} (la) 
Similarly the velocity of the particle at x is 

- Aco cos Jt -~) = 27T.4/ cos 2<jrf(t - ~] (76) 
and the acceleration 
I = -4cu 2 sin w(t - ^) = -4rr 2 4/ 2 sin 2wf(t - ^j (7c) 

Equation (7o) is the equation of a plane compressional 
wave of simple-harmonic type traveling to the right, and 
from it the displacement of any particle at any time may 
be deduced. In the present instance we have assumed a 
simple harmonic motion of the particle at the origin. We 
might have given this particle any complicated motion 
whatsoever, and, in an elastic medium, this motion would 
be transmitted, so that each particle would perform this 
same motion with a phase retardation of a>x/c radians. In 
other words, the reasoning given applies to the general 
case of a disturbance of any type set up in an elastic medium 



NATURE AND PROPERTIES OF SOUND 19 

in which the velocity of propagation is independent of the 
frequency of vibration. 

Wave Length and Frequency. 

Now the wave length of the sound has been defined as 
the distance the disturbance travels during a complete 
vibration of a single particle, for example, the distance P 
to P 4 o (Fig- 5a). The phase difference between two 
particles one wave length apart is 2ir radians. Letting x 
= X, we have 



A = c (8) 

This important relationship makes it possible to compute 
the frequency of vibration, the wave length, or the velocity 
of sound if the two other quantities are known. Since c, 
the velocity of sound in air at any temperature, is known, 
Eq. (8) gives the wave length of sound of any given fre- 
quency. Table I of Appendix A gives the frequencies and 
wave lengths in air at 20 C. (68 F.) of the tones of one 
octave of the tempered and physical scales. 

The frequencies and wave lengths given are for the first 
octave above middle C (C 3 ). To obtain the frequencies 
of tones in the second octave above middle C we should 
multiply the frequencies given in the table by 2. In the 
octave above this we should multiply by 4, and in the next 
octave by 8, etc. For the octaves below middle C we 
should divide by 2, 4, 8, etc. 

Density and Pressure Changes in a Compressional Wave. 

In the preceding paragraphs, we have followed the pro- 
gressive change in the motion of the individual particles. 
Figure 5a also indicates the changes that occur in the con- 
figuration of the particles with reference to each other. 
Initially, there is a crowding together of the particles at 
F , with a corresponding separation at P 2 o- One-quarter 
of a period later, the maximum condensation is at PI O , 
and the rarefaction is at P 30 . At the end of a complete 



20 ACOUSTICS AND ARCHITECTURE. 

period, there is again a condensation at F - A wave length 
therefore as defined above includes one complete condensa- 
tion and one rarefaction of the medium. 

The condensation, denoted by the letter s, is defined as 
the ratio of the increment of density to the undisturbed 
density: 

.-* 

P 

Thus if the density of the undisturbed air is 1.293 g: per 
liter and that in the condensation phase of a particular 
sound wave is 1.294 g. per liter, the maximum condensation 
is 1/1,293. It is apparent that a condensation results 
from the fact that the displacement of each particle at any 
instant is slightly different from that of an adjacent particle. 
Referring once more to Fig. 5a, one sees that if all the 
particles were displaced by equal amounts at the same 
time there would be no variation in the spacing of the 
particles, that is, no variation in the density. It can be 
easily shown that in a plane wave the condensation is equal 
to minus the rate per unit distance at which the dis- 
placement varies from particle to particle. Expressed 
mathematically, 

s = Sp = -I* (9o) 

p OX ^ ' 

Differentiating Eq. (la) with regard to x, we have 

?)= (96) 



We thus arrive at the interesting relation that the conden- 
sation in a progressive wave is equal to the ratio of the 
particle velocity to the wave velocity. Further, it appears 
that the condensation is in phase with the velocity and 
7T/2 radians in advance of the particle displacement. 

At constant temperature, the density of a gas is directly 
proportional to the pressure. As was indicated in Chap. 
I, for the rapid alternations of pressure in a sound wave, 
the temperature rises in compression and the pressure 



NATURE AND PROPERTIES OF SOUND 21 

increases more rapidly than the density, so that the frac- 
tional change in pressure equals y (1.40) times the frac- 
tional change in density. Whence we have 

- * - *! < 

The maximum pressure increment, which may be called 
the pressure amplitude, is therefore 2iryPAf/c. 

Energy in a Compressional Wave. 

We have seen that the total energy, kinetic and potential, 
of a particle of mass m vibrating with a frequency / and 
amplitude A is 2ir 2 mA 2 f 2 . If there are N of these particles 
per cubic centimeter, the total energy in a cubic centimeter 
is 2Nmir 2 A 2 f 2 . The product Nm is the weight per cubic 
centimeter of the medium, or the density. The total 
energy per cubic centimeter is, therefore, 2w 2 pf 2 A 2 y of which, 
on the average, half is potential and half kinetic. 

The term " intensity of sound" may be used in two ways: 
either as the energy per unit volume of the medium or as 
the energy transmitted per second through a unit section 
perpendicular to the direction of propagation. The former 
would more properly be spoken of as the "energy density/' 
and the latter as the "energy flux." We shall denote the 
energy density by symbol / and the energy flux by symbol 
/. If the energy is being transmitted with a velocity of c 
cm. per second, then the energy passing in 1 sec. through 1 
sq. cm. is c times the energy per cubic centimeter, or 

J = cl = 27r 2 pA 2 f 2 c (11) 

Equation (11) and the expression for the pressure 
amplitude given above may be combined to give a simple 
relationship between pressure change and flux intensity. 

2 A 2 / 2 _ 2y 2 P 2 
~ 



_ _ 

J "" 2w 2 pA 2 f 2 ~c* ~ pc 3 

Using the relationship c = VyP/p, we have 



22 ACOUSTICS AND ARCHITECTURE 

Now it can be shown that the average value of the square 
of dP over one complete period is one-half the square 
of its maximum value. Hence, if we denote the square root 
of the mean square value of the pressure increment by p, 
Eq. (12) may be put in the very simple form 

J = 2! = P.' =J^ = Pl Q 3) 

pc VyPp Vep r ^ 

in which 

r = Vep and e = yP. 

The expression \/cp has been called the "acoustic resist- 
ance" of the medium. Table II of Appendix A gives the 
values of c and r for various media. 

Comparison of Eq. (13) with the familiar expression 
for the power expended in an electric circuit suggests the 
reason for calling the expression r, the acoustic resistance of 
a medium. It will be recalled that the power W expended 
in a circuit whose electrical resistance is R is given by the 
expression 



where E is the electromotive force (e.m.f.) applied to the 
circuit. In the analogy between the electrical -transfer of 
power and the passage of acoustical energy through a 
medium E, the e.m.f. corresponds to the effective pressure 
increment, and the electrical resistance to r, the "acoustic 
resistance" of the medium. The analogue of the electrical 
current is the root-mean-square (r.m.s.) value of the 
particle velocity. 

The mathematical treatment of acoustical problems from 
the standpoint of the analogous electrical case is largely due 
to Professor A. G. Webster, 1 who introduced the term 
"acoustical impedance" to include both the resistance and 
reactance of a body of air in his study of the behavior of 
horns. For an extension of the idea and its application 
to various problems, the reader is referred to Chap. IV 

1 Proc. Nat. Acad. Sci., vol. 5, p. 275, 1919. 



NATURE AND PROPERTIES OF SOUND 23 

of Crandall's " Theory of Vibrating Systems and Sound" 
and to the recently published " Acoustics" by Stewart and 
Lindsay. 

Equation (13) gives flux intensity in terms of the r.-m.-s. 
pressure change and the physical constants of the medium 
only. It will be noted that it does not involve the fre- 
quency of vibration. This leads to the very important fact 
that if we have an instrument that will measure the pressure 
changes, the flux intensities of sounds of different fre- 
quencies may be compared directly from the readings of 
such an instrument. For this reason, instruments which 
record the pressure changes in sound waves are to be 
preferred to those giving the amplitude. 

Temperature Changes in a Sound Wave. 

We have seen that to account for the measured velocity 
of sound, it is necessary to assume that the pressure and 
density changes in the air take place adiabatically, that is, 
without transfer of heat from one portion of the medium to 
another. This means that at any point in a sound wave 
there is a periodic variation of temperature, a slight rise 
above the normal when a condensation is at the point in 
question, and a corresponding fall in the rarefaction. The 
relation between the temperature and the pressure in an 
adiabatic change is given by the relation 

P+dP 



where 7 is the ratio of the specific heats of the gas, equal to 
1.40, and 6 is the temperature on the absolute scale. 
Now 60/0 will in any case be a very small quantity, 

*y 

and the numerical value of ^ is 3.44. Expanding the 

second member of (14) by the binomial theorem and 
neglecting the higher powers of 50/0 we have 



24 ACOUSTICS AND ARCHITECTURE 

Obviously the temperature fluctuations in a sound wave 
are extremely small too small, in fact, to be measurable; 
but the fact of thermal changes is of importance when we 
come to consider the mechanism of absorption of sound by 
porous bodies. 

Numerical Values. 

The qualitative relationships between the various phe- 
nomena that constitute a sound wave having been 'dealt 
with, it is next of interest to consider the order of magnitude 
of the quantities with which we are concerned. All values 
must of course be expressed in absolute units. For this 
purpose we shall start with the pressure changes in a sound 
of moderate intensity. At the middle of the musical 
range, 512 vibs./sec., the r.m.s. pressure increment of a 
sound of comfortable loudness would be of the order of 5 
bars (dynes per square centimeter), approximately five- 
millionths of an atmosphere. From Eq. (13) the flux 
intensity J for this pressure is found to be p* -f- 41.5 = 0.6 
erg per second per square centimeter or 0.06 microwatt per 
square centimeter. From the relation that J = HH; 2 max. 
we can compute the maximum particle velocity, which is 
found to be 0.17 cm. per second. This maximum velocity 
is 27T/A, and the amplitude of vibration is therefore 
0.000053 cm. A moment's consideration of the minuteness 
of the physical quantities involved in the phenomena of 
sound suggests the difficulties that are to be encountered 
in the direct experimental determination of these quantities 
and why precise direct acoustical measurements are so 
difficult to make. As a matter of fact, it has been only 
since the development of the vacuum tube as a means of 
amplifying very minute electrical currents that quantitative 
work on many of the problems of sound has been possible. 

Complex Sounds. 

In the preceding sections we have dealt with the case of 
sound generated by a body vibrating with simple harmonic 
motion. The tone produced by such a source is known as 



NATURE AND PROPERTIES OF SOUND 25 

pure tone, and the most familiar example is that produced by 
a tuning fork mounted upon a resonator. The phono- 
graphic record of such a tone would be a sinusoidal curve, as 
pictured in Fig. 3. If we examine records of ordinary 
musical sounds or speech, we shall find that instead of the 
simple sinusoidal curves produced by pure tones, the traces 
are periodic, but the form is in general extremely compli- 
cated. Figure 7 is an oscillograph record of the sound of a 
vibrating piano string, and it will be noted that it consists of 
a repetition of a single pattern. The movement of the 
air particles that produces this record is obviously not the 
simple harmonic motion considered above. 

However, it is possible to give a mathematical expression 
to a curve of this character by means of Fourier's theorem. 1 




FIG. 7. Osoillogram of sound from a piano string. (Courtesy of William Braid 

White.} 

In general, musical tones are produced by the vibrations 
either of strings, as in the piano and the violin, or by those 
of air columns, as in the organ and orchestral wind instru- 
ments. When a string or air column is excited so as to 
produce sound, it will vibrate as a whole and also in seg- 
ments which are aliquot parts of the whole. The vibration 
as a whole produces the lowest or fundamental tone, and the 
partial vibrations give a series of tones whose frequencies 
are integral numbers 2, 3, 4, etc., times this fundamental 
frequency. The motion of the air particles in the sound 
thus produced will be complex, and the gist of Fourier's 
theorem is that such a complex motion may be accurately 
expressed by a series of sine (or cosine) terms of suitable 

1 FOURIER, J. B. J., "La ThSorie Analytique de la Chaleur," Paris, 1822; 
CARSE and SHEARER, "Fourier's Analysis," London, 1915. 



26 ACOUSTICS AND ARCHITECTURE 

amplitude and phase. Thus the displacement of a point on 
the vibrating body at any time t is given by an equation of 
the form 



= AI sin (wt + (pi) + At sin 

A 3 sin (3co + <f>z) +, etc. 

That property of musical sounds which makes the difference 
between the sounds of two different musical instruments 
producing tones of the same pitch is called " quality " or 
"timbre," and its physical basis lies in the relative ampli- 
tudes of the simple harmonic components of the two com- 
plex sounds. 

Harmonic Analysis and Synthesis. 

The harmonic analysis of a periodic curve consists in 
the determination of the amplitude and phase of each 
member of the series of simple harmonic components. 
This may be done mathematically, but the process is 
tedious. Various machines have been devised for the 
purpose, the best known being that of Henrici, in 1894, 
based on the rolling sphere integrator. This and other 
mechanical devices are described in full in Professor Dayton 
C. Miller's book, "The Science of Musical Sounds," 1 to 
which the reader is referred for further details. 

The reverse process, of drawing a periodic curve from its 
harmonic components, is called "harmonic synthesis." A 
machine for this purpose consists essentially of a series of 
elements each of which will describe a simple harmonic 
motion and means by which these motions may be com- 
bined into a single resultant motion which is recorded 
graphically. Perhaps the simplest mechanical means of 
producing S.H.M. is that indicated in Fig. 5, in which a pin 
carried off center by a revolving disk imparts the component 
of its motion in a single direction to a member free to move 
back and forth in this direction and in no other. 

In his book, Professor Miller describes and illustrates 
numerous mechanical devices for harmonic analysis and 

1 The Macmillan Company, 1916. 



NATURE AND PROPERTIES OF SOUND 



27 



synthesis including the 30 element synthesizer of his own 
construction. In Fig. 8 are shown the essential features of 




FIG. Sa. Disc, pin and slot, and chain mechanism of the Riverbank harmonic 

synthesizer. 

the 40-element machine built by Mr. B. E. Eisenhour of the 
Riverbank Laboratories. 

The rotating disks are driven by a common driving shaft, 
carrying a series of 40 helical gears. Each driving gear 




FIG. 86. Helical gear drive of the Riverbank synthesizer 

engages a gear mounted on a vertical shaft on which is 
also mounted one of the rotating disks. The gear ratio 



28 ACOUSTICS AND ARCHITECTURE 

for the first element is 40:1, while that for the fortieth is 
1:1, so that for 40 rotations of the main shaft the disks 
revolve 1, 2, 3, 4 ... 40 times respectively. Each disk 
carries a pin which moves back and forth in a slot cut in a 
sliding member free to move in parallel ways. Each of 
these sliding members carries a nicely mounted pulley. 
The amplitude of motion of each sliding element can be 
adjusted by the amount to which the driving pin is set off 
center, and this is measured to 0.01 cm. by means of a % scale 
and vernier engraved on the surface of the disk. The phase 
of the starting position of each disk is indicated on a circular 




Fia. 9. Forty-element harmonic synthesizer of the Riverbank Laboratories. 

scale engraved on the periphery by reference to a fixed line 
on the instrument. It is thus possible to set to any desired 
values both the amplitude and the phase of each of the 40 
sliding elements. 

The combined motion of all the sliding members is trans- 
ferred to the recording pen, by means of an inextensible 
chronometer fusee chain, threaded back and forth around 
the pulleys and attached to the pen carriage. The back- 
and-forth motion of each element thus transmits to the 
pen an amplitude equal to twice that of the element. The 
pen motion is recorded on the paper mounted on a traveling 
table which is driven at right angles to the motion of the 
pen carriage by the main shaft by means of a rack-and- 



NA TURE AND PROPERTIES OF SOUND 



29 



pinion arrangement. The chain is kept tight by means of 
weights suspended over pulleys at each end of the syn- 
thesizer. An ingenious arrangement allows continuous 
adjustment of the length of table travel for 40 revolutions 
of the main shaft over a range of from 10 to 80 cm. ; so that a 
wave of any length within these limits may be drawn. 




Fio. 10. Analysis and re-synthesis of 30 elements of complex sound wave by 
harmonic synthesizer. Re-synthesis of 40 elements gives original curve. 

With this instrument we are able to draw mechanically 
any wave form, which may be expressed by an equation of 
the form 

= 2} A k sin (ko)t + <pk) 

Moreover, the instrument may be used as an analyzer 
to determine the amplitudes and phases of a Fourier series of 



30 ACOUSTICS AND ARCHITECTURE 

40 terms that will be an approximate representation of any 
given wave form as shown by Dr. F. W. Kranz. 1 

Figure 10 shows the simple harmonic components of an 
oscillograph record of the vowel sound 0, spoken by a 
masculine voice, as determined by Kranz's method of 
analysis. A resynthesis of these components reproduces 
the original wave form with an exactness that leaves no 
doubt as to the reliability of the instrument or the correct- 
ness of the analysis. The method thus affords a means of 
studying and expressing quantitatively the quality or 
timbre of musical sounds. 

Properties of Musical Sounds. 

A musical sound as distinguished from noise is charac- 
terized by having a fairly definite pitch and quality, sus- 
tained for an appreciable length of time. Pitch is expressed 
quantitatively by specifying the lowest frequency of vibra- 
tions of the sounding body. Quality is expressed by giving 
the relative amplitudes or intensities of the simple har- 
monic components into which the complex tone may be 
analyzed. In the recorded motion of the sounding body or 
of the air itself, the length of the wave is the criterion of the 
pitch of the sound, while the shape of the wave determines 
the quality. The record of a musical sound consists of a 
definite pattern repeated at regular intervals. 

Noise is sound without definite pitch characteristics and 
an indeterminate quality. On the physical side, the line 
of distinction between musical sounds and noises is not 
sharp. Many sounds ordinarily classified as noises will 
upon careful examination be found to have fairly definite 
pitches. Thus striking a block of wood with a hammer 
would commonly be said to make a noise. But if we assem- 
ble a series of blocks of wood of the proper lengths, we find 
that the tones of the musical scale can be produced by 
striking them, and we have the xylophone, though whether 
the xylophone is really a musical instrument is perhaps a 
matter of opinion. 

1 Jour. Franklin Inst., pp. 245-262, August, 1927. 



NATURE AND PROPERTIES OF SOUND 31 

Speech sounds are a mixture of musical sounds and 
noises. In the vowels, the musical characteristics pre- 
dominate, although both pitch and quality vary rapidly. 
The consonant sounds are noises that begin and terminate 
the sounds of the vowels. In singing and intoning, the 
pitch of each vowel is sustained for a considerable length 
of time, and only the definite pitches of the musical scale 
are produced that is, in good singing. 

For the most recent and complete treatment of this 
subject the reader should consult Dr. Harvey Fletcher's 
book on " Speech and Hearing" and Sir Richard Paget's 
" Human Speech." 

Summary. 

Starting with the case of a body performing simple 
harmonic motion, we have considered the propagation of 
this motion as a plane wave in an elastic medium. To 
visualize the physical changes that take place when a 
sound wave travels through the air, we fix our attention on a 
single small region in the transmitting medium. We see 
each particle performing a to-arid-fro motion through its 
undisplaced position similar to that of the sounding body. 
Accompanying this periodic motion is a corresponding 
change in its distance from adjacent particles, resulting in 
changes in the density of the medium and consequently a 
pressure which oscillates about the normal pressure. 
Accompanying these pressure changes are corresponding 
slight changes in the temperature. 

Viewing the progress of the plane wave through the air, 
we see all the foregoing changes advancing from particle 
to particle with a velocity equal to the square root of the 
ratio of the elasticity to the density. In time, each set of 
conditions is repeated at any point in the medium once in 
each cycle. In space, the conditions prevailing at any 
instant at a given point are duplicated at points distant 
from it by any integral number of wave lengths. 
. We have also seen that any complex periodic motion 
may be closely approximated by a series of simple harmonic 



32 ACOUSTICS AND ARCHITECTURE 

motions whose relative frequencies are in the order of 
1, 2, 3, 4, etc., and whose amplitudes and phase may be 
determined by a Fourier analysis of the curve showing the 
complex motion. The derivation of the relationship for 
a single S.H.M. may be considered as applying to the 
separate components of the complex sound. In other 
words, the propagation of a disturbance of any type in a 
medium in which the velocity is independent of the fre- 
quency will take place as in the simple harmonic * case 
treated. The assumption of a plane wave, while simplify- 
ing the mathematical treatment, does not alter the physical 
picture. For a more general and a more rigorous mathe- 
matical treatment, standard treatises such as Lord Ray- 
leigh's "Theory of Sound" or Lamb's "Dynamic Theory 
of Sound" should be consulted. Among recent texts, 
"Vibrating Systems and Sound," by Crandall; "A Text- 
book on Sound," by A. B. Wood; "Acoustics," by Stewart 
and Lindsay, will be found helpful. 



CHAPTER III 

SUSTAINED SOUND IN AN INCLOSURE 

In the preceding chapter, we have considered the 
phenomena occurring in a progressive plane wave, that is, a 
wave in which any particle of the medium repeats the 
movement of any other particle with a time lag between 
the two motions of x/c. Within an inclosure, sound 
reflection occurs at the bounding surfaces, so that the 
motion of any particle in the inclosure is the resultant of the 
motion due to the direct wave and to waves that have 
suffered one or more reflections. The three-dimensional 
case is complicated, and the general mathematical solution 
of the problem of the distribution of pressures and velocities 
throughout the space has not yet been effected. This 
distribution within a room is called the " sound pattern " 
or the " interference pattern," and the variation of intensity 
from point to point within rooms with reflecting walls is one 
of the chief sources of difficulty in making indoor acoustical 
measurements. 

Stationary Wave in a Tube. 

We may clarify our ideas as to the general sort of thing 
taking place with sound in an inclosed space by a detailed 
elementary study of the one-dimensional case of a plane 
wave within a tube closed at one end by a perfectly reflect- 
ing surface, that is, a surface at which none of the energy of 
the wave is dissipated or transmitted to the stopping barrier 
in the process of reflection. This condition is equivalent to 
saying that there is no motion of the barrier and correspond- 
ingly no motion of the air molecules directly adjacent to it. 

By Newton's third law of motion, the force of the reaction 
of the reflecting surface is exactly equal in magnitude and 
opposite in direction to the force under which the vibrating 

33 



34 ACOUSTICS AND ARCHITECTURE 

air particle adjacent to it moves. In other words, the 
reflected motion is the same as would be imparted to sta- 
tionary particles by a simple harmonic motion generator 
180 deg. out of phase with the motion in the direct wave. 
This reaction generator is indicated in Fig. 56. Its motion 
is given by the equation 

= -Asinut (16) 

Considered alone, this motion gives rise to a reflected wave, 
and the equation for the displacement in the reflected wave 

is 



= -Arin^ + ^ (17) 

It is to be noted that the sign of x/c is positive, since the 
reflected wave is advancing in the opposite direction from 
that in which x increases. The progress of the reflected 
wave considered alone is shown in Fig. 56. 

The resultant particle displacement due to the super- 
position of both the direct and the reflected waves is 

d+r = A[sin (* - Jj - sin (* + *JJ = 

-2^|cos wt sin 1 (18) 

In Fig. 6, the particle positions at quarter-period intervals 
are shown, each displacement being the resultant of the 
two displacements due to the direct and reflected waves 
shown in Figs. 5a and 56 respectively. One notes imme- 
diately that particles P , P-20, and P 40 remain stationary 
throughout the whole cycle, while particles Pi and P 30 
have a maximum amplitude of 2A. The first are the nodes 
of the "stationary wave 7 ' spaced at intervals of half a wave 
length. The second are the antinodes. At the nodes, 
there is a maximum variation with time in the condensation 
and hence in the pressure; while at the antinodes, there is 
no variation in the pressure. (Note that the distance 
between particles 9 and 11 is constant.) In a stationary 



SUSTAINED SOUND JN AN INCLOSURE 35 

wave, the nodes are points of no motion and maximum 
pressure variation; while at the internodes, there is maxi- 
mum motion and zero pressure change. It is to be observed 
further that within the half wave length between the nodes, 
all the particles move together, while corresponding particles 
on opposite sides of a node are at any instant equally 
displaced but moving in opposite directions. 

It is easy to see, both from the concept of the stationary 
wave as the resultant of two waves of equal amplitudes 
moving in opposite directions and also from the fact that 
all the particles between nodes move together, that there is 
no transfer of energy from particle to particle in either direc- 
tion, so that the energy flux in a stationary wave is zero. 

We may derive all these facts from consideration of 
Eq. 18. Thus for a given value of , the displacement 
varies as the sine of cox/c, being a maximum at the points 
for which wx/c = 2irx/\ = x/2, 37T/2, 5?r/2, etc., that is, at 
points for which x = X/4, 3X/4, 5X/4, etc. The displace- 
ment is always zero for all points at which sin &x/c = 0, 
that is, at values of x = 0, X/2, X, 3X/2, etc. 

The particle velocity is obtained by differentiating 
Eq. (18) with respect to the time 

= 2 Aco sin coZ sin (19) 

c 

while the condensation is given by differentiating with 
respect to x 

s = ~- = 2A- cos co cos (20) 

c/x c c 

The kinetic-energy density 

1 0)X 

U f = - P * = 2pA 2 co 2 sin 2 cousin 2 

= 8p7r 2 A 2 / 2 sin 2 co* sin 2 
c 

The maximum kinetic-energy density is at the mid- 
point between the nodes for values of i = J4, %, %, etc., 
times the period of one vibration, that is, where sin co and sin 



36 



ACOUSTICS .AND ARCHITECTURE 



are both unity; hence 



This, it will be noted, is four times the maximum kinetic 
energy in the direct progressive wave, a result which is to 
be expected, since the amplitude of the stationary wave at 
this point is twice that of the direct wave and the energy is 
proportional to the square of the amplitude. However, the 
kinetic-energy density of the stationary wave averaged Over 
an entire wave length may be shown to be only twice the 
energy of the direct wave. 
Total kinetic energy per wave length equals 



2pA 2 w 2 sin 2 a>t 



. O o># , 
sin 2 ax = 
c 



(21) 



The kinetic-energy density, averaged over a wave length, is 
A*pu 2 sin 2 cot. The maximum kinetic energy exists in the 
medium at the times when ut = ?r/4, 3?r/4, 5?r/4 and 
sin &t is unity. At these times, its value is pA 2 co 2 or twice 
the energy density of the direct wave. 

The results of the foregoing considerations may be 
summarized in the following tables of the analytical 
expressions for the various quantities involved in the 
progressive and stationary waves pictured in Figs. 5a 
and 6. 

TABLE II 





Progressive wave 


Stationary wave 


W^ave equation 


(x\ 
t I 


CtfiC 

= 2 A cos o)t sin 


Particle velocity 
Particle acceleration . . . 
Condensation 
Pressure 


c) 

= Aa) cos w( t J 
V c) 

t 38 ,Aco 2 sin.co( t J 
\ cj 

o> / X\ 
s = A- cos oj( t I 
c \ cj 


= 2 A to sin tat sin 
* c 

= 2Aw 2 cos u>t sin 

CO toX 

s 2 A- cos <at cos 
c c 

sp 9ruP/l W /Tkffl / 1/ ma 


Energy density 


I =**y 2 pA* 


C C 


Energy flux 


J = Y A z <a?c 


/ = 









SUSTAINED SOUND IN AN IN CLOSURE 37 

It should be said that the form of the expressions for the 
various quantities considered depends upon the convention 
adopted as to sign and phase of the motion at the origin. 
The convention here used is such' that in the condensation 
phase the particle motion coincides with the direction of 
propagation. 

Vibrations of Air Columns. 

Referring again to Fig. 6, we note that the particles at 
PO and at P 2 o under the action of the direct and reflected 
waves are at all times stationary. Accordingly, after the 
actual source has performed two complete vibrations, giving 
a complete wave down the tube to the reflecting end and 
back, thus setting up the column vibration, we may suppose 
the source removed and a rigid wall placed at the point 
PO (or P 2 o) which will reflect the particle motion, and, in the 
absence of dissipative forces, the air column as a whole will 
continue its longitudinal vibration indefinitely just as does 
a plucked string or a struck tuning fork. It is plain, 
therefore, that the term " stationary wave" is something 
of a misnomer. What we have is the compressional vibra- 
tion of the air column as a whole. Since the length of the 
vibrating column is one-half or any integral number of 
halves of the wave length, it appears that a given column of 
air closed at both ends may vibrate with a series of fre- 
quencies, 1, 2, 3, 4, etc., times the lowest frequency, that 
at which the length of the column is one-half the wave 
length. 

Algebraically, if m is any integer, / a possible frequency 
of vibration, and / the length of the inclosed column of air, 

7 1 , 1 c 
1 = 2 mX = 2 m ~f 

i - T, <m 

The series of frequencies obtained by giving successive 
integral values 1, 2, 3, 4, etc., to m are the natural fre- 
quencies or the resonant frequencies of the air column. 




38 ACOUSTICS AND ARCHITECTURE 

The point will be further considered in connection with the 
resonance in rooms. 

Closed Tube with Absorbent Ends. 

In the foregoing discussion of the standing wave set up 
by the reflection of a plain wave, we have assumed that none 

of the vibrational energy is 
dissipated in the process of 
reflection and also that there 
_ ""T: I7~7 ~ ! 77 " is no dissipation of energy in 

FIG. 11. Stationary wave in a tube with ^ ^ 

reflecting ends. the passage of sound along 

the tube. 

In Fig. 11, the maximum displacement of the particles 
is pictured at right angles to the direction of propagation, 
and the envelope of the excursions of the particles is 
shown. Suppose now that the ideal perfectly reflecting 
barrier is replaced by one 
at which a part of the in- 
cident energy is absorbed, 
so that the amplitude of 
the reflected wave is not A 

Fi<;. 12. Stationary wave m a tube with 
but kA, k being leSS than absorbent ends. 

unity. 

Referring to Eq. 18, we may write in the case of an 
absorbent barrier 




? (d+r) = A[sin co(< - 5) - k sin co(* + 

\(l - k) sin wt cos - (1 + k) cos *>* sin 1 (23o) 
L c c J 

= -2fc A cos art sin ^ -tA(l - *) sin u(t - -} (236) 



At the intcrnodes, cos is zero, and sin - = 1; hence 
J c c 



at these points the displacement amplitude is A(l + k} t 
while at the nodes cos 
amplitude is A (1 k). 



while at the nodes cos = 1, and sin = 0, and the 

c c 



SUSTAINED SOUND IN AN INCLOSURE 39 

We note from the form of Eq. (236) that the particle 
displacement due to the direct and reflected waves repre- 
sents a progressive wave of amplitude A (I k) super- 
imposed upon a stationary wave of amplitude 2k A. 
Thus the state of affairs set up in a tube with partially 
absorbing ends is not a true stationary wave, since there is a 
transfer of energy in the direct portion which represents 
the energy absorbed at the end of the tube. 

The foregoing is in a very elementary way the basis of 
the so-called " stationary-wave method" of measuring the 
sound-absorption coefficients of materials, first proposed 
and used by H. O. Taylor 1 and adopted in modified form at 
the Bureau of Standards and the National Physical 
Laboratory. The development of the theory of the method 
from this point on will be given in Chap. VI. 

Intensity Pattern in a Room. Resonance. 

We have considered somewhat at length the one- 
dimensional case of a sound wave in a tube with reflecting 
ends. If we extend the two dimensions which are at right 
angles to the axis of the tube, the tube becomes a room, and 
the character of the standing wave system becomes much 
more complicated. We no longer assume that the particle 
motion occurs in a single direction parallel to the axis of 
the tube. As a result, the simple distribution of nodes and 
loops in the one-dimensional case gives place to an intricate 
pattern of sound intensities, a pattern which may be 
radically altered by even slight changes in the position of 
reflecting surfaces in the room and of the source of sound. 
The single series of natural or resonance frequencies 
obtained for the one-dimensional case by putting integral 
values 1, 2, 3, etc., for m in Eq. (22) is replaced by a trebly 
infinite series with the number of possible modes -of vibra- 
tion greatly increased. The simplest three-dimensional 
case would be that of the cube. For the sake of com- 
parison, the first five natural frequencies of a tube 28 ft. 

1 TAYLOR, H. O., Phys. Rev., vol. 2, p. 270, 1913. 



40 ACOUSTICS AND ARCHITECTURE 

long and of a room 28 by 28 by 28 ft., assuming a velocity 
of sound of 1,120 ft. per second, are given. 1 

Tube Room 

20 20 

40 28.3 

80 34.7 

160 40 

320 44 6 

The term " resonance" is somewhat loosely used to mean 
the vibrational response of an elastic body to any periodic 
driving force. Thus, the enhancement of sound produced 
by the body of a violin or the sounding board of a piano over 
the sound produced by the string vibrating alone is usually 
spoken of as " resonance." With strict nomenclature we 
should use the term "forced vibration," reserving the 
term " resonance" to apply to the enhanced response of a 
vibrating body to a periodic driving force whose frequency 
coincides with the frequency of one of the natural modes 
of vibration of the vibrating body. We shall use the term 
in this latter sense. 

It is apparent that resonance in a room of ordinary 
dimensions, that is, the pronounced response of the inclosed 
volume of air to a tone of one particular frequency corre- 
sponding to one of its natural modes of vibration, cannot 
be very marked, since the frequencies of the possible modes 
of vibration are so close together, and that moreover these 
frequencies for the lower terms of the series are very low 
in actual rooms of moderate size. In small rooms, reso- 
nance may frequently be observed, but the phenomenon is 
not an important factor in the acoustic properties of rooms 
in general. 2 

Survey of Intensity Pattern. 

The mathematical solution of the problem of the distri- 
bution of sound intensities even in the simple case of a rec- 

1 The theoretical treatment of the problem of the vibration in a rectangular 
chamber with reflecting walls is given in Rayleigh's "Theory of Sound," 
vol. 2, p. 69, Macmillan & Co., Ltd., 1896. 

2 An interesting study of the effects of resonance in small rooms is reported 
by V. O. Knudsen in the July, 1932 number of the Journal of the Acous- 
tical Society of America. 



SUSTAINED SOUND IN AN INCLOSURE 



41 



tangular room, has not, to the writer's knowledge, yet been 
effected. There are clearly two distinct problems: first, 
assuming that there is a sustained source of sound within 
the room, in which case the solution would depend upon 
the location of the source and the degree to which its 
motion is affected by the reaction of the resulting stationary 
wave on the source; and second, assuming that the sta- 
tionary wave has been set up and the source then stopped. 
Stopping the source produces a sudden shift from one 
condition to the other, a fact which accounts for the fre- 
quently observed phenomenon of a sudden rise of intensity 
at certain places in a reverber- 
ant room with the stopping of 
the source of sound. 

In 1910, Professor Wallace 
Sabine made an elaborate series 
of experimental investigations 
of the sound pattern in the 
constant-temperature room of 
the Jefferson Physical Labora- 
tory. This room is wholly of 
brick, rectangular in plan, 23.2 
by 16.2 ft., with a cylindrical 
ceiling. The axis of curvature 
of the ceiling arch is at the Ji 

floor level. The height Of the distribution of sound intensity in 

room is 9.75 ft. at the sides and a r m ' 
12 ft. at the center of the arch. A detailed account of 
these experiments has never been given, although in a paper 
published in March, 1912, Professor Sabine gave the general 
results of this study and stated that the subject would be fully 
treated in a forthcoming paper "now about, ready for 
publication." This paper never appeared, probably for 
the reason that with his passion for perfection, Professor 
Sabine felt that the work was still incomplete and awaited 
opportunity for further study. From his notes of the 
period it is possible to give the experimental details of the 
investigation, and the importance of the results in the light 




42 ACOUSTICS AND ARCHITECTURE 

which they throw on the distribution of sound within an 
inclosure is offered as a justification for so doing. 

Figure 13 shows the experimental arrangement. The 
source of sound was an electrically driven tuning fork, 248 
vibs./sec., associated with a Helmholtz resonator mounted 
at a height of 4.2 ft. above the floor. The amplitude of 
vibration of the tuning fork was measured by projecting 
the image of an illuminated point on the fork on to a distant 
scale. An interrupted current of the same frequency as 
that of the fork was supplied from a direct-current source 
interrupted by a second fork of this same frequency. This 
current was controlled so as to give any desired amplitude 
of the fork by means of a rheostat in the circuit. 






FIG. 14, Record of sound amplitude at different points in a room. 

The intensity pattern was explored by means of a travel- 
ing telephone receiver associated with a resonator mounted 
on a horizontal arm, supported by a vertical shaft, which 
was driven at a uniform speed by means of a weight-driven 
chronograph drum belted to the vertical shaft. The 
exploring telephone was supported by a carriage which was 
pulled along the horizontal shaft by a cord wound up on a 
fixed sleeve around the vertical shaft as the latter turned. 
The exploring telephone thus described a spiral path in 
the room in a horizontal plane, the pitch of the spiral 
being the circumference of the fixed sleeve. 

The current generated in the exploring telephone passed 
through the silvered fiber of an Einthoven string dynamom- 
eter, the magnified image of which was focused by means 
of a magnifying optical system upon a moving film. This 
film was driven by a shaft geared to the rotating shaft 
bearing the exploring telephone which also carried a finger 
that opened a light shutter on to the moving film at each 
revolution of the vertical shaft, so as to mark the position 



SUSTAINED SOUND IN AN INCLOSURE 43 

f the exploring telephone corresponding to any point on 
he film. 

Figure 14 is a reproduction of one of the films. Dis- 
ances along the film are proportional to distances along the 
piral. The width of the light band produced by the image 




FIG. 15. Spiral paths followed in acoustical survey of a room. 



>f the dynamometer string gives the relative sound 
implitudes at points along the spiral. Employing two 
>arallel spirals of the same pitch gave a fairly fine-grained 
lurvey of the sound-amplitude pattern. These spiral 
>aths are shown on Fig. 15, while Fig. 16 shows a map of a 
lection of the room with the amplitudes noted at various 
>oints of the exploring spirals. In Fig. 17 we have the 



44 



ACOUSTICS AND ARCHITECTURE 



results expressed by a map of what may be called "iso- 
phonic" lines or lines of equal intensities after the manner 
of contour lines of a topographic map. 

The distribution shown is that in a single horizontal 
plane. To map out the sound field in a vertical plane, one 




FIG. 16. Relative amplitudes of sound in a room with a steady source. 

may work from spiral records taken in a series of horizontal 
planes. Figure 18 is a series of records made in planes at 
four different levels above the mouth of the source resona- 
tor. The upper series shows the sound pattern with no 
absorbent material present, and the lower series shows it 
when the floor of the room is covered with absorbent 
material. It is obvious that the distribution in three 



SUSTAINED SOUND IN AN INCLOSURE 



45 



dimensions does not admit of easy representation either 
graphically or mathematically and that the intensity at 
any point of the room is a function of the position of the 
source and of every reflecting surface in the room. For 
these reasons, intensity measurements made within rooms 
are extremely difficult of interpretation. The point will be 
further considered in Chap. VI in connection with the 
problem of the reaction of the 
room on the source of sound. 
We are now in a position to vis- 
ualize the condition that exists in 
a room in which a constant 
source of sound is operative. We 
have seen that by actual measure- 
ments the intensity of the sound 
does not fall off with the distance 
from the source according to the 
inverse-square law but that the 
intensity is much more nearly uni- 
form than would be predicted 
from this law. At any point and 

j. j j_i i i -t , it Fio. 17. Distribution of sound 

at any time, there is added to the amplitudes at points in a single 
direct sound from the source the horizontal plane. 
combined effect of waves emitted earlier, which arrive, 
after a series of reflections from the bounding surfaces, 
simultaneously at the given point. The intensity at any 
point is thus the resultant of a large number of separate 
waves and varies in a most complicated manner from point 
to point. It is an interesting experience to walk about an 
empty room in which a pure tone is being produced and note 
this point-to-point variation. It is not slight. With a 
fairly powerful source, movement of only a foot or so or 
even a few inches, with a high-pitched sound, will change 
the intensity from a very faint to a very intense sound. 

Moreover, this distribution shifts with changes in the 
positions of objects within the room and with any shift in 
the pitch of the sound. One may easily note the effect 
of the movement of a second person in an empty room by 




46 ACOUSTICS AND ARCHITECTURE 

the shift produced in the intensity pattern. Fortunately, 
however, this very complicated phenomenon is a rather 
infrequent source of acoustical difficulty, since most of 
the sounds in both music and speech are not prolonged in 










Sound Pcdtern With Absorbing Material 

FIG. 18. Sound patterns at four different levels. Upper figure is for the empty 
room, lower figure, with entire floor covered with sound-absorbent material. 

time or constant in pitch, so that in listening to music or 
speech in ordinary audience rooms, it is the direct sound 
which plays the predominating role. Only when one 
comes to make instrumental measurements of the intensity 
of pure sustained tones of constant pitch do interference 
phenomena become troublesome. Here the difficulties 
of quantitative determination are almost insuperable. 
It is for this reason that progress in the scientific treatment 
of acoustical properties of rooms has been made by a method 
which, on its face, seems almost primitive, namely, the 
reverberation method, which will be considered in the next 
chapter. 



CHAPTER IV 

REVERBERATION (THEORETICAL) 

In Chap. Ill, we have noted that with a sustained source 
of sound within an inclosure, there is set up an intensity 
pattern in which the intensity of the sound energy varies 
from point to point in a complicated manner and that in the 
absence of dissipative forces, the vibrational energy of the 
air within the inclosure tends to persist after the source has 
ceased. This prolongation of sound within an inclosed 
space is the familiar phenomenon of reverberation which 
has come to be recognized as the most important single 
factor in the acoustic properties of audience rooms. 

Reverberation in a Tube. 

We shall now proceed to the consideration of this 
phenomenon, first in the one-dimensional case of a plane 
wave within a tube and then in the more complicated 
three-dimensional case of sound within a room. In this 
consideration, the variation of sound intensity from point 
to point mentioned in the preceding chapter will be 
ignored, and it will be assumed that there is a uniform 
average intensity throughout the inclosure. It will further 
be assumed that the dissipation of the energy occurs 
wholly at the bounding surface of the inclosure and that 
dissipation throughout the volume of the inclosure is 
negligibly small, in comparison with the absorption at the 
boundaries. We shall first, following Sabine, consider 
that the dissipation of acoustic energy takes place con- 
tinuously and develop the so-called reverberation formula 
on this assumption and then give the analysis recently 
presented by Norris, Eyring, and others, treating both the 
growth and the decay of sound within an inclosed space as 
discontinuous processes. 

47 



48 ACOUSTICS AND ARCHITECTURE 

Growth of Sound Intensity in a Tube. 

In Fig. 19, we represent a tube of length I and cross 
section S with absorbent ends, whose coefficient of absorp- 
tion is a, and with perfectly reflecting walls. At one end 
we set up a source of sound which sends into the tube E 
units of sound energy per second. For simplicity we shall 




FIG. 19. Tube with partially absorbent ends and source at one end. 

assume that this is in the form of a plane wave train. The 
energy density of the direct wave that has undergone no 
reflections will be E/c. 
Let 

A = time required for sound to travel length of tube 
m = number of reflections per second = c/l = 1/A2 

We desire to find the total energy in the tube t( = n&t) 
sec. after the source was started. In the interval between 

El 
reflections, the source emits units of sound energy. 

c i 

The total energy in the tube therefore at the end of any 
interval nAt is this energy plus the residues from those 
portions of the sound which have undergone 1, 2, 3 . . . 
n 1 reflections respectively. Of the sound reflected 
El 

once, (1 a) units remain. From the twice-reflected 
c 

TjlJ 

sound, there is -^-(1 a) 2 ; and of the sound that has been 
c 

F*l 

reflected (n 1) times, the residue is (1 a) n " 1 . 

c 

Summing up the entire series, we have 

El 
Total energy = [1 + (1 - a) + (1 - a) 2 + (1 - a) 3 

- (1 - a)"- 1 ] 
Note that in the analysis given, we have tacitly assumed 



REVERBERATION (THEORETICAL} 



49 



that the sound is emitted discontinuously in instantaneous 
puffs or quanta of El/c units each and that it is absorbed 
instantaneously and discontinuousiy at the two ends of the 
tube. In Fig. 20, the broken line shows the building up of 
the intensity in the tube according to this analysis, for a 
value of a = 0.10. 

If we assume that n is very large that is, if sound is 
produced for a long time we may take the limiting value 




1,000 

900 

800 

XTOO 

5 600 

500 

c 400 

300 

200 

100 



1 2 3456 7 8 9 10 11 12 13 14 IS 16 17 ia 19 20 

vf 

FIG. 20. Growth of average intensity of sound in tube with partially absorbent 
ends. (1) Emission and absorption assumed continuous. (2) Discontinuous 
emission and absorption. 

of the sum of the series as n increases indefinitely, which is 

I/a, and we have 

JTJ 
Total energy in steady state = 

and for the steady-state energy density we have this total 
energy divided by the volume, El/caV. 

We may arrive at the same result for the average value 
of the steady-state intensity by assuming that the processes 
of emission and absorption are both continuous. 

The rate at which the intensity increases due to emission 
from the source is E/V. al is the energy absorbed at each 
reflection from an end of the tube, and mal is the energy 
absorbed per second. Call the net rate of intensity 
change dl/dt. The net rate of change of intensity is 
given by the equation 



dl 

'ft 



E 
F 



E 
V 



(24) 



50 ACOUSTICS AND ARCHITECTURE 

As / increases, the energy absorbed each second increases 
and approaches a final steady state in which absorption 
and emission take place at the same rate, and hence the 
change of intensity becomes vanishingly small, as time goes 
on. Call this final steady-state intensity /i; then 

E caI 1 ___ 
V~ I ~ 
whence 



Equation (24) is easily solved by integration. The 
solution gives the familiar form 



--- 

I = Ml -e l (25) 

or, expressed logarithmically, 



The continuous line in Fig. 20 shows the development of 
the average intensity upon the assumption of a continuous 
emission and absorption of sound energy at the source and 
the absorbing boundaries respectively. The broken line 
represents the state of affairs assuming that the sound is 
emitted instantaneously and absorbed instantaneously at 
the end of each interval. The height of each step repre- 
sents the difference between the total energy emitted 
during each interval and the total energy absorbed. The 
energy absorbed increases with time, since we have assumed 
that it is a constant fraction of the intensity. We note 
that the broken line and the curve both approach asymptot- 
ically the final steady-state value /i = El/acV. Further, 
it is also to be observed that neither the broken line nor 
the curve represents the actual condition of growth of 
intensity in the tube. The important point is that both 
approximations give the same value for the final steady 
state after the source has been operating for a long time. 



REVERBERATION (THEORETICAL) 51 

Decrease of Intensity. Sabine's Treatment. 

After the sound has been fed into the tube for a time 
long enough for the intensity to reach the final steady 
state, let us assume that the source is stopped. Consider 
first the case assuming that the absorption of energy at the 
ends of the tube takes place continuously. Then in Eq. 
(24), E 0, and we have, if T is the time measured from 
the moment of cut-off, 

ad _ dl 

T - ~ dT (27) 

with the initial condition that / = /i = El/<xcV, when 
T = 0. The solution of (27) gives 

ctcT 

I = / ie -~ r = I ie l (28a) 

or, in the logarithmic form, 

IOK. = ~ (286) 

It is well to keep in mind the assumptions made in the 
derivation of (28o) and (286), namely, that we have an 
average uniform intensity throughout the tube at the 
instant that the source ceases and that, although the dissi- 
pation of energy takes place only at the absorbing surfaces 
placed at the ends of the tube, yet the rate of decay of this 
average intensity at any time is the same at all points in 
the tube. With these assumptions, (28a) and (286) tell 
us that during the decay process, in any given time interval, 
the average intensity decreases to a constant fraction of 
its value at the beginning of this interval. 

Now the " reverberation time" of a room was defined by 
Sabine as the time required for the average intensity of 
sound in the room to decrease to one millionth of its 
initial value. Denote this by T Q , and we have for the tube, 

considered as a room, y- = log e 1,000,000 = 13.8 
or 

r. - 



52 ACOUSTICS AND ARCHITECTURE 

That is, the reverberation time for a tube varies inversely 
as the absorption coefficient of the absorbing surface 
and directly as the length of the tube. When we come to 
extend the analysis to the three-dimensional case, we shall 
see what other factors enter into this quantity. 

Decay Assumed Discontinuous. 

The foregoing is essentially the method of treatment, 
given by Wallace C. Sabine, of the problem of the growth 
and decay of sound in a room, applied to the simple one- 
dimensional case. Before proceeding to the more general 
three-dimensional problem, we shall apply the analysis 
given by Schuster and Waetzmann, 1 Eyring, 2 and Norris 3 
to the case of the tube in order to bring out the essential 
difference in the assumptions made and in the final equa- 



___ - 

a b 

Fia. 21. Tube with end reflection replaced by image sources. 

tions obtained. We shall follow Eyring in method, though 
not in detail, in this section. This analysis is based on the 
assumption that image sources may replace the reflecting 
and absorbing walls in calculating the rate of decay of 
sound in a room. 

Suppose that sound originating at a point P is reflected 
from a surface S. The state of affairs in the reflected 
sound is the same as though the reflecting surface were 
removed and a second source were set up at the point P', 
which is the optical image of the point P formed by the 
surface S, acting as a mirror. Hence, for the purposes of 
this analysis, we may think of the length of the tube simply 
as a segment without boundaries of an infinite tube, with 
a series of image sources that will give the same distribution 
of energy in the segment as is given by reflections in the 
actual tube. 

1 Ann. Physik, vol. 1, pp. 671-695, March, 1929. 

2 Jour. Acous. Soc. Amer., vol. 1, No. 2, p. 217. 

3 Ibid., p. 174. 



REVERBERATION (THEORETICAL) 53 

The energy in the actual tube due to the first reflection 
is the same as would be produced in the imagined segment 
by a source whose output is (1 a)E located at a distance 
/ to the right of 6 (Fig. 21). That due to the second reflec- 
tion may be thought of as coming from a second image 
source of power E(l a) 2 located at a distance 21 from a. 
The third reflection contributes the same energy as would 
be contributed by a source E(l a) 3 located a distance 31 
to the right of 6; and so on. The number of images will 
correspond to the number of reflections which we think of 
as contributing to the total energy in the tube. 

In the building-up process, we think of all the image 
sources as being set up at the instant the sound is turned on. 
Their contribution to the energy in Z, however, will be 
recorded in each case only after a lapse of time sufficient 
for sound to travel the distance from the particular image 
source to the boundaries of the segment. We thus have 
the discontinuous building-up process shown in the broken 
line of Fig. 20. The final steady state then would be that 
produced by the original source E and an infinite series of 
mage sources and would be represented by the equation 

El 
/! = [1 + (1 - ) + (1 - a) 2 + (1 - a ) 



Now when this steady state has been reached, the 
source of sound is stopped. All the image sources are 
stopped simultaneously, but the stoppage of any image 
source will be recorded in the segment only after a time 
interval sufficient for the last sound in the space between 
the image source and the nearer boundary of the tube to 
reach the latter. Thus the intensity measured at any 
point in the tube will decrease discontinuously. For 
example, suppose we measure the instantaneous value at 
the point 6, I meters from the source. When the source is 
stopped, the intensity at 6 will remain the constant steady- 
state intensity I\ for the time l/c required for the end of 



54 



ACOUSTICS AND ARCHITECTURE 



the train of waves from E to travel the length of the tube. 
At the end of this interval, the recording instrument at b 
will note the absence of the contribution from E, and at the 
same time it will note the drop due to the cessation of 
Ei = E(l a), which is equally distant from b. No 
further change will be recorded at this point until after the 
lapse of a second interval 2Z/c, at which time the effect of 




I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 

t'f 

FIG. 21a. Decay of sound in a tube. (1) Average intensity, absorption 
assumed continuous. (2) Intensity at source end, absorption discontinuous. 
(3) Intensity at end away from source, absorption discontinuous. 

stopping E 2 and E^ both at the distance 3Z meters from fc, 
will be recorded. A succession of drops, separated by 
time intervals of 2Z/c, will be recorded at b as the sound 
intensity in the segment decreases, as pictured in the solid 
line 3 of Fig. 21a. 

If instead of setting up our measuring instrument at 6, 
the most distant point from the source, we had observed 
the intensity at a, near the source, the first drop would have 
been recorded at the instant the source was turned off, and 
the history of the decay would be that shown by the broken 
line 2 of Fig. 21a. 

If now we could set up at the mid-point of the tube an 
instrument which by some magic could record the average* 
intensity in the tube, its readings would be represented by 
the series of diagonals of the rectangles formed by the 
solid and dotted lines of Fig. 21a. The reading of this 
instrument at the exact end of any time T = nl/c is 



REVERBERATION (THEORETICAL) 55 

1 = 1,- ^[1 + (1 - a) + (1 - a) 2 + (1 - a) 8 ... 

(1 - a)"-'] 
The sum of the series for a finite number of terms is 

1 (1 - )" _ 1 (1 )" 
1 - (1 - a) ~ " a 
and 

- - m " _ (i _ ).) /( i _ a)" 



and 



= (1 - a) (30) 



Taking the logarithm of both sides of Eq. (30), we have 
log* y = n log e (1 a) or 

rT 
log. 7! - log e 7 = -~- log. (1 - OL) (31) 

If, as in Eq. (29), we substitute for I\ and / the values 
of 1,000,000 and 1 respectively, we have 

- log e (1 - a) = 13.8 



Comparison of the Two Methods. 

We note that the analysis based upon the picture of the 
decay of sound as a continuous process and that considering 
it as a step-by-step process led to two different expressions 
for the reverberation time. The first contains the term 
a where the second has the term log e (1 a). Now for 
small values of a. the difference between a. and log c (1 a) 
is not great, as is shown in the following table, but increases 
with increasing values of a. In this same table we have 
given the number of reflections necessary to reduce the 
intensity in the ratio of 1,000,000:1 and the times, assuming 



56 



ACOUSTICS AND ARCHITECTURE 
TABLE III 



a 


- log, (1 - a) 


Percentage 


n for 
il - 10 


To = 

10 Q^ 


T = 
13.8K" 








j -10 


JLO.O 
Ct 


log, (1 - a) 


0.01 


010 




1,380 


10 00 


10.00 


0.02 


0.020 




690 


5 00 


5.00 


0.05 


0.051 


2.0 


271 


2.00 


1.96 


0.10 


105 


5 


131 


1.00 


0.95 


15 


0.163 


8 7 


85 


67 


0..61 


0.20 


223 


11 5 


62 


50 


0.45 


25 


288 


15.2 


48 


0.40 


35 


30 


0.357 


19 


39 


0.33 


28 


40 


0.511 


27.8 


27 


25 


195 


50 


093 


38 6 


20 


20 


0.145 



that the time is 10 sec., when the absorption coefficient of 
the ends of the tube is 0.01, first using the Sabine formula 
and second using the later formula developed by Eyring and 
others. 

We note in the example chosen, that while the percentage 
difference in the values of T Q computed by the two formulas 
increases with increasing values of the assumed coefficient 
of absorption, yet the absolute difference in T does not 
increase. 

From the theoretical point of view, it should be said 
that the later treatment is logically more rigorous in the 
particular example chosen. In this instance, we are dealing 
with a special case in which the sound travels only back and 
forth along a given line. Reflections occur at definite 
fixed intervals of time, so that, in both the growth and 
decay processes, the rate of change of intensity must alter 
abruptly at the end of each of these intervals. Further, 
we note that in the assumption of a continuously varying 
rate of building up and decay, we also assume that the 
influence of the absorbing surface affects the intensity of a* 
train of waves both before and after reflection. Thus 
when the source starts, the total energy actually in the tube 
increases linearly with the time, since the absorbing process 
does not begin until the end of the first interval. In 



REVERBERATION (THEORETICAL) 57 

setting down the differential equation, however,, we assume 
that absorption begins at the instant the sound starts. 
We arrive at the same final steady state by the two methods 
of analysis only by assuming that the source operates so 
long that the number of terms in the series may be con- 
sidered as infinite. But in the decay, we are concerned 
with the decrease of intensity in a limited time, and n is a 
finite number. The divergence in the results of the final 
equations increases with increasing values of a and cor- 
responding decrease in n. 

Growth of Sound in a Room : Steady State. 

In the case of the tube, we have assumed that the sound 
travels only back and forth as a plane wave along the tube, 
so that the path between reflections is /, the length of the 
tube. To take the more general case of sound in a room, 
we assume that the sound is emitted in the form of a spheri- 
cal wave traveling in all directions from the source, that it 
strikes the various bounding surfaces of the room at all 
angles of incidence, and that hence after a comparatively 
small number of reflections the various portions of the 
initial sound will be traveling in all directions. The 
distance traveled between reflections will not be any 
definite length. Its maximum value will be the distance 
between the two most remote points of the room, while its 
minimum value approaches zero. To form a picture of the 
two-dimensional case, one may imagine a billiard ball shot 
at random on a table and note the varying distance it will 
travel between its successive impacts with the cushions. 
If we think of the billiard ball as making a large number of 
these impacts and take the average distance traveled, we 
shall have a quantity corresponding to what has been called 
the "mean free path" of a sound element in a room. We 
can extend the picture of the building-up process in a tube, 
where the mean free path is the length of the tube, to the 
three-dimensional case, simply by substituting for I, the 
length of the tube, p, the mean free path of a sound element 
in the room, and for a the coefficient of the ends, a a , the 



8 ACOUSTICS AND ARCHITECTURE 

average coefficient of the boundaries. In the case of a 
room, then, with a volume F, a source of sound emitting E 
units of sound energy per second, and an average coefficient 
of absorption of , we should have by analogy the average 
steady-state intensity 

/i - -ifr (83a) 

where 



+ 2>S>2 + a 3 S 3 + etc. 

am = -- _ - _ 

i, <* 2 , 3, etc., are the absorption coefficients of the sur- 
faces whose areas are Si, S 2 , S 3 , etc. 

S is the total area of the bounding surfaces of the room, 
and the summation of the products in the numerator 
includes all the surfaces at which sound is reflected. This 
sum has been called the "total absorbing power" of the 
room and is denoted by the letter a. Hence 

a 

a a = s 

and Eq. (33a) becomes 



A further simplification is effected if we can express 
p, the mean free path, in terms of measurable quantities. 
As a result of his earlier experiments, Sabine arrived at a 
tentative value for p of 0.62 F 3 *. He recognized that this 
expression does not take account of the fact that the mean 
free path will depend upon the shape as well as the size of 
the room and, subsequently, as a result of experiment put 
his equations into a form in which p is involved in another 
constant k. Franklin first showed in a theoretical deriva- 
tion of Sabine's reverberation equation 1 that 

4F 



1 For the derivation of this relationship, see Franklin, Phys. Rev., vol. 16, 
p. 372, 1903; Jaeger, Wiener Akad. Ber. Math-Nature Klasse, Bd. 120, Abt. 
Ho, 1911; Eckhardt, Jour. Franklin Inst., vol. 195, pp. 799-814, June, 1923; 
Buckingham, Bur. Standards Sci. Paper 506, pp. 456-460. 



REVERBERATION (THEORETICAL) 59 

Putting this value of p in Eq. (336), we have 

/! = (33c) 

ac v x 

This expression for the steady-state intensity has been 
derived by analogy from the case of sound in the tube. 
Since the result of the analysis for the tube is the same 
regardless of whether we treat the growth of the sound as a 
continuous process or as a series of steps, Eq. (33c) gives the 
final intensity on either hypothesis. 

Decay of Intensity in a Room. Sabine's Treatment. 

We proceed to derive the equation for the intensity at 
any time in the decay process in terms of the time, measured 
from the instant of cut-off, in a manner quite similar to that 
used in the case of the tube. On the average, the number of 
reflections that will occur in each second is c/p. At each 
reflection, the intensity / is decreased on the average by the 

fraction <x a l = o/. The rate of change per second is then 

c a 
pS IoT 

dl _ ad _ acl 

dt ~ ~~ Sp ~ ~47 



Integrating, and using the initial condition that / = /i 
when T = 0, we have 

i acT f*A ^ 

= - (34a) 



or, in the exponential form, 



For the reverberation time T , the time required for the 
sound to decrease from an intensity of 1,000,000 to an 
intensity of 1, we have 



60 ACOUSTICS AND ARCHITECTURE 

1,000,000 acT Q 

log* -- J - = ^ 



(35) 



Taking the velocity of sound as 342 m. per second, we 
have K equal to 0.162, when a is expressed in square meters 
and V in cubic meters (or 0.0495 in English units). 
Sabine's experimental value is 0.164, a surprisingly close 
agreement when one considers the difficulties of precise 
quantitative determination under the conditions in which 
he worked. 

Process Assumed Discontinuous. 

We shall consider next the dying away of sound in a 
room, picturing the process from the image point of view. 
In the case of a room, where p y the mean free path, is a 
statistical mean of a number of actual paths, ranging in 
magnitude from the distance between the two most remote 
points of the room to zero, the picture of the decay taking 
place in discrete steps is not so easy to visualize as in the 
case of the tube already considered, where the distance 
between reflections is definitely the length of the tube. 
In his analysis, Eyring assumes that the image sources 
which replace the reflecting walls can be located in discrete 
zones, surfaces of concentric spheres, whose radii are 
p, 2p, 3p, etc., from the source. Then p/c is the time 
interval between the arrival of the sound from any two 
successive zones. The energy supplied by the source 
E in this interval is E&t = Ep/c and by the first zone of 

Ep Ev 

images is (1 ex a ). That by the second zone is .- 
c c 

(1 a ) 2 . Hence the total energy in the steady state 

* 

^[1 + (1 - ) + (1 - a a ) 2 (1 - .)-] = g (36) 
and 

T - J& 
11 ~ 



REVERBERATION (THEORETICAL} 61 

We then consider that the source is stopped and with 
it all the sources in the image zones. If our point of 
observation is taken near the source, the average intensity 
will drop by an amount Ep/cV at the instant the source 

Ep Ep 

stops, and there will be drops of 7^(1 ~~ <*)> ~7y 

(1 a ) 2 , etc., at the beginning of each succeeding p/c 
interval. The total diminution in the intensity at the 
end of the time T = np/c will be 

|?[1 + (1 - a a ) + (1 - a a ) 2 + (1 - aj' + 

(1 - a,,)"- 1 ] 
and we have 

Ep(\ - (1 - a a )\ 

7 = 7i - ^KT^O--- j y = /l(1 ~ a] = 

/,(! - a.)? 

Taking the logarithm of both sides of this equation, we 
have 

T i'T r^T 

log e y = -y log, (1 - a a ) - -~ log, (1 - a ) (37) 

For the reverberation time T = T Q , /i/7 = 1,000,000 
we have 

_ 13.8 X 4F 
' - - 



Comparing (35) and (38), we note, as in the case of 
the tube, that the final equations arrived at by the two 
analyses differ only in the fact that for a in the Sabine 

formula we have S log c ( 1 ]o ) in the later formula. 

We have also seen that for values of a a = a/S, less than 
0.10 the computed values of the reverberation time will 
be the same by the two formulas. We have also seen that 
while the percentage difference increases with increasing 
values of the average coefficient, yet due to the decrease 



62 ACOUSTICS AND ARCHITECTURE 

in the absolute value of the reverberation time the numer- 
ical difference does not increase proportionately. 

Eyring's argument in favor of the new formula is much 
more detailed and carefully elaborated than the foregoing, 
but one has the suspicion that as applied to the general 
case it may not be any more rigorous. Certainly in the 
case of an ordered distribution of direction and intensity 
of sound in an in closure, as would be the condition, for 
example, of sound in a tube, or from a source located at 
the center of a sphere, the assumptions of a diffuse distribu- 
tion and a continuously changing rate of decay do not hold. 
But in a room of irregular shape, where the individual 
paths between reflections vary widely, replacing the average 
reflections by average image sources all located at a fixed 
distance from the point of observation introduces a dis- 
continuity which does not exist in actuality. It is true 
that the earlier analysis implicitly assumes that each 
element of an absorbing surface effects a reduction in any 
particular train of waves both before and after reflection, 
whereas the later treatment recognizes that, viewed from 
an element of the reflecting surface, there will be a constant- 
energy flow toward the surface from any given direction 
during the time required for sound to travel the distance 
from the preceding point of reflection in this direction. 
The latter reasoning, however, is valid, on the assumption 
that the actual paths of sound from all directions may be 
replaced by the mean free path. The point at issue seems 
to be whether or not the introduction of the idea of the 
mean free path makes valid the concept of the decay as a 
continuous process and the use of the differential equation 
as a mathematical expression of it. For "dead" rooms 
where n y the total number of reflections in the time To, 
is small the older treatment is not rigorous. On the other 
hand, there is the question whether, in such a case, the 
assumption of a mean free path is compatible with the 
fact of a small number of reflections. 

In this dilemma, we shall in the succeeding treatment 
adhere to the older point of view, since the existing values 



sion 



REVERBERATION (THEORETICAL) 63 

of the absorption coefficients have been obtained using the 
Sabine equations, and the criterion of acoustical excellence 
has been established on the basis of Sabine coefficients and 
the 0.05 V/a formula. We can, if desired, shift to the 
later point of view by substituting for a in the older expres- 

S log e f 1 -gj. It must be remembered, however, 

that the absorption coefficients of materials now extant 
are all derived on the older theory, so that to shift would 
involve a recalculation of all absorption coefficients based 
on the later formula. 

Experimental Determination of Reverberation Time. 

We have developed a simple usable formula for com- 
puting the reverberation time T from the values of volume 
and absorbing power. We next have to consider the 
question of the experimental measurement of this quantity. 
To do this directly we should need some means of direct 
determination of the average intensity of the sound in a 
room at two instants of time during the decay process 
and of precise measurement of the intervening time. We 
have already seen the inequality that exists in the intensities 
at different points due to interference, a phenomenon 
that in the theoretical treatment we have ignored, by 
assuming an average uniform intensity throughout the 
room. Moreover, there has not as yet been developed any 
simple portable apparatus by which the required intensity 
measurement can be made. l We are thus forced to indirect 
methods for experimental determination. 

The method employed by Sabine involved the minimum 
audible intensity or the threshold of hearing, for the lower 
of the two intensities used in defining reverberation time. 

1 Very recently, apparatus has been devised for making direct measure- 
ments of this character. In the Jour. Soc. Mot. Pict. Eng., vol. 16, No. 3, 
p. 302, Mr. V. A. Schlenker describes a truck-mounted acoustical laboratory 
for studying the acoustic properties of motion-picture theaters, in which an 
oscillograph is used for recording the decay of sound within a room. The 
equipment required is elaborate. The spark chronograph of E. C. Wente 
described in Chap. VI has also been used for this purpose. 



64 ACOUSTICS AND ARCHITECTURE 

The absolute value of this quantity varies with the pitch 
of the sound and is by no means the same at any given 
pitch for two observers or, indeed, for the two ears of any 
one observer. Fortunately, it does remain fairly constant 
over long periods of time for a given observer, so that with 
proper precautions it may be used in quantitative work. 
Another quantity which Sabine assumed could be con- 
sidered as constant under different room conditions is 
the quantity E, the energy emitted per second by an organ 
pipe blown at a definite wind pressure. We note that 
neither E, the power of the source, nor i, the intensity at 
the threshold of hearing, is known independently, but we 
may put our equations into a form that will involve only 
their ratio. 

We shall, in the following, denote by TI the time that 
sound from a source of output E, which has been sustained 
until the steady state I\ has been reached, remains audible 
after the source is stopped. Let i be the threshold intensity 
(ergs per cubic meter) for a given observer. Then, by 
Eq. (34a), we have 



Putting in the value of /i given by Eq. (336), we have 



TI is a directly measurable quantity. The same mecha- 
nism that stops the pipe may simultaneously start the 
timing device, which, in turn, the observer stops at the 
moment he judges the decaying sound to be inaudible. 
The quickness and simplicity of the operation allow 
observations to be made easily, so that by averaging a 
large number of such observations made at different parts 
of the room, the differences due to interference and the 
personal error of a single observation may be made reason- 
ably small. However, in Eq. (39), there are two unknown 
quantities, a and E/i. Obviously, some means of deter- 
mining one or the other of them independently of Eq. 



REVERBERATION (THEORETICAL) 65 

(39) is necessary. Thus, if the output of a given source of 
sound is known in terms of the observer's threshold inten- 
sity, the measured value of the duration of audible sound 
from this source within a room allows us to compute the 
total absorbing power of the room. 

The extremely ingenious method which Sabine employed 
for experimentally determining one of the two unknown 
quantities of Eq. (39) with no apparatus other than the 
unaided ear, a timing device, and organ pipes will be con- 
sidered in the next chapter. 



CHAPTER V 

REVERBERATION (EXPERIMENTAL) 

In the preceding chapter, we have treated the question 
of the growth and decay of sound within a room from the 
theoretical point of view. Starting with the simple one- 
dimensional case, in which visualization of the process is 
easy, we proceeded by analogy to the three-dimensional 
case and derived the relations between the various quanti- 
ties involved. 

A connected account of Professor Sabine's series of 
experiments, on which he based the theory of reverberation, 
should prove helpful to the reader who is interested in 
something more than a theoretical knowledge of the 
subject. 

In the first place, it is well to remind ourselves of the 
situation which confronted the investigator in the field 
of architectural acoustics at the time this work was begun 
nearly forty years ago. In undertaking any problem, 
the first thing the research student does is to go through 
the literature of the subject to find what has already been 
done and what methods of measurement are available. 
In Sabine's case, this first part of the program was easy. 
Aside from one or two small treatises dealing with the prob- 
lem purely from the observational side and an occasional 
reference to the acoustic properties of rooms in the general 
texts on acoustics, there were no signposts to point out the 
direction which the proposed research should follow. It 
was a new and untouched field for research. To offset 
the attractiveness of such a field, however, there were no 
known means available for direct measurement of sound 
intensity. Tuning forks and organ pipes were about the 
only possible sources of sound. These do not lend them- 

66 



REVERBERATION (EXPERIMENTAL) 



67 



selves readily to variation in pitch or intensity. Study 
of the problem thus involved the invention of a wholly 
new technique of experimental procedure. 

Reverberation and Absorbing Power. 

The immediate occasion for the research was the neces- 
sity of correcting the very poor acoustic properties of the 



40 80 120160200240 
y Meters of, Cushions 




FIG. 22. W. C. Sabiiie'a orig- 
inal single-pipe apparatus for 
reverberation measurements. 



40 80 120160200240280320360400 



Fio. 2,3. Reciprocal of re- 
verberation time, constant 
source, plotted against length 
of seat cushion introduced. 



then new lecture room of the Fogg Art Museum. This 
room was semicircular in plan, not very different in general 
design from Sanders Theater, a much larger room, but 
one which was acoustically excellent. 

To even casual observation, one outstanding difference 
between the two rooms was in the matter of the interior 
finish and furnishing. There was a considerable area of 
wood paneling in Sanders Theater. Aisles were carpeted 
and the pews were furnished with heavy cushions. Rever- 
beration in the smaller room of the Fogg Art Museum was 
markedly greater. 

The first thing to try, therefore, was the effect of the 
seat cushions upon the reverberation in the new lecture 
hall. 



68 ACOUSTICS AND ARCHITECTURE 

The apparatus was simple. An organ pipe blown by 
air from the constant-pressure tank shown in Fig. 22 was 
the source of sound. The time of stopping the source 
and the time at which the reverberant sound ceased to be 
audible to the observer were recorded on a chronograph. 
Seat cushions were introduced and the duration of the audi- 
ble sound with varying lengths of seat cushion present was 
measured. The relation found between these two variables 
is shown on the graph (Fig. 23), on which is plotted not 
T but l/T as a function of the length of cushions. 

T - l (Lr + Lc) (40) 

where A: is a constant, the reciprocal of the slope of the 
straight line; L c is the length of seat cushions brought 
in; and L r is obviously the length of seat cushion that 
is the equivalent in absorbing power of the walls and 
contents of the room before any cushions were introduced. 
If we include in the term a c the total absorbing power of 
the room and its contents, measured in meters of seat 
cushion, the expression assumes the now familiar form 

a c T = k c (41) 

The numerical value of a c and consequently of k c is 
determined by the number which finally is to be attached 
to the absorbing power of one meter of cushions as measured 
by its effect on the reverberation time. 

We note that the experimental points of Fig. 23 show 
a tendency to fall further and further from the straight 
line as the total absorbing power is increased. So carefully 
was the work done that Sabine concluded that this depar- 
ture was more than could be ascribed to errors of experiment. 
The significance which he attached to it will be considered 
later. For the time being we shall retain Eq. (41) as an 
approximate statement of the facts so far adduced. 

Reverberation and Volume. 

The next step was to extend the experiment to a number 
of different rooms, first to establish the generality of the 



REVERBERATION (EXPERIMENTAL} 69 

ation found for the single room and, also, to ascertain 
at meaning is to be ascribed to the constant k. Accord- 
;ly the laborious task of performing the seat-cushion 
jeriment in a number of different rooms of different 
ipes and sizes was undertaken. These ranged from a 
all committee room with a volume of 65 cu. m. to a 
xater seating 1,500 persons and having a volume of 
00 cu. m. 

[n order to secure the necessary conditions of quiet, the 
rk had to be done in the early hours of the morning, 
e mere physical job of handling the large number of 
,t cushions necessary for the experiments in the larger 
uns was a formidable task. There was, however, no 
ter way. The results of this work well repaid the effort, 
ey showed first that the approximate constancy of the 
>duct of absorbing power and time held for all the rooms 
which the experiment was tried. More important 
1, it appeared that the values of this product for different 
uns are directly proportional to their volumes. Equa- 
n (41) may therefore be written in the form 

a c T = K C V, (42) 

ere K c is a new constant, approximately the same for 
rooms. Again its numerical value will depend upon the 
iiber which is to be attached to the absorbing power of 
\ cushions. 

en-window Unit of Absorbing Power. 

>abine recognized that in order to give K c something 
ier than a purely empirical significance, it was necessary 
express the total absorbing power of rooms in some more 
idamental and reproducible unit than meters of a 
ticular kind of seat cushion. He proposed, therefore, 
treat the area presented by an open window as a surface 
which all the incident sound energy is transmitted to 
/side space with none returned to the room. In other 
rds, at this stage of the investigation the open window 
3 to be considered as an ideal perfectly absorbing surface 



70 ACOUSTICS AND ARCHITECTURE 

with a coefficient of absorption of unity. A comparison 
between the absorbing power of window openings and 
seat cushions as measured by the change in the reverbera- 
tion time was accordingly made in a room having seven 
windows, each 1.10 m. wide. The width of the openings 
was successively 0.20, 0.40, and 0.80 m. The experiment 
showed that within the limits of error of observation the 
increase in absorbing power was proportional to the 
width of the openings. It should be noted, in passing, 
that in this experiment the increase in absorbing power was 
secured by a number of comparatively small openings. 
Later experiments have shown that had the increased 
absorption been secured by increasing the dimensions of a 
single opening, the results would have been quite different. 
That Professor Sabine 1 recognized this possibility is shown 
by his statement "that, at least for moderate breadths, the 
absorbing power of open windows as of cushions is accu- 
rately proportional to the area." 

Experiments comparing seat cushions and open windows 
in several rooms gave the absorbing power of the cushion 
as 0.80 times that of an equal area of open window. 

This figure gave him the data for evaluating the con- 
stant K, using a square meter of open window as his unit 
of absorbing power simply by multiplying the parameter 
K c as obtained by experiment with cushions by the factor 
0.80. This operation yielded the equation 

aT = 0.1717 (43) 

Here T is the duration of sound produced by the par- 
ticular organ pipe used in these experiments, a gemshorn 
pipe one octave above middle C, 512 vibs./sec., blown 
at a constant pressure. V is the volume of the room in 
cubic meters, and a is the total absorbing power in square 
meters of open window, measured as outlined above. 

Logarithmic Decay of Sound in a Room. 

At this point in his study of reverberation, Sabine 
was confronted with two problems. The first was to give 

1 "Collected Papers on Acoustics," p. 23. 



REVERBERATION (EXPERIMENTAL) 71 

a coherent theoretical treatment of the phenomenon that 
would lead to the interesting experimental fact expressed 
in Eq. (43). The second was to explain the departure from 
strict linearity in the relation between the reciprocal of 
the observed time of reverberation, using a presumably 
constant source, and the total absorbing power. Obviously 
Eq. (43) does not involve the acoustical power of the 
source. It is equally evident that the time required after 
the source has ceased for the sound to decrease to the 
threshold intensity should depend upon the initial inten- 
sity, and this, in turn, upon the sound energy emitted per 
second by the source. The next step then in the investiga- 




FIG. 24. W. C. Sabine's four-organ apparatus. 

tion is to find the relation between the acoustical power of 
the source and the duration of audible sound. Here, again* 
the experimental procedure was simple and direct. The 
apparatus is shown in Fig. 24. Four small organs were 
set up in a large reverberant room. They were spaced at a 
distance of 5 m. from each other so that the output of 
one might not be affected by the close proximity of another 
pipe speaking at the same time. They were supplied 
with air pressure from a common source. Each pipe was 
controlled by an electropneumatic valve, and the electric 
circuits were arranged so that each pipe could be made to 
speak alone or in all possible combinations with any or all 
of the remaining pipes. The time between stopping the 



72 



ACOUSTICS AND ARCHITECTURE 



pipe and the moment at which it ceased to be audible was 
recorded on a chronograph placed in an adjoining room. 
By sounding each pipe alone and then in all possible com- 
binations with the remaining pipes, allowance was made 
for slight inequalities in the acoustic powers of the indi- 
vidual pipes. 

The results of one experiment of this sort were as follows :* 



ti = 8.69 
* 2 = 9.14 
t* = 9.36 
U = 9.55 



t 2 



= 0.45 

- h = 0.67 

- ^ = 0.86 

- ^3 = 0.19 

- t 2 = 0.22 



The fact that the difference in time for one and two 
pipes is very nearly one-half of the difference for one and 

four pipes suggests that the 
difference in times is propor- 
tional to the logarithm of the 
ratio of the number of pipes, 
a relation that is clearly 
brought out by graph 1 of 
Fig. 25. 

Let us now assume that the 
average intensity of sound 
in the room, that is, the 
average sound energy per 
cubic meter set up by n 

. . ,. ,, 

pipes, is n times the average 
-Results of the four-organ intensity produced by a single 

experiment. . TTT 

pipe. We may write 



Four Organ experiment 
1. Room bare . 




0.7 04 06 08 

Difference m Time 



FIG. 25.- 



In - log e /I = 



= A(t n - 
1 1 



(44) 



wherein A is the slope of the line in Fig. 25. Its numerical 
value in the example given is 1.59. 

Suppose, now, that we have a pipe of such minute power 
that the average intensity which it sets up is i, the minimum 
audible intensity. The sound from such a pipe would, of 

1 "Collected Papers on Acoustics," p. 36. 



REVERBERATION (EXPERIMENTAL] 73 

course, cease to be audible the instant the pipe stops. 
Therefore T for such a pipe is zero. If the relation given 
in Eq. (44) is a general one, then we may write 

log^, 1 = A(T, - 0) (45o) 

t 

In other words, the time required for sound of any given 
initial intensity to decrease to the threshold of audibility 
is proportional to the logarithm of the initial intensity 
measured in terms of the threshold intensity. 

The next step in the investigation was to relate the 
quantity A, which is the change per second in the natural 
logarithm of the intensity of the reverberant sound, to the 
total absorbing power of the room. To do this, a large 
quantity of a highly absorbent felt was installed upon the 
walls of the room, and the four-organ experiment was 
repeated. The following results were obtained: 

t\ = 3.65 t' t - t\ = 0.20 

t\ = 3.85 ' 3 - t'i = 0.31 

t\ = 3.96 t\ - t'i = 0.42 

*' 4 = 4.07 i\ - t' 9 = 0.11 

t' 9 - if = 0.11 

Plotting all the possible relations between the difference 
in times and the ratio of the number of pipes, we have the 
straight line 2, in Fig. 25. The slope of this line is 3.41. 

It is apparent that the logarithmic decrement A in the 
intensity increases as the absorbing power of the room 
increases. Now the experiment with the seat cushions gave 
the result that the product of absorbing power and time for 
a given room is nearly constant. The results of the four- 
organ experiments give the same sort of approximate rela- 
tion between the logarithmic decrement and the time. 
For 

1.59 X 8.69 = 13.80 = log e l 

i 

3.41 X 3.65 = 12.45 = log,^ 



74 ACOUSTICS AND ARCHITECTURE 

Solving for / and /', we have 

/ = 1,000,000;, and I' = 250,000i. 

We have here the explanation of the departure from a 
constant value of the product a X T in the experiments 
with seat cushions. It lies in the fact that with a source that 
generates sound energy at a constant rate, independently 
of room conditions, the level to which the intensity (sound 
energy per cubic meter) rises is less in the absorbent tha'n in 
the reverberant room. The presence of absorbent material 
thus acts in two ways to reduce the time of reverberation 
from a fixed source: first, by increasing the rate of decay 
and, second, by lowering the level of the steady-state 
intensity. The complete theory must take account of both 
of these effects. 

Total Absorbing Power and the Logarithmic Decrement. 

It still remains to connect the quantities a, the total 
absorbing power of the room in the equation aT = 0.171F, 
and A, the change per second in the natural logarithm 

of the decaying sound in the equation ATi = log e - 1 - This 

% 

latter equation suggests that the rate at which the intensity 
decreases is proportional to the intensity at that instant, 
that is, 

^= -AI 
dt A1 

Integrating, we have 

- log e 7 + C = AT. 

When T is 0, that is, at the moment of cut-off of the source, 

/ = /i. So that - log e I\ + C = 

whence 

log./!- log,/ = AT 

If T = TI, the time required for the reverberant sound to 
decrease to the threshold intensity i, we have 

AT, = log.^ (456) 

V 



REVERBERATION (EXPERIMENTAL} 75 

Thus A, the slope of the straight line in the four-organ 
experiment, is the instantaneous rate of change of intensity 
per unit intensity, when no sound is being produced in the 
room. 

Now a, the total absorbing power of a room in the 
reverberation equation, is the area of perfectly absorbing 
surface (open window) that would, in an ideal room that is 
otherwise perfectly reflecting, produce the same rate of 
decay as that of the actual room. If we divide this area by 
S, the total area of the exposed surfaces in the room, we 
have the average absorption coefficient of these surfaces or 
the fraction by which the intensity of the sound is decreased 
at each reflection. Call this average coefficient a a . Then 

a a -o = change of intensity per unit intensity per reflection 

A = change of intensity per unit intensity per second 

Then A = ma a where ra is the average number of 
reflections per second which any single element of the 
sound undergoes in its passage back and forth across the 
room during the decay process. In terms of the mean free 
path, already referred to in Chap. IV, 

A = ~a a = - - 1 (46) 

We may now express the results of the four-organ 
experiment in terms of the total absorbing power as 
measured by the cushion and open-window experiments. 
We have 

o f) t 

whence 



aTl = SP log. k (47) 

C if 



Power of Source. 



The experimentally determined value of A makes it 
possible to express the acoustical power of the source in 
terms of the threshold intensity. 



76 ACOUSTICS AND ARCHITECTURE 

Let E be the number of units of sound energy supplied 
by the source to the room in each second. The rate then 
at which the sound energy density / is increased by E is 
E/V. The rate at which the energy decreases due to 
absorption is AI. The net rate of increase while the source 
is operating is therefore given by the equation 

dl _E _ A 

~3JL V Al 

The steady-state intensity I\ is that at which the rates 
of absorption and emission are just balanced, so that 

/J T P 1 

when 7 = /i, -,- = 0, and we have ^ AIi = 0, whence 



and 
and 



AT, = log,' 1 = log, ~ (48) 



log.f = A I 7 ! + log. VA 
if 



Approximate Value of Mean Free Path. 

Equations (43) and (46) make it possible to arrive at 
an approximate experimental value of the mean free path. 
To a first approximation 

ac (UTlcF ,._. 

p = AS = - (49) 



By performing the four-organ-pipe experiment in two 
rooms of the same relative dimensions but of different 
volumes, Sabine arrived at a tentative value of p = 0.62V M . 
This clearly is not an exact expression, since it is apparent 
that p, the average distance between reflections, must 
depend upon the shape as well as the volume. For this 
reason, it was desirable to put the expression for p in a form 
which includes this fact. This was done by a more exact 
experimental determination of the constant K making use 
of the results of the four-organ experiment. 



REVERBERATION (EXPERIMENTAL) 77 

Precise Evaluation of the Constant K. 

The experiment with the seat cushions and open windows 
led to the approximate relation aTi = 0.171F, while the 

four-organ experiment gave the equation aTi = log e * 

c % 

The complete solution of the problem then involves an 

exact correlation of the expressions 0.171F and log e -^- 

c % 

We note immediately that since p is proportional to 
F M and the surface S is proportional to F*' 1 , their product 
must be proportional to V itself. We may therefore write 
Eq. (47) in the form 

aTi - kV log, 7 . 1 = kV log, (50) 



In the rooms in which his experiments had been con- 
ducted, the steady-state intensities set up by the particular 
pipe employed were of the order of 10 6 X i. Sabine there- 
fore adopted this as a standard intensity, and upon the 
assumption of this fixed steady-state intensity, the results 
of the four-organ experiment reduce to the form given by 
the cushion experiments, namely, 

aT = kV log, 10 6 = k X 13.8V 

where T n is the time required for a sound of initial intensity 
10 6 X i to decrease to the threshold intensity, and k is a 
new constant whose precise value is desired. 

One is compelled to admire the skill with which Sabine 
handled the various approximate relations in order to 
arrive at as precise a value as possible for his fundamental 
constant. His procedure was as follows: From the four- 
organ-pipe experiment he derived the value of J&7, the acous- 
tic power of his organ pipe. He then measured the times 
with this pipe as a source in a number of rooms of widely 
varying sizes and shapes, with windows first closed and 
then open. He assumed that the open windows produced 
the same change in reverberation as would a perfectly 
absorbing surface of equal area. 



78 



ACOUSTICS AND ARCHITECTURE 



Let w equal the open- window area, and T'\ the time under 
the open-window condition. Then 



Dividing (50) by (51) gives 
aT l lo * 






(a + w)I", 



(51) 



(52) 



In this equation we have two quantities a and p, both 
of which are known approximately, a = 0.171V /T and 
p = 0.62 F* 8 are both approximations. In the right-hand 
member of (52), these quantities are involved only in 
logarithmic expressions, so that slight departures from 
their true values will not materially affect the numerical 
value of the right-hand member of the equation. Evalu- 
ating the right-hand member of (52) in this way, the equa- 
tion was solved for a. Using this value of a, Eq. (50) may 
be solved for k. The constant K of Sabine's simple rever- 
beration equation is 13.8&. 

The following are the data and the results of the experi- 
ment for the precise determination of K. 

TABLE IV 



Places of experiment 


V 


/i/t 


w 


K 


Lobby, Fogg Art Museum: 
1 uipe 


96 


8 800 000 


1 86 


159 


16 pipes. . 


96 


67 , 000 , 000 


1 86 


164 


Jefferson Laboratory : 
Room 15 
Room 1 
Room 41 . 


202 
1,630 
1,960 


1,000,000 
390,000 
300 000 


5 10 
12.0 
14 6 


0.169 
167 
161 




















Ave. 
0.164 



Experimental Value of Mean Free Path. 

Equation (47) and Sabine's experimental value for K 
enable us to arrive at an expression for p, the mean free 



REVERBERATION (EXPERIMENTAL) 79 

path in terms of the volume and bounding surface of an 
inclosure. If in Eq. (47) I/i is put equal to 1,000,000, we 
have 

aT Q = ^ i oge (1,000,000) = 13.8^ = 0.164F 
c c 

With a velocity of sound at 20 C. of 342 m. per second, 
this gives 

4.067 ,. Q v 

P = g (53) 

The theoretical derivation of the relation p = 4F '/S is 
given in Appendix B. The close agreement between the 
experimental and theoretical relations furnishes abundant 
evidence of the validity of the general theory of reverbera- 
tion as we have it today. We shall hereafter use the value 
of K = 0.162, corresponding to the theoretically derived 
value of p, where V is expressed in cubic meters and a is 
expressed in square meters of perfectly absorbing surface. 
If these quantities are expressed in English units, the 
reverberation equation becomes 

aT Q = 0.0494F 

Complete Reverberation Equation. 

In Chap. IV, the picture of the phenomenon of reverbera- 
tion was drawn from the simple one-dimensional case of 
sound in a tube and extended by analogy to the three- 
dimensional case of sound within a room, considering the 
building up and decay of the sound. In the preceding 
paragraphs, we have followed the experimental work, as 
separate processes, by which the fundamental principles 
were established and the necessary constants were evaluated. 
We shall now proceed to the formal derivation of a single 
equation giving the relation between all the quantities 
involved. 

The underlying assumptions are as follows : 
1. The acoustic energy generated per second by the 
source is constant and is not influenced by the reaction of 
sound already in the room. 



80 ACOUSTICS AND ARCHITECTURE 

2. The sustained operation of the source sets up a final 
steady-state intensity. In this steady state, the sound 
energy in the room is assumed to be "diffuse"; that is, its 
average energy density is the same throughout the room, 
and all directions of energy flow at any point are equally 
probable. 

3. In the steady state, the total energy per second 
generated at the source equals the total energy dissipated 
per second by absorption at the boundaries. Dissipation 
of energy throughout the volume of the room is assumed to 
be negligibly small. 

4. The time rate of change of average intensity at any 
instant is directly proportional to the intensity at that 
instant, provided the source is not operating. 

5. At any given surface whose dimensions are large 
in comparison with the wave length, a definite fraction of 
the energy of the incident diffuse sound is not returned by 
reflection to the room. This fraction is a function of the 
pitch of the sound, but we shall assume it to be independent 
of the intensity. It is the absorption coefficient of the 
surface. 

6. The absorbing power of a surface is the product of the 
absorption coefficient and the area. The " total absorbing 
power of the room" is the sum of the absorbing powers of 
all of the exposed surfaces in the room. 

While the source is operating, the rate of change of 
energy density is the difference between the rates per unit 
volume of emission and absorption or 

dl E AT 

dt = v~ AI 

Remembering that in the steady state, the rate of 
change of intensity dl/dt = and that therefore All = 
fl/V, we may write 



Integrating and supplying the constant of integration, we 
have 



REVERBERATION (EXPERIMENTAL) 81 

I t = A(l - e-") = -(\ - e- A ') (54) 



Tf after the source has been operating for a time t it is 
suddenly stopped, the sound begins to die away. During 
this stage E = 0, and, denoting the time measured from 
the instant of cut-off by !T, we have for the decay process 

dl - -AT 
dT~ AI 

Integrating, and supplying the constant of integration from 
the fact that at the moment of cut-off T = 0, I T = It, we 
have 

IT = Ifi- AT 

and 

It AT K(l - e-") AT 
- = e AT or - jrf r - = e AT 
IT A VI T 

Taking the logarithm of both sides, we have 

AT = log e (l - -*) (55a) 



In order not to complicate matters unduly, let us simply call 
T the time from cut-off required for sound to decrease 
to the threshold intensity i. Putting I T = i, (55a) then is 
written 

/ 7v 7 \ 

(556) 



This may be expressed in terms of the total absorbing power 
a, by the relations already deduced, namely, A = ac/Sp 

ac/4F, 

acT 4E/ jn*\ 

whence we have -^ = log e ^1 - e* v J 

and 

Here t is the time interval during which the source speaks, 
while T is the period from cut-off to the instant at which the 
sound becomes inaudible 



82 



ACOUSTICS AND ARCHITECTURE 



The expression 1 e * v is the fraction of the final 
steady intensity which the intensity of the sound in the 
building-up process attains in the time L The greater the 
ratio of absorbing power to volume the shorter the time 
required to reach a given fraction of the steady state. As 

-act 

t increases indefinitely, e 4F approaches zero, and I t 
approaches /i, the steady-state intensity. If Ti equals the 
time required for the intensity to decrease from this steady 
state to the threshold, we have 



4F . 
= log, 



(57) 



which reduces to the Sabine equation aT Q = 0.162y, if 
4E/aci is set equal to 10 6 . 

The whole history of the building-up and decay of sound 
in a room is shown graphically in Fig. 26. 1 It must be 

i.o 




0123401234 
Time From Start Time From Cuf-off 

Fio. 26. Statistical growth and decay of sound in a room absorption assumed 

Continuous. 

remembered that what is here shown is the theoretical value 
of the average intensity (sound-energy density). In both 
the building-up and decay processes, the actual intensity at 
any point fluctuates widely as the interference pattern 
referred to in Chap. Ill shifts in an altogether undetermined 
way. Figure 27 is an oscillograph record of the decay of 

1 For an instructive series of curves, showing many of the implications of 
the general reverberation equation as related to the acoustic properties of 
rooms, the reader is referred to an article by E. A. Eckhardt, Jour. Franklin 
Inst., vol. 195, p. 799, 1923. 



REVERBERATION (EXPERIMENTAL} 83 

sound in a room, kindly furnished by Mr. Vesper A. 
Schlenker. 1 

Extension of Reverberation Principles to Other Physical 
Phenomena. 

Reference should be made here to an extremely interesting 
theoretical paper by Dr. M. J. O. Strutt, 2 in which from 




FIG. 27. Oscillogram of decay of sound at a single point in a room. (Courtesy 
ofV.A. Schlenker.) 

the most general hydrodynamic considerations he deduced 
Sabine's law. This he states as follows: 

The duration of residual sound in large rooms measured by the time 
required for the intensity to decrease to 1/1,000,000 of the steady-state 
intensity is proportional to the volume of the room over the total 
absorbing power but does not depend upon the shape of the room, the 
places of source, and experimenter, while the frequency does not change 
much after the source has stopped. 

He shows that the law holds even without the assumption 
of a diffuse distribution in the steady state. Further, he 
shows that the law does not hold unless the frequency of 
the source is much higher than the lowest resonance fre- 

1 A Truck-mounted Laboratory, Jour. Soc. Mot. Pict. Eng., vol. 16, No. 3, 
p. 302. 

2 Phil Mag., vol. 8, pp. 236-250, 1929. 



84 



ACOUSTICS AND ARCHITECTURE 



quencies of the room, that is, unless the dimensions of 
the room are considerably greater than the wave length 
of the sound. This has an important bearing upon the 
problem of the measurement of sound absorption coeffi- 
cients by small scale methods. 

It is also shown that a similar law holds for other physical 
phenomena, as, for example, radiation within a closed space 
and the theory of specific heats of solid 
bodies. 

Perhaps the most interesting fact 
brought out by Strutt is the proof that 
the amplitude at any point in the room 
at any time in the building-up process 
is complementary to the corresponding 
amplitude in the decay process; that 
FIG. 28. Osciiiograms is, at a given point, the amplitude at 

showing that the building ,, , . -, - ,, j, 

up and dying out of sound the time t measured irom tne moment 
T of startin S P lus the amplitude at the 
time T measured from stopping the 
source after the steady state has been reached equals the 
amplitude in the steady state. This is shown in Fig. 28 
taken from Strutt's paper. That this is true for the average 
intensities follows from the fact that in the building-up 
process, /i I t decreases according to the same law as 
that followed by I T in the decay process. That it should 
be true for building-up and decaying intensities at every 
point of the room is a fact not brought out by the ele- 
mentary treatment here given. 




Summary. 

In the present chapter we have followed Sabine's attack 
on the problem of reverberation from the experimental 
side to the point of a precise evaluation of the constant 
in his well-known equation. This equation has been shown 
to be a special case of the more general relation given in 
Eq. (56). He expressed all the relations with which we 
are concerned in terms of this constant K with correction 



REVERBERATION (EXPERIMENTAL) 85 

terms to take care of departures from the assumption of a 
steady-state intensity of 10 6 X i. The simplicity of 
Sabine's equation makes it extremely useful in practical 
applications of the theory. It is well, however, to hold 
in mind the rather special assumptions involved in its 
derivation. 



CHAPTER VI 

MEASUREMENT OF ABSORPTION COEFFICIENTS 

In order to apply the theory of reverberation developed 
in Chaps. IV and V to practical problems of the control of 
this phenomenon, it is necessary to have information as to 
the sound-absorptive properties of the various materials 
that enter into the finished interiors of rooms. These 
properties are best expressed quantitatively by the numer- 
ical values of the sound-absorption coefficients of materials. 
In this chapter, we shall be concerned with the precise 
significance of this term and the various methods employed 
for its determination. 

Two Meanings of Absorption Coefficient. 

In the study of the one-dimensional case of sound in a 
tube, a (the absorption coefficient) was defined as the 
fraction of itself by which the incident energy is reduced at 
each reflection from the end of the tube. Here we were 
dealing with a train of plane waves, incident at right angles 
to the absorbing surface. If Id is the energy density in the 
direct train and I r that in the reflected train, then 

a = I *- Ir = 1 - f (58a) 

J. d *- d 

In going to the three-dimensional case, diffuse sound was 
assumed to replace the plane unidirectional waves in the 
tube. In a diffuse distribution, all angles of incidence from 
to 90 deg. are assumed to be equally probable. Now it is 
quite possible that the fraction of the energy absorbed at 
each reflection will depend upon the angle at which the wave 
strikes the reflecting surface, so that the absorption coeffi- 
cient of a given material measured for normal incidence may 
be quite different from that obtained by reverberation 

86 



MEASUREMENT OF ABSORPTION COEFFICIENTS 87 

methods. It is not easy to subject the question to direct 
experiment. In view of the fact that there is not a very 
close agreement between measurements made by the two 
methods, we shall for the sake of clarity refer to coefficients 
obtained by measurements in tubes as " stationary-wave 
coefficients." Coefficients obtained by reverberation meth- 
ods we shall call "reverberation coefficients." 

Measurement of Stationary-wave Coefficients. Theory. 

In Eq. (23a), Chap.- Ill, let the origin be at the absorbing 
surface. Then the equation for the displacement at any 
point at a distance x from this surface due to both the direct 
and reflected waves is 

A o sin co(t H J ft sin o>(t j = 

AO (1 ft) sin ut cos + (1 + ft) cos o)t sin (586) 

If we elect to express the condition in the tube in terms 
of the pressure, we have 

&x 
ft) sin cot sin + 

(1 + k) cos ut cos 

For values of cox/c = 0, TT, 2?r, etc., dP max . = 5(1 + ft), 
while for the values of wx/c = ?r/2, 3?r/2, etc., dP max . = 
B(l ft), where J? is a constant. 

Let M be the value of dP max at the pressure internodes 
Let N be the value of rfP max at the pressure nodes 

M = B (1 + ft) 
N = B (I - ft) 
Adding, 

M + N = 2B 
Subtracting, 

M - N = 2ftJ5 
whence 



88 ACOUSTICS AND ARCHITECTURE 

For a fixed frequency, the intensity is proportional to the 
square of the amplitude. Then 



By definition, 

1 Ir 1 ,2 ! M ~ N2 

a = 1 - Td , - 1 - P - 1 - 

or 



" (M + TV) 2 

The absorption coefficient then can be determined by 
the use of any device the readings, of which are proportional 
to the alternating pressure in the stationary wave. It is 
obvious that the absolute value of the pressures need not be 
known, since the value of a depends only upon the relative 
values of the pressures at the maxima and minima. One 
notes further that the percentage error in a is almost the 
same as the percentage error in N, the relative pressure at 
the minimum. For materials which are only slightly 
absorbent N will be small, and for a given absolute error 
the percentage error will be large. For this reason the 
stationary-wave method is not precise for the measurement 
of small absorption coefficients. It is also to be observed 
that a velocity recording device may be used in the place of 
one whose readings are proportional to the pressure. M 
and N would then correspond to velocity maxima and 
minima respectively. 

The foregoing analysis assumes that there is no change 
of phase at reflection from the absorbent surface. A more 
detailed analysis shows 1 that there is a change of phase 
upon reflection. The effect of this, however, is simply to 
shift the maxima and minima along the tube. 2 If, in the 
experimental procedure, one locates the exact position of 
the maxima and minima by trial and measures the pressure 
at these points, no error due to phase change is introduced. 

1 PARIS, E. T., Proc. Phys. Soc. London, vol. 39, No. 4, pp. 269-295, 1927. 

2 DAVIS and EVANS, Proc. Roy. Soc., Ser. A, vol. 127, pp. 89-110, 1930. 



MEASUREMENT OF ABSORPTION COEFFICIENTS 89 

Dissipation along the Tube. 

Eckhardt and Chrisler 1 at the Bureau of Standards 
found that due to dissipation of energy along the tube there 
was a continuous increase in the values of both the maxima 
and minima with increasing distance from the closed end. 
Thus the oncoming wave diminishes in amplitude as it 
approaches the reflecting surface, while the reflected wave 
diminishes in amplitude as it recedes from the reflecting 
surface. Taking account of this effect, Eckhardt and 
Chrisler give the expression 



a = 1 - 



Af i 

- Ni + 



(60) 



Nz Ni is the difference between the pressures measured 
at two successive minima. 

Davis and Evans give this correction term in somewhat 
different form. Assuming that as the wave passes along 
the tube the amplitude decreases, owing to dissipation at 
the walls of the tube, according to the law 



the values for the nth maximum and minimum respectively 
are 

M n = M + y 2 (n - 1)AXAT (61) 



N n = N + -AXM (62) 

In order to make the necessary correction, a highly 
reflecting surface was placed in the closed end of the tube, 
and successive maxima and minima were measured. From 
these data, the value of A was computed, which in Eqs. 
(61) and (62) gave the values for M and N to be used in 
Eq. (59). 

It is to be noted that since N is small in comparison with 
M, the correction in the former will have the greater 

1 Bur. Standards Sci. Paper 526, 1926. 



90 ACOUSTICS AND ARCHITECTURE 

effect upon the measured value of a. For precision of 
results therefore it is desirable to have A, the attenuation 
coefficient of the tube, as small as possible. This condition 
is secured by using as large a tube as possible, with smooth, 
rigid, and massive walls. The size of the tube that can be 
used is limited, however, since sharply defined maxima and 
minima are difficult to obtain in tubes of large diameter. 
Davis and Evans found that with a pipe 30 cm. in diameter, 
radial vibrations may be set up for frequencies greater 
than 1 ,290 cycles per second. This of course would vitiate 
the assumption of a standing-wave system parallel to the 
axis of the tube, limiting the frequencies at which absorp- 
tion coefficients can be measured in a tube of this size. 

Standing-wave Method (Experimental). 

H. O. Taylor 1 was the first to use the stationary-wave 
method of measuring absorption coefficients in a way to 
yield results that would be comparable with those obtained 
by reverberation methods. 2 The foregoing analysis is 
essentially that given by him. For a source of sound he 
used an organ pipe the tone of which was freed from 
harmonics by the use of a series of Quincke tubes, branch 
resonators, each tuned to the frequency of the particular 
overtone that was to be filtered out. The tube was of 
wood, 115 cm. long, with a square section 9 by 9 cm. The 
end of this tube was closed with a cap, in which the test 
sample was fitted. A glass tube was used as a probe for 
exploring the standing-wave system. This was connected 
with a Rayleigh resonator and delicately suspended disk. 
The deflections of the latter were taken as a measure of the 
relative pressures at the mouth of the exploring tube and 
gave the numerical values of M and N in Eq. (59). 

Tube Method at the Bureau of Standards. 

Figure (29) illustrates the modification of Taylor's 
experimental arrangement developed and used for a time 

1 Phys. Rev., vol. 2, p. 270, 1913. 

2 The method was first proposed by Tuma (Wien. Ber., vol. Ill, p. 402, 
1902), and used by Weisbach (Ann. Physik, vol. 33, p. 763, 1910). 



MEASUREMENT OF ABSORPTION COEFFICIENTS 91 



The source of sound was a 
alternating current from a 



Loudspeaker 



at the Bureau of Standards. 1 
loud-speaker, supplied with 
vacuum-tube oscillator. The 
standing-wave tube was of 
brass and was tuned to reso- 
nance with the sound produced 
by the loud-speaker. The 
exploring tube was terminated 
by a telephone receiver, and 
the electrical potential gener- 
ated in the receiver by the 
sound was taken as a measure FlG 29 ._ Apparatus for measuring 

of the pressure in the Standing absorption coefficients formerly used 
T , at the Bureau of Standards. 

wave. In order to measure 

the e.m.f. generated by the sound, the current from the 
telephone receiver after amplification and rectification 
was led into a galvanometer. The deflection of the 
galvanometer produced by the sound was then duplicated 
by applying an alternating e.m.f. from a potentiometer 
whose current supply was taken from the oscillator. The 
potentiometer reading gave the e.m.f. produced by the 



Absorbing 
surface\ 




FIQ. 30. Stationary-wave apparatus for absorption measurements used at the 
National Physical Laboratory, Teddington, England. 

sound. These readings at maxima and minima gave 
the M' s and N's of Eq. (59), from which the absorption 
coefficients of the test samples were computed. 

1 ECKHARDT and CHRISLEB, Bur. Standards Sci. Paper 526, 1926. 



92 



ACOUSTICS AND ARCHITECTURE 



Dr. E. T. Paris, 1 working at the Signals Experimental 
Establishment at Woolwich, England, has carried out 
investigations on the absorption of sound by absorbent 
plasters using the standing-wave method. His intensity 
measurements were made with a hot-wire microphone. 

Work at the National Physical Laboratory. 

Some extremely interesting results have been obtained 
by the stationary-wave method by Davis and Evans, at the 
National Physical Laboratory at Teddington, 2 England. 





FIG. 31. Absorption coeffi- 
cient as a function of the thick- 
ness. (After Davis and Evans.') 



Distance from Plate in Fractions of Wave Lengtti A 



Fio. 32. Variation in absorption 
affected by position of test sample in 
stationary wave. 



This apparatus is shown in Fig. 30 and is not essentially 
different in principle from that already described. Experi- 
mental details were worked out with a great deal of care, 
and tests of the apparatus showed that its behavior was 
quite consistent with what is to be expected on theoretical 
grounds. A loud-speaker source of sound was used. 
The exploring tube was of brass 1.2 cm. in diameter, which 
communicated outside the stationary-wave tube with a 
moving-coil loud-speaker movement. The pressure meas- 
urements were made by determining the e.m.fs. generated 
in a manner similar to that employed at the Bureau of 
Standards. 

1 Proc. Phys. Soc., vol. 39, p. 274, 1927. 

2 DAVIS and EVANS, Proc. Roy. Soc. London, Ser. A, vol. 127, 1930. 



MEASUREMENT OF ABSORPTION COEFFICIENTS 



93 



Among the facts brought out in their investigations was 
the experimental verification of a prediction made on 
theoretical grounds by Crandall, 1 that at certain thicknesses 
an increase of thickness will produce a decrease of absorp- 
tion. The phenomenon is quite analogous to the selective 
reflection of light by thin plates with parallel surfaces. In 
Fig. 31, the close correspondence between the theoretical 
and experimental results is shown. A second interesting 
fact is presented by the graph of Fig. 32, in which the 
absorption coefficient of J^-in. felt is plotted against the 
distance expressed in fractions of a wave length at which it 
is mounted from the reflecting end plate of the tube. We 



To amplifier ana/ 
potentiometer 




To oscillafor 



Fia. 33. Measurement of absorption coefficient by impedance method. 

Wente.) 



(After 



note that the absorption is a minimum at points very close 
to what would be the pressure internodes and the velocity 
nodes of the stationary wave if the sample were not present. 
In other words, the absorption is least at those points 
where the particle motion is least. The points of maximum 
absorption are not at what would be the points of maximum 
motion in the tube if the sample were not present. The 
presence of the sample changes the velocity distribution in 
the tube. 

The very great increase in the absorption coefficient when 
the sample is mounted away from the backing plate is to 
be noted. It naturally suggests the question as to how we 
shall define the absorption coefficient as determined by 
standing-wave measurements. Its value obviously depends 
upon the position of the sample in the standing-wave 
pattern. As ordinarily measured, with the sample mounted 

1 " Theory of Vibrating Systems and Sound," D. Van Nostrand Company, 
p. 195 



94 



ACOUSTICS AND ARCHITECTURE 



at the closed end of the tube, the measured value is that 
for the sample placed at a pressure node, almost the 
minimum. We shall consider this question further when 
we come to compare standing-wave and reverberation 
coefficients. 

Absorption Measurement by Acoustic Impedance Method. 

This method was devised by E. C. Wente of the Bell 
Telephone Laboratories. 1 The analysis is based upon the 
analogy between particle velocity and pressure in the 
standing-wave system and the current and voltage in an 
electrical-transmission line. Acoustic impedance is defined 
as the ratio of pressure to velocity. The experimental 




30 



60 125 250 500 1000 2000 4000 
Frequency^Cycles per Second 



FIG. 34. Absorption coefficients of felt by impedance method. (After Wcntc.) 

arrangement is indicated in Fig. 33. The tube was 3-in. 
internal diameter Shelby tubing, 9 ft. long, with j^-in. 
walls. The test sample was mounted on the head of a 
nicely fitting piston which could be moved back and forth 
along the axis of the tube. The source of sound was a 
heavy diaphragm 2% in. in diameter, to which was attached 
the driving coil lying in a radial magnetic field. The 
annular gap between the diaphragm and the interior of the 
tube was filled with a flexible piece of leather. 

Instead of measuring the maximum and minimum 
pressures at points in a tube of fixed length, the pressure at a 
point near the source, driven at constant amplitude, is 
measured for different tube lengths. Wente's analysis 

1 Bell System Tech. Jour., vol. 7, pp. 1-10, 1928. 



MEASUREMENT OF ABSORPTION COEFFICIENTS 



95 



gives for the absorption coefficient in terms of the maximum 
and minimum pressures measured near the source 



ot 



(63) 



The pressures were determined by measuring with an 
alternating-current potentiometer the voltages set up in a 
telephone transmitter, connected by means of a short tube, 
to a point in the large tube near the source. Figure 34 
shows the absorption coefficients of felt of various thick- 
nesses as measured by this method. Figure 35 shows the 
effect on the absorption produced by different degrees of 



t.oo 



080 



^60 



3 
< 



5 0.40 



0.70 



1" l" Hair felt- normal thickness 
~2- I" Hair felt -expanded to 2" thickness ~ 




125 250 500 1000 

Frequency-Cycles per Second 



2000 



4000 



30 60 

Fia. 35. Effect of compression of hair felt on absorption coefficients. 



packing of the hair felt. The three curves were all obtained 
upon the same sample of material but with different degrees 
of packing. They are of a great deal of significance in the 
light which they throw upon the discrepancy of the results 
obtained by different observers in the measurement of 
absorption coefficients of materials of this sort. Both 
thickness and degree of packing produce very large differ- 
ences in the absorption coefficients, so that it is safe to say 
that differences in the figures quoted by different authorities 
are due in part at least to actual differences in the samples 
tested but listed under the same description. 



96 



ACOUSTICS AND ARCHITECTURE 



S 




I 

w 
B 



o 
O 



i 
i 



> 




GO 

> : 







alll-a 



o I 



COCOrHiOQQ OQCO COCO 
rt CD CO "*< O5 O5 COt*- t*-iO 

odddod do do 



<N r- 

t-H o 

d o 



8: 



O CO <N O O OQ COOO O*O 

d d d d d d do" do 



:: 

O 



CO i-< -CO 

eq rt< -co 
O o : o 



O '"^ O5 00 O O 00 CO 

o d - d o o' o d ' d 



o o o o d d d d d 



s a a a s s 

o o o o o 

.. CO O5 O iO C O 

^ ^H ** <N <M' <M' c 



:l 



X X 

<x> o> 
1=! "S 

; 11 



as.. 

x 

1C ^ 



ts 

J w 



: - ;-S 

. c3 

' ' W2 1 

PQ PQ PQ 



i 



MEASUREMENT OF ASORPTION COEFFICIENTS 97 

Comparison of Standing-wave Coefficients by Different 
Observers. 

In Table V are given values of the absorption coefficients 
of a number of materials which have been measured by the 
investigators mentioned, using some form of the standing- 
wave method. 

For values on other materials reference may be made to 
the papers cited. Unfortunately there has not been any 
standard practice in the choice of test frequencies. More- 
over, in certain cases all the conditions that affect the 
absorption coefficients are not specified, so that comparison 
of exact values is not possible. The materials here given 
were selected as probably being sufficiently alike under the 
different tests to warrant comparison with each other and 
also with the results of measurements made by reverbera- 
tion methods. Inspecting the table, one notes rather wide 
differences between the results obtained by different 
observers in cases where a fair measure of agreement is to be 
expected. 

On the whole, it has to be said that while the stationary- 
wave method has the advantage of being applicable to 
measurements on small samples and, under a certain fixed 
condition, is capable of giving results that are of relative 
significance, yet it is questionable whether coefficients of 
absorption so measured should be used in the application 
of the reverberation theory. 

Reverberation Coefficient : Definition. 

We shall define the reverberation coefficient of a surface 
in terms of the quantities used in the reverberation theory 
developed in Chaps. IV and V. We can define it best in 
terms of the total absorbing power of a room, and this in 
turn can be best defined in terms of A, the rate of decay. 
A is defined by the equation 



dt 



98 ACOUSTICS AND ARCHITECTURE 

and a in turn by the relation 

_ 4AF 
a 

c 

We shall define the absorption coefficient of a given surface 
as its contribution per unit area to the total absorbing power 
(as just defined) of a room. If, therefore, QJI, a 2 , #3, etc., 
be the reverberation coefficients of the exposed surfaces 
whose areas are s h s. 2 , s 3 , etc., then 

a = aiSi + a< 2 s-2 + 3 3 + ' ' ' 

where the summation includes all the surfaces in the room 
exposed to the sound. As will appear, this definition 
conforms to the usual practice in reverberation measure- 
ments of absorption coefficients and is based on the assump- 
tions made in the reverberation theory. 

Sound Chamber. 

Any empty room with highly reflecting walls and a 
sufficiently long period of reverberation may be used as a 
sound chamber. The calibration of a sound chamber 
amounts simply to determining the total absorbing power 
of the room in its standard condition for tones covering the 
desired frequency range. This standard condition should 
be reproducible at will. For this reason, whatever furnish- 
ing it may have in the way of apparatus and the like should 
be kept fixed in position and should be as non-absorbent as 
possible. If methods depending upon the threshold of 
audibility are employed, the room should be free from 
extraneous sounds. In any event, the sound level from 
outside sources should be below the threshold of response 
of the apparatus used for recording sound intensities. The 
ideal condition would be an isolated structure in a quiet 
place, from which other activities are excluded. If the 
sound chamber is a part of another building, it should be 
designed so as to be free from noises that originate else- 
where. The first room of this sort to be built is that at the 
Riverbank Laboratories, designed by Professor Wallace 
Sabine shortly before his death and built for him by Colonel 



MEASUREMENT OF ABSORPTION COEFFICIENTS 



99 



George Fabyan. It was fully described in the American 
Architect of July 30, 1919, but for those readers to whom 
that article may not be accessible, the plan and section of 
the room are here shown. 




18 air space 



Section A-A 
Plan and section of Riverbank sound rhamber 

The dimensions of the room are 27 ft. by 19 ft. by 19 ft. 
10 in., and the volume is 10,100 cu. ft. (286 cu. m.). To 
diminish the inequality of distribution of intensity due to 
interference, large steel reflectors mounted on a vertical 



100 



ACOUSTICS AND ARCHITECTURE 




View of sound chamber of the Riverbank Laboratories. 




Plcin 



A 



feei 
012345 




Section 
A -A 



Fro. 36. Plan and section of sound chamber at Bureau of Standards. 



MEASUREMENT OF ABSORPTION COEFFICIENTS 101 

shaft are rotated noiselessly during the course of the 
observations. The source of sound usually employed is a 
73-pipe organ unit, with provisions for air supply at con- 
stant pressure to the pipes. A loud-speaker operated by 
vacuum-tube oscillator and amplifier is also used as a 
sound source. 

Other sound chambers have been subsequently built in 
this country, notably those at the University of Michigan 
and the University of California at Los Angeles. The 
plan and section of the chamber recently built at the 
Bureau of Standards 1 are shown in Fig. 36. The dimen- 
sions of this room are 25 by 30 by 20 ft., and the double 
walls are of brick 8 in. thick with 4-in. intervening air space. 
In order to reduce the effect of the interference pattern, the 
source of sound is moved on a rotating arm, approximately 
2 ft. long, during the course of the observations. 

Sound-chamber Methods : Constant Source. 

Essentially any sound-chamber method of measuring the 
reverberation coefficients of a material is based upon 
measuring the change in total absorbing power produced 
by the introduction of the material into a room whose total 
absorbing power without the material is known. If a and 
a' be the total -absorbing power of the room, first empty and 
then with an area of s square units of absorbing material 
introduced, then (a' a)/s is the increase in absorbing 
power per square unit effected by the material. If the 
test material replaces a surface of the empty room whose 
absorption coefficient is i, then the absorption coefficient of 
the material in question is 

a = on + ^~ (64) 

The equations of Chap. V suggest various ways in which 
a and a! may be measured, either by measurements of time 
of decay or by measurements of intensities. The procedure 
developed by Professor Sabine which has been followed 

1 Bur. Standards Res. Paper 242. 



102 ACOUSTICS AND ARCHITECTURE 

for the most part by investigators since his time is as 
follows: 

1. The value of a, the absorbing power of the empty 
room, is determined by means of the four-organ experiment 
or some modification thereof, in which the times for sources 
of known relative powers are measured. 

2. From the known value of the absorbing power of the 
empty room and the time required for the reverberant 
sound from a given source to decrease to the threshold of a 
given observer, the ratio E/i for this particular source and 
observer can be computed by Eq. (39). 

3. E/i being known, and assuming that E, the acoustic 
output of the source, is not influenced by altering the 
absorbing power of the room, Eq. (39) is evoked to deter- 
mine a' from the measured value of T", the measured time 
when the sample is present. Since Eq. (39) contains both 
a' and log a' , its solution for a has to be effected by a method 
of successive approximations. As a matter of convenience, 
therefore, it is better to compute the values of T' for various 
values of a! and plot the value of a' as a function of T'. 
The values of a' for any value of T' are then read from this 
curve. 

Calibration of Sound Chamber : Four-organ Method. 

Illustrating the method of calibration and the measure- 
ment of absorption coefficients outlined above, the proce- 
dure followed at the Riverbank Laboratories will be given 
somewhat in detail. 1 Four small organs, each provided 
with six C pipes, 128 to 4,096 vibs./sec., operated by electro- 
pneumatic action, were set up in the sound chamber. 
These were operated from a keyboard in the observer's 
cabinet, wired so that each pipe of a given pitch could be 
made to speak singly or in any combination with the other 
pipes of the same pitch. Air pressure was supplied from the 
organ blower outside the room. The pressure was con- 
trolled by throttling the air supply so that the speaking 

1 Jour. Franklin Imt., vol. 207, No. 3, p. 341, 1929. 



MEASUREMENT OF ABSORPTION COEFFICIENTS 103 



pressure was the same whether one or four pipes were 
speaking. The absorbing power of the sound chamber was 
increased over that of its standard condition by the presence 
of this apparatus. Before the experiment the four pipes of 
each pitch were carefully tuned to unison. Slight varia- 
tions of pitch were found to produce a marked difference in 
the measured time. As in all sound-chamber experiments, 
the large steel reflector was kept revolving at the rate of 
one revolution in two minutes. It was found that even 
with the reflector in motion, the observed time varied 
slightly with the observer's posi- 
tion. For this reason timings 
with each combination of pipes 5 
were made in five different posi- 
tions. At the end of the series 
of readings, the time for the pipe 
of the large organ was measured, 
with the four-organ apparatus 
first in and then out of the room. 
This gave the necessary data for 
evaluating E/i for the pipes of 
the large organ used as the stand- the logarithm of the number of 
ard sources of sound and also for 

determining the absorbing power of the room in its standard 
condition from the measurements made with the four organs 
present. 

Figure 37 gives the results of the four-organ experiment 
made in 1925. The maximum departure from the straight- 
line relation called for by the theory is 0.04 sec. 

Let a! be the absorbing power at a given frequency of the 
sound chamber with the four organs present. 




a' = 



47 loge n 4 X 286 X 2.3 logio n 

c(T n - ro " 



7.7m 



where m is the slope of the corresponding line of Fig. 37. 
Computing E/i for the pipe of the large organ and the 
particular observer, we can compute a for the sound 
chamber in its standard condition. 



104 ACOUSTICS AND ARCHITECTURE 

TABLE VI. SOUND-CHAMBER CALIBRATION, FOUR-ORGAN PIPE METHOD 



Frequency 


T' 


T 


a' 


a 


E 

logio Y 


128 


12 58 


14 70 


3.56 


3 09 


8.31 


256 


12.42 


14 50 


4.67 


4.03 


10.14 


512 


11.94 


14.25 


5.38 


4.55 


11.01 


1,024 


11.32 


12.80 


5.58 


4.98 


30 91 


2,048 


9.16 


9.86 


6.34 


5.92 


10 27 


4,096* 


5 29 


5.63 


8.90 


8.40 


8 98 


4 096 f 


4 86 




12 2 




8 84 















* Relative humidity 80 per cent, 
t Relative humidity 63 per cent. 

Table VI gives the values of absorbing powers of the 
sound chamber, and logic E/i for the organ-pipe sources used 
in measuring absorption coefficients of materials. It is to be 
noted that i is the threshold of audibility for a given 
observer. Knowing the absorbing power, E/i for a second 
observer can be obtained from his timings of sound from 
the same sources. In this way, the method is made 
independent of the absolute value of the observer's thresh- 
old of hearing. Comparison of the values of E/i for two 
observers using the same sources of sound is made in Table 
VII. One notes a marked difference in the threshold of 
audibility of these two observers, both of whom have what 
would be considered normal hearing. 1 

TABLE VII. REVERBERATION TIMES BY Two OBSERVERS 



Frequency . . 
Observer. . . 


128 


256 


512 


1,024 


2,048 


A 


B 


A 


B 


A 


B 


A 


B 


A 


B 


Time 


14.70 
8.31 


12 59 
7 46 


14.58 
10.14 


13.04 
9.33 


14.25 
11.01 


13.81 
10.75 


12.80 
10.91 


13 24 
11.20 


9.80 
10 27 


9.73 
10.21 


i E 

login -r . . 


Difference . . 


-0.85 


-0.81 


-0.26 


+0.29 


-0.06 



1 For the variation in the absolute sensitivity of normal ears see FLETCHER, 
"Speech and Hearing," p. 132, D. Van Nostrand Company, 1929; KRANZ, 
F. W., Phys. Rev., vol. 21, No. 5, May, 1923. 



MEASUREMENT OF ABSORPTION COEFFICIENTS 105 



Effect of Humidity upon Absorbing Power. 

In Table VI, the values of the absorbing power at 4,096 
vibs./sec. are given for two values of the relative humidity. 
It was early noted in the research at the Riverbank Labora- 
tories that the reverberation time at the higher frequencies 
varied with the relative humidity, being greater when the 
relative humidity was high. It was at first supposed that 
this effect was due to surface changes 
in the walls, possibly an increase in 
the surface porosity of the plaster 
as the relative humidity decreased. 
Experiments showed, however, that 
changes in reverberation time fol- 
lowed too promptly the decrease in 
humidity to account for the effect as 
due to the slow drying out of the 
walls. Subsequent painting of the 
walls with an enamel paint did not 
alter this effect. Hence, it was con- 
cluded that the variation of rever- 
beration time with changes 







1 o^ 


^S 
< 


\ 


o 


**& 




i* 


V 


S 








CD*- 




/ 






a 




Xs 


40C 



















in 



Per Cent Relive Humidify, 
( Temp, 17 Deg.C.) 

, , FIG. 38. Variation of re- 

mUSt be due to the ettect vcrberation time with rela- 

of water vapor in the air upon the j^ ) umidity - (Aftcr 
atmospheric absorption of acoustic 

energy. High-frequency sound is more strongly absorbed 
in transmission through dry than through moist air. 

Erwin Meyer, 1 working at the Heinrich Hertz Institute, 
has also noted this effect. Figure 38 shows the relation, 
as given by Meyer, between reverberation time and relative 
humidity at 6,400 and 3,200 vibs./sec. 

The most recent work on this point has been done by 
V. O. Knudsen. 2 The curves of Fig. 39a taken from his 
paper show the variation of reverberation times for fre- 
quencies from 2,048 to 6,000 vibs./sec. with varying relative 
humidities. We note that the time increases linearly with 



1 Zeits. tech. Physik, No. 7, p. 253, 1930. 

2 Jour. Acous. Soc. Arner., p. 126, July, 1931. 



106 



ACOUSTICS AND ARCHITECTURE 



relative humidity up to about 60 per cent. In these 
experiments the temperature of the wall was lower than 
that of the air in the room. It was observed that condensa- 
tion on the walls began at a relative humidity of about 70 
to 80 per cent. In other experiments, conducted when the 
wall temperature was higher than the room temperature 
and there was no condensation on the walls, the reverbera- 
tion time increased uniformly with the relative humidity 
up to more than 90 per cent. Knudsen ascribes the bend 
in the curves to surface effects which increased the absorp- 
tion at the walls when moisture collected on them. 



20 30 40 50 60 70- 80 
Percentage Relative Humidity (eitoe2C) 



90 100 



FIG. 39a. Knudsen's results on variation of reverberation time with changes in 
relative humidity. 

Knudsen further found that when the relative humidity 
was maintained at 100 per cent and fog appeared in the 
room, the reverberation times became markedly lower for 
all frequencies. Thus the reverberation time at 512 vibs./ 
sec. was decreased from 12.65 sec., relative humidity 80 
per cent to 6.52 sec., relative humidity 100 per cent with 
fog present. Knudsen is inclined to ascribe this marked 
increase in absorption with fog in the air to the presence 
of moisture on the wall. The author questions this 
explanation in view of the magnitude of the effect, par- 
ticularly at the low frequencies. The decrease from 12.65 
to 6.52 sec. calls for a doubling of the coefficient of absorp- 
tion, if we assume the effect to be due only to changes in 



MEASUREMENT OF ABSORPTION COEFFICIENTS 107 



surface condition. In numerous instances, in the River- 
bank sound chamber, moisture due to excessive humidity 
has collected on floors and walls but without fog in the 
room. No marked change in the reverberation has been 
observed in such cases. It would seem more likely that 
the effect noted when fog is present is due to an increase of 
atmospheric absorption. The question is of considerable 
importance both theoretically and practically in atmos- 
pheric acoustics. 

Making reverberation measurements in two rooms of 
different volumes but with identical surfaces of painted 
concrete, Knudsen was able to separate the surface and 
volume absorption and to measure the attenuation due to 



0005 



0004 



-^0003 -- 



= 0.002 



0001 



EO 30 40 50 60 v 70 

Pencentage Relaiive Humidity (21 to 22 C) 

Fia. 396. Values of m (sec toxt) as a function of relative humidity. (After 

Knudsen.) 

absorption of acoustic energy in the atmosphere. Assume 
that the intensity of a plane wave in air decreases according 
to the equation / = I Q e ~~ ct . Then if we take account of 
both the surface and volume absorption, the Sabine 
formula in English units becomes 

0.049 V 




a + 4mV 
or, with the Eyring modification, 

, = 0.049F 

-5 log, (1 - a.) + 4mV 



108 ACOUSTICS AND ARCHITECTURE 

Figure 396 gives Knudsen's values of m in English units 
for different frequencies and different relative humidities. 
For frequencies below 2,048 vibs./sec. m is so small as to 
render the expression 4m V negligible in comparison with 
the surface absorption. 

Experiments on the effect of moisture on the viscosity 
of the atmosphere show that while there is a slight decrease 
in viscosity with increase in relative humidity, the magni- 
tude of the effect is far too small to account for observecl 
changes in atmospheric sound absorption. A greater heat 
conductivity from the compression to the rarefaction phase 
for low than for high humidity is a possible explanation. 
If this be true, then the velocity of sounds of high fre- 
quency should be greater in moist than in dry air, since, 
assuming a heat transfer between compression and rarefac- 
tion, the velocity of sound would tend to the lower New- 
tonian value. The writer knows of no experimental 
evidence for such a supposition. The point is of con- 
siderable theoretical interest and is worthy of further study. 

Calibration of Sound Chamber with Loud-speaker. 

The development in recent years of the vacuum-tube 
oscillator and amplifier together with the radio loud-speaker 
of the electrodynamic or moving-coil type gives a con- 
venient source of sound of variable output, for use in the 
measurement of sound-absorption coefficients. Experi- 
ment shows that the amplitude response of the electro- 
dynamic free edge-cone type of loud-speaker is proportional 
to the alternating-current input. 1 

Under constant room conditions, therefore, the acoustic- 
power output is proportional to the square of the input 
current, whence we may write 

E = kC* 
where A; is a constant. 

1 Recent experiments show that this relation holds over only a limited 
range for most commercial types of dynamic loud speakers. 



MEASUREMENT OF ABSORPTION COEFFICIENTS 109 
Equation (57) then may be written 

aT = 4 

1 



ac 

The equivalent of the four-organ experiment then may be 
very simply performed by measuring the reverberation 
time for different measured values of the audiofrequency 
current input of the loud-speaker source. If T z be the 
measured time with a constant input C 2 , we have 



aci 



whence 



n _ 4yrio ge Ct - loge c,i _ gVriogio(CiVCi)-| 

a - TL T, - T* \ ~ J - 2 cL T,-T t \ 



For the Riverbank sound chamber this becomes 



(65) 



In Fig. 40 the logarithm of the ratio of the current in the 
loud-speaker to the minimum current employed is plotted 



Z 



/ 



4 



y 



"7 



1-5/2 vi bs/sec. 



01234-56780 

FIG. 40. Difference in reverberation time as a function of logarithm of loud- 
speaker current ratio. 

against the difference in the corresponding reverberation 
times. 

The expression T -^T ^ or eac ^ f fec l uenc y * s ^ e 



110 



ACOUSTICS AND ARCHITECTURE 



slope of the corresponding line in Fig. 40. We note that 
with the loud-speaker we have a range of roughly 250 to 1 
in the currents, corresponding to a variation of about 
62,500 to 1 in the intensities, whereas in the experiment with 
the organ pipes the range of intensities was only 4 to 1. 
The loud-speaker thus affords a much more precise means 
of sound-chamber calibration. In Table VIII, the results 
of the four-organ calibration and of three independent 
loud-speaker calibrations are summarized. 

TABLE VIII. ABSORBING POWER IN SQUARE METERS OP RIVERBANK SOUND 

CHAMBER 





Four organs 


Loud-speaker 


Frequency 






1930 


1931* 




1925 


1928 


Steady 


Flutter 


Flutter 








tone 


tone 


tone 


128 


3 09 


4.03 






4 85 


256 


4 03 


4.87 


4.70 


4.68 


5 05 


512 


4.55 


4 56 


4 70 


4.68 


5.30 


1,024 


4 98 


5 32 


4 73 


5.08 


5 42 


2,048 


5 86 


7.02 


7.22 


7 38 


7 65 



* Room conditions slightly altered from those of 1930. 

We note fair agreement between the results with the 
four organs and the loud-speaker at 512 and 1,024 but 
considerable difference at the low and high frequencies. 
It is quite possible that for the lower tones the separation 
of the four organs was not sufficiently great to fulfill the 
assumption that the sound emitted by a single pipe is 
independent of whether or not other pipes are speaking 
simultaneously. At 2,048 the difference in times between 
one and two or more pipes was small, so that the possible 
error in the slope of the lines was considerable, in view of 
the limited range of intensities available in this means of 
calibration. 

Data are also presented in which a so-called " flutter 
tone" was employed, that is, a tone the frequency of which 
is continuously varied over a small range about a mean 



MEASUREMENT OF ABSORPTION COEFFICIENTS 1 1 1 

frequency. The flutter range in these measurements was 
about 6 per cent above and below the mean frequency, and 
the flutter frequency about two per second. This expedient 
serves to reduce the errors in timing due to interference. 

Calibration Using the Rayleigh Disk. 

A variation of the four-organ method of calibration has 
been used by Professor F. R. Watson 1 at the University of 
Illinois. He used the Rayleigh disk as a means of evaluat- 
ing the relative sound outputs of his sources of sound. 
For the latter he used a telephone receiver associated with a 



'L 
\ 

>o 

1 

-3 

-4 
-; 






























/ 




























f 




























/ 




























/x 


























< 




























^ 




























/ 




























^. 




























/ 




























/ 


























. 


' 


























. 


r 


























/ 





























































Time in Seconds 
FIG. 41. Sound-chamber calibration using Rayleigh disk. (After Watson.) 

PTelmholtz resonator. The receiver was driven by current 
from a vacuum-tube oscillator and amplifier. 

The relative outputs of the source for different values of 
the input current were measured by placing the sound 
source together with a Rayleigh disk inside a box lined with 
highly absorbent material. The intensities of the sound 
for different values of the input current were taken as 
proportional to 0/cos 26, where 6 is the angle through which 
the disk turns under the action of the sound against the 
restoring torque exerted by the fiber suspension. The 
maximum intensity was 1830 times the minimum intensity 
used. Figure 41, taken from Watson's paper, shows the 

1 Univ. III. Eng. Exp. Sta. Bull. 172, 1927. 



112 



ACOUSTICS AND ARCHITECTURE 



linear relation between the logarithm of the intensity 
measured by the Rayleigh disk and the time as measured 
by the ear. From the slope of this line and the constants 
of the room the absorbing power was computed. With 
this datum the absorption coefficients of materials are 
measured as outlined in the preceding section. 

Other Methods of Sound-chamber Calibration. 

The ear method of calibration is open to the objections 
that it is laborious and that it requires a certain amount of 
training upon the part of the observer to secure consistency 
in his timings. The objection is also raised that variation 




\Loudspeaker 
Fia. 42. Reverberation meter of Werite and Bedell. 

in the acuity of the observer's hearing may introduce 
considerable personal errors. However, the writer's own 
experience of twelve years' use of this means is that with 
proper precautions the personal error may be made less 
than errors due to variation in the actual time of decay 
due to the fluctuation of the interference pattern, as 
the reverberant sound dies away. Various instrumental 
methods either of measuring the total time required for 
the reverberant sound to die away through a given intensity 
range or of following and recording the decay process con- 
tinuously have been proposed and used. 

Relay and Chronograph Method. 

The apparatus, devised at the Bell Laboratories and 
described by Wente and Bedell, 1 is shown in Fig. 42. The 
authors describe the method as consisting of an "electro- 

1 Jour. Acous. Soc. Amer., vol. I, No. 3, Part 1, p. 422, April, 1930. 



MEASUREMENT OF ABSORPTION COEFFICIENTS 113 

acoustical ear of controllable threshold sensitivity." The 
microphone T serves as a "pick-up" and is connected to a 
Vacuum-tube amplifier provided with an attenuator which 
may control the amplification in definite logarithmic steps. 
The amplifier is terminated by a double-wave rectifier. 
The rectified current passes through the receiving windings 
of a relay, which is constructed so that when the current 
exceeds a certain value, the armature opens the contact at 
A, charging the condenser C. When the current falls 
below a certain value, the armature is released and the 
condenser discharges through the primary windings of the 
spark coil M causing a spark to pass to the rotary drum 
D at P. The drum is rotated at a constant speed and is 
covered by a sheet of waxed paper on which the passage 
of the spark leaves a permanent impression. The key k 
opens the circuit from the oscillator which supplies current 
to the loud-speaker source, thus cutting off the sound. 
This key is operated mechanically by a trigger not shown. 
This trigger is released automatically when the drum is in a 
given angular position. 

The threshold of the instrument is set at a definite value 
by adjustment of the attenuator. The sound source is 
started. After the sound in the room has reached a steady 
state, the trigger of the key k is set and is then auto- 
matically released by the rotating drum. When the 
decaying sound has reached the threshold intensity of 
the instrument, the relay A operates, causing the spark to 
jump. The distance on the waxed paper from the cut-off 
of the sound to the record of the spark gives the time 
required for the sound to decrease from the steady state 
to the threshold of the instrument. Call this initial thresh- 
old ii. The threshold of the instrument is then raised 
to a second value i' 2 . The point P is shifted on the scale S 
to the right an amount proportional to the log i 2 log f i, 
and the operation is repeated; and by repeatedly raising 
the threshold and shifting the point P, a series of dots 
giving the times for the reverberant sound to decrease 
from the steady state to known relative intensities are 



114 



ACOUSTICS AND ARCHITECTURE 



recorded on the waxed paper. Such a record is shown 
in Fig. 43. The authors state: "If the decay of a sound 
at the microphone had been strictly logarithmic, these dots 
would all lie along a straight line. This ideal will almost 
never be encountered in practice. We must therefore be 

content with drawing a line of 
best fit through these points." 
From the slope of the line and 
the peripheral speed of the 
drum the absorbing power of 
the room may be obtained, in 
the manner already described 
in the four-organ experiment. 
We note that this method is 
based on measurement of the 
times from a fixed steady- 
state intensity to a threshold, 
variable in known ratios, whereas the four-organ experiment 
and its loud-speaker variant measure the time from a 
variable steady state to a fixed threshold. Properly 
designed and freed from mechanical and electrical sources 
of error, the apparatus is not open to the objection of 




i 



i i i I 



logarithmic Gain of Amplifier 
FIG. 43. Decay of reverberant 
sound as obtained with reverberation 
meter. 



Amplifier 



May for 
sfop worfch 




FIG. 44. Vacuum-tube circuit for automatically recording reverberation times. 

(After Meyer.) 

personal error. The uncertainties due to the shifting 
interference pattern, however, still exist. 

Another arrangement for automatically determining the 
reverberation time, due to Erwin Meyer 1 of the Heinrich 
Hertz Institute, is shown in Fig. 44. Here the reverberant 
sound is picked up by a condenser microphone. The 

1 Znt*. tech. Physik, vol. 11, NO. 7, p. 253, 1930. 



MEASUREMENT OF ABSORPTION COEFFICIENTS 115 



current is amplified and rectified, and the potential drop 
of the rectified current is made to control the grid voltage 
of a short radio-wave oscillator. In the plate circuit of 
this oscillator is a sensitive relay which operates the 
stopping mechanism of a stop watch. As long as the sound 
is above a given intensity, the negative grid voltage of 
the oscillating tube is sufficiently great 20 
to prevent oscillation. As the rever- 1^ 
berant sound dies away, the negative | JO 
grid bias decreases. When the intensity j> 05 
reaches the given value, oscillations are 
set up, increasing the plate current, oper- 
ating the relay, and stopping the watch. 




10 20 30 40 50 
Time in Seconds 

FIG. 45. D e c a y of 
sound intensity, Bureau 
of Standards sound 
chamber, oscillograph 
method. 



Oscillograph Methods. 

The oscillograph has been employed for recording the 
actual decay of sound in a room. Experiments using a 
steady tone have shown that the extreme fluctuations of 
intensity due to the shifting of the interference pattern 
give records from which it is extremely difficult to obtain 

precise quantitative data. 1 
Employing a flutter tone 
and at the same time rotat- 
ing the source of sound, 
Chrisler and Snyder 2 found 
that oscillograms could be 
made on which the average 
amplitude of the decaying 
2048 4096 sound could be drawn as the 
envelope of the trace made 




128 



256 512 1024 
Frequency 

FIG. 46. Total absorption, Bureau of 
Standards sound chamber, by the oscil- by the OSClllograph mirror 
lograph and ear measurements. Qn ^ moving film Thege 

envelope curves proved to be logarithmic, and by measuring 
the ordinates for different times the rate of decay can be 
obtained. In Fig. 45, the squares of the amplitudes from one 

1 KNUDSEN, V. O., Phil. Mag., vol. 5, pp. 1240-1257, June, 1928. 

2 CHRISLER, V. L., and W. F. SNYDER, Bur. Standards. Jour. Res., vol. 5, 
pp. 957-972, October, 1930. 



116 



ACOUSTICS AND ARCHITECTURE 



of their curves are plotted against the time. In Fig. 46, the 
absorbing powers of the sound chamber at the Bureau of 
Standards as determined both by oscillograph and by the 
ear method are plotted as a function of the frequency of 
the sound. In summing up the results of their research 
with the oscillograph, Chrisler and Snyder state that for 



Amplifier 





FIG. 47. Meyer and Just's apparatus for recording decay of sound intensity. 

accuracy of results a considerable number of records must 
be made and that the time required is greater than that 
required to make measurements by ear. For these reasons 
the oscillograph method has been abandoned at the 
Bureau of Standards. 




Fia. 48. Decay of sound intensity recorded with apparatus of Meyer and Just- 

A modification of the oscillograph method has been used 
by E. Meyer and P. Just. 1 Their electrical circuit is shown 
in Fig. 47. A microphone is placed in the room in which the 
reverberation time is to be measured. The current is 
amplified first by a two-stage amplifier. Before the third 
stage is an automatic device which at the end of a given 
period increases the amplification by a factor of ten, 



1 Electr. Nachr-Techn., vol. 5, pp. 293-300, 1928. 



MEASUREMENT OF ABSORPTION COEFFICIENTS 117 

thus increasing the sensitivity in the same ratio. The 
amplified current is passed through a vacuum-tube rectifier, 
in the output of which is a short-period galvanometer. The 
deflection of the galvanometer is recorded upon a moving 
film, on which time signals are marked. A series of records 
from their paper is shown in Fig. 48. The logarithmic 
decay is computed from these records as indicated in the 
previous paragraph. 

Methods Based on Intensity Measurements. 

With a source of acoustic power E y the output of which 
is independent of room conditions, we have for the steady- 
state intensity 



whence 

acl l = 4E 

If now an absorbing area be brought into the room, the 
total absorbing power becomes a', and the steady-state 
intensity 7'i, so that 

a'c/'i = 4E 
and 



Subtracting, 

a! - a = a[ A ~ f/1 ] (66) 

Equation (66) offers an attractively simple method of 
measuring the absorbing power of the material brought 
into the room, provided one has means of measuring the 
average intensity of the sound in the room and knows the 
value of a, the absorbing power of the room in its standard 
condition. Professor V. O. Knudsen 1 at the University of 
California in Los Angeles has used this method of measuring 
change in absorbing power. The experimental arrange- 

Mag., vol. 5,pp. 1240-1257, June, 1928. 



118 



ACOUSTICS AND ARCHITECTURE 



ynent is shown in Fig. 49. A loud-speaker source driven 
by an oscillator and amplifier was used, and in one series 
of experiments the sound was picked up by four electro- 
magnetic receivers, suitably mounted on a vertical shaft 
which was rotated with a speed of 40 r.p.m. In other 
experiments, an electrodynamic type of loud-speaker was 
substituted, and the sound was received by a single-con- 
denser microphone mounted on a swinging pendulum. 




uuf^J 

Motor *sariv& 
variable inductance 

FIG. 49. Knudsen's apparatus for determining absorbing power by sound- 
intensity measurements. 

Table IX gives results taken and computed from Knud- 
sen's measurement by this method of the absorbing power of 
a room as more and more absorbing material is brought in. 

TABLE IX 



Area absorb- 
ent material, 
square feet 


Average 
galvanometer 
deflection d' 


Average 
intensity 


Absorbing 
power 


Coefficient 


None 


14.25 


5.00 X 10 7 i 


28.3 




4 


13 34 


4.68 


30 2 


0.489 


16 


11.13 


3 91 


36 2 


0.507 


36 


9 25 


3 25 


43.6 


0.439 


49 


8.19 


2.88 


49 2 


0.442 


64 


6.99 


2.46 


57 5 


0.471 


81 


6.42 


2.25 


62.8 


0.434 


100 


5.61 


1 97 


71.9 


450 


127 


5.05 


1.77 


79.8 


422 










Ave. 457 



Calibration of the amplifier showed that the galva- 
nometer deflection was proportional to the square of the 



MEASUREMENT OF ABSORPTION COEFFICIENTS 119 

voltage input of the amplifier and therefore proportional 
to the sound-energy density in the room. The values of a' 
are based upon a value of a = 28.3 units, obtained by rever- 
beration measurements in the empty room. Absorbing 
powers are given in English units. 

The mean value of the absorption coefficient of this same 
material obtained by both the reverberation method and the 
intensity method is given by the author as 0.433. It is 
obvious that the precision of the method is no greater than 
that with which a, the absorbing power of the empty room, 
can be determined, and this in turn goes back to rever- 
beration methods. 



ABSORPTION COEFFICIENT 

OF SAMPLE / 

Computed from t/'mes 
(5amjj/e in anc/sampfe 



Steady) t t' a 9' 
W 8.19 473 8.QS 



a' -4 "3.31 a'- a * 3.41 

Coefficient * 75.5 % Coefficient = 775 % 




5 10 15 

FIG. 50. Data for determination of absorption coefficients by various methods 

Absorption Coefficient Using Source with Varying Output. 

With a source of sound whose acoustic output can be 
varied in measured amounts the absorbing power of the 



120 ACOUSTICS AND ARCHITECTURE 

reverberation chamber both in its standard condition and 
with the absorbent material present can be determined 
directly. A carefully conducted experiment of this sort 
serves as a useful check on the validity of the constant- 
source method and the assumptions made and also shows 
the degree of precision that may be obtained in reverbera- 
tion methods of measurement. Figure 50 presents the 
results of such an experiment. Here the logarithm of 
the current in milliamperes in the loud-speaker is plotted 
against the duration of audible sound, first without absorb- 
ent material and then with 4.46 sq. m. (48 sq. ft.) of an 
absorbent material placed in the sound chamber. The 
data presented afford two independent means of computing 
the absorption coefficient of the material: (1) From the 
slopes of the straight lines a and a' may be determined by 
Eq. (65). Thus 

a = 15.4m 
and 

a' = 15.4w' 

ra, and m 1 being the slopes of the straight lines representing 
the experimental points without and with the absorbent 
material present. (2) Assuming equal acoustic outputs 
for equal loud-speaker currents, o! may be computed from 
the values of T and T", the times for any given current 
value before and after the introduction of the absorbent 
material, and a, the absorbing power of the empty room. 
Thus if E be the acoustical power of the source, assumed 
for the moment to be the same for a given current under 
the two room conditions, we have 

aT = 7.70 lo glo *-f = 7.7 lo glo 7 (67a) 

CiCt' L 

4/T J f 

a'T = 7.70 logio -=F-. = 7.7 logio -^ (676) 

a ci % 

Whence by eliminating 4E/ci, we have 



MEASUREMENT OF ABSORPTION COEFFICIENTS 121 



a' = y r \aT - 7.7 log -) (68) 

We note, however, that if the straight lines of Fig. 50 
be extrapolated, their intersection falls on the axis of zero 
time. This means that equal currents in the loud-speaker 
set up equal steady-state intensities throughout the room, 
whereas the assumption of equal acoustical powers for a 
given current implies a lower intensity in the more absorbent 
room. We are forced to the conclusion, therefore, that the 
sound output of a loud-speaker operating at a fixed amplitude 
is not independent of the room conditions. The data 
here presented indicate that for a fixed amplitude of 
the source the output of the speaker is directly proportional 
to the absorbing power; that is, E'/a' = E/a, I = /', and 
hence 

' = ^ (69) 

The values of the absorption coefficients for the material as 
determined by organ-pipe data and by the two methods 
from the loud-speaker data are shown in Fig. 50. The 
computations from the organ-pipe data are based on the 
assumption that the power of the pipe is constant under 
altered room conditions. The value of a' is computed from 
Eq. (69) instead of Eq. (68), since the latter would give 
different values of a', depending upon the particular current 
values for which T and T r are taken. 

Reaction of Room on the Source. 

The foregoing brings up the very important question of 
the assumption to be made as to what effect on the rate of 
emission of sound energy from a source of constant ampli- 
tude results from altering the absorbing power of the room 
in which it is placed. On this point Professor Sabine 
states: 1 

In choosing a source of sound, it has usually been assumed that a 
source of fixed amplitude is also a source of fixed intensity (power) . On 

1 " Collected Papers on Acoustics," Harvard University Press, p. 279, 
1922. 



122 ACOUSTICS AND ARCHITECTURE 

the contrary, this is just the sort of source whose emitting power varies 
with the position in which it is placed in the room. On the other hand, 
an organ pipe is able within certain limits to adjust itself automatically 
to the reaction due to the interference system. We may say briefly that 
the best standard source of sound is one in which the greatest percentage 
of emitted energy takes the form of sound. 

Sabine is here speaking of the effect of shifting the source 
of sound with reference to the stationary-wave system. Tn 
the experiment of the preceding section, the large steel 
reflectors already mentioned were kept moving, thus 
continuously shifting the stationary-wave system. The 
use of the flutter tone also would preclude a fixed inter- 
ference pattern, so that if there were a difference in the 
output of the loud-speaker brought about by the introduc- 
tion of the absorbent, this difference was due to the change 
in the total absorbing power of the room. Repeating the 
experiment for the tones 1,024 and 2,048 vibs./sec. gave 
the same results, namely, straight lines whose intersection 
was on the axis of zero times. Earlier experiments, 1 using 
a different loud-speaker and slightly different electrical 
arrangements, showed the intersection of the lines at positive 
values of the time. On the other hand, Chrisler reports 2 
a few sets of measurements in which the intersection of the 
lines was at negative values of the time, indicating that the 
loud-speaker at constant-current input acts as a source of 
constant acoustical output independent of room conditions. 
Existing data are therefore equivocal as to just how a loud- 
speaker driven at constant amplitude behaves as the absorb- 
ing power of the room is altered. 

In this connection, the results of Professor Sabine's 
acoustical survey of a room shown in Fig. 18 of Chap. Ill 
are interesting. The upper series shows the amplitudes at 
points in the empty room. The lower series gives the 
amplitudes at the same points when the entire floor is 
covered with hair felt. The amplitude of the source was 
the same in the two cases. Comparing the two, one notes 

1 Jour. Franklin InsL, vol. 207, p. 341 March, 1929. 

2 Bur. Standards Res. Paper 242. 



MEASUREMENT OF ABSORPTION COEFFICIENTS 123 

that the introduction of the felt does not materially alter 
the general distribution of sound intensity. The maxima 
and minima fall at the same points for the two conditions. 
Further, the minima in the empty room are for the most 
part more pronounced than when the absorbent is present, 
and finally we note that on the whole the amplitude is 
less in the empty than in the felted room. Taking the 
areas of the figures as measured with a planimeter, we 
find that the average amplitudes in the empty and felted 
rooms are in the ratio of 1:1.38, and this for a source in 
which the measured amplitude is the same in the two cases. 
One cannot escape the conclusions (1) that in this experi- 
ment, at least, covering the entire floor with an absorbent 
material did not shift the interference pattern in horizontal 
planes and (2) that the acoustic efficiency of the constant- 
amplitude source set up in the absorbent room was enough 
greater than in the empty room to establish a steady-state 
intensity 1.9(1. 38) 2 times as great. The same output 
should, on the reverberation theory, produce a steady-state 
intensity only about one-half as great. 

Lack of sufficient data to account for the apparently 
paradoxical character of these results probably led Professor 
Sabine to withhold their publication until further experi- 
ments could be made. Penciled notations in his notes of 
the period indicate that further work was contemplated. 
Repeating the experiment with the tremendously improved 
facilities now available both with loud-speaker and organ- 
pipe sources and with steady and flutter tones would be 
extremely interesting and should throw light on the 
question in point. 

The writer's own analysis of the problem indicates that 
with a constant shift of the interference pattern, by means 
of a flutter tone or a moving source or by moving reflectors, 
the constant-amplitude source should set up the same 
steady-state intensity under both the absorbent and the 
non-absorbent room conditions. Operated at constant 
current, a loud-speaker should thus act as a source whose 
output in a given room is directly proportional to the 



124 



ACOUSTICS AND ARCHITECTURE 



absorbing power, while at a constant power input it 
should operate as a constant-output source. Experimental 
verification of these conclusions has not yet been attained, 
so that there is still a degree of uncertainty in the determi- 
nation of absorption coefficients by the reverberation 
method using the electrical input as a measure of the 
sound output of the source. 

In Table X are given the values of the absorption 
coefficients at three different frequencies of a single material, 
computed in the manners indicated from organ-pipe and 



TABLE X 



Organ pipe, a' -r\ at> 7.7 lg ~~) 



Loud-speaker, of = , (at) 



(1) 
(2) 



Loud-speaker, variable current, a 1 = 15.4^^^ _ | gl Cii 1 ( 



Fre- 
quency 


Source 


Equa- 
tion 


Tone 


Coeffi- 
cient 


Bureau of 
Standards 


Wat- 
son 


Knud- 
sen 




Pipe 


(1) 


Steady 


f>4 7 










Loud speaker 


(2) 


Steady 


61 6 








512 


Loud speaker 


(2) 


Flutter 


57 3 


61 


70 


67 




Loud speaker 


(3) 


Steady 


60 9 










Loud speaker 


(3) 


Flutter 


57 8 










Pipe 


(1) 


Steady 


68 7 










Loud speaker 


(2) 


Steady 


76 








1,024 


Loud speaker 


(2) 


Flutter 


78 4 


72 


76 


74 




Loud speaker 


(3) 


Steady 


75.5 










Ixwd speaker 


(3) 


Flutter 


77 5 










Pipe 


(1) 


Steady 


71 2 










Loud speaker 


(2) 


Steady 


73 9 








2,048 


Loud speaker 


(2) 


Flutter 


79 2 


76 


76 


80 




Loud speaker 


(3) 


Steady 


78 5 










Loud speaker 


(3) 


Flutter 


79 5 









loud-speaker data obtained in the Riverbank sound 
chamber. For comparison, figures by the Bureau of 
Standards, by F. R. Watson and V. O. Knudsen, on the 
same material are given. The latter were all obtained by 



MEASUREMENT OF ABSORPTION COEFFICIENTS 125 

the reverberation method. The table gives a very good 
idea of the order of agreement that is to be expected in 
measurements of this sort. The fact that the coefficient 
is obtained by taking the difference between two quantities 
whose precision of measurement is not great will account 
for rather large variations in the computed value of the 
coefficient. Thus in Fig. 50, errors of 1 per cent in the 
value of a and a 1 would, if cumulative, make an error of 
4.5 per cent in the computed values of the coefficient. 
The variations that are to be expected in determining 
a and a', due to interference, are certainly as great as 1 
per cent, so that the precision with which absorption 
coefficients can be measured by existing methods is not 
great. 

Summary. 

We have seen that the standing-wave method gives 
coefficients of absorption for normal incidence only and 
for samples placed always at a motion node of the standing- 
wave system. Moreover, with materials whose absorption 
is due to inelastic flexural vibrations, the small-scale 
measurements on rigidly mounted samples fail to give 
the values that are to be expected from extended areas 
having a degree of flexural motion. On the other hand, 
reverberation coefficients are deduced on the assumptions 
made and verified in the reverberation theory as it is 
applied to the practical problems of architectural acoustics. 

We have seen also that all of the reverberation methods 
now in use go back to the determination of the rate of 
decay of sound in a reverberation chamber and the effect 
of the absorbent material on this rate of decay and that 
in the very nature of the case, the precision of such measure- 
ments is not great. The oscillograph method is equally 
laborious and requires repeated measurements in order to 
eliminate the error due to the irregularities in the decay 
curve resulting from interference. Finally, we have noted 
that there is a certain degree of uncertainty as to the 
assumptions to be made when we take the electrical input 



126 ACOUSTICS AND ARCHITECTURE 

of a telephonic source of sound as a measure of the sound 
energy which it generates under varying absorbing powers 
of the room. As will be seen in the succeeding chapter, 
there are a number of other factors that affect the measured 
values of reverberation coefficients. All things considered, 
it has to be stated that before precise agreement on meas- 
ured values can be attained, arbitrary standards as to 
methods and conditions of measurement will have to be 
adopted. 



CHAPTER VII 
SOUND-ABSORPTION COEFFICIENTS OF MATERIALS 

In this chapter, it is proposed to consider the physical 
properties that affect the sound-absorbing efficiency of 
materials and the variation of this efficiency with the 
pitch and quality of the sound. We shall also consider 
various conditions of test that affect the values of the 
absorption coefficients of materials as measured by rever- 
beration methods and finally deal with some questions that 
arise in the practical use of sound absorbents in the correc- 
tion of acoustical defects and the reduction of noise in 
rooms. 

Physical Properties of Sound Absorbents. 

The energy of a train of sound waves in the air resides 
in the regular oscillations of the molecules. The absorption 
of this energy can occur only by some process by which 
these ordered oscillations are converted into the random 
molecular motion of heat. In other words, the absorption 
of sound is a dissipative process and occurs only when the 
vibrational motions are damped by the action of the forces 
of friction or viscosity. Now experience shows that only 
materials which are porous or inelastically flexible or 
compressible absorb sound in any considerable degree. 
For a material to be highly absorbent, the porosity must 
consist of intercommunicating channels, which penetrate 
the surface upon which the sound is incident. Cellular 
products with unbroken cell walls or with an impervious 
surface do not show any marked absorptive properties. 
A simple practical test as to whether a material possesses 
absorbing efficiency because of its porosity is to attempt 
to force air into it or through it by pressure. If air cannot 
be forced into it, it will not show high absorbent properties. 

127 



128 



ACOUSTICS AND ARCHITECTURE 



By inelastically flexible and compressible materials we 
mean those in which the damping force is large in com- 
parison with the elastic forces brought into play when such 
materials are distorted. 

Absorption Due to Porosity. 

The theoretical treatment of this problem is beyond 
the scope of our present purpose. Theoretical treatment's 
given in papers by Lord Rayleigh, by E. T. Paris, and by 
Crandall. 1 In an elementary way, it can be said that the 
absorption coefficient of a porous, non-yielding material 
will depend upon the following factors : (a) the cross section 
of the pore channels, (6) their depth, and (c) the ratio of 
perforated to unperforated area of the surface. Rayleigh's 
analysis is for normal incidence and leads to the conclusion 
that the absorption increases approximately as the square 

root of the frequency. For 
a given ratio of unperforated 
to perforated area, the ab- 
sorption at low frequencies 
increases, though not linearly 
with the radius of the pores, 
considered as cylindrical 
tubes. If the radius of pores 
be greater than 0.01 cm., the 
assumptions made in the theory do not hold. For a 
coarse-grained structure, the thickness required to produce 
a given absorption at a given frequency is greater than 
with a fine-pored material. Crandall 2 has worked out 
the theoretical coefficients of absorption of an ideal wall 
of closely packed honeycomb structure (i.e., one in which 
the ratio of unperforated to perforated area is small), 
the diameter of the pores being 0.02 cm. The thick- 

1 RAYLEIGH, "Theory of Sound," vol. II, pp. 328-333. 
, Phil. Mag., vol. 39, p. 225, 1920. 

PARIS, E. T., Proc. Roy. Soc., Ser. A, vol. 115, 1927. 
CRANDALL, "Theory of Vibrating Systems and Sound," p. 186, D. Van 
Nostrand Company, 1926. 

2 CRANDALL, op. tit., p. 189. 



IUU 

030 
0.80 
070 

060 
050 
















^ 


^ 






^^^ 






* ^ 


^ 





















200 400 



3200 6400 



fiOO 
Frequency 

Fia. 51. Theoretical absorption of 
a closely packed porous material of 
great thickness. (After Crandall.) 



ROUND-ABSORPTION COEFFICIENTS OF MATERIALS 129 



ness is assumed great enough to give maximum absorp- 
tion. His values are plotted in Fig. 51. We note a maxi- 
mum of absorption at 1,600 
vibs./sec. This suggests 
selective absorption due to 
resonance; but as Crandall 
points out, it is " quite acci- 
dental, as no resonance phe- 
nomenon or selective absorp- 
tion has been implied" in the 
problem. On his analysis, we 
should expect a porous ma- 
terial always to show a maxi- 
mum of absorption at some 
frequency, this maximum 
shifting to lower frequencies 
as the coarseness of the porosity is increased. Thus, he 
states, if the cross section of the pores is doubled, the curve 
shown would be shifted one octave lower. It is to be 



noo 












f\ on 












f\ -]/\ 




x/C 




Z^ = ^ E *I 


^. 


nan 


'' 


// 




^~ 


-N 




/(? 


^ 


***-/ 


' 


^ 


040 




/A 


,/ 








/// 


/ ( 










/// 


/ 








10 


'/l*^ 
























128 256 512 1024 3048 4096 
Frequency 

FKJ. 52. Absorption coefficients of 
asbestos hair felt of different thick- 
nesses. 



IUU 
nan 














n on 














mn 








^'^ 


2.048'* 


f 


060 






4/ 




256^ 


-^"^ 


4- 


/, 


/ 




^^\ 




^4,096 


5 050 


// 


/ 


^ 






g*^- 


040 

$ O7fl 


li. 


/ 






^^ 






1 ^ 


/ 




^^ 








// / 


^ 


^~~ " 








010 

o 


ys^ 












( 


) 01 


> l( 

Tl_. 


) 1. 


5 2 


9 2 


5 3 



Thickness in Inches 
Fio. 52a. Absorption coefficients of asbestos hair as a function of thickness. 

remembered that a porous wall of great thickness is 
assumed. 

Effect of Thickness of Porous Materials. 

For limited thickness, the absorption coefficient of a 
porous material increases in general with the thickness 
approaching a maximum value as the thickness is increased. 



130 



ACOUSTICS AND ARCHITECTURE 



The curve shown in Fig. 31 shows the absorption by the 
stationary-wave method of cotton wool as a function of 
thickness. Davis and Evans state that the maximum 
absorption there shown occurs at thicknesses of one-quarter 
of the wave length of sound in the material. In Fig. 52 
are shown the reverberation coefficients of an asbestos hair 
felt of different thicknesses at different frequencies. Figure 
53 shows the absorption coefficients of a porous tile as 
given by the Bureau of Standards. We note, in all cases, 
a maximum absorption over a frequency range. This 



090 




126 



356 512 1024 
Frequency 



FIG. 53. Absorption coeffi- 
cients of porous tile. (Bureau 
of Standards.) 



4.096 



FIG. 54. Effect of scaling the surface 
of a porous compressible material. 



maximum shifts to lower frequencies as the thickness 
of the absorbing layer is increased. These facts are in 
qualitative agreement with the theory of absorption by 
porous bodies deduced by Rayleigh and Crandall. 

In soft, feltlike materials, the absorption, particularly 
at lower frequencies, is due both to porosity and to inelastic 
compressibility. The curves of Fig. 54 show the effect 
of sealing the surface of felt with an impervious membrane. 
Curve 2 may be assumed to be the absorption due to the 
compressibility of the material. We note the marked 
falling off at the higher frequencies where the porosity is 
the more important factor. 

Absorption Due to Flexural Vibrations. 

The absorption of sound by fiber boards is due very 
largely to the inelastic flexural vibration of the material 



SOUND-ABSORPTION COEFFICIENTS OF MATERIALS 131 



under the alternating pressure. The absorption of sound 
by wood paneling is of this character. Figure 55 shows the 
coefficients of pine sheathing 2.0 cm. thick as given by 
Professor Sabine. We note the irregular character of the 
curve suggesting that resonance plays an important r61e 
in absorption by this means. The difference in the 
mechanics of the absorption of sound by damped flexible 
materials and absorption due to porosity is shown by 
comparison of the curves of Fig. 56 with those of felt in 



0.12 
0,11 



go.io 



0,08 
0.07 
0.06 



050 



Frequency 

FIG. 55. Absorption coeffi- 
cients of pine sheathing 2.0 cm. 
thick. (After W. C. Sabine.) 




256 



2048 4096 



512 1024 
Frequency 

FIG. 56. Effect of thickness of 
material whose absorption is due to 
damped flexural vibrations. 



Fig. 52a. The former are for a stiff pressed board with a 
fairly impervious surface made of wood fiber. The lower 
curve is for a %-in. thickness, while the upper curve is for 
the same material %e in. thick. In contrast to the felt, 
the thinner more flexible material shows the higher absorp- 
tion. In absorption due to flexural vibration, the density, 
stiffness, and damping coefficient of the material affect 
the absorption coefficient. The mathematical theory of 
the process has not yet been worked out. The almost 
uniform value of the coefficients for different frequencies 
in Fig. 56 indicates the effect of damping in decreasing the 
effects due to resonance. 

Area Effects in Absorption Measurements. 

In an investigation conducted in 1922, upon the absorp- 
tion of impact sounds, it appeared that the increase of 
absorption of sounds of this character was not strictly 



132 



ACOUSTICS AND ARCHITECTURE 



proportional to the area of the absorbent surface, intro- 
duced into the sound chamber, small samples showing 
markedly greater absorption per unit area than large 
samples of the same material. The investigation was 
extended, using sustained tones, and the same phenomenon 
was observed as in the case of short impact sounds. In 
Fig. 57 are shown the apparent absorbing powers per unit 
area of a highly absorbent hair felt plotted against the 
area of the test sample. 

The same effect under somewhat less ideal conditions 
was observed in the case of the absorbing power of (trans- 




"0133456789 
Area 

FIG. 57. Absorbing power per unit area as a function of area. 



mission through) an opening. A large window in an empty 
room 30 by 30 by 10 ft. was fitted with a series of frames so 
that the area of the opening could be varied, the ratio of 
dimensions being kept constant. The absorption coeffi- 
cient for these openings at 512 vibs./sec. varied from 1.10 
to 0.80 as the size was increased from 3.68 to 30.2 sq. ft. 
A doorway 8 by 9 ft. in a room 30 by 30 by 9 ft. showed an 
apparent coefficient as low as 0.65. (In this case, condi- 
tions were complicated by reflection of sound from the 
ground outside.) 

In view of the fact that the earlier measurements of 
Professor Wallace Sabine were based on the open window as 
an ideal absorber with an assumed coefficient of 1.00, it was 
of interest to recompute from his data the values of the 
absorption coefficients of openings. These data were 



SOUND- ABSORPTION COEFFICIENTS OF MATERIALS 133 



used by him, assuming a coefficient of LOO in the deter- 
mination of the constant K of the simple reverberation 
formula. Fortunately the data necessary for these com- 
putations are found in his original notes, but, unfortunately 
for the present purpose, only the total open-window area is 
given and not the dimensions of the individual openings. 
The figures are shown in Table XI, where w is the total 
area of the open windows, and a and a! are computed from 
the equation 

_ 9.2 V 



TABLE XI. ABSORPTION COEFFICIENTS OF OPENINGS (512 VIBB./SEC.) 



Room 


V 


t 


t' 


a 


w 


a' 


a' a 


w 


Fogg Art Museum: 
1 pipe 


96 
96 

202 
202 
202 
1,630 
1,960 


4.59 
5.26 

1.78 
1.78 
1.78 
3.88 
3.41 


3.00 
3.43 

1.66 
1.56 
1.41 
3.26 
2 88 


3.99 

3.86 

19.15 
19.15 
19.15 
65.0 
87 


1 86 
1 86 

1.34 
2.67 
5.28 
12.2 
14.9 


5.85 
5 80 

20 5 
21.7 
23.7 
76.2 
101 2 


1.00 
1.04 

1.01 
0.96 
0.86 
92 
0.88 


16 pipes (7.6 X 1 pipe)-. 
Jefferson Physical Laboratory : 
Room 15 
Room 15 
Room 15 . 


Room 1 . . 


Room 41. .. 





* It will be noted that there is a marked variation in 
the values of the coefficients for the open window. The 
data for Room 15 show a decrease in the apparent absorb- 
ing power as the area of the individual openings is increased, 
quite in agreement with the results obtained in this 
laboratory. 

One finds an explanation of these facts in the phenomenon 
of diffraction and the screening effect of an absorbent area 
upon adjacent areas In the reverberation theory, we 
assume a random distribution of the direction of propaga- 
tion of sound energy. Thus, on the average, two-thirds of 
the energy is being propagated parallel to the surface of 
the absorbent material. Neglecting diffraction, in such a 
distribution, only that portion traveling at right angles 



134 



ACOUSTICS AND ARCHITECTURE 



to the absorbent surface would be absorbed by a very large 
area of a perfectly absorbent material. Due to diffraction, 
however, the portion traveling parallel to the surface is 
also absorbed over the entire area but more strongly at the 
edges. 

The following experiment illustrates this "edge effect/' 
Strips of felt 12 in. wide were laid on the sound-chamber floor, 
forming a hollow rectangle 8 by 5 ft. (2.44 by 1.53 m.). 
The increase in the absorbing power due to the sample Was 
measured. The space inside was then filled and the 
increase produced by the solid rectangle measured. The 
absorbing power per unit area of the peripheral and central 
portions is given below : 



Area 


C 2 


<\ 


c 4 


C 5 


C 6 


Peripheral 


12 


42 


69 


0.88 


0.68 


Central 


0.05 


27 


24 


29 


30 















The screening effect of the edges is obvious and serves to 
explain the decrease in absorbing power per unit area 
shown in Fig. 57. In a similar manner, long, narrow 
samples show more effective absorbing power than equal 
areas in square form, as shown below : 



Area, square 


Dimensions, 


Coefficient 


meters 


centimeters 














C 3 


(\ 


c b 


c, 


1 


330 X 33 


62 


78 


1 01 


73 


1 


100 X 100 


42 


0.62 


0.86 


70 


2 


666 X 30 


0.53 


76 


0.93 


0.71 


2 


141 X 141 


0.37 


0.55 


0.73 


59 



The fact that small samples show an apparent absorption 
coefficient greater than unity calls for notice. It is to be 
remembered that we are here dealing with linear dimensions 
that are of the order of or even less than the wave length 
of the sound. The mathematics of the problem involves 
the same considerations as that of radiation of sound from 



SOUND-ABSORPTION COEFFICIENTS OF MATERIALS 135 

the open end of an organ pipe or the amplification by a 
spherical resonator. Diffraction plays an important role, 
and just as in the case of the resonator energy is drawn 
from the sound field around the resonator to be reradiated, 
so the presence of the small absorbent sample or small 
opening affects a portion of the wave front greater than its 
own area. Qualitatively, the effect is shown by the sound 
photograph of Fig. 58, which shows a sound pulse reflected 




FIG. 58. Sound pulse incident, upon a barrier with an opening. A-B marks the 
limits of the opening. 

from the surface of a barrier with an opening. The cross 
section of the portion cut out of the reflected pulse is clearly 
greater than that of the opening. 

Absorption Coefficients of Small Areas. 

Some very interesting results flowing from the phe- 
nomena described in the preceding section were brought 
out in a series of experiments conducted at the Riverbank 
Laboratories for the Johns-Manville Corporation and 
reported by Mr. John S. Parkinson. 1 

A fixed area 48 sq. ft. (4.46 sq. m.) of absorbent material 
was cut up into small units, which were distributed with 
various spacings and in various patterns. The addition 
to the total absorbing power of the room was measured by 
the reverberation method. In Fig. 59, the apparent 
absorption coefficients of 1-in. hair felt are shown, under the 
conditions indicated. The explanation of the increase in 

l Jour. Acous. Soc. Amer., vol. 2, No. 1, pp. 112-122, July, 1930. 



136 



ACOUSTICS AND ARCHITECTURE 



absorbing power as the units are separated lies in the 
screening effect of an absorbent surface on adjacent 
surfaces. We note that the increase in absorption with 

separation is a function both 
of the absorbing efficiency and 
of the wave length of the sound. 
The experiment does not per- 
mit us to separate the effects 
of these two factors. We* do 
note, however, that for the 
two lower frequencies, where 
the absorption coefficient and 
the separation measured in 
wave lengths are both small, 



120 
110 

too 

090 
060 
Q70 
Q60 
030 
CMfl 
















/ 


\ 








y 


~\ 


N 






>*/- 


^-^ 


^S> 






^ 


/ 


V 




/ 


/^ i 








// 


^ 




\ 


















030 
020 

QIO 
12 


/ 


\-Areao 
2- - 
3- - - 
4- - 
1 


f unit 6x8 

- 1*8 
- M 
' 1x6 


wtdthofs, 
'" - 


paangO 

- r 


/ 


P 


" j*~ 


8 256 51Z 1,024 2,048 409< 
Frequency 



the effect of spacing the units 

FIG. 59. Effect of spacing on is Small. 

absorption by small units. A i i r j_ i i L 

An interesting fact is brought 

out by the curves of Fig. 60, taken from Parkinson's 
paper. Here the absorbing power per unit area of the 
total pattern is plotted as ordinate against the ratio 



Absorption per Sq Ft of Total FWtern 
o 5 8 g g S g 










512 
CP.S. 








"^ 


^ 














,-' 


/ 














/ 


/S 
>" 


/ 












/ 


^ 


/ 












s 




/^.s 

































wo aa 030 0.40 oso aeo o?o o.w ago IDO 

Ratio of Treated to TotaUrea 



FIG. 60. Absorbing power per square unit of distributed material plotted as a 
function of ratio of absorbent area to total area of pattern. (After Parkinson.) 

of the actual area of felt to the total area over which 
it is distributed. In any one case, the units were all 
of one size and uniformly spaced, but the different 
points represent units ranging in size from 1 by 1 ft. to 2 
by 8 ft. spaced at distances of from 1 to 4 ft. Diamond- 



SOUND-ABSORPTION COEFFICIENTS OF MATERIALS 137 



laped and hexagonal units are also represented. The 
ict that points so obtained fall upon a smooth curve shows 
lat with a given area of absorbent material cut into units 
f the same size and uniformly distributed over a given 
rea, the total absorption is independent of the size and 
tiape of the units and of the particular pattern in which 
tiey are distributed. The difference between the ordinate 
f the straight line and the curve for any ratio of treated to 
3tal area gives the increase in absorbing power per square 
)ot due to spreading the material. 

rffect of Sample Mounting on Absorption Coefficients. 

Figure 32 (Chap. VI) shows the very marked increase in 
bsorption, as measured by the stationary-wave method, 
f shifting a porous material from a motion node to a 



Vibrations 
per second 


Mounted on 
1-in. furring 


Mounted on 
2 by 4-in. 
studs 


128 




19 


256 


16 


0.14 


512 


0.22 


0.13 


1,024 


0.20 


0.14 


2,048 


0.16 


14 


4,096 


15 


16 



lotion loop of the standing wave. In this case, the change 
f position of the test sample was of the order of half a wave 
mgth. Professor Sabine measured the effect of mounting 
-in. hair felt at distances of 2, 4, and 6 in., respectively, 
[*om the wall of the sound chamber. The absorption 
oefficient at 512 vibs./sec. was increased from 0.57 to 0.67 
rith an air space of 6 in. between the felt and the wall, with 
orresponding increases at lower frequencies. At higher 
requencies the increase was negligibly small. That the 
ffect of the method of mounting flexible, non-porous 
naterials may be very pronounced is shown by the above 
gures for a pressed-vegetable fiber board >- in. thick, 
n the first instance, it was nailed loosely on 1-in. furring 



138 ACOUSTICS AND ARCHITECTURE 

strips laid on the floor of the sound chamber, while in the 
second case, it was nailed firmly to a 2 by 4-in. wood-stud 
construction, with studs set 16 in. apart. The more rigid 
attachment to the heavier structure, allowing less freedom 
of motion, accounts for the difference. 

In the table of absorption coefficients given in Appendix 
C y we note the marked difference in the absorbing efficiency 
of draperies resulting from hanging at different distances 
from the wall and from different amounts of folding. 

Effect of Quality of Test Tones. 

It is well known that the tones produced by organ pipes 
are rich in harmonic overtones. In using an organ pipe as 
a source of sound for absorption measurements, one assumes 
that the separate component frequencies of the complex 
tone are absorbed independently and that the absorption 
coefficient measured is that for the fundamental frequency. 
It is obvious that if the strength of any given harmonic 
relative to the fundamental is so great that this particular 
harmonic persists longer than the fundamental itself in the 
reverberant sound or if the conditions are such that the rate 
of decay of this harmonic is less than that of the funda- 
mental under the two conditions of the sound chamber, 
then the absorption coefficient obtained by reverberation 
measurements will be that for the frequency of this har- 
monic rather than for the fundamental. The fact that the 
reverberation time of a sound chamber decreases as the 
pitch of the sound is raised makes for purification of the tone 
as the reverberant sound dies away; that is, the higher- 
pitched components die out first, leaving the fundamental 
as the tone for which the times are measured. If this 
condition exists, and if the assumption of the independent 
absorption of the harmonic components is correct, then the 
absorption coefficient obtained from a complex source 
should be the same as that for a pure tone of the same 
frequency. 

The following data bear evidence on the question of the 
effect of tone quality upon the measured values of absorp- 



SOUND-ABSORPTION COEFFICIENTS OF MATERIALS 139 



tion coefficients. The absorption at 512 vibs./sec. of four 
different materials was measured using seven organ pipes 
of different tone qualities. These measurements on each 
material with the different pipes were all made under 
identical conditions. Three of the pipes were of the open 
diapason stop, from different manufacturers. The tone 
qualities may be roughly described as follows : 

1. Tibia clausa Stopped wooden pipe, strong fundamental, 

weak octave and twelfth 

2. Melodia Open wood pipe, strong' fundamental, weak 

octave and twelfth 

Strong fundamental and strong octave 
Strong fundamental, weak octave, strong 
higher harmonics 

7. Gamba Weak fundamental, weak octave, strong higher 

harmonic 

The materials were chosen so as to represent a wide- 
diversity of characteristics as regards the variation of 
absorption with pitch. 

TABLE XII. COEFFICIENTS AT 512 VIBS./SEC. 



3, 4, 5. Open diapason. 
6. Gemshorn . . . 



Source 


1 


2 


3 


4 


5 


6 


7 


Loud-speaker 


1. ^ -in. acoustical plas- 
ter 


0,26 
36 
0.52 

0.54 


0.28 
0.37 
52 

55 


0.31 
0.37 
0.65 

65 


0.27 
0.36 
0.57 

60 


0.30 
0.39 
0.61 

61 


0.29 
0.40 
0.56 

58 


0.30 
0.33 
0.52 

57 


62 0.61 


2. 1-in. wood-fiber tile. . 
3. 1-in. asbestos hair felt 
4. H-in. perforated fi- 
ber tile 





It is to be said that for materials 1, 3, and 4 the coeffi- 
cients for the open diapason tones are higher than for the 
other pipes. All of these materials have higher coefficients 
at the octave above 512, so that it seems fair to conclude 
that the presence of the strong octave in these tones tends 
to raise the coefficient at 512. The wood-fiber tile, on the 
other hand, has a nearly uniform coefficient over the whole 
frequency range, and we note no very marked difference 
which can be ascribed to the difference in quality. On the 
whole, it may be said that the presence of strong harmonics 
in the source of sound will occasion somewhat higher 



140 ACOUSTICS AND ARCHITECTURE 

measured coefficients for materials that are decidedly more 
absorbent at the harmonic than at the fundamental 
frequency. With the exceptions just noted, the differences 
shown in Table XII are little if any greater than can be 
accounted for by experimental error. 

Absorbing Power of Individual Objects. 

In practical problems of computing the reverberation 
time in rooms, we must include the addition to the total 
absorbing power made by separate articles of furniture, 
such as seats and chairs and more particularly the absorbing 
power of the audience. The experimental data are best 
expressed not as absorption coefficients pertaining to the 
exposed surface but as units of absorbing power contributed 
to the total by each of the objects in question. Thus, in 
the case of chairs, for example, we should measure the 
change in the total absorbing power by bringing a number 
of chairs into the sound chamber and divide this change by 
the number. 

Absorbing Power of Seats. 

The wide variation in the absorbing power of seats of 
different sorts is shown in Table I (Appendix C), compiled 
from various sources. It is a difficult matter to specify all 
of the factors which affect the absorbing power, so that the 
figures given are to be taken as representative rather than 
exact. In the table, the absorbing power is expressed in 
English units, that is, as the equivalent area in square feet 
of an ideal surface whose coefficient is unity. The values 
in metric units may be obtained by dividing by 10.76, the 
number of square feet in a square meter. 

Absorbing Power of an Audience. 

By far the largest single contributor to the total absorbing 
power of an auditorium is the audience itself, and for this 
reason the reverberation will be markedly influenced by 
this factor. The usual procedure in estimating the total 
absorbing power of an audience is to find by measurement 



SOUND-ABSORPTION COEFFICIENTS OF MATERIALS 141 

the absorbing power per person and multiply this by the 
number of persons. The data generally accepted for the 
absorbing power per person are those published by Wallace 
C. Sabine in 1906. Expressed in English units they are as 
follows : 





Absorbing 


Frequency 


Power 


128 


3.6 


256 


4.2 


512 


4.6 


1,024 


4.7 


2,048 


4 9 


4,096 


4.9 



The audience on which these measurements were made con- 
sisted of 77 women and 105 men, and the measurements 
were made in the large lecture room of the Jefferson Physi- 
cal Laboratory. These values are considerably higher than 
those given by the results of more recent measurements 
made under more ideal conditions as regards quiet. The 
effect of the inevitable noise created by this number of 
persons as well as disturbing sounds from without would 
tend toward higher values. Moreover, the much lighter 
clothing, particularly of women, at the present time, is not 
an inappreciable factor in reducing the coefficient from these 
earlier figures. The fact that Sabine's auditors were 
seated in the old ash settees with open backs, shown in 
his " Collected Papers/' thus exposing more of the clothing 
to the sound, would give higher absorbing powers than are 
to be expected with an audience seated in chairs or pews 
with solid, non-absorbent backs. In view of the results 
already noted on the effect of spacing on the effective 
absorbing power of materials, it is apparent that the 
seating area per person also is a factor in the absorbing 
power of an audience. 

The table shown on p. 142 gives some recent data on the 
measured absorbing power of people. 

As will be noted, these figures are considerably lower 
than those of Sabine. There is another way of treating the 
absorbing power of an audience, and that is to regard the 



142 



ACOUSTICS AND ARCHITECTURE 
TABLE XIII 















Area per 
























Audience 


128 


256 


512 


1,024 


2,048 


square 


Date 


Authority 














feet 






20 women 


1 2 


1 9 


2 7 


3 3 


3 3 


3.9 


1931 


P. E. S. 


28 women 




2.2 


3.4 


4 


3 8 


4.6 


1920 


P. E. S. 


20 mon 




1.9 


3.2 


3.7 


3 9 


3.9 


1931 


P. E. S. 


15 women 


7 


1.3 


2 3 


3.6 


4 5 




1930 


V. L. q.* 


15 men. . 


1 4 


2.2 


4 1 


5 3 


7 2 




1930 


V. L. C.* 


Average. 


1 1 


2 1 


3.2 


4 


4 5 









* CHHIMLEH, V L , Jour Aeons Soc. Amer,, p. 126, July, 1930. 

audience as an absorbing surface and to express the absorb- 
ing power per unit area. The following are the results so 
expressed obtained by the writer compared with the earlier 
figures. 



Audience 


256 


512 


1,024 


2,048 


Authority 


Women, 1920 . . . 
Women, 1931 
Men, 1931. . 


0.48 
48 
48 


74 
0.69 
82 


87 
84 
95 


83 
0.84 
1 00 


P. E. S. 
P. E. S. 
P. E. S. 














Average ... 


48 


75 


89 


89 




Mixed, 1906 


88 


96 


98 


99 


W. C. S. 



The materially higher values of the earlier measurements 
can be accounted for partly by the change of style in cloth- 
ing and partly by the difference in conditions as regards 
quiet. 

Chrisler 1 gives the following figures for a mixed audience 
of six men and six women occupying upholstered theater 
chairs : 





Absorbing 


Frequency 


Power 


256 


3.6 


512 


4 1 


1,024 


4.8 


2,048 


4.2 



l Jour. of Acou*. Soc. Awcr., Vol. 2, No. 1, p. 127, July, 1930. 



SOUND-ABSORPTION COEFFICIENTS OF MATERIALS 143 

These values are more nearly in accord with those that are 
universally used in practice. 

It is. apparent from the foregoing that the sound-absorb- 
ing power of an audience is a quantity which depends upon 
a number of variable and uncontrollable factors and cannot 
be specified with any great degree of scientific precision. 
Up to the present time, the universal practice has been to 
use the values given by Professor Sabine, and the criteria 
of acoustical excellence are all based on these figures. 
We shall therefore, in considering the computation of the 
reverberation time of rooms in Chap. VIII, use his values, 
even while recognizing that they are higher than those 
which in scientific accuracy apply to present-day audiences. 

Absorption Coefficients of Materials. 

The last ten years have seen the commercial development 
and production of a very large number of materials designed 
for use as absorbents in reducing the reverberation in 
auditoriums and the quieting of noise in offices, hospitals 
and the like. The problems to be solved in the develop- 
ment of such materials are threefold: (1) the reduction of 
cost of material and application, (2) the production of 
materials that shall meet the practical requirements of 
appearance, durability, fireproofness, and cleanliness, and 
(3) the securing of materials with sufficiently high absorp- 
tion coefficients to render them useful for acoustical 
purposes. 

In the earliest application of the method, hair felt was 
extensively used. This was surfaced with fabric of 
various sorts stretched on furring over the felt. Painting 
of this fabric in the usual manner was found materially to 
lessen the absorbing efficiency. The development of better 
grades of felt mixed with asbestos, the gluing of a perfo- 
rated, washable membrane directly to the felt, and finally 
the substitution of thin, perforated-metal plates for the 
fabric mark the evolution of this form of absorbent treat- 
ment. Boards made from sugar-cane fiber show moder- 
ately high absorption coefficients. These have been 



144 ACOUSTICS AND ARCHITECTURE 

markedly increased by the expedient of increasing their 
thickness and boring holes at equally spaced intervals. 
Professor Sabine early developed a porous tile composed of 
granular particles bonded only at their points of contact, 
which has found extensive use in churches and other rooms 
where a tile treatment is applicable. In 1920, the writer 
began a series of investigations looking to the development 
of a plaster that should be much more highly absorbent 
than are the usual plaster surfaces. These investigations 
have resulted in a practicable commercial product of con- 
siderable use. Recently a highly absorbent tile of which 
the chief ingredient is mineral wool has been extensively 
used. Materials fabricated from excelsior, flax fiber, wood 
wool, and short wood fiber are also on the market. 

The more widely used of these materials are listed 
under their trade names in the table of absorption coeffi- 
cients given in Appendix C. As a result of development 
research, many manufacturers are effecting increases in 
the absorption coefficients of their materials, so that the 
date of tests is quoted in each instance. Earlier published 
data from the Riverbank Laboratories were based on the 
four-organ calibration. The figures given in the table are 
corrected to the more precise values given by loud-speaker 
methods of calibration. 



CHAPTER VIII 
REVERBERATION AND THE ACOUSTICS OF ROOMS 

Having developed the theory of reverberation and its 
application to the measurement of the absorption coeffi- 
cients of materials, we are now in a position to apply the 
theory to the prediction and control of reverberation. 
In this chapter, we shall consider first the detailed methods 
of calculating the reverberation time from the architect's 
plans for an audience room and, second, the question of the 
desired reverberation time considered both from the 
standpoint of the size of the room and also as it depends 
upon the uses for which the room is intended. 

Calculation of Reverberation Time: Rectangular Room. 

While it is theoretically possible to calculate precisely 
the reverberation time of any given room from a knowledge 
of volume and the areas and absorption coefficients of all 
the absorbing surface, yet in any practical case, certain 
approximations will have to be made, and the prediction 
of the reverberation in advance of construction is a matter 
of enlightened estimate rather than precise calculation. 
The nature of these approximations will be indicated in 
the two numerical examples given. Fortunately the limits 
between which the reverberation time may lie without 
materially affecting hearing conditions are rather wide, so 
that such an estimate will be quite close enough for practical 
purposes. 

We shall take as our first example a simple case of a small 
high-school auditorium, rectangular in plan and section, 
without balcony. From the architect's plans the following 
data are secured: 

Dimensions, 100 by 50 by 20 ft. 
Walls, gypsum plaster on tile, smooth finish 
Ceiling, gypsum on metal lath, smooth finish 

145 



146 



ACOUSTICS AND ARCHITECTURE 



Stage opening, 30 by 12 ft., velour curtains 

Wood floor 

Wood-paneled wainscot, 6 ft. high, side and rear walls 

700 unupholstered seats. 

The common practice is to figure the reverberation for 
the tone 512 vibs./sec. The absorption coefficients are 
taken from Table II in Appendix C. 

Volume = 100 X 50 X 20 = 100,000 cu. ft. 



Material 


Area 


Coefficients 


Absorbing 








power 


Plaster on tile 


4,140 


020 


82 


Plaster on lath 


5,000 


032 


160 


Wood paneling . . 


1,500 


10 


150 


Wood floor 


5,000 


04 


200 


Stage opening 


360 


44 


159 


700 seats 




25 


175 


Total 






926 



In computing the absorption due to the audience, it is 
the common practice to assume that the additional absorp- 
tion due to each person is 4.6 minus the absorbing power of 
the seat which he occupies. We have then the following 
for different audiences: 





Additional 


Total 




Audience 


absorption 


absorbing 


T = 0.05 V/a 




n(4.6 - 0.25) 


power 




None 




926 * 


5 40 


200 


870 


1,796 


2 78 


400 


1,740 


2,666 


1.87 


600 


2,610 


3,536 


1 41 


700 


3,045 


3,961 


1 26 



In passing, the preponderating role which the audience 
plays in the total absorbing power of the room is worth 
noting. In this case, 77 per cent of the total for the occu- 
pied room is represented by the audience. It is apparent 
that the absorption characteristics of an audience consid- 



REVERBERATION AND THE ACOUSTICS OF ROOMS 147 



ered as a function of pitch, will in large measure determine 
the absorption frequency characteristics of audience rooms. 
We may for the sake of comparison calculate the rever- 
beration times using the later formula 

T 0-05 F 

lo ~ -5 log. (1- a) 

where a is the average coefficient of all reflecting surfaces, 
and S is the total surface. In this very simple case, the 
average coefficient may be obtained by dividing the total 
absorbing power by the total area of floors, walls, and ceil- 
ing. In more complicated problems of rooms with bal- 
conies and recesses, arbitrary judgments will have to be 
made as to just what are to be considered as the bound- 
ing surface. In the present example, S = 16,000, and 
0.05 FAS = 0.312. We have the following values: 



Audience 


a 
a ~ 16,000 


- log, (1 - ) 


0.312 


- log e (1 - ) 


None 


058 


0585 


5.33 


200 


1136 


1178 


2 65 


400 


1690 


1845 


1.69 


600 


2242 


2540 


1 23 


700 


2520 


2910 


1.07 



We shall refer to this difference in the results of computa- 
tion by the two formulas in considering the question of 
desirable reverberation times. 

Empirical Formula for Absorbing Power. 

Estimating the areas of the various surfaces in a room 
when the design is not simple may be a tedious process. 
Since, in audience rooms, the contribution to the total 
absorbing power of the empty room exclusive of the seats 
is usually only about one-fourth or one-fifth of the total 
when the room is filled, it is apparent that extreme precision 
in estimating the empty room absorption is not required. 
Thus in the example given, an error of 10 per cent in the 
empty-room absorption would make an error of only 3.4 



148 ACOUSTICS AND ARCHITECTURE 

per cent in the reverberation time for an audience of 400 
and 2.3 per cent for the full audience. 

The following empirical rule was arrived at by computing 
from the plans the absorbing powers of some 50 rooms 
ranging in volume from 50,000 to 1,000,000 cu. ft. Assum- 
ing only the usual interior surfaces of masonry walls and 
ceilings, wood floors, and having seats with an absorbing 
power of 0.3 unit each, the absorbing power exclusive of 
carpets, draperies, or other furnishings is given approxi- 
mately by the relation 

a (empty) = 0.3 V* 

Illustrating the use of this empirical formula, we estimate 
the total absorbing power of the empty room of the preced- 
ing example. 

Bare room, masonry walls throughout, wood seats, 
0.3^(100^000p 630 

For 1,500 sq. ft. wood-paneling coefficient 0.10 in place of 
masonry coefficient 0.02, add 1,500 (0.10 0.02) 120 

Stage opening 159 

Total 909 

The total of 909 units is quite close enough for practical 
purposes to the 926 units given by the more detailed esti- 
mate. In more complicated cases, the rule makes a very 
useful shortcut in estimating the empty-room absorbing 
power. We shall use it in estimating the reverberation 
time of a theater with balconies. 

Reverberation Time in a Theater. 

Figure 61 shows the plan and longitudinal section of 
the new Chicago Civic Opera House. The transverse 
section is rectangular, so that the room as a whole presents 
a series of expanding rectangular arches. In figuring the 
total volume, the volumes of these separate sections were 
computed, and deductions made for the balcony and box 
spaces. The necessary data, taken from the plans and 
preliminary specifications, are as shown on page 150. 



REVERBERATION AND THE ACOUSTICS OF ROOMS 149 





Fia. 01. Plan and section, Chicago Civic Opera. 



150 ACOUSTICS AND ARCHITECTURE 

Total volume, from curtain line and including spaces under balcony, 842,000 

cu. ft. 

Walls and ceiling of hard plaster 
Floors of cement 
All aisles carpeted 

Boxes carpeted and lined with plush 
Heavy velour draperies in wall panels 
3,600 seats, upholstered, seat and back 
Velour curtain, 36 by 50 ft. 

The figures for the total absorbing power follow: 

Units 

1. Absorbing power of bare room, 0.3^842,000* 
(assuming wood seats 0.3) .... 2 , 630 

2. Boxes,* 8 by 112ft, X 1.0 896 

3. Stage, 1,800 sq. ft. X 0.44 ... . 792 

4. Carpets, 3,200 sq. ft. X 0.25 800 

5. Seats, f 3,600 X (2.6 - 0.3)... . . 8,300 

6. Wall drapes, 2,400 sq. ft. X 0.44. . 1,050 



Total absorbing power of empty room . . . . 14 , 468 

_ T 0.5 X 842,000 

r '- 14,468 = 2 - 91s . 

* On account of the heavy padding of the boxes, the total opening of the boxes into the 
main body of the room was considered as an area from which no sound was reflected. 

t Seats with absorbing power of 0.3 are assumed in the formula for the bare room; hence 
the deduction of 3 from 2.6, the absorbing power of the seats used. 

Upon completion, careful measurements were made to 
determine the reverberation time. An organ pipe whose 
acoustic output was measured by timing in the sound 
chamber was used for this purpose. From the known 
value of a, the total absorbing power of the sound chamber, 
the value of E/i for this particular pipe and observer was 
determined from Eq. 39. With this value known, and the 
measured value of TI in the completed room, the value 
of the total absorbing power of the latter was computed 
using the same equation. The total absorbing power thus 
measured turned out to be 13,830 units, giving for the 
reverberation time a value of 3.05 sec., as compared with 
the estimated value of 2.91 sec. 

Noting the effect of an audience occupying the more 
highly absorbent upholstered seats in comparison with the 
less absorbent chairs of the first example, we have the 
following reverberation times: 



REVERBERATION AND THE ACOUSTICS OF ROOMS 151 



Audience 


Added 
absorption 


Total 
absorption 


To 


None 




14,400 


2 93 


1,200 X 2.0 


2,400* 


16,800 


2 51 


2,400 X 2.0 


4,800 


19,200 


2.19 


3,600 X 2.0 


7,200 


21,600 


1 95 



* The added absorbing power per person is assumed to be 4 6. The absorbing power of 
the person less 2 6, the absorbing power of the seat, which he is assumed to replace, is 2 0. 

The value of the upholstered chairs in minimizing the 
effect of the audience upon the reverberation time is well 
brought out by the two examples chosen. 

Allowance for Balcony Recesses. 

In the foregoing, we have treated the recessed spaces 
under the balconies as a part of the main body of the room, 
contributing both to the volume and to the absorbing 
power terms of the reverberation equation. We may also 
consider these spaces as separate rooms coupled to the main 
body of the auditorium. 1 Assuming, for the moment, that 
the average coefficient of absorption of the surfaces of the 
under-balcony spaces is the same as that of the main room, 
it is apparent that the rate of decay of sound intensity 
in the former will be greater due to the fact that the mean 
free path is smaller and the number of reflections per second 
is correspondingly greater. In a recent paper, Eyring 2 
reports the results of some interesting experiments on the 
reverberation times as measured in an under-balcony space 
27 ft. long and 12 ft. deep, with a ceiling height of 11 ft. 
All the surface in this space except the floor was covered 
with sound-absorbent material. His measurements showed 
that at the rear of the space, there were two distinct rates 
of decay for tones of frequencies above 500 vibs./sec., the 
more rapid rate taking place during the first part of the 
decay process, while the slower rate at the end corresponds 

1 For a recent account of experimental research on this question the 
reader is referred to Reverberation Time in Coupled Rooms, by Carl F. 
Eyring, Jour. Acous. Soc. Amer., vol. 3, No. 2, p. 181, October, 1931. 

2 Jour. Soc. Mot. Pict. Eng., vol. 15, pp. 528-548, October, 1930. 



152 ACOUSTICS AND ARCHITECTURE 

very closely to that in the main body of the room. At 
the front, there was only a single rate of decay, except 
for the highest frequency of 4,000 vibs./sec. This single 
rate was nearly the same as the slower rate measured in 
the main body of the room. 

The initial higher rate at the rear is thus the rate of 
decay of sound in the under-balcony space considered as 
a separate room. During this stage of the decay, the 
balcony opening feeds some energy back into the 'main 
body of the room and hence, looked at from the large space, 
does not act as an open window. Later on, however, the 
sound originally under the balcony having been absorbed, 
energy is fed in through the opening, and the opening 
behaves more like a perfectly absorbent surface for the main 
body of the room. 

It is apparent that there is no precise universal rule by 
which allowance can be made for the effect of a recessed 
space upon the reverberation time. It will depend upon 
the average absorption coefficient of the recessed portion, 
the wave length of the sound, and the depth of the recess 
relative to the dimensions of the opening. A common- 
sense rule, and one which in the writer's experience works 
very well in practice, is the following: 

Compute the total absorbing power of the space under the balcony. 
If this is less than the absorption supplied by treating the opening as a 
totally absorbing surface with a coefficient of unity, then consider this 
space as contributing to the volume and absorbing power of the room. 
Otherwise neglect both the volume and the absorbing power of the 
recessed space and consider the opening into the recessed portion as 
contributing to the main body of the room, an absorbing power equal to 
its area. 

We shall, as an illustration, estimate the reverberation 
time of the Civic Opera House treating the balcony open- 
ings as perfectly absorbing surfaces. In the space under 
the balconies, there were 1,150 seats and 1,200 sq. ft. of 
carpet. These are to be deducted from the figures for the 
room considered as a whole. 



REVERBERATION AND THE ACOUSTICS OF ROOMS 153 

Volume (excluding space under boxes and in 
the first balcony) 765,000 cu. ft. 

Units 



1. Absorbing power of bare room O.S^?" . . 2,400 

2. Boxes, 8 by 112 ft. X 1.0 .............. 896 

3. Stage, 1,800 sq. ft. X 0.44 .............. 792 

4. Carpets, 2,000 sq. ft. X 0.25 ............. 500 

5. Seats, 2,450 X (2.6 - 0.3) .......... 5,650 

6. Wall drapes, 2,400 sq. ft. X 0.44 ......... 1 ,050 

7. Balcony opening, 15 by 112 ft. X 1.0.. .. 1,680 

8. Balcony opening, 15 by 114 ft. X 1.0 .... 1,710 

Total ........................... 14,678 



This we note is decidedly less than the measured value of 
3.05 sec. Upon comparison of the data with that con- 
sidering the under-balcony spaces as part of the main body 
of the room, it appears that the total absorbing power under 
the balconies is 3,190 units while the area of the openings 
is 3,390. According to the rule given above, we should 
expect the results of the first calculation to agree more 
nearly with the measured value, as they do. It turns out, 
in general, that if the depth of the balcony is more than 
three times the height from floor to ceiling at the front, 
then calculations based on the "open-window" concept 
agree more closely with measured values. It also appears 
that when the space under the balcony has a total absorbing 
power considerably greater than that of the opening con- 
sidered as a surface with a coefficient of unity, the results of 
computing the reverberation times on the two assumptions 
do not differ materially. 

This last follows from the fact that we reduce both the 
assumed volume and the total absorbing power when we 
treat the balcony opening as a perfectly absorbing portion 
of the boundary of the main body of the room. This 
decrease in both V and a will not materially alter their 
ratio, which determines the computed reverberation time. 
The interested reader may satisfy himself on this point 
by computing the reverberation time for the full-audience 



154 ACOUSTICS AND ARCHITECTURE 

condition, treating the balcony opening as a perfectly 
absorbing surface. 

Effect of Reverberation on Hearing. 

Referring to the conditions for good hearing in audience 
rooms, as given by Professor Sabine, we note that the last 
of these is that "the successive sounds in rapidly moving 
articulation, either of music or speech, shall be free from 
each other. " The effect of the prolongation of individual 
sounds by reverberation Obviously militates against this 
requirement for good hearing. For example, the sound 
of a single spoken syllable may persist as long as 4 sec. 
in a reverberant room. During this interval, a speaker 
may utter 16 or more syllables. The overlapping that 
results will seriously lessen the intelligibility of sustained 
speech. With music, the effect of excessive reverberation 
is quite similar to that of playing the piano with the sus- 
taining pedal held down. On the other hand, common 
experience shows that in heavily damped rooms, speech, 
while quite distinct, lacks apparent volume, and music is 
lifeless and dull. The problem to be solved therefore is to 
find the happy medium between these two extremes. Two 
lines of attack suggest themselves. By direct experiment 
one may vary the reverberation time in a single room to 
what is considered to be the most satisfactory condition 
and measure, or calculate this time; or one may measure 
the reverberation time in rooms which have an established 
reputation for good acoustical properties. The first method 
was employed by Professor Sabine for piano music in small 
rooms. 1 His results showed a remarkably precise agree- 
ment by a jury of musicians upon a value of 1.08 sec. 
as the most desirable reverberation time for piano music, 
for rooms with volumes between 2,600 and 7,400 cu. ft. 
(74 and 210 cu. m.) His contemplated extension of this 
investigation to larger rooms and different types of music 
was never carried out. 

1 "Collected Papers on Acoustics," p. 71. 



REVERBERATION AND THE ACOUSTICS OF ROOMS 155 

Reverberation : Speech Articulation, Knudsen's Work. 

V. O. Knudsen 1 has made a thoroughgoing investigation 
of the effect of reverberation upon articulation. The 
method employed was that used by telephone engineers in 
testing speech transmission by telephone equipment. 
The " percentage articulation " of an auditorium is the 
percentage of the total number of meaningless syllables 
correctly heard by an average listener in the auditorium. 
Typical monosyllabic speech sounds are called in groups of 
three by a speaker. Observers stationed at various parts 
of the room record what they 
think they hear. The number lioo 
of syllables correctly recorded 5 so 
expressed as a percentage of the Jf 60 




number spoken is the percentage ^ 40 o io"2o 30 40 so Jo TO so 9 > 



articulation for the auditorium. Reverbe*,*,- seconds 

FIG. 62. Relation between 
1 he CUrveS OI I 1 Ig. 52 taken trom percentage articulation and 

Knudsen's paper show the per- ^J^ e ) rafcion timc ' (Afi " r 
centage articulation in a number 

of rooms similar in shape and with volumes between 200,000 
and 300,000 cu. ft. The lower curve gives the best fit 
with the observed data, while the upper curve shows the 
percentage articulation assuming that the level of intensity 
in all cases was 10 7 X i. (It will be recalled that for a 
source of given acoustical power, the average intensity 
set up is inversely proportional to the absorbing power.) 
We note that increasing the reverberation time from 1.0 to 
2.0 sec. lowers the percentage articulation from 90 to 86 
per cent, while if the reverberation is still further increased 
to 3.0 sec., the articulation becomes about 80 per cent. 

Reverberation and Intelligibility of Speech. 

These figures, however, do not give the real effect of 
reverberation upon the intelligibility of connected speech. 
In listening to the latter, the loss of an occasional syllable 
produces a very slight decrease in the intelligibility of con- 

1 Jour. Acous. Soc. Amer., vol. 1, No. 1, p. 57. 1929. 



156 ACOUSTICS AND ARCHITECTURE 

nected phrases and sentences. The context supplies the 
meaning. Tests made on this point by Fletcher 1 at the 
Bell Laboratories show the following relation between the 
percentage articulation and the intelligibility of connected 
speech. 

Percentage Intelligibility, 

Articulation Per Cent 

70 98 

80 99 

90 99 + 

In these tests, the intelligibility was the percentage of 
questions correctly understood over telephone equipment, 
which gave the single-syllable articulation shown. These 
figures taken with Knudsen's results indicate that rever- 
beration times as great as 3 sec. may not materially affect the 
intelligibility of connected speech. 

Further, it seems fair to say that with a reverberation 
time of 2 sec. or less in rooms of this size (200,000 to 300,000 
cu. ft.) connected speech of sufficient loudness should be 
quite intelligible. This conclusion should be borne in 
mind when we come to consider the reverberation time 
in rooms that are intended for both speech and music. 
Viewed simply from the standpoint of intelligibility, the 
requirements of speech do not impose any very precise 
limitation upon the reverberation time of auditoriums 
further than that it should be less than 2.0 sec. While 
in Knudsen's tests the decrease in reverberation time below 
2.0 sec. produced measurable increase in syllable articula- 
tion, yet the improvement in intelligibility of connected 
speech is negligibly small. 

Reverberation and Music. 

In order to arrive at the proper reverberation for rooms 
intended primarily for music, the procedure has been to 
secure data on rooms which according to the consensus 
of musical taste as well as of popular approval are acousti- 
cally satisfactory. Obviously both of these are, in the very 

1 "Speech and Hearing," D. Van Nostrand Company, p. 266, 1929. 



REVERBERATION AND THE ACOUSTICS OF ROOMS 157 

ture of the case, somewhat difficult to arrive at. With 
me rare and refreshing exceptions the critical faculties 
musicians do not extend to the scientific aspects of their 
b, while public opinion seldom becomes articulate in 
icing approval. 

However, the data given in Table XIV, compiled from 
rious sources, are for rooms which enjoy established 



ABLE XIV. REVERBERATION TIMES OP ACOUSTICALLY GOOD ROOMS 



Auditorium 


Volume, 
cubic feet 


Seats 


To 


Ti 


Authority 


sic Rooms, New England Con- 
srvatory 
ill auditorium, Moscow. . . . 
scow Conservatory 


2,600-7,400 
4,450 
90,000 


550 


1.08 
1 06 
1 30 


1 07-1 19 
1.15 
1 19 


W. C. Sabine 
Lifshitz 
Lifshitz 


aikvereinssaal, Vienna 
?zig Cewandhaus 


290,000 
363,000 


1,800 
1,560 


1.62* 
1 90 


1.39 
1 64 


Knudsen 


jzig Gewanahaus 


363,000 


] ,560 


1 50* 




Ivnudsen 


demy of Music, Philadelphia 
hestra Hall, Detroit. . 
demy of Music, Brooklyn 
at Theater, Moscow 
ton Opera House 


400,000 
400,000 
430,000 
486,000 
500,000 


2,800 
2,200 
2,200 
2,300 
2,350 


1.76 
1.44 
1.60 
1.55f 
1 51 


.49 
.20 
.31 
.25 
23 


P. E. Sabine 
P. E. Sabine 
Tallant 
Lifshitz 
P E Sabine 


trance Hall Cleveland 


554,000 


1 840 


1 85 


61 


D C Miller 


at Hall, Moscow Conservatory 
iphony Hall, Boston 


600,000 
650,000 


2,150 
2,600 


2 OOj 
1 93 


.63 
55 


Lifshitz 
P E Sabine 


negie Hall, New York 


737,000 


2,600 


1 76 


1 38 


P. E. Sabine 


Memorial, Ann Arbor, Michi- 
in .... 


795,000 


5,000 


1 70 


1 33 


Tallant 


tman Theater, Rochester 
c Opera House, Chicago 
''ago Auditorium 


790,000 
842,000 
925,000 


3,340 
3,600 
3,640 


2.08 
1 95 
1 90 


1.65 
1 48 
1 48 


Watson 
P. E. Sabine 
P E Sabine 















r. - 



00083V,- , . 
(9.1 - log, a). 



* Computed by formula TQ = 



t Reverberation considered too low 
t Reverberation considered too high. 



0.05V 



0-1) 



mutations for good acoustics. In Fig. 63, the reverbera- 
n times computed by the formula T == 0.05 V/a are 
)tted as a function of the volume in cubic feet. So 
)tted, these data seem to justify two general statements. 
rst, the reverberation time for acoustically good rooms 
3ws a general tendency to increase with the volume in 
bic feet and, second, for rooms of a given volume there is a 



158 



ACOUSTICS AND ARCHITECTURE 



Acceptable Reverberation Times 



Concert halls 
and auditoriums 
Motion picture *rj 




fairly wide range within which the reverberation time 
may fall. 

In Fig. 64 is shown an empirical curve given by Professor 
Watson 1 of what he has called the " optimum time of 

reverberation," also a curve 
partly empirical and partly 
theoretical proposed by 
Lifshitz. 2 

Watson's curve is de- 
duced from the computed 
reverberation times of 
acoustically good rooms, 
i i i 1" I~ s i" s Lifshitz finds experimental 
volume m Cub.c Peer ~~ verification f or his curve in 

FIG. 03. Acceptable reverberation times the judgment of trained 
for rooms of different volumes. mus i c i a ns flfl to the best 

time of reverberation in a small room (4,500 cu. ft.) and the 
reverberation times of a number of auditoriums in Moscow. 
Recently MacNair 3 has undertaken to arrive at a theo- 
retical basis for the increase -g2 
in desirable reverberation time 1 2.0 - 
with the volume of the room 3i&- 
by assuming that the loudness 
of the reverberant sound .i l>4 
integrated over the total time w l ' 2 
of decay shall have a constant 
value. This implies that both 
the maximum intensity and 
the duration of a sound con- 

PIG. 64. Optimum reverberation 
tribute to the magnitude OI times proposed by Watson and 

the psychological impression. Llfshltz - 

MacNair's theoretical curve for optimum reverberation 
time does not differ materially from those shown in Figs. 
63 and 64, a fact which would seem to give weight to his 
assumption. Experience in phonograph recording also 

1 Architecture, vol. 55, pp. 251-254, May, 1927. 

2 Phys. Rev., vol. 3, pp. 391-394, March, 1925. 

3 Jour. Acous. Soc. Amcr., vol. 1, No. 2, p. 242, January 1930. 




Volume in Cubic Feet 



REVERBERATION AND THE ACOUSTICS OF ROOMS 159 

aows that a certain amount of reverberation gives the effect 
f increased volume of tone to recorded music, an effect 
r hich apparently cannot be obtained by simply raising the 
>vel of the recorded intensity. 

Optimum Reverberation Time." 

The practical question arises as to whether, in the light 
f present knowledge, given a proposed audience room 
f given volume, we are justified in assigning a precise 
alue to the reverberation time in order to realize the best 
earing conditions. (The author objects strenuously to 
le frequently used term "perfect acoustics.") There 
i also the other question as to whether this specified time 
f reverberation should depend upon the use to which the 
3om is to be put, whether speech or music, and, if the latter, 
pon the particular type of music contemplated. 

The data on music halls given above suggest a negative 
nswer to the first question, if the emphasis is laid upon the 
r ord precise. 

To draw a curve of best fit of the points shown and say 
lat the time given by this curve for an auditorium of any 
iven volume is the optimum time would be straining for a 
jientific precision which the approximate nature of our 
stimate of the reverberation time does not warrant. Take 
le best-known case, that of the Leipzig Gewandhaus. 
abine's estimate of the reverberation time, based on the 
iformation available to him, was 2.3 sec. Bagenal, 1 
*om fuller architectural data obtained in the hall itself, 
stimates the time at 1.9 sec.; while Knudsen, from 
3verberation measurements in the empty room, estimates 
time of 1.5 sec. for the full-audience condition. Further, 
icre is a question as to the precision of musical taste as 
pplied to different rooms. Several years ago, a question- 
aire was sent to the leading orchestra conductors in 
jnerica in an attempt to elicit opinions as to the relative 

1 BAGENAL, HOPE, and BURSAR GODWIN, Jour. R.I.B.A., vol. 36, pp. 
56-763, Sept. 21, 1929. 



160 ACOUSTICS AND ARCHITECTURE 

acoustical merits of American concert halls. Only five 1 
of the gentlemen replied. 

Rated both by the number of approvals and the unquali- 
fied character of the comments of these five, of the rooms 
for which we have data the Academy of Music (Phila- 
delphia), Carnegie Hall (New York), the Chicago Audito- 
rium, Symphony Hall (Detroit), and Symphony Hall 
(Boston) may be taken as outstanding examples of satis- 
factory concert halls. Referring to Fig. 63, we note that 
all of these, with the exception of the Chicago Auditorium, 
fail to satisfy our desire for scientific precision by refusing 
to fall exactly on the average curve. The shaded area 
of Fig. 63 represents what would seem to be the range 
covered by the reverberation times of rooms which are 
acoustically satisfactory for orchestral music. The author 
suggests the use of the term " acceptable range of reverbera- 
tion times" in place of " optimum reverberation time," 
as more nearly in accord with existing facts as we know 
them. There can be only one optimum, and the facts do 
not warrant us in specifying this with any greater precision 
than that given by Fig. 63. 

Speech and Musical Requirements. 

Turning now to the question of discriminating between 
the requirements for speech and music, we recall that 
on the basis of articulation tests a reverberation time as 
great as 2.0 seconds does not materially affect the intelli- 
gibility of connected speech. Since the requirements for 
music call for reverberation times less than this, there 
appears to be no very strong reason for specifying condi- 
tions for rooms intended primarily for speaking that are 
materially different from those for music. As a practical 
matter, auditoriums in general are designed for a wide 
variety of uses, and, except in certain rare instances, a 
reasonable compromise that will meet all requirements 

1 It is a pleasure to acknowledge the courteous and valuable information 
supplied by Mr. Frederick Stock, Mr. Leopold Stokowski, Mr. Ossip 
Gabrilowitsch, Mr. Willem Van Hoogstraten, and Mr. Eric De Lamarter. 



REVERBERATION AND THE ACOUSTICS OF ROOMS 161 

is the more rational procedure. The range of reverberation 
times shown in Fig. 63 represents such a compromise. 
Thus the Chicago Auditorium and Carnegie Hall have 
long been considered as excellent rooms for public addresses 
and for solo performances as well as for orchestral music, 
while the new Civic Opera House in Chicago, intended 
primarily for opera, has received no criticism when used for 
other purposes. Tests conducted after its completion 
showed that the speech of very mediocre speakers was 
easily understood in all parts of the room. 1 

European Concert Halls. 

V. O. Knudsen 2 gives an interesting and valuable account 
of some of the more important European concert halls. 
His findings are that reputations for excellent acoustics 
are associated with reverberation times lying between 1 
and 2 sec., with times between 1.0 and 1.5 sec. more com- 
monly met with than are times in the upper half of the 
range. The older type of opera house of the conventional 
horseshoe shape with three or four levels of boxes and 
galleries all showed relatively low reverberation times, 
of the order of 1.5 sec. in the empty room. This is to be 
expected in view of the fact that placing the audience in 
successive tiers makes for relatively small volumes and large 
values of the total absorbing power. Custom would 
account, in part, for the general approval given to the low 
reverberation times in rooms of this character. The 
desirability of the usual, no doubt, is also responsible for 
the general opinion that organ music requires longer 
reverberation times than orchestral music. Organ music 
is associated with highly reverberant churches, while the 
orchestra is usually heard in crowded concert halls. 

In the paper referred to, Knudsen gives curves showing 
reverberation times which he would favor for different 

1 Severance Hall in Cleveland may be cited as a further instance. In the 
design of this room, primary consideration was given to its use for orchestral 
music. Its acoustic properties prove to be eminently satisfactory for speech 
as well. 

2 Jour. Acous. Soc. Amer., vol. 2, No. 4, p. 434, April, 1931. 



162 ACOUSTICS AND ARCHITECTURE 

types of music and for speech. The allowable range which 
he gives for music halls is considerably greater than that 
shown in Fig. 63, although the middle of the range coincides 
very closely with that here given. He proposes to distin- 
guish between desirable reverberation for German opera 
as contrasted with Italian opera, allowing longer times 
for the former. He advocates reverberation times for 
speech, which are on the average 0.25 sec. less than the 
lowest allowable time for music. These times are con- 
siderably lower than those in existing public halls, and their 
acceptance would call for a marked revision downward 
from present accepted standards. The adoption of these 
lower values would necessitate either marked reduction 
in the ratio of volume to seating capacity or the adoption 
of the universal practice of artificially deadening public 
halls. Whether the slight improvement in articulation 
secured thereby would justify such a revision is in the mind 
of the author quite problematical. 1 

Formulas for Reverberation Times. 

In the foregoing, all computations of reverberation times 
have been made by the simple formula T 0.05F/a, 
which gives the time for the continuous decay of sound 
through an intensity range of 1,000,000 to 1. Since the 

publication of Eyring's paper giving the more general 

Q 05 y 
relation T = ~i 71 x > some writers on the subject 

-ASlog e (1 - Ota) '' 

have preferred to use the latter. As we have seen, this 
gives lower values for the reverberation time than does the 
earlier formula, the ratio between the two increasing with 
the average value of the absorption coefficient. Therefore, 
there is a considerable difference between the two, partic- 
ularly for the full-audience conditions. As long as we 
adhere to a single formula both in setting up our criterion 
of acoustical excellence on the score of reverberation and 

1 Since the above was written, the author has been informed by letter that 
Professor Knudsen's recommended reverberation times are based on the 
Eyring formula. This fact materially lessens the difference between his 
conclusions and those of the writer. 



REVERBERATION AND THE ACOUSTICS OF ROOMS 163 

also in computing the reverberation time of any proposed 
room, it is immaterial which formula is used. Consistency 
in the use of one formula or the other is all that is required. 
In view of the long-established use of the earlier formula 
and the fact that the criterion of excellence is based upon 
it, and also because of its greater simplicity, there appears 
to be no valid reason for changing to the later formula in 
cases in which we arc interested only in providing satisfac- 
tory hearing conditions in audience rooms. 

Variable Reverberation Times. 

Several writers have proposed that, in view of the 
supposedly different reverberation requirements of different 
types of music, means of varying the reverberation time of 
concert halls is desirable. This plan has been employed in 
sound-recording rooms and radio broadcasting studios. 
Osswald of Zurich has suggested a scheme whereby the 
volume term of the reverberation equation may be reduced 
by lowering movable partitions which would cut off a part 
of a large room when used by smaller audiences and for 
lighter forms of music. Knudsen intimates the possible 
use of suitable shutters in the ceiling with absorptive 
materials behind the shutters as a quick and easy means of 
adapting a room to the particular type of music that is 
to be given. In connection with Osswald's scheme, one 
must remember that in shutting off a recessed space, we 
reduce both volume and absorbing power and that such a 
procedure might raise instead of lower the reverberation 
time. Knudsen's proposal is open to a serious practical 
objection in the case of large rooms on the score of the 
amount of absorbent area that would have to be added 
to make an appreciable difference in the reverberation 
time. Take the example given earlier in this chapter. 
Assume that 2.0 sec. is agreed upon as a proper time for 
"Tristan and Isolde " and 1.5 sec. for "Rigoletto." The 
absorbing power would have to be increased from 21,600 
to 28,800 units, a difference of 7,200 units. With a material 
whose coefficient is 0.72 we should need 10,000 sq. ft. of 



164 ACOUSTICS AND ARCHITECTURE 

shutter area to effect the change. Even if he were willing 
to sacrifice all architectural ornament of the ceiling, the 
architect would be hard put to find in this room sufficient 
ceiling area that would be available for the shutter treat- 
ment. One is inclined to question whether the enhanced 
enjoyment of the average auditor when listening to "Rigo- 
letto" with the shutters open would warrant the expense of 
such an installation and the sacrifice of the natural archi- 
tectural demand for normal ceiling treatment. 

Waiving this practical objection, one is inclined to 
wonder if there is valid ground for the assumption that 
there is a material difference in reverberation requirements 
for different types of music. Wagnerian music is asso- 
ciated with the tradition of the rather highly reverberant 
Wagner Theater in Bayreuth. Italian opera is associated 
with the much less reverberant opera houses of the horse- 
shoe shape and numerous tiers of balconies and boxes. 
Therefore, we are apt to conclude that Wagner's music 
demands long reverberation times, while the more florid 
melodic music of the Italian school calls for short reverbera- 
tion periods. Organ music is usually heard in highly 
reverberant churches and cathedrals, while chamber music 
is ordinarily produced in smaller, relatively non-reverberant 
rooms. All of these facts are tremendously important in 
establishing not musical taste but musical tradition. 

On the whole it would appear that the moderate course 
in the matter of reverberation, as given in Fig. 63, will lead 
to results that will meet the demands of all forms of music 
and speech, without imposing any serious special limitations 
upon the architect's freedom of design or calling for elabo- 
rate methods of acoustical treatment. 

Reverberation Time with a Standard Sound Source. 

In 1924, the author 1 proposed a formula for the calcula- 
tion of the reverberation time based on the assumption of 
a fixed acoustic output of the source, instead of a fixed 
value of the steady-state intensity of 10 6 X i. For a source 

1 Am&r. Architect, vol. 125, pp. 57&-S86, June 18, 1924. 



REVERBERATION AND THE ACOUSTICS OF ROOMS 165 

of given acoustic output E, the steady-state intensity is 
given by the equation 

4E 
II ~ 

so that the steady-state intensity for a given source will 
vary inversely as the absorbing power. This decrease of 
steady-state intensity with increasing absorbing power 
accounts in part for the greater allowable reverberation 
time in large rooms, so that it would seem that the rever- 
beration in good rooms computed for a fixed source would 
show less variation with volume than when computed for a 
fixed steady-state intensity. The acoustical output of the 
fundamental of an open-diapason organ pipe, pitch 512 
vibs./sec., is of the order of 120 microwatts. This does not 
differ greatly from the power of very loud speech. Taking 
this as the fixed value of the acoustic output of our sound 
source, we have, by the reverberation theory, 

-1 _ logjoa) (7Q) 

The bracketed expression is the logarithm of the steady- 
state intensity set up by a source of the specified output in a 
room whose absorbing power is a. When this equals 6.0 
(logarithm of 10 6 ), Eq. (70) reduces to the older formula. 
The expression log a thus becomes a correction term for 
the effect of absorbing power on initial intensity. The 
values of T\ 9 computed on the foregoing assumptions, are 
included in Table XIV. We note a considerably smaller 
spread between the values of the reverberation times for 
good rooms when computed by Eq. (70) and a less marked 
tendency toward increase with increasing volume. For 
rooms with volumes between 100,000 and 1,000,000 cu. ft. 
we may lay down a very safe working rule that the rever- 
beration time computed by Eq. (70) should lie between 
1.2 and 1.6 sec. For smaller rooms and rooms intended 
primarily for speech the reverberation time should fall 
in the lower half of the range, and for larger rooms and 
rooms intended lor music it may fall in the upper half. 



166 ACOUSTICS AND ARCHITECTURE 

Reverberation for Reproduced Sound. 

An important difference between original speech and 
music and that reproduced in talking motion pictures lies 
in the fact that the acoustical output of an electrical loud- 
speaker may be and usually is considerably greater than 
that of the original source. This raises the average steady- 
state intensity set up by the loud-speaker source above that 
which would be produced in the same room by natural 
sources and hence for a given relation of volume to absorb- 
ing power produces actually longer duration of the residual 
sound. For this reason it has generally been assumed that 
the desirable computed reverberation should be somewhat 
less than for theaters or concert halls. R. K. Wolf 1 has 
measured the reverberation in a large number of rooms that 
are considered excellent for talking motion pictures and 
gives a curve of optimum reverberation times. Interest- 
ingly enough, his average curve coincides almost precisely 
with the lower limit of the range given in Fig. 63. Due 
to the directive effect of the usual types of loud-speaker, 
echoes from rear walls are sometimes troublesome in 
motion-picture theaters. What would be an excessive 
amount of absorption, reverberation alone considered, is 
sometimes employed to eliminate these echoes. By raising 
the reproducing level, the dullness of overdamping may be 
partly eliminated, so that talking-motion-picture houses 
are seldom criticised on this score even though sometimes 
considerably overabsorbent. At the present time, there 
is a tendency to lay many of the sins of poor recording 
and poor reproduction upon the acoustics of the theater. 
The author's experience and observation would place the 
proper reverberation time for motion-picture theaters in the 
lower half of the range for music and speech, given in Fig. 
63, with a possible extension on the low side as shown. 
Sound recording and reproduction have not yet reached a 
stage of perfection at which a precise criterion of acoustical 
excellence for sound-motion-picture theaters can be set up. 

1 Jour. Soc. Mot. Pict. Eng., vol. 45, No. 2, p. 157. 



REVERBERATION AND THE ACOUSTICS OF ROOMS 167 

Reverberation in Radio Studios and Sound-recording 
Rooms. 

Early practice in the treatment of broadcasting studios 
and in phonograph-recording rooms was to make them 
as "dead" (highly absorbent) as possible. For this pur- 
pose, the walls and ceilings were lined throughout with 
extremely heavy curtains of velour or other fabric, and the 
floors were heavily carpeted. Reverberation was extremely 
low, frequently less than 0.5 sec., computed by the 0.05 V/a 
formula. The best that can be said for such treatment 
is that from the point of view of the radio or phonograph 




Early radio broadcasting studio. All surfaces were heavily padded with absorb- 
ent material. 

auditor the effect of the room was entirely negative one 
got no impression whatsoever as to the condition under 
which the original sound was produced. For the performer, 
however, the psychological effect of this extreme deadening 
was deadly. It is discouraging to sing or play in a space 
where the sound is immediately " swallowed up" by an 
absorbing blanket. 

One of the best university choirs gave up studio broad- 
casting because of the difficulty and frequent failure to 
keep on the key in singing without accompaniment. 

Gradually the tendency in practice toward more rever- 
berant rooms has grown. In 1926, the author, through the 
courtesy of Station WLS in Chicago, tried the effect on 



168 



ACOUSTICS AND ARCHITECTURE 



the radio listeners of varying the reverberation time in a 
small studio, with a volume of about 7,000 cu. ft. The 
reverberation time could be quickly changed in two succes- 
sive steps from 0.25 to 0.64 sec. The same short program 
of music was broadcast under the three-room conditions, 
and listeners were asked to report their preference. There 
were 121 replies received. Of these, 16 preferred the least 
reverberant condition, and 73 the most reverberant con- 
dition. It was not possible to carry the reverberation to 
still longer times. J. P. Maxfield states that in sound 




Modern broadcasting studio. Absorption can be varied by proper disposal of 

draperies. 

recording for talking motion pictures,, the reverberation 
time of the recording room should be about three-fourths 
that of a room of the same size used for audience purposes. 
This, he states, is due to the fact that in binaural hearing, 
the two ears give to the listener the power to distinguish 
between the direct and the reverberant sound and that 
attention is therefore focused on the former, and the latter is 
ignored. With a single microphone this attention factor 
is lacking, with the result that the apparent reverberation 
is enhanced. 

On this point, it may be said that phonograph records 
of large orchestras are often made in empty theaters. 



REVERBERATION AND THE ACOUSTICS OF ROOMS 169 

Music by the Philadelphia Symphony Orchestra is recorded 
in the empty Academy of Music in Philadelphia, where 
accordirg to the author's measurements, the measured 
reverberation time is 2.3 sec. It is easy to note the rever- 
beration in the records, but this does not in any measure 
detract from the artistic quality or naturalness of the 
recorded music. 

The Sunday afternoon concerts of the New York Philhar- 
monic Orchestra are broadcast usually from Carnegie Hall 
or the Brooklyn Academy of Music. Here the reverbera- 
tion of these rooms under the full-audience condition is not 
noticeably excessive for the radio listener. All these facts 
considered, it would appear that reverberation times 
around the lower limit given for audience rooms in general 
will meet the requirements of phonograph recording and 
radio broadcasting. 

In view of the fact that in sound recording the audience 
is lacking as a factor in the total absorbing power, it follows 
that this deficiency will have to be supplied by the liberal 
use of artificial absorbents. It is at present a mooted 
question as to just what the frequency characteristics of such 
absorbents should be. If the pitch characteristics of the 
absorbing material do have an appreciable effect upon the 
quality of the recorded sound, then these characteristics 
are more important than in the case of an auditorium. In 
the latter, the audience constitutes the major portion of the 
total absorbing power, and its absorbing power considered 
as a function of pitch will play a preponderant role. This 
r61e is taken by the artificial absorbent in the sound stage. 
A " straight-line absorption, " that is, uniform absorption at 
all frequencies, has been advocated as the most desirable 
material for this use. As has already been indicated, 
porous materials show marked selective absorption when 
used in moderate thickness. Thus hair felt 1 in. thick is 
six times as absorbent at high as at low frequencies, and 
this ratio is 2.5:1 for felt 3 in. thick. Certain fiber boards 
show almost uniform absorption over the frequency range, 
but the coefficients are relatively low as compared with 



170 ACOUSTICS AND ARCHITECTURE 

those of fibrous material. The nearest approach to a 
uniform absorption for power over the entire range of which 
the writer has any knowledge consists of successive layers of 
relatively thin felt, } $ in. thick, with intervening air space. 
A rather common current practice in sound-picture stages 
is the use of 4 in. of mineral wool packed loosely between 
2 by 4-in. studs and covered with plain muslin protected by 
poultry netting. The absorption coefficients of this mate- 
rial are as follows: 

Frequency Coefficient 

128 46 

256 61 

512 82 

1,024 82 

2,048 64 

4,096 60 

Here the maximum coefficient is less than twice the mini- 
mum, and this seems to be as near straight-line absorption 
as we are likely to get without building up complicated 
absorbing structures for the purpose. 

Whether or not uniform absorption is necessary to give 
the greatest illusion of reality is a question which only 
recording experience can answer. If the effect desired is 
that of out of doors, it probably is. But for interiors, 
absorption characteristics which simulate the conditions 
pictured would seem more likely to create the desired 
illusion. For a detailed treatment of the subject of rever- 
beration in sound-picture stages the reader is referred to 
the chapter by J. P. Maxfield in "Recording Sound for 
Motion Pictures. ' M 

1 McGraw-Hill Book Company, Inc., 1931. 



CHAPTER IX 
ACOUSTICS IN AUDITORIUM DESIGN 

The definition of the term auditorium implies the neces- 
sity of providing good hearing conditions in rooms intended 
for audience purposes. The extreme position that the 
designer might take would be to subordinate all other 
considerations to the requirements of good acoustics. In 
such a case, the shape and size of the room, the contours of 
walls and ceiling, the interior treatment, both architectural 
and decorative, would all be determined by what, in the 
designer's opinion, is dictated by acoustical demands. The 
result, in all probability, would not be a thing of architec- 
tural beauty. Acoustically, it might be satisfying, assum- 
ing that the designer has used intelligence and skill in 
applying the knowledge that is available for the solution 
of his problem. 

Fortunately, good hearing conditions do not impose any 
very hard and fast demands that have to be met at the 
sacrifice of other desirable features of design. Rooms are 
good not through possession of positive virtues so much as 
through the absence of serious faults. The avoidance of 
acoustical defects will yield results which experience shows 
are, in general, quite as satisfying as are attempts to secure 
acoustical virtues. The present chapter will be devoted to 
a consideration of features of design that lead to undesired 
acoustical effects and ways in which they may be avoided. 

Defects Due to Curved Shapes. 

Figure 65 gives the plan and section of an orchestral 
concert hall, very acceptable in the matter of reverberation 
but with certain undesirable effects that are directly 
traceable to the contours of walls and ceilings. These 
effects are almost wholly confined to the stage. The 
conductor of the orchestra states that he finds it hard to 

171 



172 



ACOUSTICS AND ARCHITECTURE 



secure a satisfactory balance of his instruments and that the 
musical effect as heard at the conductor's desk is quite 
different from the same effect heard at points in the audi- 
ence. The organist states that at his bench at one side and 
a few feet above the stage floor, it is almost impossible to 
hear certain instruments at all. A violinist in the front 
row on the left speaks of the sound of the wood winds on the 




FIG. 65. Plan and section, Orchestra Hall, Chicago. 

right and farther back on the stage as "rolling down on his 
head from the ceiling." A piano-solo performance heard 
at a point on the stage is accompanied by what seems like 
a row of pianos located in the rear of the room. In listening 
to programs broadcast from this hall, one is very conscious 
of any noise such as coughing that originates in the audi- 
ence, as well as an effect of reverberation that is much 
greater than is experienced in the hall itself. None of 
these effects is apparent from points in the audience. 



ACOUSTICS IN AUDITORIUM DESIGN 



173 



The sound photographs of Fig. 66 were made using plaster 
models of plan and section of the stage. In A and B, the 
source was located at a position corresponding to the 




FIG. 66. Reflection from curved walls, Orchestra Hall stage. 

conductor's desk. Referring to the plan drawing, we note 
that this is about halfway between the rear stage wall and 
the center of curvature of the mid-portion of this wall. 
In the optical case, this point is called the principal focus of 
the concave mirror, and a spherical wave emanating from 



174 



ACOUSTICS AND ARCHITECTURE 



this point is reflected as a plane wave from the concave 
surface, a condition shown by the reflected wave in A. 
In B, we note that the sound reflected from the more 
sharply curved portions is brought to two real focuses at 
the sides of the stage. In C, with the source near the 
back stage wall, the reflected wave front from the main 
curvature is convex, while the concentrations due to the 




FIG. 67. Concentration from curved rear wall. 

coved portions at the side are farther back than when the 
source is located at the front of the stage. In ''Collected 
Papers," 1 Professor Sabine shows the effect of a cylindrical 
rear stage wall upon the distribution of sound intensity 
on the stage, with marked maxima and minima, and an 
intensity variation of 47 fold. The example here cited is 
somewhat more complicated by the fact that there are two 
regions of concentration at the sides instead of a single 
region as in the cylindrical case. But it is apparent that 

1 "Collected Papers on Acoustics," Harvard University Press, p. 167, 1922. 



ACOUSTICS IN AUDITORIUM DESIGN 



175 



the difficulty of securing a uniformly balanced orchestra 
at the conductor's desk is due to the exaggerated effects of 
interference which these concentrations produce. The 
effect of the ceiling curvature in concentrating the reflected 
sound is apparent in Z>, which accounts for the difficulty 
reported by the violinist. This photograph also explains 
the pronounced effect at the microphone of noise originating 




FIG. 08. Plane roiling of stage prevents concentration. 

in the audience. Such sounds will be concentrated by 
reflection from the stage wall and ceiling in the region 
of the microphone. The origin of the echo from the rear 
wall of the room noted on the stage is shown in Fig. 67. 
Here the reflected wave was plotted by the well-known 
Huygens construction. We note the wave reflected from 
the curved 'rear wall converging on a region of concentra- 
tion, from which it will again diverge, giving to the listener 
on the stage the effect of image sources located in the rear 



176 ACOUSTICS AND ARCHITECTURE 

of the room. Again, a second reflection from the back 
stage wall will refocus it near the front of the stage. The 
conductor's position thus becomes a sort of acoustical 
" storm center/' at which much of the sound reflected once 
or twice from the principal bounding surfaces tends to be 
concentrated. Figure 68 shows the effect of substituting 
plain surfaces for the curved stage ceiling. One notes 
that the reflected wave front is convex, thus eliminating 
the focusing action that is the source of difficulty. 

Allowable Curved Shapes. 

If we were to generalize from the foregoing, we should 
immediately lay down the general rule that all curved 
shapes in wall and ceiling contours of auditoriums should be 
avoided. This rule is safe but not eminently sane, since 
there are many good rooms with curved walls and ceilings. 
Moreover, the application of such a rule would place a 
serious limitation upon the architectural treatment of 
auditorium interiors. In the example given, we note that 
the centers of curvature fall within the room and either 
near the source of sound or near the auditors. We may 
borrow from the analogous optical case a formula by which 
the region of concentration produced by a curved surface 
may be located. If s is the distance measured along a 
radius of curvature from the source of sound to the curved 
reflecting surface, and R the radius of curvature, then the 
distance x from the surface (measured along a radius) at 
which the reflected sound will be concentrated will be given 
by the equation 

x - sR 

X ~~ 2s - R 

The rule is only approximate, but it will serve as a means 
of telling us whether or not a curved surface will produce 
concentrations in regions that will prove troublesome. 

Starting with the source near the concave reflecting 
surface, s < J^jR, x comes out negative, indicating that 
there will be no concentration within the room. This is 
the condition pictured in C (Fig. 66). When s = J^/2, x 



ACOUSTICS IN AUDITORIUM DESIGN 



177 



becomes infinite; that is, a plane wave is reflected from the 
concave surface as shown in A. When s is greater than %R 
and less than /?, x comes out greater than 72, while for any 
value of s greater than /?, x will lie between %R and R. 
When s equals R } x comes out equal to R. Sound originat- 
ing at the center of a concave spherical surface will be 
focused directly back on itself. This is the condition 
obtaining in the well-known whispering gallery in the Hall 
of Statuary in the National Capitol at Washington. 1 There 
the center of curvature of the spherical segment, com- 
prising a part of the ceiling, falls at nearly head level 
near the center of the room. A whisper uttered by the 




i - i I 

FIG. 69. A. Vaulted ceiling which will produce concentration. B. Flattening 
middle portion of vault will prevent concentration. 

guide at a point to one side of the axis of the room is heard 
with remarkable clearness at a symmetrically located point 
on the other side of the axis. If the speaker stands at 
the exact center of curvature of the domed portion of the 
ceiling, his voice is returned with striking clearness from 
the ceiling. Such an effect, while an interesting acoustical 
curiosity, is not a desirable feature of an auditorium. 

From the foregoing, one may lay down as a safe working 
rule that when concave surfaces are employed, the centers or 
axes of these curvatures should not fall either near the source 
of sound or near, any portion of the audience. 

Applied to ceiling curvatures this rule dictates a radius of 
curvature either considerably greater or considerably less 
than the ceiling height. Thus in Fig. 69, the curved ceiling 

1 See "Collected Papers on Acoustics," p. 259. 



178 



ACOUSTICS AND ARCHITECTURE 



shown in A would result in concentration of ceiling-reflected 
sound with inequalities in intensity due to interference 
enhanced. At B is shown a curved ceiling which would 
be free from such effects. Here the radius of curvature 
of the central portion is approximately twice the ceiling 
height. No real focus of the sound reflected from this 
portion can result. The coved portions at the side have 

a radius so short that the real 
focuses fall very close to the 
ceiling without producing any 
difficulty for either the per- 
formers or the auditors. 

Figure 70 shows the plan 
and longitudinal section of a 
large auditorium, circular 
in plan, surmounted by a 
spherical dome the center of 
curvature of which falls about 
15 ft. above the floor level. 
In this particular room, the 
general reverberation is well 
within the limits of good hear- 
ing conditions, yet the focused 
echoes are so disturbing as 
to render the room almost 
It is to be noted 
surface 

produces much more marked 
Focusing action due to the concentration in two planes, 
whereas cylindrical vaults give concentrations only in 
the plane of curvature. Illustrating the local character 
of defects due to concentrated reflection, it was observed 
that in the case just cited, speech from the stage was 
much more intelligible when heard outside the room in the 
lobby through the open doors than when the listener was 
within the hall. It is to be said, in passing, that while 
absorbent treatment of concave surfaces of the curvatures 
just described may alleviate undesired effects, yet, in 




Fio. 70. Auditorium in which Unusable. 
circular plan and spherical dome that a Spherical 
produce focused echoes. 



ACOUSTICS IN AUDITORIUM DESIGN 179 

general, nothing short of a major operation producing radi- 
cal alterations in design will effect a complete cure. 

In talking-motion-pieture houses, the seating lines 
and rear walls are frequently segments of circles which 
center at a point near the screen. Owing to the directive 
action of the loud-speakers the rear-wall reflection not 
infrequently produces a pronounced and sometimes trouble- 
some echo in the front of the room which absorbent treat- 
ment of rear-wall surface will only partially prevent. The 
main radius of curvature of the back walls in talking- 
motion-picture houses should be at least twice the distance 
from the curtain line to the rear of the room. This rule 
may well be observed in any room intended for public 
speaking, to save the speaker from an annoying "back 
slap" when speaking loudly. The semicircular plan of 
many lecture halls with the speaker's platform placed 
at the center is particularly unfortunate in this regard, 
unless, as is the case, for example, in the clinical amphi- 
theater, the seating tiers rise rather sharply from the 
amphitheater floor. In this case, the absorbent surface 
of the audience replaces the hard reflecting surface of the 
rear wall, so that the speaker is spared the concentrated 
return of the sound of his own voice. In council or legisla- 
tive chambers, where the semicircular seating plan is 
desirable for purposes of debate, the plan of the room itself 
may be semioctagonal rather than semicircular, with panels 
of absorbent material set in the rear walls to minimize 
reflection. Rear-wall reflection is almost always acousti- 
cally a liability rather than an asset. 

Ellipsoidal Shapes : Mormon Tabernacle. 

The great Mormon Tabernacle in Salt Lake City has a 
world-wide reputation for good acoustics, based very largely 
on the striking whispering-gallery effect, which is daily 
demonstrated to hundreds of visitors. This phenomenon 
is treated by Professor Sabine in his chapter on whispering 
galleries in "Collected Papers," together with an interesting 
series of sound photographs illustrating the focusing effect 



180 ACOUSTICS AND ARCHITECTURE 

of an ellipsoidal surface. As a result of the geometry of the 
figure, if a sound originates at one focus of an ellipsoid, 
it will after reflection from the surface all be concentrated 
at the other focus. In the Tabernacle, these two focuses are 
respectively near the speaker's desk and at a point near the 
front of the rear balcony. A pin dropped in a stiff hat at 
the former is heard with considerable clarity at the latter 
a distance of about 175 ft. (Parenthetically it may be said 




An old illustration of the concentrated reflection from the inner surface of an 
ellipsoid. The two figures are at the foci. (Taken from Neue Hall-und Thon- 
Kunst, by Athanasius Kircher, published in 1684.) 

that under very quiet conditions, this experiment can be 
duplicated in almost any large hall.) 

Based on this well-known fact and the fact that the Salt 
Lake tabernacle is roughly elliptical in plan and semi- 
elliptical in both longitudinal and transverse section, 
superior acoustical virtues are sometimes ascribed to the 
ellipsoidal shape. That the phenomenon just described 
is due to the shape may well be admitted. It does not 
follow, however, that this shape will always produce 
desirable acoustical conditions or that even in this case the 
admittedly good acoustic properties are due solely to the 
shape. In 1925, through the courtesy of the tabernacle 
authorities, the writer made measurements in the empty 
room, which gave a reverberation time of 7.3 sec. for the 
tone 512 vibs./sec. Mr. Wayne B. Hales, in 1922, made a 



ACOUSTICS IN AUDITORIUM DESIGN 181 

study of the room. From his unpublished paper the 
following data are taken : 

Length 232 ft. 

Width. 132 ft. 

Height 63.5ft. 

Computed volume 1,242,400 cu. ft. 

Estimated seating capacity 8,000 

From these data one may compute the reverberation 
time as 1.5 sec. with an audience of 8,000 and as 1.8 sec. 
with an audience of 6,000. 

One notes here a low reverberation period as a con- 
tributing factor to the good acoustic properties of this 
room. 

In 1930, Mr. Hales published a fuller account of his 
study. 1 Articulation tests showed a percentage articula- 
tion of about that which would be expected on the basis 
of Knudsen's curves for articulation as a function of 
reverberation. That is to say, there is no apparent 
improvement in articulation that can be ascribed to the 
particular shape of this room. Finally he noted echoes 
in a region where an echo from the central curved portion 
of the ceiling might be expected. 

Taken altogether, the evidence points to the conclusion 
that apart from the whispering-gallery effect there are no 
outstanding acoustical features in the Mormon Tabernacle 
that are to be ascribed to its peculiarity of shape. A 
short period of reverberation 2 and ceiling curvatures which 
for the most part do not produce focused echoes are suffi- 
cient to account for the desirable properties which it 
possesses. The elliptical plan is usable, subject to the 
same limitation as to actual curvatures as are other curved 
forms. 3 

l Jour. Acous. Soc. Amer., vol. 1, pp. 280-292, 1930. 

2 This is due to the small ratio of volume to seating capacity about 155 
cu. ft. per seat. A balcony extending around the entire room and the 
relatively low ceiling height for the horizontal dimension yields a low value 
of the volume-absorbing power ratio. 

8 The case of one of the best known and most beautiful of theaters in New 
York City may be cited as an example of undesirable acoustical results from 
elliptical contours. 



182 



ACOUSTICS AND ARCHITECTURE 




FIQ. 71. Rays originating at 
the focus of a paraboloid mirror 
are reflected in a parallel beam. 



Paraboloidal Shapes : Hill Memorial. 

Figure 71 shows the important property of a paraboloidal 
mirror of reflecting in a parallel beam all rays that originate 
at the principal focus of the paraboloid. Hence if a source 

of sound be placed at the focus 
of an extended paraboloid of 
^rr^i rxr-h^ revolution, the reflected wave 

/,/' I X ^V will be a plane wave traveling 
fi j ^^ parallel to the axis. The action 

/I | I \ is quite analogous to the directive 

1 action of a searchlight. This 

property of the paraboloid has 
from time to time commended it to 
designers as an ideal auditorium shape, from the standpoint 
of acoustics. The best known example in America of a 
room of this type is the Hill Memorial Auditorium of the 
University of Michigan, at Ann Arbor. The main-floor 
plan and section are shown in Fig. 72. A detailed descrip- 
tion of the acoustical design is given by Mr. Hugh Tallant, 
the acoustical consultant, in The Brickbuilder of August, 
1913. The forward surfaces of the room are paraboloids 
of revolution, with a common focus near the speaker's 
position on the platform. The axes of these paraboloids 
are inclined slightly below the horizontal, at angles such 
as to give reflections to desired parts of the auditorium. 
The acoustic diagram is shown in Fig. 73. Care was 
taken in the design that the once-reflected sound should 
not arrive at any point in the room at an interval of more 
than )f 5 sec. after the direct sound. This was taken as 
the limit within which the reflected sound would serve to 
reinforce the direct sound rather than produce a perceptible 
echo. Tallant states that the final drawings were made 
with sufficient accuracy to permit of scaling the dimen- 
sions to within less than an inch. In 1921, the author made 
a detailed study of this room. The source of sound was set 
up at the speaker's position on the stage, and measurements 
of the sound amplitude were made at a large number of 



ACOUSTICS IN AUDITORIUM DESIGN 



183 




Longitudinal Section A-A 



Ticket 
Office 




-" ir 



Room 



Main Floor Plan 



FIG. 72. Paraboloidal plan and section of the Hill Memorial Auditorium, Ann 
Arbor, Michigan. 



184 



ACOUSTICS AND ARCHITECTURE 



points throughout the room. The intensity proved to be 
very uniform, with a maximum value at the front of the 
first balcony and a value throughout the second balcony 
a trifle greater than on the main floor, a condition which 
the acoustical design would lead one to expect. This 
equality of distribution of intensity was markedly less 
when the source was moved away from the focal point. In 
the empty room, pronounced echoes were observed on the 
stage from a source on the stage, but these were not app&r- 
ent in the main body of the room. The measured rever- 

Gradem-KT 




lmf>05t Grade 100-0- 

FIG. 73. Ray reflections from parabolic surface, Hill Memorial Auditorium. 

beration time in the empty room was 6.1 instead of 4.0 
sec. computed by Mr. Tallant. The difference is doubtless 
due to the value assumed for the absorbing power of the 
empty seats. The reverberation for the full audience, 
figured from the measured empty-room absorbing power, 
agreed precisely with Tallant's estimate from the plans. 

The purpose of the design was skillfully carried out, 
and the results as far as speaking is concerned fully meet 
the designer's purpose, namely, to provide an audience room 
seating 5,000 persons, in which a speaker of moderate voice 
placed at a definite point upon the stage can be distinctly 
heard throughout the room. 

For orchestral and choral use, however, the stage is 
somewhat open to the criticisms made on the first example 
given in this chapter. Only when the source is at the 
principal focus is the sound reflected in a parallel beam. 
If the source be located closer to the rear wall than this, 



ACOUSTICS IN AUDITORIUM DESIGN 



185 



the reflected wave front will be convex, while if it is outside 
the focus, the reflected wave front will be concave. This 





FIQ. 74. Plan and section of Auditorium Theater, Chicago. 

fact renders it difficult to secure a uniform balance of 
orchestral instruments as heard by the conductor. This 
is particularly true for chorus accompanied by orchestra, 



186 



ACOUSTICS AND ARCHITECTURE 



with the latter and the soloists placed on an extension of 
the regular stage. 

This serves to indicate the weakness of precise geometri- 
cal planning for desired reflections from curved forms. 
Such planning presupposes a definite fixed position of the 
source. Departure of the source from this point may 
, materially alter the effects 

\ /',\--:_ -' '7^i r produced. Properly dis- 
posed plain surfaces may 
be made to give the same 
general effect in directing 
the reflected sound, with- 
out danger of undesired 
results when the position 
of the source is changed. 

A second objection lies 
in the fact that just as 
sound from a given point 
on the stage is distributed 
uniformly by reflection to 

Fio. 75.- Plan and section of Carnegie n , ,- 

Hull, New York. (Courtesy of Johns- a11 P arts ot tne room, SO 

Manviiic Corporation.) sound or noise originating 

at any part of the room tends to be focused at this point 
on the stage. This fact has already been noted in the first 
example given. 

Finally it has to be said that acoustically planned curved 
shapes are apt to betray their acoustical motivation. 
The skill of the designer will perhaps be less seriously 
taxed in evolving acoustical ideas than in the rendering of 
these ideas in acceptable architectural forms. 

By way of comparison and illustrating this point Fig. 74 
shows the plan and section of the Chicago Auditorium. 
By common consent this is recognized as one of the very 
excellent music rooms of the country. We note, in the 
longitudinal section, the same general effect of the ceiling 
rising from the proscenium; while in plan, the plain splays 
from the stage to the side walls serve the purpose of reflect- 
ing sound toward the rear of the room instead of diagonally 




ACOUSTICS IN AUDITORIUM DESIGN 187 

across it. Figure 75 gives plan and section of Carnegie 
Hall in New York. Here is little, if any, trace of acoustical 
purpose, and yet Carnegie Hall is recognized as acoustically 
very good. 

All of which serves to emphasize the point originally 
made, that the acoustical side of the designer's problem 
consists more in avoiding sources of difficulty than in 
producing positive virtues. 

Reverberation and Design. 

Since reverberation can be controlled by absorbent 
treatment more or less independently of design, it is all too 
frequently the practice to ignore the extent to which it 
may be controlled by proper design. It is true that in 
most cases any design may be developed without regard to 
the reverberation, leaving this to be taken care of by 
absorbent treatment. While this is a possible procedure, 
it seems not to be the most rational one. Since reverbera- 
tion is determined by the ratio of volume to absorbing 
power, it obviously is possible to keep the reverberation in 
a proposed room down to desired limits quite as effectively 
by reducing volume as by increasing absorbing power. 
As has already appeared, the greater portion of the absorb- 
ing power of an audience room in which special absorbent 
treatment is not employed is represented by the audience. 
It is therefore apparent that, without any considerable 
area of special absorbents, the ratio of volume of the room 
to the number of persons in the audience will largely^ 
determine the reverberation time. 

Table XV gives the ratio of volumes to seating capacities 
for the halls listed in Table XIV. In none of these are 
there any considerable areas of special absorbents, other 
than the carpets and the draperies that constitute the 
normal interior decorations of such rooms. We note that, 
with few exceptions, a range of 150 to 250 cu. ft. per person 
will cover this ratio for these rooms. A similar table 
prepared for rooms of the theater type with one or two 
balconies gave values ranging from 150 to 200 cu. ft. per 



188 



ACOUSTICS AND ARCHITECTURE 



person. One may then give the following as a working rule 
for the relation of volume to seating capacity : 



Volume 


Use 


Volume 
per seat 


100, 000 to 500,000 


( Speech 
I Music 


150 to 175 
150 to 200 


500, 000 to 1,000,000 


J Speech 
) Music 


175 to 200 
200 to 250 



TABLE XV. RATIO OF VOLUME TO SEATING CAPACITY IN ACOUSTICALLY 

GOOD ROOMS 



Auditorium 


Time 
(full audience) 


Volume 
per soat 


Moscow Conservatory. . 


1 30 


163 


Musikvereinssaal 


1 62 


161 


Leipzig Gewandhaus. ... ... 


1 90 


232 


Academy of Music, Philadelphia. . 
Orchestra Hall, Detroit 
Academy of Music, Brooklyn 
Great Theater, Moscow 
Boston Opera House 
Great Hall, Moscow Conservatory. 
Symphony Hall, Boston 


1.76 
1.44 
1.60 
1.55 
1.51 
2 00 
1 93 


143 
182 
195 
210 
212 
279 
250 


Carnegie Hall 


1 75 


274 


Hill Memorial .... 
Eastman Theater, Rochester 
Civic Opera House, Chicago . 
Chicago Auditorium 


1.70 
2.08 
1.95 
1.90 


159 
236 
234 
254 



If the requirements of design allow ratios of volume to 
seating capacity as low as the above, then for the full- 
audience condition additional absorptive treatment will 
not in general be needed. On the other hand, there are 
many types of rooms such as high-school auditoriums, 
court rooms, churches, legislative halls, and council 
chambers in which the capacity audience is the exception 
rather than the rule. In all such rooms the total absorbing 
power should be adjusted to give tolerable reverberation 



ACOUSTICS IN AUDITORIUM DESIGN 189 

times for the average audiences, rather than the most 
desirable times for capacity audiences. In such rooms, a 
compromise must be effected so as to provide tolerable 
hearing under all audience conditions. 

Adjustment for Varying Audience. 

For purposes of illustration, we shall take a typical case 
of a modern high-school auditorium. A room of this sort 
is ordinarily intended for the regular daily assembly of the 
school, with probably one-half to two-thirds of the seats 
occupied. In addition, occasional public gatherings, with 
addresses, or concerts or student theatrical performances 
will occupy the room, with audiences from two-thirds to 
full seating capacity. In general, the construction will 
be fireproof throughout, with concrete floors, hard plaster 
walls and ceiling, wood seats without upholstery, and a 
minimum of absorptive material used in the normal 
interior finish of the room. 

The data for the example chosen, taken from the pre- 
liminary plans, are as follows : 

Dimension, 127 by 60 by 48 ft. or 366,000 cu. ft. 

Floors, cement throughout, linoleum in aisles 

Walls, hard plaster on clay tile 

Ceiling, low paneled relief, hard plaster on suspended metal lath 

Stage opening, velour curtains 36 by 20 ft. 

1,600 wood seats, coefficient 0.3 

ABSORBING POWER 

__. Units 

Empty room including seals, 0.3 \/V* . 1,480 

Stage, 36 by 20 X 0.44 316 



Absorbing power of empty room 1 , 796 

Curve 1 (Fig. 76) shows the reverberation times plotted 
against the number of persons in the audience, assuming 
the room to be built as indicated in the preliminary design. 
We note that the ratio of volume to seating capacity is 
somewhat large 230 cu. ft. per person. This, coupled 
with the fact that there is a dearth of absorbent materials 
in the normal furnishings, gives a reverberation time that 



190 



ACOUSTICS AND ARCHITECTURE 



is great even with the capacity audience present. In this 
particular example, it was possible to lower the ceiling 

height by about 8 ft., giving a 
volume of 305,000 cu. ft. Curve 2 
shows the effect of this alteration 
in design. This lowers the rever- 
beration time to 1.75 sec. with the 
maximum audience, a value some- 
what greater than the upper limit 
for a room of this volume shown in 
Fig. 63. In view of the probably 
frequent small audience use of the 
room, the desirability of artificial 
absorbents is apparent. The ques- 
tion as to how much additional 
absorption ought to be specified 
should be answered with the par- 
ticular uses in mind. At the daily school assembly, 900 to 
1,000 pupils were expected to be present. A reverberation 
time of 2 sec. for a half audience would thus render 
comfortable hearing for assembly purposes. The total 
absorbing power necessary to give this time is 
0.05X305,000 




1,600 



FIG. 76. Effect of volume, 
character of seats, size of audi- 
ence, and added absorption on 
reverberation time. 



Without absorbent treatment the total absorbing power 
with 800 persons present is as follows : 



Empty room, *\/V* 
Stage, 36 by 20 X 0.44. 
800 persons X 4.3 . . . . 



Units 

1,330 

316 

3,440 



Total 5,086 

Additional absorption necessary to give reverberation 
of 2.0 sec. with half audience 7,630 - 5,086 = 2,544 units. 
Curve 3 gives the reverberation times with this amount of 
added absorbing power. It is to be noted that with this 
amount of acoustical treatment, the reverberation time 
with the capacity audience is 1.4 sec., which is the lower 



ACOUSTICS IN AUDITORIUM DESIGN 



191 



limit for a room of this volume given by Fig. 63, while 
with a half audience it is not greater than 2.0 sec. a 
compromise that should meet all reasonable demands. 

An even more satisfactory adjustment can be effected, 
if upholstered seats be specified. In curve 4, we have the 
reverberation times under the same conditions as curve 3, 
except that the seats are upholstered in imitation leather 
and have an absorbing power of 1.6 per seat instead of the 
0.3 unit for the wood seats. The effect of this substitution 
is shown in the following comparison: 



Items of absorption 


Absorbing power 


Wood seats 


Upholstered 
seats 


Empty room . 


1,330 
316 
2,544 


1,330 
316 
2,544 
2,080 


Stago 


Added absorption . 


Upholstered scats (1,600 X 1.3) 
Total absorbing powor (without audience) . . 


4,190 


6,270 



Seated in the wood seats, the audience adds 4.3 units per 
person and in the upholstered seat 3.0 units per person, so 
that we have: 



Absorbing power 



Audience 


Wood seats 


Upholstered 
seats 


None 


4,190 


6,270 


300 


5,480 


7,170 


600 


6,770 


8,070 


900 


8,060 


8,970 


1,200 


9,350 


9,870 


1,600 


11,070 


11,070 



The effect of the upholstered seats in reducing the 
reverberation for small audience use is apparent. With 
the upholstered seats and the added absorbing power the 



192 ACOUSTICS AND ARCHITECTURE 

reverberation is not excessive with any audience greater 
than 400 persons. 

Choice of Absorbent Treatment. 

The area of absorbent surface which is required to give 
the additional absorbing power desired will be the number 
of units divided by the absorption coefficient of the material 
used. Thus with a material whose absorption coefficient 
is 0.35, the area needed in the preceding example would be 
2,466 0.35 = 7,000 sq. ft., while with a material twice 
as absorbent, the required area would be only half as great. 
*As a practical matter, it is ordinarily more convenient to 
apply absorbent treatment to the ceiling. In the example 
chosen, the area available for acoustical treatment was 
approximately 7,000 sq. ft. in the soffits of the ceiling 
panels. In this case, the application of one of the more 
highly absorbent of the acoustical plasters with an absorp- 
tion coefficient between 0.30 and 0.40 would have been a 
natural means of securing the desired reverberation time. 
Had the available area been less, then a more highly 
absorbent material applied over a smaller area would be 
indicated. In designing an interior in which acoustical 
treatment is required, knowledge, in advance, of the amount 
of treatment that will be needed and provision for working 
this naturally into the decorative scheme is an essential 
feature of the designer's problem. The choice of materials 
for sound absorption should be dictated quite as much 
by their adaptability to the particular problem in hand as 
by their sound-absorbing efficiencies. 

Location of Absorbent Treatment. 

As has been noted earlier, sound that has been reflected 
once or twice will serve the useful purpose of reinforcing 
the direct sound, provided the path difference between 
direct and reflected sound is not greater than about 70 ft. 
producing a time lag not greater than about J^ 6 sec. For 
this reason, in rooms so large that such reinforcement is 
desirable, absorbent treatment should not be applied on 



ACOUSTICS IN AUDITORIUM DESIGN 193 

surfaces that would otherwise give useful reflections. In 
general, this applies to the forward portions of side walls 
and ceilings, as well as to the stage itself. Frequently 
one finds stages hung with heavy draperies of monk's 
cloth or other fabric. Such an arrangement is particularly 
bad because of the loss of all reflections from the stage 
boundaries thus reducing the volume of sound delivered 
to the auditorium. Further, the recessed portion of the 
stage acts somewhat as a separate room, and if this space is 
"dead," the speaker or performer has the sensation of 
speaking or playing in a padded cell, whereas the reverber- 
ation in the auditorium proper may be considerable. 

Professor F. R. Watson 1 gives the results of some 
interesting experiments on the placing of absorbent 
materials in auditoriums. As a result of these experiments 
he advocated the practice of deadening the rear portion of 
audience rooms by the use of highly absorbent materials, 
leaving the forward portions highly reflecting. Carried to 
the extreme, in very large rooms this procedure is apt to 
lead to two rates of decay of the residual sound, the more 
rapid occurring in the highly damped rear portions. In 
one or two instances within the author's knowledge, this 
has led to unsatisfactory hearing in the front of the room, 
while seats in the rear prove quite satisfactory. As a 
general rule, the wider distribution of a moderately absorb- 
ent material leads to better results than the localized 
application of a smaller area of a highly absorbent material. 

In general, the application of sound-absorbent treatment 
to ceilings under balconies is not good practice. Properly 
designed, such ceilings may give useful reflection to the 
extreme rear seats. Moreover, if the under-balcony space 
is deep, absorbents placed in this space are relatively 
ineffective in lowering the general reverberation in the 
room, since the opening under the balcony acts as a nearly 
perfectly absorbing surface anyway. Rear-wall treatment 
under balconies may sometimes be needed to minimize 
reflection back to the stage. 

1 Jour. Amer. Inst. Arch., July, 1928. 



194 ACOUSTICS AND ARCHITECTURE 

Wood as an Acoustical Material. 

There is a long-standing tradition that rooms with a 
large amount of wood paneling in the interior finish have 
superior acoustical merits. Very recently Bagenal and 
Wood published in England a comprehensive treatise on 
architectural acoustics. 1 These authors strongly advocate 
the use of wood in auditoriums, particularly those intended 
for music, on the ground that the resonant quality of wood 
"improves the tone quality," "brightens the tone." In 
support of this position, they cite the fact that wood is 
employed in relatively large areas in many of the better 
known concert halls of Europe. The most noted example 
is that of the Leipzig Gewandhaus, in which there is about 
5,300 sq. ft. of wood paneling. The acoustic properties 
of this room have assumed the character of a tradition. 
Faith in the virtues of the wood paneling is such that its 
surface is kept carefully cleaned and polished. 

The origin of this belief in the virtue of wood is easily 
accounted for. It is true that wood has been extensively 
used as an interior finish in the older concert halls. It is 
also true that the reverberation times in these rooms are 
not excessive. For the Leipzig Gewandhaus, Bagenal and 
Wood give 2.0 sec. Knudsen estimates it as low as 1.5 
sec. with a full audience of 1,560 persons. It is not impos- 
sible that the acoustical excellence which has been ascribed 
to the use of wood may be due to the usually concomitant 
fact of a proper reverberation time. Figure 55 shows that 
the absorption coefficient of wood paneling is high, roughly 
0.10 as compared with 0.03 for plaster on tile. . Small 
rooms in which a large proportion of the surface is wood 
naturally have a much lower reverberation time than 
similar rooms done in masonry throughout, particularly 
when empty. Noting this fact, a musician with the 
piano sounding board and the violin in mind would 
naturally arrive at the conclusion that the wood finish 
as such is responsible for the difference. 

1 "Planning for Good Acoustics," Methucn & Co., London, 1931. 



ACOUSTICS IN AUDITORIUM DESIGN 195 

That the presence of wood is not an important factor 
in reducing reverberation is evidenced by the fact that the 
5,300 sq. ft. of paneling accounts for only about 5.5 per 
cent of the total absorbing power of the Leipzig Gewand- 
haus when the audience is present. Substituting plastered 
surfaces for the paneled area would not make a perceptible 
difference in the reverberation time, nor could it change 
the quality of tone in any perceptible degree, in a room of 
this size. It is possible that in relatively small rooms, in 
which the major portion of the bounding surfaces are of a 
resonant material, a real enhancement of tone might result. 
In larger rooms and with only limited areas, the author 
inclines strongly to the belief that the effect is largely 
psychological. 

, For the stage floor, and perhaps in a somewhat lesser 
degree for stage walls in a concert hall, a light wood con- 
struction with an air space below would serve to amplify 
the fundamental tones of cellos and double basses. These 
instruments are in direct contact with the floor, which 
would act in a manner quite similar to that of a piano 
sounding board. This amplification of the deepest tones 
of an orchestra produces a real and desired effect. One 
may note the effect by observing the increased volume of 
tone when a vibrating tuning fork is set on a wood table 
top. 1 

Orchestra Pit in Opera Houses. 

Figure 77 shows the section of the orchestra pit of the 
Wagner Theater in Bayreuth. This is presented to call 
attention to the desirability of assigning a less prominent 

1 Of interest on this point is some recent work by Eyring (Jour. Soc. Mot. 
Pict. Eng., vol. 15, No. 4, p. 532). At points near a wall made of fKe-in. 
ply-wood panels in a small room, two rates of decay of reverberant sound 
were observed. The earlier rate, corresponding to the general decay in the 
room, was followed by a slower rate, apparently of energy, reradiated from 
the panels. This effect was observed only at two frequencies. Investiga- 
tion proved that the panels were resonant for sound of these frequencies and 
that the effect disappeared when the panels were properly nailed to supports 
at the back. 



196 



ACOUSTICS AND ARCHITECTURE 



place to the orchestra in operatic theaters. The wide 
orchestra space in front of the stage as it exists in many 
opera houses places the singer at a serious disadvantage 
both in the matter of distance from the audience and in the 
fact that he has literally to sing over the orchestra. Wag- 
ner's solution of the problem was to place most of the 
orchestra under the stage, the sound emerging through a 
restricted opening. Properly designed, with resonant floor 
and highly reflecting walls and ceiling, ample sound can 
be projected into the room from an orchestra pit of this 




FIG. 77. Section of orchestra pit of Wagner Theater, Bayreuth. 

type. With the usual orchestra pit, the preponderance 
of orchestra over singers for auditors in the front seats 
is decidedly objectionable. 

Acoustical Design in Churches. 

No type of auditorium calls for more careful treatment 
from the acoustical point of view than does that of the 
modern evangelical church in America. Puritanism gave 
to ecclesiastical architecture the New England meeting 
house, rectangular in plan, often with shallow galleries on 
the sides and at the rear. With plain walls and ceilings 
and usually with the height no greater than necessary to 
give sufficient head room above the galleries, the old New 
England meeting house presented rio acoustical problems. 
The modern version is frequently quite different. Sim- 
plicity of design and treatment is often coupled with 
horizontal dimensions much greater than are called for by 
the usual audience requirements and with heights which are 
correspondingly great. Elimination of the galleries and 



ACOUSTICS IN AUDITORIUM DESIGN 197 

the substitution of harder, more highly reflecting materials 
in floors, walls, and ceilings often render the modern 
church fashioned on New England colonial lines highly 
unsatisfactory because of excessive reverberation. Fre- 
quently the plain ceiling is replaced with a cylindrical 
vault with an axis of curvature that falls very close to the 
head level of the audience, with resultant focusing and 
unequal distribution of intensity due to interference. 1 
These difficulties have only to be recognized in order to 
be avoided. Excessive reverberation may be obviated 
by use of absorbent materials. A flattened ceiling with 
side coves will not produce the undesired effects of the 
cylindrical vault. Both of these can be easily taken care 
of in the original design. They are extremely difficult to 
incorporate in a room that has once been completed in a 
type of architecture whose excellence consists in the 
simplicity of its lines and the perfect fitness of its details. 

The Romanesque revival of the seventies and eighties 
brought a type of church auditorium which is acoustically 
excellent, but which has little to commend it as ecclesiastical 
architecture. Nearly square in plan, with the pulpit and 
choir placed in one corner (Tallmadge has called this 
period the " cat-a-corner age") and with the pews circling 
about this as a center, the design is excellent for producing 
useful reflections of sound to the audience. Add to this 
the fact that encircling balconies are frequently employed, 
giving a low value to the volume per person and hence of 
the reverberation time, and we have a type of room which 
is excellent for the clear understanding of speech. This 
type of auditorium is admirably adapted to a religious 
service in which liturgy is almost wholly lacking and of 
which the sermon is the most important feature. 

The last twenty-five years, however, have seen a marked 
tendency toward ritualistic forms of worship throughout 
the Protestant churches in America. Concurrently with 
this there has been a growing trend, inspired largely by 
Bertram Grosvenor Goodhue and Ralph Adams Cram, 
toward the revival of Gothic architecture even for non- 



198 ACOUSTICS AND ARCHITECTURE 

liturgical churches. Now, the Gothic interior, with its 
great height and volume, its surfaces of stone, and its 
relatively small number of seats, is of necessity highly 
reverberant. This is not an undesirable property for a 
form of service in which the clear understanding of speech 
is of secondary importance. The reverberation of a great 
cathedral adds to that sense of awe and mystery which is 
so essential an element in liturgical worship. 

The adaptation of the Gothic interior to a religious 
service that combines the traditional forms of the Roman 
church with the evangelical emphasis upon the words of 
the preacher presents a real problem, which has not as yet 
had an adequate solution. Apart from reverberation, the 
cruciform plan is acoustically bad for speech. The usual 
locations of the pulpit and lectern at the sides of the chancel 
afford no reinforcement of the speaker's voice by reflection 
from surfaces back of him. The break caused by the 
transepts allows no useful reflections from the side walls. 
Both chancel and transepts produce delayed reflections, 
that tend to lower the intelligibility of speech. The great 
length of the nave may occasion a pronounced echo from 
the rear wall, which combined with the delayed reflections 
from the chancel and transepts makes hearing particularly 
bad in the space just back of the crossing. The sound 
photograph of Fig. 78 show the cause of a part of the 
difficulty in hearing in this region. Finally the attempt 
by absorption to adjust the reverberation to meet the 
demands of both the preacher and the choirmaster usually 
results in a compromise that is not wholly satisfactory to 
either. 

The chapel of the University of Chicago, designed by 
Bertram Goodhue, is an outstanding example of a modern 
Gothic church intended primarily for speaking. The plan 
is shown in Fig. 79. The volume is approximately 900,000 
cu. ft., and 2,200 seats are provided. The ratio of volume 
to seating capacity is large. In order to reduce reverber- 
ation, all wall areas of the nave and transept are plastered 
with a sound-absorbing plaster with a coefficient of 0.20. 



ACOUSTICS IN AUDITORIUM DESIGN 199 

The groined ceiling is done in colored acoustical tile with a 
coefficient of 0.25. The computed reverberation time for 
the empty room is 3.6 sec., which is reduced to 2.4 sec. 
when all the seats are occupied. The reverberation 
measured in the empty room is 3.7 sec. The rear-wall echo 
is largely eliminated by the presence of a balcony and, 
above this, the choir loft in the rear. Speakers who use a 
speaking voice of only moderate power and a sustained 




FIG. 78. Sound pulse in modified cruciform plan. Hearing conditions much 

better in portion represented by lower half of photograph. 
0, Pulpit; 1, direct sound; 2, reflected from side wall A; 3, diffracted from P\ 
4, reflected and focused from curved wall of sanctuary. 

manner of speaking can be clearly heard throughout the 
room. Rapid, strongly emphasized speech becomes diffi- 
cult to understand. Dr. Gilkey, the university minister, 
has apparently mastered the technique of speaking under 
the prevailing conditions, as he is generally understood 
throughout the chapel. In this connection, it may be said 
that the disquisitional style of speaking, with a fairly 
uniform level of voice intensity and measured enunciation, 
is much more easily understood in reverberant rooms than 
is an oratorical style of preaching. In fact it is highly 
probable that the practice of intoning the ritualistic service 



200 



ACOUSTICS AND ARCHITECTURE 



had its origin in part at least in the acoustic demands of 
highly reverberant medieval churches. 



FIG. 79. Plan of the University of Chicago Chapel 




Interior of the University of Chicago Chapel. (Bertram Goodhue Associates, 

Architects.) 

The choirmaster, Mr. Mack Evans, finds the chapel 
very satisfactory both for his student choir and for the 
organ. The choir music is mostly in the medieval forms, 



ACOUSTICS IN AUDITORIUM DESIGN 



201 



sung a capella, with sustained harmonic rather than rapid 
melodic passages. 

All things considered, it may be fairly said that this 
room represents a reasonable compromise between the 
extreme conditions of great reverberation existing in the 
older churches of cathedral type and proportions and that 
of low reverberation required for the best understanding of 
speech. It has to be said that, like all compromises, it 
fails to satisfy the extremists on both sides. The conditions 




Fm. 80. Plan of Riverside Church, New York. 

inherent in the cruciform plan, of the long, narrow seating 
space and consequently great distances between the speaker 
and the auditor, and the lack of useful reflections are 
responsible for a considerable part of what difficulty is 
experienced in the hearing of speech. These difficulties 
would be greatly magnified were the room as reverberant 
as it would have been without the extensive use of sound- 
absorbent materials. 

Riverside Church, New York. 

This building is by far the most notable example of 
the adaptation of the Gothic church to the uses of Protes- 
tant worship. We note from the plan 1 (Fig. 80) the 

1 Acknowledgment is made of the courtesy of the architects Henry C. Pel- 
ton and Allen and Collens, Associates, in supplying the plan and photograph. 



202 



ACOUSTICS AND ARCHITECTURE 



rectangular shape, the absence of the transepts, and a 
comparatively shallow chancel. The height of the nave 
is very great 104 ft. to the center of the arch, giving a 
volume for the main cell of the church of more than 1,000,- 
000 cu. ft. With the seating capacity of 2,500, we have 
over 400 cu. ft. per person. To supply the additional 
absorption that was obviously needed, the groined ceiling 
vault of the nave and chancel, the ceiling above the aisles, 
and all wall surfaces above a 52-ft. level were finished in 
acoustical tile. 




Interior of Riverside Church, New York. (Henry C. Pclton, Allen & Collens 
Associates, Architects.) 

The reverberation time measured in the empty room 
is 3.5 sec., which with the full audience present reduces 
to 2.5 sec. While the room is still noticeably reverberant 
even with a capacity audience, yet hearing conditions 
are good even in the most remote seats. A very successful 
public-address system has been installed. The author is 
indebted to Mr. Clifford M. Swan, who was consulted on 
the various acoustical problems in connection with this 
church, for the following details of the electrical amplifying 
system : 



ACOUSTICS IN AUDITORIUM DESIGN 203 

Speech from the pulpit is picked up by a microphone in the desk, 
amplified, and projected by a loud-speaker installed within the tracery 
of a Gothic spire over the sounding board, directly above the preacher's 
head. Thu voices of the choir are picked up by microphones concealed 
in the tracery of the opposing choir rail and projected from a loud- 
speaker placed at the back of the triforium gallery in the center of the 
apse. The voice of the reader at the lectern is also projected from this 
position. The amplification is operated at as low a level of intensity 
as is consistent with distinct hearing. 

It is worth noting that the success of the amplifying 
system is largely conditioned upon the reduced reverber- 
ation time. Were the reverberation as great as it would 
have been had ordinary masonry surfaces been used 
throughout, amplification would have served little in the 
way of increasing the intelligibility of speech. Loud 
speaking in a too reverberant room does not diminish the 
overlapping of the successive elements of speech. The 
judicious use of amplifying power and the location of the 
loud-speakers near the original sources also contribute to 
the success of this scheme. It would appear that with the 
rapid improvement that has been made in the means for 
electrical amplification and with highly absorbent masonry 
materials available, the acoustical difficulties inherent in 
the Gothic church can be overcome in very large measure. 



CHAPTER X 
MEASUREMENT AND CONTROL OF NOISE IN BUILDINGS 

It has been noted in Chap. II that a musical sound as 
distinguished from a noise is characterized by having 
definite and sustained pitch and quality. This does not 
tell the whole story, however, for a sound of definite pitch 
and quality is called a noise whenever it happens to be 
annoying. Thus the hum of an electrical motor or gener- 
ator has a definite pitch and'quality, but neighbors adjacent 
to a power plant may complain of it as noise. The desira- 
bility or the reverse of any given sound seems to be a 
determining factor in classifying it either as a musical 
tone or as a noise. Hence the standardization committee 
of the Acoustical Society of America proposes to define 
noise as "any unwanted sound/' Obviously then, the 
measurement of a noise should logically include some means 
of evaluating its undesirability. Unfortunately this psy- 
chological aspect of noise is not susceptible of quantitative 
statement. Moreover, neglecting the annoyance factor, 
the mere sensations of loudness produced by two sound 
stimuli depend upon other factors than the physical 
intensity of those stimuli. Therefore to deal with noise 
in a quantitative way, it is necessary to take account of the 
characteristics of the human ear as a sound-receiving 
apparatus, if we want our noise measurements to correspond 
with the testimony of our hearing sense. 

To go into this question of the characteristics of the ear 
in any detail would quite exceed the scope of this book. 
For our quantitative knowledge of the subject we are 
largely indebted to the extensive research at the Bell 
Telephone Laboratories, and the reader is referred to the 
comprehensive account of this work given in Fletcher's 
"Speech and Hearing." For the present purpose certain 
general facts with regard to hearing will suffice. 

204 



MEASUREMENT AND CONTROL OF NOISE 205 

Frequency and Intensity Range of Hearing. 

There is a wide variation among individuals in the range 
of frequencies which produce the sensation of tone. In 
general, however, this range may be said to extend from 
20 to 20,000 vibs./sec. Below this range, the alternations 
of pressure are recognized as separate pulses; while above 
the upper limit, no sensation of hearing is produced. 

The intensity range of response of the ear is enormous. 
Thus for example, at the frequency 1,024 vibs./sec. the 
physical intensity of a sound so loud as to be painful is 
something like 2.5 X 10 13 times the least intensity which 
can be heard at this frequency. One cannot refrain from 
marveling at the wonder of an instrument that will register 
the faintest sound and yet is not wrecked by an intensity 
25 trillion times as great. Illustrating the extreme sensi- 
tivity of the ear to small vibrations, Kranz 1 has calculated 
that the amplitudes of vibration at the threshold of 
audibility at higher frequencies are of the order of one- 
thirtieth of the diameter of a nitrogen molecule and one 
ten-thousandth of the mean free path of the molecules. 

Another calculation shows that the sound pressure at 
minimum audibility is not greater than the weight of a 
hair whose length is one-third of its diameter. 

The intensity range varies with the frequency. It 
is greatest for the middle of the frequency range. The 
data for Table XVI are taken from Fletcher and show the 
pressure range measured in bars between the faintest and 
most intense sounds of the various pitches and also the 
intensity ratio between painfully loud and minimum- 
audible sound at each pitch. 

Decibel Scale. 

The enormous range of intensities covered by ordinary 
auditory experience suggests the desirability of a scale 
whose readings correspond to ratios rather than to differ- 
ences of intensity. This suggests a logarithmic scale, since 

1 Phys. Rev., vol. 21, No. 5, May, 1923. 



206 



ACOUSTICS AND ARCHITECTURE 
TABLE XVI 



Frequency 


Pressure range 
in bars 


Intensity ratio, 
maximum divided 






by minimum 


64 


0.12 to 200 


1 7 X 10 6 


128 


021 to 630 


9 X 10 8 


256 


0039 to 2, 000 


2 6 X 10 11 


512 


0.0010 to 3, 200 


1 X 10 13 


1,024 


0.00052 to 2, 500 


2.5 X 10 13 


2.048 


0.00041 to 1,000 


6 X 10 12 


4,096 


0.00042 to 320 


6 4 X 10 11 



the logarithm of the ratio of two numbers is the difference 
between the logarithms of these numbers. The decibel 
scale is such a scale and is applied to acoustic powers, 
intensities, and energies and to the mechanical or electrical 
sources of such power. The unit on such a logarithmic 
scale is called the bel in honor of Alexander Graham Bell. 
The difference of level expressed in bels between two 
intensities is the logarithm of the ratio of these intensities. 
The intensity level expressed in bels of a given intensity 
is the number of bels above or below the level of unit 
intensity or, since the logarithm of unity is zero, simply 
the logarithm of the intensity. Thus if the microwatt per 
square centimeter is the unit of intensity, then 1,000 
microwatts per square centimeter represents an intensity 
level of 3 bels or 30 db. (logic 1,000 = 3). An intensity of 
0.001 microwatt gives an intensity level of 3 bels or 
30 db. Two intensities have a difference of level of 1 db. 
if the difference in their logarithm, that is, the logarithm 
of their ratio, is 0.1. Now the number whose logarithm 
is 0.1 is 1.26, so that an increase of 26 per cent in the 
intensity corresponds to a rise of 1 db. in the intensity 
level. The table on page 207 shows the relation between 
intensities and decibel levels above unit intensity: 

Obviously, we can express the intensity ratios given in 
Table XVI as differences of intensity levels expressed in 
decibels. Thus, for example, at the tone 512 vibs./sec., 
the intensity ratio between the extremes is 1 X 10 13 . The 



MEASUREMENT AND CONTROL OF NOISE 



207 



/ 


log/ 


Intensity level, 
decibels 


1.00 





' 


1.26 


1 


1 


1.58 


2 


2 


2.00 


3 


3 


2.55 


4 


4 


3 16 


5 


5 


4 00 


6 


6 


5 03 


7 


7 


6 30 


8 


8 


8 00 


9 


9 


10 00 


1 


10 


100 00 


2 


20 


1,000 00 


3.0 


30 


1,000,000 00 


6 


60 



logarithm of 10 ia is 13, and the difference in intensity levels 
between a painfully loud and a barely audible sound of 
this frequency is 13 bels or 130 db. 

Sensation Level. 

The term " sensation level ' ' is used to denote the intensity 
level of any sound above the threshold of audibility of 
that sound. Thus in the example just given, the sensation 
level of the painfully loud sound is 130 db. We note that 
levels expressed in decibels give us not the absolute values 
of the magnitudes so expressed but simply the logarithms 
of their ratios to a standard magnitude. The sensation 
level for a given sound is the intensity level minus the 
intensity level of the threshold. 

Threshold of Intensity Difference. 

The decibel scale has a further advantage in addition 
to its convenience in dealing with the tremendous range of 
intensities in heard sounds. The Weber-Fechner law in 
psychology states that the increase in the intensity of a 
stimulus necessary to produce a barely perceptible increase 
in the resulting sensation is a constant fraction of the 
original intensity. Applied to sound, the law implies 



208 ACOUSTICS AND ARCHITECTURE 

that any sound intensity must be increased by a constant 
fraction of itself before the ear perceives an increase of 
loudness. If A/ be the minimum increment of intensity 
that will produce a perceptible difference in loudness, then 
according to the Weber-Fechner law 

-j- = k (constant) 

Hence, if /' and 7 be any two intensities, one of which is 
just perceptibly louder than the other, then 
log /' log / = constant 

On the basis of the Weber-Fechner law as thus stated, 
we could build a scale of loudness, each degree of which is 
the minimum perceptible difference of loudness. The 
intensities corresponding to successive degrees on this 
scale would bear a constant ratio to each other. They 
would thus form a geometric series, and their logarithms 
an arithmetical series. 

The earlier work by psychologists seemed to establish 
the general validity of the Weber-Fechner law as applied to 
hearing. However, more recent work by Knudsen, 1 and 
subsequently by Riesz at the Bell Laboratories, has shown 
that the Weber-Fechner law as applied to differences of 
intensities is only a rough approximation to the facts, that 
the value of A/// is not constant over the whole range of 
intensities. For sensation levels above 50 db. it is nearly 
constant, but for low intensities at a given pitch it is much 
greater than at higher levels. Moreover, it has been found 
that the ability to detect differences of intensities between 
two sounds of the same pitch depends upon time between 
the presentations of the two sounds. For these reasons, 
a scale based on the minimum perceptible difference of 
intensity would not be a uniform scale. However, over 
the entire range of intensities the total number of threshold- 
of -difference steps is approximately the same as the number 
of decibels in this range, so that on the average 1 db. is the 
minimum difference of sensation level which the ear can 

1 Phys. Rev., vol. 21, No. 1, January. 1923. 



MEASUREMENT AND CONTROL OF NOISE 209 

detect. For this reason, the decibel scale conforms in a 
measure to auditory experience. However, there has not 
yet been established any quantitative relation between 
sensation levels in decibels and magnitudes of sensation." 
That is to say, a sound at a sensation level of 50 db. is not 
judged twice as loud as one at 25 db. 1 

Reverberation Equation in Decibels. 

It will be recalled that the reverberation time of a room 
has been defined as the time required for the average energy 
density to decrease to one-millionth of its initial value and 
also that in the decay process the logarithm of the initial 
intensity minus the logarithm of the intensity at the time 
T is a linear function of the time. The logarithm of 
1,000,000 is 6, so that the reverberation time is simply the 
time required for the residual intensity level to decrease 
by 60 db. Denote the rate of decay in decibels per second 
by 5; then 

_ 60 60a a 

8 = ri = .057 = 1 > 2 V 

The rate of decay in decibels per second is related to A, 
the absolute rate of decay, by the relation 

d = 4.35A 

Recent writers on the subject sometimes express the 
reverberation characteristic in terms of the rate of decrease 
in the sensation level. The above relations will be useful 
in connecting this with the older mode of statement. 

Intensity and Sensation Levels Expressed in Sound 
Pressures. 

In Chap. II (page 22), we noted that the flux intensity 
of sound is given by the relation J = p 2 /r, where p is the 

1 Very recently work has been done on the quantitative evaluation of the 
loudness sensation. The findings of different investigators are far from 
congruent, however. Thus Ham and Parkinson report from experiments 
with a large number of listeners that a sound is judged half as loud when the 
intensity level is lowered about 9 db. Laird, experimenting with a smaller 
group, reports that an intensity level of 80 db. appears to be reduced one- 
half in loudness when reduced by 23 db. It is still questionable as to just 
what we mean, if anything, when we say that one sound is half as loud as 
another. 



210 ACOUSTICS AND ARCHITECTURE 

root mean square of the pressure, and r is the acoustic 
resistance of the medium. Independently of pitch, the 
intensity of sound is thus proportional to the square of 
the pressure, so that intensity measurements are best made 
by measuring sound pressures. The intensity ratio of two 
sounds will be the square of this pressure ratio, and the 
logarithm of the intensity ratio will be twice the logarithm 
of the pressure ratio. The difference in intensity level in 
decibels between two sounds whose pressures are pi and fa 

p\ 
is therefore 20 Iogi * 

f>2 

Loudness of Pure Tones. 

"Loudness" refers to the magnitude of the psychological 
sensation produced by a sound stimulus. The loudness of a 
sound of a given pitch increases with the intensity of the 
stimulus. We might express loudness of sound of a given 
pitch in terms of the number of threshold steps above 
minimum audibility. As has already been pointed out, 
such a loudness scale would correspond only roughly to 
the decibel scale. Moreover, it has been found that two 
sounds of different pitches, at the same sensation levels, 
that is, the same number of decibels above their respective 
thresholds, are not, in general, judged equally loud, so 
that such a scale will not serve as a means of rating the 
loudness either of musical tones of different pitches or 
of noise in which definite pitch characteristics are lacking. 
It is apparent that in order to speak of loudness in quantita- 
tive terms, it is necessary to adopt some arbitrary conven- 
tional scale, such that two sounds of different characters 
but judged equally loud would be expressed by the same 
number and that the relative ratings of the loudness of a 
series of sounds as judged by the ear would correspond at 
least qualitatively to the scale readings expressing their 
loudness. 

In "Speech and Hearing," Fletcher has proposed to use 
as a measure of the loudness of a given musical tone of 
definite pitch the sensation level of a 1,000-cycle tone which 



MEASUREMENT AND CONTROL OF NOISE 



211 



normal ears judge to be of the same loudness as the given 
tone. This standard of measurement seems to be by way 
of being generally adopted by various engineering and 
scientific societies which are interested in the problem. 
The use of such a scale calls for the experimental loudness 
matching of a large number of frequencies with the standard 
frequency at different levels. To allow for individual 
variation such matching has to be done by a large number 
of observers. Kingsbury of the Bell Telephone Labora- 



60 



40 



-80 



dOO 



\ 



Contour Lines of Equal Loudness for Pyre Tones 



p ooog Q 2 Q 

(\j <3*CPoo<-> JP S Q 

~ Fr? qU5 nV = W * **o- 

FIG. 81. Each ourved line shows the intensity levels at different frequencies 
which sound as loud as a 1,000-cycle tone of the indicated intensity level above 
threshold. (After Kingsbvry.) 

tories has made an investigation of this sort using 22 
different observers. The results of this work are given in 
Fig. 81. Here the zero intensity level is one microwatt per 
square centimeter. The lines of the figure are called con- 
tour lines of equal loudness. The ordinates of any contour 
line give the intensity levels at the different frequencies 
which sound equally loud. The loudness level assigned 
to each contour is the sensation level of the 1,000-cycle tone 
at the intensity shown on the graph. Illustrating the use 



212 



ACOUSTICS AND ARCHITECTURE 



of the diagram, take, for example, the contour marked 60. 
We have for equal loudness the following: 



Frequency 


Intensity level 
measured from 1 
microwatt/sq. cm. 


Intensity, 
micro watts/sq. cm. 


125 


-24 


004 


250 


-28 


0016 


500 


-30 


0.001 


1,000 


-32 


00063 


2,000 


-35 


00032 


4,000 


-34 


00040 



According to Kingsbury's work these frequencies at 
the respective intensity levels shown sound as loud as a 
1,000-cycle tone 60 db. above threshold. 

To the layman this doubtless sounds a bit complicated, 
but it is typical of the sort of thing to which the physicist 
is driven whenever he attempts to assign numerical meas- 
ures to psychological impressions. 

Loudness of Noises. 

Numerical expression of the loudness of noises becomes 
even more complicated by virtue of the fact that we have 
to deal with a medley of sounds of varying and indiscrimi- 
nate pitches. As we have seen, the sensation of loudness 
depends both upon the pitch and upon the intensity of 
sound. Moreover, the annoyance factor of noise may 
depend upon still other elements which will escape evalua- 
tion. The sound pressures produced by noises may be 
measured, but their relative loudness as judged by the ear 
will not necessarily follow the same order as these measured 
pressures. At the present time, the question of evaluating 
noise levels is the subject of considerable discussion 
among physicists and engineers. Standardization com- 
mittees have been appointed by various technical societies, 
notably the American Institute of Electrical Engineers, 
the American Society of Mechanical Engineers, and the 
Acoustical Society of America. 



MEASUREMENT AND CONTROL OF NOISE 213 

The whole subject of noise measurement is still young, 
and much research still is needed before we are sure that our 
quantitative values correspond to the testimony of our 
ears with regard to the loudness of noises. For example, 
one might ask whether the ear forms any quantitative 
judgments at all as to the relative loudness of noise. 
Further, does the ear rate the noise of 10 vacuum cleaners 
as ten times that of one? The probable answer to the 
latter question is in the negative, but merely asking the 
question serves to show the inherent difficulties in express- 
ing the loudness of noise in a way that will be meaningful 
in terms of ordinary experience. 

Comparison of Noises. Masking Effect. 

While the exact measurement of the loudness level 
of noise is still a matter of considerable uncertainty, yet 
it is possible to make quantitative comparisons that will 




ffotse fobe 

measured enters 

same ear 

FIG. 82. Buzzer audiometer. 



serve a great many useful purposes. Various types of 
audiometers, devised primarily for the testing of the acuity 
of hearing, have been employed. That most commonly 
used is the Western Electric 3-A audiometer (Fig. 82) 
of the so-called buzzer type. In this, a magnetic inter- 
rupter interrupts an electric current which passes through a 
resistance network called an attenuator, consisting of 
parallel- and series-connected resistances arranged to reduce 
the current by definite fractions of itself. 

The attenuator dial is graduated to read the relative 
sound-output levels of the ear phone in decibels. The 
zero of the instrument for a given observer is determined 
by setting the attenuator so as to produce a barely audible 
sound in a perfectly quiet place. The reading above this 



214 ACOUSTICS AND ARCHITECTURE 

zero is the sensation level of the sound produced by this 
setting for this particular observer. 

In comparing noises by this type of instrument, one 
makes use of the so-called masking effect of one sound 
upon another. Suppose that a sound is of such intensity 
as to be barely audible in a quiet place. Then in a noisy 
place it will not be heard due to the masking effect of the 
noise. The rise in intensity level which must be effected! 
in the sound before it can be heard in the presence of a given 
noise is the masking effect of that noise. In using the 3- A 
audiometer, the instrument is taken to a perfectly quiet 
place (which, by the way, is seldom easy to find), and the 
attenuator dial is adjusted so that the buzz is just heard 
in the receiver by the observer. The difference between 
this and the setting necessary to produce an audible sound 
in the receiver in a noisy place is the masking effect of the 
noise on the sound from the audiometer. Different noisesji 
are compared by measuring their masking effects on the? 
noise from the audiometer. Measurements may also be 
made by matching the unknown noise with that made ^ 
by the audiometer, but experience shows that judgments of 
equal loudness are apt to be less precise than are those on 
masking. No very precise relation can be given between 
the matching and the masking levels. Gait 1 states that 
for levels between 20 and 70 db. above the threshold, thej 
matching level is 12 db. higher than the masking level. 
For very loud noises such as that of the cheering of a large 
crowd the difference is apparently greater, probably 20 db. , ; 
for levels of around 100 db. 

Tuning-fork Comparison of Noises. 

A. H. Davis 2 has proposed and used an extremely 
simple method of comparing noises by means of tuning 
forks. The dying away of a sound of a struck tuning fork 
is very approximately logarithmic, so that time intervals 

1 GALT, R. H., Noise Out of Doors, Jour. Acous. Soc. Amer., vol. 2, No 1, 
pp. 30-58, July, 1930. 
* Nature, Jan. 11, 1930. 



MEASUREMENT AND CONTROL OF NOISE 215 

measured during the decay of the fork are proportional 
to the drop in decibels of the intensity level of the sound. ' 
The decibel drop per second can be determined by measur- 
ing in a quiet place the times required for the sound of the 
fork to sink to the masking levels for two different settings 
of an audiometer. An alternative method of calibration 
is to measure by means of a microscope with micrometer 
eyepiece the amplitude of the fork at any two moments 
in the decay period. Twice the logarithm of the ratic^ 
of the two amplitudes divided by the intervening time{ 
interval gives the drop of intensity level in bels per second. 
(The intensity of the sound is proportional to the square 
of the amplitude of the fork, hence the factor 2.) 

In use, one measures in a perfectly quiet place the time 
required for the sound of the fork to decrease to the 
threshold. In strict accuracy, 
the fork should always be struck 
with the same force. However, 
considerable variation in the force 
of the blow will make only slight 
difference in the time. Measure- 
ment is then made of the time 
required for the sound of the 
fork to become inaudible in the 
presence of the noise to be meas- 
ured. The difference in time , 
between the quiet and noisy con- ! 
ditions multiplied by the number 

i_ j FlG ' 83 ' Rivcrbank tuning 

OI decibels drop per Second gives forks designed for noise measure- 

the masking effect of the noise oh ments - 
the sound of the fork. Different noise levels may be 
directly compared in this manner. Obviously, the pitch 
of the fork will make a difference, but with forks between 
500 and 1,000 vibs./sec. this difference is less than the 
fluctuation in levels which commonly occurs in ordinary 
noises. Figure 83 shows the type of fork designed by 
Mr. B. E. Eisenhour of the Riverbank Laboratories 
for noise measurements of this sort. The particular shape 




216 ACOUSTICS AND ARCHITECTURE 

is such that the damping of the fork is about 2 db. per 
second and the duration of audible sound is approximately 
60 sec. By means of Gradenigo figures etched on the prong 
of the fork it is possible to measure the time from a fixed 
amplitude. This makes an extremely simple means of 
rating noise levels. 




FIG. 84. A single sound pulse is shown in A. In D, the pulse is reflected 
from a hard surface with slight decrease in energy. In (7, the energy reflected 
from the absorbent surface is too small to be photographed. D shows the 
repeated reflections inside an iiiclosure. 

Measured Values of Noise Levels. 

The most comprehensive survey of the general noise 
conditions in a large city is that conducted in 1930 by the 
Noise Abatement Commission under the Department of 
Public Health of the city of New York. A complete 
report is published under the title "City Noise/' issued by 
the New York Health Department. In Appendix D are 
given average values of both indoor and outdoor noise 
levels above the threshold of hearing as measured in a 



MEASUREMENT AND CONTROL OF NOISE 



217 



large number of places and under varying conditions in 
that survey. Such figures help very materially in relating 
noise-level values to auditory experience. The value of a 
decibel rating is shown when we consider that the range of 
physical intensities is from 1 to 10 10 . 

Reverberation and Noise Level in Rooms. 

The sound photographs of Fig. 84 will serve to show 
qualitatively how the repeated reflection of sound to and 
fro will increase the general noise level within a room and 
also the effect of absorbent materials in reducing this 



09 
08 
0.7 
06 
OB 
04 
03 
08 
0.1 



\ 


\ 


\ 


\ 


\ 


0.69 

047 

0.33 

Q23 
016 
288 


\ 


\ 


\ 


\ 


\ 


\ 


\ 




\ 


\ 


\ 




\-~ 


k 


V 


V 






\ N 


\ s 


V 






^\ 


s. ^ 


v, - - --- 










\. 


^\ 


Increase in Noise 
Due to 
Reverberation 






^^^^, 









I 2 3 4 b 

Time in Fifth Seconds 

FIG. 85. Increase in noise due to reverberation. 



level. A shows the pulse generated by a single impact 
sound. B shows the reflection of this pulse from a hard, 
highly reflecting wall or ceiling with only a slight diminu- 
tion of its intensity. C shows its dissipation at a highly 
absorbing surface. In D we have the state of affairs within 
a room after the sound has undergone one or two reflections 
from non-absorbent surfaces. In a room 30 by 30 ft., 
D would represent conditions 0.05 sec. after the sound was 
produced. Assume that the absorption coefficient of the 
bounding surfaces is 0.03 and that the mean time between 
reflections is 0.05 sec. Then at the end of this time 97 per 
cent of the energy would still be in the room. At the end 
of two such intervals, the residual energy would be (0.97) 2 
or 94.1 per cent of the original. If at the end of 0.1 



218 ACOUSTICS AND ARCHITECTURE 

sec. the impact producing the sound is repeated, then the 
total sound energy in the room due to the two impacts 
would be 1.94 times the energy of a single impact. If we 
imagine the impacts repeated at intervals of 0.1 sec., the 
sound of each one requiring a considerable length of time 
to be dissipated, the cumulative effect on the general noise 
level is easily pictured. The effect of introducing absorbing 
material in reducing this cumulative action is also obvious. 

Figure 85 shows quantitatively the effect of reverber- 
ation on the noise produced by the click of a telegraph 
sounder in the sound chamber of the Riverbank Labora- 
tories. In this room, the sound of a single impact persisted 
for about 7.5 sec. Experiments showed that the intensity 
decreased at the rate of about 8 db. per second. 

In the figure, it is assumed that the impacts occur at 
the rate of five per second, and the total sound energy 
in the room due to one second's operation of the sounder 
is, as shown, 2.88 times that produced at each impact. 
Under continuous operation, this would be further increased 
by the residual sound from impacts produced in preceding 
seconds. 

An analysis similar to that given in Chap. IV for the 
steady-state intensity set up by a sustained tone is easily 
made. 

Let e = sound energy of a single impact 
N = number of impacts per second 
ct average coefficient of absorption 
p = mean free path = 4V /S 

Then we may treat the source of noise as a sustained source 
whose acoustic output is Ne. The total energy in the room 
under sustained operation is 



Total energy = ~[l + (1 - a) + (1 - a) 2 + (I - a) 3 

- . (1 - a) 1 ] 
Making n large, the sum of the series is I/a; hence 



m . , A NeV A NeV _ ON 

Total energy = - = 4 ^- = 4 - (72) 

&J ca aSc ac ^ J 



MEASUREMENT AND CONTROL OF NOISE 



219 



Total Absorbiri 
8 fi $ 



Power 



The average energy density of the sound in the room under 
continued operation is Ne/ac. 

It is to be noted that the analysis is based on the assump- 
tion that the repeated impact sounds may be treated as a 
sustained source whose acoustic output is Ne. This 
assumption holds if the interval between impacts is short 
compared with the duration of the sound from the impact. 

It appears from the foregoing that with a given amount 
of noise generated in a room, the average intensity due to 
diffusely reflected sound is inversely proportional to the 
total absorbing power of the room. Now the noise level is 
roughly proportional to the logarithm of the intensity 
Hence with a given source of sound in a room, the nois- 
level will decrease linearly not 
with the absorbing power but with 
the logarithm of the absorbing 
power. This fact should be borne 
in mind in considering the amount 
of quieting in interiors that can be 
secured by absorbent treatment. 

Figure 86 shows the variation of 
noise intensity and noise level with 
total absorbing power in a typical 
case of office quieting. We have 
assumed an office space 100 by 40 
by 10 ft., occupied by 50 typists 
and initially without any absorbent 
material other than that of the 
usual office furniture and the clothing of the occupants. The 
absorbing power of such a room would normally be about 500 
square units. We shall consider that this is initially a 
fairly noisy office, with a noise level of 50 db. above thresh- 
old (7 = 100,0000, and that the ceiling originally has an 
absorption coefficient of 0.03. The curves show the effect 
upon the intensity and the noise level of surfacing the 4,000 
sq. ft. of ceiling successively with materials of increasingly 
greater coefficient. 




003 013 023 033 043 Q53 063 073 083 
Absor pti on Coefficient of Ce 1 1 1 nq 

FIG. 86. Noise level in a 
room as a function of absorp- 
tion coefficient of ceiling. 



220 ACOUSTICS AND ARCHITECTURE 

We note here what may be called a law of diminishing 
returns. Each additional increment of absorption yields 
a smaller reduction in the intensity and noise level than 
does the preceding. Thus the first 10-point increase of 
absorption effects a reduction of 2.5 db. ; the last 10-point 
increase, a reduction of only 0.5 db. Since the threshold 
of intensity difference is approximately 1 db., it is plain 
that an increase of from 0.73 to 0.83 in the absorption 
coefficient of the ceiling treatment would not make a 
perceptible difference in the noise level, while an increase 
from 0.63 to 0.83 would effect only a barely perceptible 
difference. 

Measured Reduction Produced by Absorption. 

The most important practical use of the reduction 
of noise by absorbent treatment is in the quieting of business 
offices and hospitals. The use of absorbent treatment in 
large business-office units is a matter of common practice 
in this country, and the universal testimony is in favor of 
this means of alleviating office noise. There are, however, 
comparatively few published data on the actual reduction 
so effected. The writer at various times has made audi- 
ometer measurements both before and after the application 
of absorbent treatment in the same offices. The most 
carefully conducted of these was under conditions approxi- 
mately described in the example of the preceding section. 
The average noise level before treatment, using a Western 
Electric 3-A audiometer, was 55.3 db. A material with an 
absorption coefficient of 0.65 was applied to the entire 
ceiling, and the measured value was reduced to 48 db. The 
introduction of the absorbent treatment increased the 
total absorbing power about fivefold. The theoretical 
reduction in decibels would be 10 logio 5 = 7.0 db. as 
compared with the measured 7.3 db. R. H. Gait 1 has 
measured, under rather carefully controlled conditions, the 
effect of absorbent treatment upon the noise level generated 

1 Method and Apparatus for Measuring the Noise Audiogram, Jour. 
Acous. Soc. Amer., vol. 1, No. 1, pp. 147-157, October, 1929. 



MEASUREMENT AND CONTROL OF NOISE 221 

by a fixed source of noise. A phonograph record of the 
noise in a large office was made. The amplified sound 
from this record was admitted to a test room by way of 
two windows from a smaller room in which the record was 
reproduced. The absorbing power of the test room was 
varied by bringing in absorbent panels. Gait found that 
a fivefold increase in the total absorbing power reduced 
the intensity to one-fifth, corresponding to a reduction 
of 7 db. in the noise level. He found that closing the win-? 
dows of a tenth-floor office room reduced the noise from 
the street by approximately this same amount. Perhaps 
this comparison affords a better practical notion of what 
is to be expected in the way of office quieting by absorbent 
treatment than do the numerical values given. Tests 
conducted by the author for the Western Felt Works 
showed a reduction of 7 db. in the noise of passing street 
cars as a result of closing the windows. 

Computation of Noise Reduction. 

By Eq. (72) we have for the intensity of the noise 



ac 



If the absorbing power of the room is increased to a', 
then the intensity is reduced to 



ac 
whence 

L = ?_' = Z 

/' a T'o 

The reduction in bels is the logarithm of the ratio of the 
two intensities; hence 

Reduction (db) = 10 log - = 10 log ^ 

a 1 o 

It is apparent that the degree of quieting effected by 
absorbent treatment depends not upon the absolute value 



222 ACOUSTICS AND ARCHITECTURE 

of the added absorbing power but upon its ratio to the 
original. Adding 2,000 units to a room whose absorbing 
power is 500 increases the total absorbing power fivefold, 
giving a noise reduction of 10 log 5 = 7 db. Adding the 
same number of units to an initial absorbing power of 1,000 
units increases the total threefold, and the reduction is 
10 log 3 = 4.8 db. 

Quieting of Very Loud Noise. 

It is evident that the reduction in the noise level secured 
by the absorbent treatment of a room is the same inde- 
pendently of the original level. Thus if the introduction 
of sound-absorbing materials into a room reduces the noise 
level from 50 to 40, let us say, then the same treatment 
would produce the same reduction 10 db. if the original 
noise level were raised to 80 or 90 db. This suggests the 
comparative ineffectiveness of absorption as a means of 
quieting excessively noisy spaces. A treatment that would 
lower the general noise level in an office from 50 to 40 db. 
would make a marked difference in the working conditions. 
In its effect on speech, for example, the 40-db. level is 
10 db. below the level of quiet speech, and its effect on 
intelligibility is decidedly less than that of the original 
50-db. noise level. In a boiler factory, however, with an 
original noise level of 100 db. this 10-db. reduction to 90 
db. would still necessitate shouting at short range in order 
to be heard. Moreover, it must be remembered that 
absorption can affect only the contribution to the general 
noise level made by the reflected sound. The operator 
working within a few feet of a very noisy machine where 
the direct sound is the preponderating factor is not going 
to be helped very much by sound-absorbent treatment 
applied to the walls. 

While these common-sense considerations are obvious 
enough, yet one finds them frequently ignored in attempts 
to achieve the impossible in the way of quieting extremely 
noisy conditions. The terms " noisy" and '* quiet" are 
relative. A given reduction of the noise level in an office, 



MEASUREMENT AND CONTROL OF NOISE 223 

where the work requires mental concentration on the part 
of the workers, may change working conditions from noisy 
to quiet. The same reduction in a boiler factory would 
produce no such desired results. 

Effect of Office Quieting. 

In addition to the actual physical reduction of the 
noise level from the machines in a modern business office, 
there are certain psychological factors that are operative in 
promoting the sense of quiet in rooms in which sound 
reflection is reduced by absorbent treatment. We have 
noted the effect of reverberation upon the intelligibility of 
speech. In reverberant rooms, there is therefore the tend- 
ency for employees to speak considerably above the ordi- 
nary conversational loudness in order to meet this handicap. 
The effect is cumulative. Employees talk more loudly 
in order to be understood, thus adding to the sense of noise 
and confusion. Similarly, the reduction of reverberation 
operates cumulatively in the opposite sense. Further, 
there is evidence to indicate that under noisy conditions 
the attention of the worker is less firmly fixed upon the work 
in hand. Hence his attention is more easily and frequently 
distracted, and he has the sense of working under strain. 
A relief of this strain produces a heightened feeling of well- 
being which increases the psychological impression of relief 
from noise. If these and other purely subjective factors 
could be evaluated, it is altogether probable that they would 
account for a considerable part of the increased efficiency 
which office quieting effects. 

Precise evaluation of the increased efficiency due to 
office quieting is attended with some uncertainty. Perhaps 
the most reliable study of this question is that made in 
certain departments of the Aetna Life Insurance Company 
of Hartford, Connecticut, the results of which were reported 
by Mr. P. B. Griswold of that company to the National 
Office Management Association, in June, 1930. The study 
was made in departments in which the employees worked 
on a bonus system, the bonuses being awarded on the basis 



224 ACOUSTICS AND ARCHITECTURE 

of the efficiency of the department as a whole. Records of 
the efficiency were kept for a year prior to the installation 
of absorbent treatment and for a year thereafter. All 
other conditions that might affect efficiency were kept 
constant over the entire period. The reduction in noise as 
measured by a 3-A audiometer was from 41.3 to 35.3 db. 
(These were probably masking levels.) The observed 
over-all increase in efficiency in three departments was 
8.8 per cent. The quieting was found to produce a decrease 
of 25 per cent in the number of errors made in the depart- 
ment. A large department store reports a reduction in 
the number of errors in bookkeeping and in monthly state- 
ments from 118 to 89 per month a percentage reduction of 
24 per cent. 

Physiological Effects of Noise. 

In psychological literature, there is a considerable body 
of information on the effect of noise and music on mental 
states and processes. A bibliography of the subject is given 
by Professor Donald A. Laird in the pamphlet " City Noise " 
already referred to. A paper by Laird on the physiological 
effects of noise appeared in the Journal of Industrial 
Hygiene. 1 This paper gives the results of measurements 
on the energy consumption as determined by metabolism 
tests on typists working under extreme conditions of noise, 
in a small room (6 by 15 by 9 ft.), with and without absorb- 
ent material. Laird states that the absorbent treatment 
" reduced the heard sound in the room by about 50 per 
cent." Four typists working at maximum speed were the 
subject of the experiment. The outstanding result was 
that the working-energy consumption under the quieter 
conditions was 52 per cent greater than the consumption 
during rest, while under the noisier condition it was 71 
per cent greater. Further, the tests showed that the slow- 
ing up of speed of the various operators was in the order 
of their relative speeds; that is, the " speedier the worker the 
more adversely his output is affected by the distraction of 

1 October, 1927. 



MEASUREMENT AND CONTROL OF NOISE 



225 



noise. " If fatigue is directly related to oxygen consump- 
tion, it would appear that these tests furnish positive 
evidence of the increase of fatigue due to noise. 

Absorption Coefficients for Office Noises. 

Due to the nondescript character of noise in general 
and to the fact that most of the published data on the 
absorbing efficiencies of material are for musical tones of 
definite pitch, it is not possible to assign precise values 
to the noise-absorbing prop- 
erties of materials. In 1922, 
the writer made a series of 
measurements on the relative 
absorbing efficiencies of vari- 
ous materials for impact 
sounds, such as the clicks of 
telegraph sounders and type- 
writers. 1 The experimental 
procedure was quite anal- 
ogous to that used in the 
measurement 



2.0 

1.7 
1.6 
1,5 
1.4 

1.1 






























f 














f 








' 














~y 




















t 


f 




















/ 






i 


f 2 












J 


\ 






f 














( 








J 












j 








/ 













A 


















( 


































u> <o to vd i 

Time in Seconds 
afvanrrktirm Flo> 87. Reverberation time of 

aosorption sound of a sing j impact> (1) Room 

Coefficients by the reverbera- empty. (2) With absorbent material 

tion method for musical tones. 

As a source of variable intensity a telegraph sounder was 
employed. The intensity of the sound was varied by alter- 
ing the strength of the spring which produces the upstroke 
of the sounder bar. In Fig. 87, the logarithm of the energy 
of the impact of the bar is plotted against the measured 
reverberation times. We note the straight-line relation ( 
between these two quantities, quite in agreement with 1 
the results obtained using a musical tone of varying initial 
intensities. It follows therefore that the sound energy> 
generated by an impact is proportional to the mechanical 
energy of the impact. From the slope of the straight line, 
we determine the absorbing power of the empty room, and 
proceeding in a manner similar to that for musical tones, 

1 Nature and Reduction of Office Noise, Amer. Architect, May 24; June 7, 
1922. 



226 



ACOUSTICS AND ARCHITECTURE 



we may measure the absorption coefficients of materials for 
this particular noise. The experiment was extended to 
include other impact sounds including noise of typewriters 
of different makes. It appeared that the absorbing power/ 
of the empty sound chamber was the same as that for' 
musical tones in the range from 1,024 to 2,048 vibs./sec. 
and that the relative absorbing efficiencies of a number of 
different materials were the same as their efficiencies for 
musical tones in this range. 

If we include voice sounds as one of the components of 
office noise, it would appear that the mean coefficient over 



u 
20 

40 

60 
12 

reqi 


"T 1 


Quirt 


Office- 




/ 


H 






r 


L | 


_g 








_F 


nyeroofnin 




j 








i 










rr i 










" ^.J Room witti 










fe/eg raph sounders 


8 256 512 1024 2048 4096 8192 
Frequency 

iiency distribution of office uoisc. (After Gait.) 



the range from 512 to 2,048 vibs./sec. should be taken as the 
measure of the efficiency of a material for office quieting. ' 
Figure 88 gives the noise audiogram of the noise in offices as 
given by Gait. 

Here we note that the range of maximum deafening for 
the large office in active use extends from 256 to 2,048, 
while in the room with telegraph sounders, the maximum 
occurs in the range 500 to 3,000 vibs./sec. 

Quieting of Hospitals. 

Nowhere is the necessity for quiet greater than in the 
hospital, and perhaps no type of building, under the usual 
conditions, is more apt to be noisy. The accepted notions 
of sanitation call for walls, floors, and ceiling with hard, non- 
porous, washable surfaces. The same ideas limit the 
furnishings of hospital rooms to articles having a minimum 
of sound absorption. The arrangement of a series of 



MEASUREMENT AND CONTROL OF NOISE 227 

patients' rooms all opening on to a long corridor with! 
highly reflecting walls, ceiling, and floor makes for the easy i 
propagation of sound over an entire floor. Sanitary 
equipment suggests basins and pans of porcelain or enamel! 
ware that rattle noisily when washed or handled. Modern 
fireproof steel construction makes the building structure a 
solid continuous unit through which vibrations set up by 
pumps or ventilating machinery and elevators are trans- 
mitted with amazing facility. In the city, the necessity for 
open-window ventilation adds outside noises to those of 
internal origin. 

The solution of the problem of noise in hospitals is a 
matter both of prevention and of correction. Prevention 
calls for attention to a host of details. The choice of a site, 
the layout of plans, the selection, location, and installation 
of machinery so that vibrations will not be transmitted to 
the main building structure, acoustical isolation of nurs- 
eries, delivery rooms, diet kitchens, and service rooms, 
provision for noiseless floors, quiet plumbing, doors that 
will not slam, elevators and elevator doors that will operate 
with a minimum of noise attention to all of these in the 
planning of a hospital will go far toward securing the desired 
degree of quiet. The desirability of a quiet site is obvious. 
City hospitals, however, must frequently be located iity 
places where the noise of traffic is inevitable. In such 
cases, a plan in which patients' rooms face an inside court[ 
with corridors and service rooms adjacent to the street, 
will afford a remarkable degree of shielding from traffic 
noise. Laundries and heating plants should, whenever^ 
possible, be in a separate building. Failing this, location in 
basement rooms on massive isolated foundations with 
proper precautions for the insulation of air-borne sounds to 
the upper floors can be made very effective in prevent- 
ing the transmission of vibrations. Properly designed 
spring mountings afford an effective means of reducing 
structural vibrations that would otherwise be set up by 
motors and other machinery that have to be installed in 
the building proper. 



228 ACOUSTICS AND ARCHITECTURE 

In addition to the prevention of noise, there is the 
possibility of minimizing its effects by absorbent treatment. 
There has been a somewhat general feeling among those 
responsible for hospital administration against the use of 
sound-absorbent materials on grounds of sanitation. It 
has been assumed that soft or porous materials of plaster, 
felts, vegetable fiber, and the like are open to objections as 
harboring and breeding places for germs and bacteria. 
Hospital walls and ceilings come in for frequent washings 
and for surface renewals by painting. 

The whole question of the applicability of sound- 
absorbent treatment to the reduction of hospital noises, 
including the important item of cost of installation and 
upkeep, has been gone into by Mr. Charles F. Neergaard of 
New York City. The results of his investigations have 
been published in a series of papers which should be con- 
sulted by those responsible for building and maintenance 
of a hospital. 1 

A number of acoustical materials were investigated, 
both as to the viability of bacteria within them and as to the 
possibilities of adequately disinfecting them. The general 
conclusion reached was that certain of these materials are 
well adapted for use in hospitals, that the sanitary hazard 
is more theoretical than real, and that the additional cost 
of installation and upkeep is more than compensated for 
by the alleviation of noise which they afford. 

On the strength of these findings, one feels quite safe in 
urging the use of sound-absorbent treatment on the ceilings 
and upper side walls of hospital corridors, in diet kitchens, 
service rooms, and nurseries, as well as in private rooms 
intended for cases where quiet and freedom from shock are 
essential parts of the curative regime. 

1 How to Achieve Quiet Surroundings in Hospitals, Modern Hospital, 
vol. 32, Nos. 3 and 4, March and April, 1929; Practical Methods of Making 
the Hospital Quiet, Hospital Progress, March, 1931; Are Acoustical Materials 
a Menace in the Hospital? Jour. Acous. 8oc. Amer., vol. 2, No. 1, July, 1930; 
Correct Type of Hardware, Hospital Management, February, 1931. 



MEASUREMENT AND CONTROL OF NOISE 229 

Noise from Ventilating Ducts. 

A frequent source of annoyance and difficulty in hearing 
in auditoriums is the noise of ventilating systems. There 
may be several sources of such noise; the more important 
are (1) that due to the fluctuations in air pressure as the 
fan blades pass the lip of the fan housing; (2) noise from the 
motor or other driving machinery; (3) noise produced by 
the rush of air through the ducts, and particularly at the 
ornamental grills covering the duct opening. It is not 
uncommon to find cases in which complaints are made of 
poor acoustical conditions, which upon investigation show 
that the noise level produced by the ventilating system is 
responsible. The noise from the motor and fan is trans- 
mitted in two ways: as mechanical vibrations along the 
walls of the duct and as air-borne sound inside the ducts. 
It is good practice to supply a short-length flexible rubber- 
lined canvas coupling between the fan housing and the duct 
system. This will obviate the transfer of mechanical 
vibrations. 

Lining the duct walls with sound-absorbent material 
measurably reduces the air-transmitted noise. Such treat- 
ment is more effective in small than in large conduits. 
Both duct and fan noises, however, increase with the speed 
of the fan and the velocity of air flow, so that in absorbent- 
lined ducts, the preference as between low speeds through 
large ducts and high speeds through small ducts is doubtful. 

Larson and Norris 1 have reported the results of a valuable 
study on the question of noise reduction in ventilating 
systems. The tests were made on a 30-ft. section of 10 
by 10-in. galvanized iron duct made up in units 2 ft. long. 
Air speeds ranged from about 300 to 5,000 ft. per minute, 
and fan speeds from 100 to 1,300 r.p.m. The noise level 
was measured by means of an acoustimeter, with the receiv- 
ing microphone set up 2 ft. from the duct opening. Two 

1 Some Studies on the Absorption of Noise in Ventilating Ducts, Jour. 
Heating, Piping, and Air Conditioning, Amer. Soc. Heating Ventilating Eng. t 
January, 1931. 



230 



ACOUSTICS AND ARCHITECTURE 



types of absorbent lining were used. The absorbent 
material was a wood-fiber blanket 1 in. thick. In one case 
it was bare and, in the other, covered with thin, perforated 
sheet metal. In the lined duct, the outer casing was 12 
by 12 in. giving the same section 10 by 10 in. for the air 
flow through both the lined and the unlined ducts. 

The following are some of the more important facts 
deduced : 

1. The perforated metal covering over the absorbent 
material produced no reduction in the air flow for a given 
fan speed. The bare absorbent reduced the air flow by 
about 1 1 per cent. 

2. The reduction of noise was the same with as without 
the perforated metal covering over the absorbent material. 

3. The reductions in decibels in the noise level produced 
by lining the entire duct are given below. 



Air speed, 
feet per minute 


Noise level 


Reduction, dhs. 


Unlined 


Lined 


1,000 


33 


23 





2,000 


44 


32 


12 


3,000 


54 


41 


13 


4,000 


63 


48 


15 


5,000 


68 


53 


15 



4. Absorbent lining placed near the inlet end of the duct 
produces a somewhat greater reduction than the same 
length at the outlet end. Thus 6 ft. at the intake produced 
a greater reduction than 12 ft. at the outlet. 

5. The authors state that a 41-db. level at the outlet 
end does not materially affect the hearing in an auditorium. 
At this level, lining the duct throughout would allow an 
increase of 75 per cent in the air speed over that which 
would produce this level in an unlined duct. 

6. The noise reduction increases with the length of duct 
that is lined. The increment in the reduction per unit of 
lining decreases as the lining already present increases. 



MEASUREMENT AND CONTROL OF NOISE 231 

These results serve to show the general effect on noise 
reduction of lining ducts. The reduction effected will also 
depend upon the size of the duct, decreasing as the cross sec- 
tion increases. Data on this point are lacking. Another 
source is that due to the rush of air through the grill work 
covering the opening. This may be expected to increase 
markedly with the air velocity. All things considered, it 
would appear that a high-velocity system is apt to produce 
more noise for a given delivery of air to the room than a 
low-speed system. 



CHAPTER XI 

THEORY AND MEASUREMENT OF SOUND 
TRANSMISSION 

Nature of the Problems. 

Two distinct problems arise in the study of the trans- 
mission of sound from room to room within a building. 
The first is illustrated by the case of a motor or other 
electric machinery mounted directly upon the building 
structure. Due to inevitable imperfections in the bearings 
and to the periodic character of the torque exerted on the 
armature, vibrations are set up in the machine. These are 
transmitted directly to the structure on which it is mounted 
and hence by conduction through solid structural members 
to remote parts of the building. These vibrations of the 
extended surfaces of walls, floors, and ceiling produce sound 
waves in the air. Consideration of this aspect of sound 
transmission in buildings will be reserved for a later chapter. 

The second problem is the transmission of acoustic 
energy between adjacent rooms by way of intervening 
solid partitions, walls, floors, or ceiling. Transmission of 
the sound of the voice or of a violin from one room to 
another is a typical case. The transmission of the sound of 
a piano or of footfalls on a floor to the room below would 
come under the first type of problem. 

Mechanism of Transmission by Walls. 

Let us suppose that A and B (Fig. 89) are two adjacent 
rooms separated by a partition P and that sound is pro- 
duced by a source S in A. The major portion of the sound 
energy striking the partition will be reflected back into A, 
but a small portion of it will appear as sound energy in B. 
There are three distinct ways in which the transfer of 
energy from A to B by way of the intervening partition is 

232 



MEASUREMENT OF SOUND TRANSMISSION 



233 



effected: (1) If P is perfectly rigid, compressional air waves 
in A will give rise to similar waves in the solid structure P, 
which in turn will generate air waves in B. (2) If P is a 
porous structure such as to allow the passage of air through 
it, the pressure changes due to the sound in A will set up 
corresponding changes in B by way of the pore channels in 
the partition. A part of the energy entering the pore 
channels will be dissipated by friction; that is, there will be 
a loss of energy by absorption in transmission through a 
porous wall. (3) If P is non-porous and not absolutely 



FIG. 89. 

rigid, the alternating pressure changes on the surface will 
set up minute flexural vibrations of the partition, which 
will in turn set up vibrations in B. 

Of these three modes of transmission the first is of 
negligible importance in any practical case, in comparison 
with the two others. When sound in one medium is 
incident upon the surface of a second medium, a part of its 
energy is reflected back into the first medium and part is 
refracted. The ratio of this refracted portion to the 
incident energy is equal to the ratio of the acoustic resist- 
ances of the first medium to that of the second. From the 
values of acoustic resistances in various media given in 
Table II of Appendix A it will be seen that for solid 
materials the acoustic resistance is very high as com- 
pared with that of air, so that the sound entering a solid 
partition is a very small fraction of the incident sound, and 
hence very little sound is transmitted through solid walls in 
this manner. Davis and Kaye state that a mahogany 



234 



ACOUSTICS AND ARCHITECTURE 



board two inches thick, if rigid, would transmit only 20 
parts in 1, 000,000. 1 

We shall consider the transmission of energy in the two 
other ways in Chap. XII. 

Measurement of Sound Transmission : Reverberation 
Method. 

The first serious attempt at the measurement of the 
sound transmitted by walls was made by W. C. Sabine 
prior to 1915. The results of these earliest measure- 




FIG. 90. W. C. Sabine's experimental arrangement for sound transmission 

measurements. 

ments for a single tone are described in his " Collected 
Papers." The method used was based on the reverberation 
theory, already employed in absorption measurements. 
The experimental arrangements are shown in Fig. 90 
taken from the " Collected Papers." Sound was produced 
by organ pipes located in the constant-temperature room 
of the Jefferson Physical Laboratory. The test panels were 
mounted in the doorway constituting the only means of 
entrance into the room, so that when the panel was in place 
the experimenter had to be lowered into the room by means 

1 "Acoustics of Buildings," p. 179, George Bell & Sons, 1927. 



MEASUREMENT OF SOUND TRANSMISSION 235 

of a rope through a manhole in the ceiling. The experi- 
mental procedure was to measure the time of decay of 
sound heard in the constant-temperature room and then 
the time as heard through the test panel in a small vestibule 
outside. If the average steady-state intensity set up by 
the source in the larger room is /i, and / is the intensity t sec. 
after the source is stopped, then the reverberation equation 
gives 

, 1 1 A act act 

loge - r = At = = TT/ 

/ sp 4V 

If ti be the time required for the intensity in the source room 
to decrease to the threshold intensity i, then 

, /i acti 
Iog < T = 4F 

Let k, the reduction factor of the partition, be the ratio 
at any moment of the average intensity of the sound in 
the source room to the intensity at the same moment on the 
farther side of the partition and close to it. Then when 
the sound heard through the partition has just reached 
the threshold intensity, the intensity / in the source room 
is given by the relation 

I = Id 

If a sound of initial intensity /i can be heard for t z sec. 
through the test partition, we have 

1 1 actz 
ft - 4F 

whence, by subtraction, we have 



log e k = ~(ti - 



and 



logic t = ~g ~(ti - W (73) 

The value of the total absorbing power of the source room 
is known from ti, the measured reverberation time in the 
source room and the initial calibration as described in 
Chap. V. 



236 ACOUSTICS AND ARCHITECTURE 

This method is simple and direct and involves no assump- 
tions save those of the general theory of reverberation. 

Using this method, Professor Sabine made measurements 
upon a considerable number of materials and structural 
units such as doors and windows of various types. His 
only paper on the subject of sound transmission, however, 
gives the results only for hair felt of various thicknesses and 
a complex wall of alternate layers of sheet iron and felt, and 
these at only a single frequency. The experiments with 
felt showed a strictly linear relation between the thickness 
and the logarithm of k as defined above. The intervention 
of the World War and Professor Sabine's untimely death 
just at its close prevented the carrying out of the extensive 
program of research along this line which he had planned. 

Experimental Arrangement at Riverbank Laboratories. 

The sound chamber of the Riverbank Laboratories was 
built primarily to provide for Professor Sabine the facilities 
for carrying on the research to which reference has just been 
made. A general description together with detailed draw- 
ings have already been given in Chap. VI (page 98). To 
that description it is necessary to add only the details of 
construction of the test chambers rooms corresponding 
to the small vestibule of the constant-temperature room at 
Harvard. As will be noted, the sound chamber is built 
upon a separate foundation and is structurally isolated from 
the rest of the building as well as from the test chambers. 
The openings from the two smaller test chambers into the 
sound chamber are each 3 by 8 ft., while that from the 
larger is 6 by 8 ft. The test chambers are protected from 
sounds of outside origin by extremely heavy walls of brick, 
and entrance is effected by means of two heavy ice-box doors 
through a small vestibule. Tests on small structural units 
such as doors and windows are made with these mounted in 
the smaller opening, screwed securely to heavy wooden 
frames set in the openings. All cracks are carefully sealed 
with putty. Leads for electrically operating the organ are 
supplied to each of the test chambers. Originally, experi- 



MEASUREMENT OF SOUND TRANSMISSION 237 

ments were made to determine the effect of the size and 
absorbing power of the receiving space upon the measured 
duration of sound from the sound chamber as heard through 
a test wall. It was found that both of these factors pro- 
duced measurable effects, so that the standard practice 
was adopted of closing off a small space adjacent to the test 
wall by means of highly absorbent panels. Reverberation 
in the test rooms was thus rendered negligibly small in 
comparison with that in the sound chamber. The effect of 
this will be considered somewhat in detail later. During 
the twelve years of the laboratory's operation, test condi- 
tions have been maintained constant so that all results 
might be comparable. 

All hitherto published results from this laboratory have 
been based upon the values of absorbing power obtained by 
the four-organ calibration of the sound chamber. It will be 
noted in Eq. (73) that the value of the logarithm of k 
varies directly as the value assigned to a, the absorbing 
power of the sound chamber. In view of the commercial 
importance attached to many of the tests, these earlier 
values have been adhered to for the sake of consistency, 
even though the work of recent years using a loud-speaker 
source indicated that for certain frequencies these values 
were somewhat too low. Correction to these later values 
for the sound-chamber absorption are easily made, however, 
and in the results hereinafter given these corrections are 
applied. 

Bureau of Standards Method. 

Figure 91 shows the experimental arrangements for 
sound-transmission measurements at the Bureau of Stand- 
ards. 1 The source of sound is placed in a small room AS, 
provided with two openings in which the test panels are 
placed. The source room is structurally isolated from the 
receiving rooms Ri and 72 2 in the manner shown. The walls 

1 ECKIIARDT and CHRISLER, Transmission and Absorption of Sound by 
Some Building Materials, Bur. Standards Sci. Paper 526. 



238 



ACOUSTICS AND ARCHITECTURE 



are of concrete 6 in. thick, separated from the receiving 
room walls by a 3-in. air space. Lighter constructions are 
prepared as panels and mounted in the ceiling opening, 
while heavier walls are built directly into the vertical 
window. 

The source, supplied with alternating current from a 
vacuum-tube oscillator, is mounted on an arm approxi- 
mately 2 ft. long, which rotates at a speed of about one 
revolution per second. The frequency of the tone is 
varied cyclically over a limited range of frequencies. The 
width of the frequency band can be controlled within 
limits by varying the capacity of a rotating condenser 




FIG 91. Room for sound transmission measurements at the Bureau of Standards. 

which is associated with the fixed capacity of the oscillator 
circuit. The frequency variation and the rotation of the 
source serve to produce a constantly shifting interference 
pattern in the source room. The intensity measurements 
are made by means of a long-period galvanometer, which 
tends to give an averaged value of the intensity at any 
position of the pick-up device. 

To determine relative intensities, a telephone receiver is 
placed in the position at which the intensity is to be 
measured. The e. m. f. generated in the receiver windings 
is taken as proportional to the amplitude of the sound. 
The alternating current produced, after being amplified and 
rectified, is passed through a long-period galvanometer, and 
the deflection noted. Shift is then made from the telephone 
pick-up to a potentiometer which derives oscillating current 



MEASUREMENT OF SOUND TRANSMISSION 239 

from the oscillator which operates the sound source, and 
the potentiometer is adjusted to give the same deflection 
of the ga^anometer as was produced by the sound. The 
potentiometer reading is taken as a measure of the sound 
amplitude, and the relative intensities of two sounds are 
to each other as the squares of these values. The validity 
of this method of measurements rests upon the strict 
proportionality between the amplitude of the sound and 
the e. m. f. generated in the field windings of the pick-up. 
In measuring the reduction of intensity, the procedure is 
as follows : Without the panel in place, a series of intensity 
measurements is made at points along a line through the 
middle point of the opening and perpendicular to its plane. 
Denote the average values of these readings in the trans- 
mitting room by T 7 , and in the receiving room by R. The 
panel is then placed and the readings repeated. It is found 
that T is usually increased to T + t, let us say, while R is 
reduced to R r. The apparent ratio of the intensities in 
the receiving room without and with the panel in place is 
R/(R r). But the placing of the panel has increased the 

T -\ j 
intensity in the transmitting room in the ratio of - > so 

that for the same intensity in the transmitting room under 
the two conditions the ratio of the intensities in the receiv- 
ing room is R(T + t)/T(R - r). 

This expression Eckhardt and Chrisler have also called 
the reduction factor. Obviously it is not the same thing 
as the reduction factor as defined above. (T + f)/(R r) is 
the ratio of the intensities in the two rooms with the parti- 
tion in place, which is the reduction factor k as defined by 
the writer. The Bureau of Standards reduction factor 
is therefore kR/T. R/T is the ratio of the intensities in the 
receiving and transmitting rooms without the partition. 
This is less than unity, so that values for the reduction 
factor given by the Bureau of Standards should, presuma- 
bly, be less than those by the reverberation method as 
outlined in the preceding section. 



240 



ACOUSTICS AND ARCHITECTURE 



Other Methods of Transmission Measurements. 

Figure 92 shows the experimental arrangement proposed 
and used by Professor F. R. Watson 1 for the measurement 
of sound transmission. An organ pipe was mounted at the 
focus of a parabolic mirror, which is presumed to direct a 
beam of sound at oblique incidence upon the test panel 
mounted in the opening between two rooms. The inten- 
sity of the sound was measured by means of a Rayleigh 
disk and resonator in the receiving room, first without and 
then with the transmitting panel in the opening. The 




FIQ. 92. Arrangement for sound transmission measurements at the University 
of Illinois. (After Watson.) 

relative sound reductions by various partitions were 
compared by comparing the ratios of the deflections of the 
Rayleigh disk under the two conditions. Observations 
were made apparently at only a single point of the sound 
beam, so that the effect of the presence of the panel on the 
stationary-wave system in the receiving room was ignored. 
A modification of the Watson method has been used at 
the National Physical Laboratory in England. 2 Two 
unusually well-insulated basement rooms with double walls 
and an intervening air space were used for the purpose. 
The opening in which the test panels were set was 4 by 5 
ft. Careful attention was paid to the effects of inter- 
ference, and precautions taken to eliminate them as far as 
possible by means of absorbent treatment. The measure- 

1 Univ. 111., Eng. Exp. Sta. Bull. 127, March, 1922. 

2 DAVIS and LITTLER, Phil. Mag., vol. 3, p. 177, 1927; vol. 7, p. 1050, 1929. 



MEASUREMENT OF SOUND TRANSMISSION 



241 



ments were made by apparatus not essentially different 
from that used at the Bureau of Standards. The general 
set-up is chown in Fig. 93. Some interesting facts were 
brought out in connection with these measurements. For 
example, experiments showed that with an open window 
between the two rooms there was a considerable variation 



y^w#^^ 




Feet 



Fia. 93. National Physical Laboratory rooms for sound transmission measure- 
ments. (After Davis and Littler.) 

along the path of the beam sent out by the parabolic 
reflector but that the distribution of intensity in the receiv- 
ing room was practically the same with a felt panel in 
place as when the sound passed through the unobstructed 
opening. This implies that the oblique beam is transmitted 
through the felt unchanged in form. It was further found 
that even with a 4^-in. brick wall interposed there was a 



242 



ACOUSTICS AND ARCHITECTURE 



pronounced beam in the receiving room and that the 
intensity of sound outside this beam was comparatively 
small. In actual practice, measurements were made at a 
number of points, and the average values taken. 

Davis and Littler used the term " reduction factor" for 
the ratio between the average intensity along the sound 
beam in the receiving room without the partition to the 
intensity at the same points with the partition. This is 
clearly not quite the same thing as either the Riverbank or 
Bureau of Standards definition. It differs from both 
in the fact that it assumes transmission at a single angle 
of incidence instead of a diffuse distribution of the incident 
sound. Further, it is assumed that the intensity on the 
source side is the same regardless of the presence of the 
partition. 

Sound transmission measurements have also been made 
by Professor H. Kreuger of the Royal Technical University 
in Stockholm. His experimental 
arrangement is taken from a paper 
by Gunnar Heimburger 1 and is shown 
in Fig. 94. A loud-speaker is set up 
very close to one face of the test wall, 
and a telephone receiver is similarly 
placed on the opposite side. The 
loud-speaker and the telephone pick- 
up are both inclosed in absorbent- 
lined boxes. The current generated 
Feet in the pick-up is amplified and meas- 

FIG. 94. Experimental ured in a manner similar to that 
^ UP JSl^ ta t ^3 I t employed at the Bureatf of Standards. 
Kreuger. The intensity as recorded by the 

pick-up when the sound source is directly in front of the 
receiver divided by the intensity when the two are placed on 
opposite sides of the test wall is taken as the reduction 
factor of the wall. It is fairly obvious that this method of 
measurement, while affording results that will give the 
relative sound-insulating properties of different partitions, 

1 Amer. Architect., vol. 133, pp. J25-128, Jan. 20, 1928. 




MEASUREMENT OF SOUND TRANSMISSION 243 

will not give absolute values that are comparable with 
values obtained when the alternating pressures are applied 
over the entire face of the wall or to the sound reduction 
afforded by the test wall when in actual use. 

Here the driving force is applied over a limited area, but 
the entire partition is set into vibration. The amplitude 
of this vibration will be very much less and the recorded 
intensity on the farther side will be correspondingly lower 
than when the sound pressure is applied to the entire wall as 
is the case in the preceding methods. 




Audiometer receiver] 
with offset cap 

FIQ. 95. Audiometer method of sound transmission measurements. (After 

Waterfall.') 

An audiometric method of measurement has been used 
b*y Wallace Waterfall. 1 The experimental arrangement is 
shown in Fig. 95. The sound is produced by a loud- 
speaker supplied with current from a vacuum-tube oscil- 
lator and amplifier. The output of the amplifier is fed into 
a rotary motor-driven double-pole double-throw switch, 
which connects it alternately to the loud-speaker and to an 
attenuator and receiver of a Western Electric 3-A audiome- 
ter. The loud-speaker is set up on one side of the test 
partition, and the attenuator is adjusted so that the tone 
as heard through the receiver is judged to be equally loud 
with the sound from the speaker direct. The observer 
moves to the opposite side of the partition, and a second 
loudness match is made. The dial of the attenuator being 
graduated to read decibel difference in the settings on the 
two sides gives the reduction produced by the wall. The 
apparatus is portable and has the advantage of being 
applicable to field tests of walls in actual use. The 
results of measurements by this method are in very good 

1 Jour. Acorn. Soc. Amer., p. 209, January, 1930. 



244 



ACOUSTICS AND ARCHITECTURE 



agreement with those obtained by the reverberation 
method. 

Not essentially different in principle is the arrangement 
used by Meyer and Just at the Heinrich Hertz Institute and 
shown in Fig. 96. Two matched telephone transmitters 
are set up on the opposite sides of the test wall. Number 1 
on the transmitting side feeds into an attenuator and 
thence through a rotary double-pole switch into a poten- 
tiometer, amplifier, and head set. Number 2, on the 
farther side, feeds directly into the potentiometer. The 
attenuator is adjusted for equal loudness of the sounds from 




Fiu. 96. Method of sound transmission measurements used by Meyer and Just. 

the two transmitters as heard alternately in the head set 
through the rotating switch. The electrical reduction 
produced by the attenuator is taken as numerically equal 
to the acoustical reduction produced by the wall. The 
reduction as thus measured corresponds closely to the 
reduction factor as defined under the reverberation method. 

Resonance Effects in Sound Transmission. 

As has been indicated, sound is transmitted from room 
to room mainly by virtue of the flexural vibrations set up 
in the partition by the alternating pressure of the sound 
waves. The amplitude of such a forced vibration is 
determined by the mass, flexural elasticity or stiffness, and 
the frictional damping of the partition as a whole, as well 
as by the frequency of the sound. A partition set in an 
opening behaves as an elastic rectangular plate clamped at 
the edges. Such a plate has its own natural frequencies 
of vibrations which will depend upon its physical properties 



MEASUREMENT OF SOUND TRANSMISSION 245 

of mass, thickness, stiffness and its linear dimensions. 
Under forcing, the amplitude of the forced vibrations at 
a given frequency will depend upon the proximity of the 
forcing frequency to one of the natural frequencies of the 
plate. For example, it can be shown that a plate of K-in. 
glass 3 by 8 ft. may have 37 different natural modes of 
vibration and the same number of natural frequencies 
below 1,000 vib./sec. 

Its response to any of these frequencies would theoreti- 
cally be much greater than to near-by frequencies, and its 
transmission of sound at these natural frequencies cor- 
respondingly greater. As an experimental fact, these 
effects of resonance do appear in transmission measure- 
ments, so that a thorough study of the properties of a 
single partition would involve measurements at small 
frequency intervals over the whole sound spectrum. To 
rate the relative over-all sound-insulating properties of 
constructions on the basis of tests made at a single fre- 
quency would be misleading. Thus, for example, in tests 
on plaster partitions it was found that, at a particular 
frequency, a wall 1J^ in. thick showed a greater reduction 
than a similar wall 2j^ in. thick, although over the entire 
tone range the thicker wall showed markedly greater 
reduction. 

For this reason, we can scarcely expect very close 
agreement in the results of measurements on a given 
construction at a single frequency under different test 
conditions. In addition to this variation with slight 
variations in frequency, most constructions show generally 
higher reductions for high than for low frequencies. It 
appears, therefore, that in the quantitative study of sound 
transmission, it is necessary to make measurements at a 
considerable number of frequencies and to adopt some 
standard practice in the distribution of these test tones in 
the frequency range. 

Up to the present time, there has been no standard 
practice among the different laboratories in which trans- 
mission measurements have been made in the selection 



246 ACOUSTICS AND ARCHITECTURE 

of test frequencies. At the Riverbank Laboratories, with 
few exceptions, tests have been made at 17 different 
frequencies ranging from 128 to 4,096 vibs./sec., with 4 
frequencies in each octave from 128 to 1,024, and 2 each in 
the octaves above this. This choice in the distribution of 
test tones was made in view of the fact that variation in the 
reduction with frequency is less marked for high- than for 
low-pitched sounds and also because in practice the more 
frequent occurrence of the latter makes them of more 
importance in the practical problem of sound insulation. 
At the Bureau of Standards, most of the tests give more 
importance to frequencies above 1,000, while the total 
frequency range covered has been from 250 to 3,300. The 
National Physical Laboratory has used test tones of 300, 
500, 700, 1,000, and 1,600. Kreuger employed a series of 
tones at intervals of 25 cycles ranging from 600 to 1,200 
vibs./sec. This lack of uniformity renders comparisons at 
a single frequency impossible and average values over the 
whole range scarcely comparable. 

Coefficient of Transmission and Transmission Loss. 

As has been noted above, different workers in the field 
of sound transmission have used the term " reduction 
factor" for quantities which are not identically defined. 
In any scientific subject, it is highly desirable to employ 
terms susceptible of precise definition and to use any given 
term only in its strict sense. The term reduction factor was 
originally employed by the author as a convenient means of 
expressing the results of transmission measurements as 
conducted at the Riverbank Laboratories. 1 Since its sub- 
sequent use, with a slightly different significance, by other 
investigators leaves its precise definition by usage somewhat 
in doubt, the introduction of another term seems advisable. 
It was early recognized that the volume and absorbing 
power of the receiving room had a measurable effect upon 
the reverberation time as heard through the partition 
and that therefore the computed value of k would depend 

1 Amcr. Architect, vol. 118, pp. 102-108, July 28, 1920. 



MEASUREMENT OF SOUND TRANSMISSION 247 

upon the acoustic conditions of the receiving room as well as 
upon the sound-transmitting properties of the test wall. 
Accordingly the standard practice was adopted of measur- 
ing the time in a small, heavily padded receiving room, 
with the observer stationed close to the test partition. 
This procedure minimizes the effect of reflected sound in 
the receiving room and gives a value of k which is a function 
only of the transmitting panel. 

In 1925, Dr. Edgar Buckingham 1 published a valuable 
critical paper on the interpretation of sound-transmission 
measurements, giving a mathematical treatment of the 
effect of reflection of sound in the receiving room upon the 
measured intensity. Following his analysis, 

Let J{ = energy incident per second per unit of surface 

of test wall. 
J t = energy per second per unit surface entering 

receiving room 

r = coefficient of transmission = J t /Ji 
S = area of test wall 

/i = average sound density in transmitting room 
7 2 = average sound density in receiving room 
Vij ai = volume and absorbing power of transmitting 

room 

F 2 , 02 = volume and absorbing power of receiving room 
Buckingham showed that for a diffuse distribution Jt the 
incident energy flux is given by the relation 



hence, 

T 

Jt = 



and the total energy per second E 2 entering through the 
entire surface is 



E, = SJ t = (74) 

1 Bur. Standards Sci. Paper 506. 



248 ACOUSTICS AND ARCHITECTURE 

In the steady state, the wall acts for the receiving room 
as a sound source whose acoustic output is J? 2 , and the 
average steady-state sound density 7 2 is given by the 
equation 

T 4E 2 4 



dzC a^c 4 0,2 

whence 

1 IlS 



The coefficient of transmission r is a quantity which 
pertains alone to the transmitting wall and is quite inde- 
pendent of the acoustic properties of the rooms which it 
separates. The intensity reductions produced by parti- 
tions separating the same two rooms will be directly pro- 
portional to the reciprocals of their transmission coefficients, 
so that 1/r may be taken as a measure of the sound-insulat- 
ing merits of a given partition. Knudsen 1 has proposed 
that the term " transmission loss in decibels" be used to 
express the sound-insulating properties of partitions and 
that this be defined by the relation 

T. L. (transmission loss) = 10 logio - (76) 

This would seem to be a logical procedure. Thus 
defined and measured, the numerical expression of the 
degree of sound insulation afforded by partitions does not 
depend upon conditions outside the partitions themselves. 
Moreover, this is expressed in units which usage in other 
branches of acoustics and telephony has made familiar. 
Illustrating its meaning, if experiment shows that the 
energy transmitted by a given wall is Kooo of the incident 

energy r = 0.001, - = 1,000, log - = 3, and the transmis- 

T T 

sion loss is 30 db. The relation between " reduction f actor " 
as measured and transmission loss as just defined remains to 
be considered. 

1 Jour. Acous. Soc. Amer., vol. 11, No. 1, p. 129, July, 1930. 



MEASUREMENT OF SOUND TRANSMISSION 249 

Transmission Loss and Reduction Factor. 

We note in Eq. (75) that l/r is equal to the ratio of the 
average intensity in the transmitting room to that in the 
receiving room, multiplied by the expression S/a 2 . This 
equation applies to the steady-state intensity set up while 
the source is in operation. It would appear therefore that 
in any method based on direct measurement of the steady- 
state intensities in two rooms on opposite sides of a parti- 
tion, the reduction factor, defined as the ratio of the 
measured value of /i//2, multiplied by S/a 2 gives the value 
of l/r. Hence 

T. L. = 10 log (k . ^] = loftog k + log ~1 (77) 
\ fib/ L #2 J 

Obviously, if the area of the test panel is numerically equal 
to the total absorbing power of the receiving room, then 

IV ~~~ 

r 

In the case of the Riverbank measurements, we have 
not steady-state intensities but instantaneous relative 
values of decreasing intensities. This case calls for 
further analytical consideration, which Buckingham gives. 
From his analysis, and assuming that the coefficient of 
transmission is small, so that the amount of energy trans- 
mitted from the sound chamber to the test chamber and 
then back again is too small to have any effect on the inten- 
sity in the sound chamber, and that the reverberation time 
in the test chamber is very small compared with that of the 
sound chamber, we have 

logio - = logic k + lo glo - - logic (l - ^/] (78) 
T 2 \ ttaKi/ 

Here logio k is taken as {pnr(h *"" k)> the logarithm of the 

y.z y i 

reduction factor of the Riverbank tests. 



250 ACOUSTICS AND ARCHITECTURE 

If the sound chamber is a large room with small absorbing 
power, and the test chamber is a small room with a rela- 
tively large absorbing power, the expression a 1 F 2 /a 2 7i is 
numerically small, and the third term of the right-hand 
member of Eq. (78) becomes negligibly small. We then 
have for the transmission loss 

T. L. - 10 logic Pj\ = loflogio* + logio 

The relation between reduction factor and transmission loss 
is sensibly the same when measured by the reverberation 
method as when measured by the steady-intensity method. 
Using the values of S and a 2 for the Riverbank test 
chambers, the corrections that must be added to 10 log k 
to give transmission loss in decibels are as follows: 

Frequency Correction 

128 2 5 

256 1.2 

512 0.5 

1,024 0.5 

2,048 1 3 

Average 1 

This correction is scarcely more than the experimental 
errors in transmission measurements. 

Total Sound Insulation. 

It is sometimes desirable to know the over-all reduction 
of sound between two rooms separated by a dividing struc- 
ture composed of elements which have different coefficients 
of transmission. 1 

Suppose that a room is located where the average inten- 
sity outside is I\ and that Si, s 2 , $3, etc., sq. ft. of the inter- 
vening wall have coefficients of TI, r 2 , r 3 , etc., respectively. 

Let the average intensity of the incident sound be /i. 
Then the rate at which sound strikes the bounding wall is 
c/i/4 per square unit of surface. Calling E% the rate at 
which transmitted sound enters the room, we have 

1 The theoretical treatment that follows is due to Knudsen. 



MEASUREMENT OF SOUND TRANSMISSION 251 

cl 



Then 7 2 , the intensity inside the room, will be 

/, = 4^ = b [8lTl + S2T2 + S3T3 +... 



where T is the total transmittance of the boundaries, and 

/i = 02 
/t T 
The reduction of sound level in decibels is 

10 log f 1 = 10 log % (79) 

*2 1 

Illustrating by a particular example: Suppose that a 
hotel room with a total absorbing power of 100 units is 
separated from an adjacent room by a 2>-in. solid-plaster 
partition whose area is 150 sq. ft., with a communicating 
door having an area of 21 sq. ft. The transmission loss 
through such a wall is about 40 db., and for a solid-oak 
door \% in. thick, about 25 db. 

For wall: 



For door: 



40 = 10 log -, log - = 4, and r = 0.0001 



25 = 10 log - , log - = 2.5, and r -= 0.0032 



For wall and door, total transmittance is 

T = (150 X 0.0001) + (21 X 0.0032) = 0.015 + 0.0672 = 0.082 

100 
Reduction in decibels between rooms ... ... 10 log n n = 30.9 



If there were no door, the reduction would be 36.8 
db. If we substitute for the heavy oak door a light paneled 
door of veneer, with a transmission loss of 22 db., the reduc- 
tion becomes 28.2, while with the door open we calculate a 
reduction of only 6.8 db. 

The illustration shows in a striking manner the effect 
on the over-all reduction of sound between two rooms of 
introducing even a relatively small area of a structure 
having a high coefficient of transmission. We may carry 
it a step further and compute the reduction if we substitute 



252 



ACOUSTICS AND ARCHITECTURE 



an 8-in. brick wall with a transmission loss of 52 db. for 
the 2>^-in. plaster wall. The comparison of the sound 
reductions afforded by two walls, one of 8-in. brick and the 
other of 2>-in. plaster, is given in the following table : 



Wall 


T.L. 


Reduction, in decibels 


No 
open- 
ing 


1%-in. door 
T.L. 25 


Light veneer 
door T.L. 22 


Opening 
T.L. = 


8-in. brick . . . 


52 
40 


47 
36.8 


31.7 
30.9 


28.7 
28.2 


6.8 
6.8 


2^-in. plaster 





This comparison brings out the fact that any job of 
sound insulation is little better than the least efficient 
element in it and explains why attempts at sound insulation 
so frequently give disappointing results. Thus in the 
above illustration, we see that to construct a highly sound- 
insulating partition between two rooms with a connecting 
doorway is of little use unless we are prepared to close 
this opening with an efficient door. The marked effect 
of even very small openings in reducing the sound insulation 
between two rooms is also explained. 

Equation (79) shows that the total absorbing power in 
the receiving room plays a part in determining the relative 
intensities of sound in the two rooms. With a given wall 
separating a room from a given source of sound the general 
level of sound due to transmission can be reduced by 
increasing the absorbing power of the room, just as with 
sound from a source within the room the intensity produced 
from outside sources is inversely proportional to the total 
absorbing power. The general procedure then to secure 
the minimum of noise within an enclosure, from both 
inside and outside sources, is to use walls giving a high 
transmission loss or low coefficients of transmission and 
inner surfaces having high coefficients of absorption. 



CHAPTER XII 
TRANSMISSION OF SOUND BY WALLS 

Owing to the rather wide diversity not only in the 
methods of test but in the experimental conditions and the 
choice of test frequencies, it is scarcely possible to present 
a coordinated account of all the experimental work that has 
been done in various laboratories on the subject of sound 
transmission by walls. We shall therefore, in the present 
chapter, confine ourselves largely to the results of the 
twelve years' study of the problem carried on at the 
Riverbank Laboratories, in which a single method has 
been employed throughout, and where all test conditions 
have been maintained constant. The study of the problem 
has been conducted with a threefold purpose in mind: 
(1) to determine the various physical properties of parti- 
tions that affect the transmission of sound and the relative 
importance of these properties; (2) to make quantitative 
determination of the degree of acoustical insulation afforded 
by ordinary wall constructions; and (3) to discover if 
possible practicable means of increasing acoustic insulation 
in buildings. 

Statement of Results. 

Reference has already been made to the fact that due 
to resonance, the reduction of sound by a given wall may 
vary markedly with slight variations in pitch. For this 
reason, tests at a single frequency or at a small number of 
frequencies distributed throughout the frequency range 
may be misleading, and difficult to duplicate under slightly 
altered conditions of test, such, for example, as variations 
in the size of the test panel. It appears, however, that 
for most of the constructions studied there is a general 
similarity in the shape of the frequency-reduction curve, 

253 



254 ACOUSTICS AND ARCHITECTURE 

namely, a general increase in the reduction with increasing 
frequency, so that the average value of the logarithm of 
the reduction will serve as a quantitative expression of the 
sound-insulating properties of walls. For any frequency, 
we shall call ten times the logarithm of the ratio of the 
intensities on the two sides of a given partition under the 
conditions described in Chap. XI the "reduction in 
decibels" produced by the partition or, simply, the " reduc- 
tion." The " average reduction 7 ' is the average of these 
single-frequency reductions over the six-octave range from 
128 to 4,096 vibs./sec., with twice as many test tones in 
the range from 128 to 1,024 as in the two upper octaves. 
This average reduction can be expressed as average "trans- 
mission loss" as defined by Knudsen by simply adding 1 db. 

Doors and Windows. 

A door or window may be considered as a single structural 
unit, through which the transmission of acoustic energy 
takes place by means of the minute vibrations set up in the 
structure by the alternating pressure of the incident sound. 
Therefore the gross mechanical properties of mass, stiffness, 
and internal friction or damping of these constructions 
determine the reduction of sound intensity which they 
afford. Of these three factors it appears that the mass per 
unit area of the structure considered as a whole is the most 
important. As a general rule, the heavier types of doors 
and windows show greater sound-insulating properties. 

Table XVII is typical of the more significant of the 
results obtained on a large number of different types of 
doors and windows that have been tested. These tests 
were made on units 3 by 7 ft., sealed tightly, except as 
otherwise noted, into an opening between the sound 
chamber and one of the test chambers. Numbers 2, 6, 
and 7 show the relative ineffectiveness of filling a hollow 
door with a light, sound-absorbing material. Comparison 
of Nos. 4 and 5 indicates the order of magnitude of the 
effect of the usual clearance necessary in hanging a door. 
Inspection of the figures for the windows suggests that the 



TRANSMISSION OF SOUND BY WALLS 



255 



cross bracing of the sash effects a slight increase in insu- 
lation over that of larger unbraced areas of glass. On the 
whole, it may be said that ordinary door and window 
constructions cannot be expected to show transmission 
losses greater than about 30 db. 

Sound-proof Doors. 

Various types of nominally " sound-proof " doors are 
now on the market. These are usually of heavy construc- 
tion, with ingenious devices for closing the clearance 
cracks. 

Table XVIIa presents the results of measurements made 
on a number of doors of this type. It is interesting to note 
the increase in the sound reduction with increasing weight, 
and this regardless of whether the increased weight is due 
to the addition of lead or steel sheets incorporated within 
the door or simply by building a door of heavier con- 
struction. The significance of this will be considered in a 
later section. 

TABLE XVII. SOUND REDUCTION BY DOORS AND WINDOWS 



Number 



Description 



Average 

reduction, 

decibels 



1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 

14 
15 
16 



>-in. steel door 

Refrigerator door, 5>2- m - yellow pine filled with cork 

Solid oak, 1? in. thick 

Hollow flush door, 12 in. thick ... .... 

No. 4, as normally hung 

No. 4, with 2 layers of } 2-in. Celotex in hollow space 
No. 4 with 1-in. balsam wool in hollow space . 

Light- veneer paneled door 

Two- veneer paneled doors, with 2-in. separation, 

normally hung in single casement 

Window single pane 79 X 30 in. >-in. plate glass . . 
Window, 4 panes each 15 X 39 in. }-m. plate glass 
Window 2 panes each 31 X 39 in. ?f 6- in - plate glass. 
Same as 12, but double glazed, glass set in putty both 

sides, 1-in. separation 

Same as 13, but with glass set in felt 

Diamond-shape leaded panes, %Q-m. glass 

Window, 12 panes 10 X 19 in., >-in. glass 



34 7 
29.4 
25.0 
26.8 
24.1 
26.8 
26.6 
21.8 

30.0 
26.2 
29.2 
22.8 

26.6 
28.9 
28.4 
24.7 



256 



ACOUSTICS AND ARCHITECTURE 



TABLE XVIIa. REDUCTION BY SOUND-PROOF DOORS* 





Thick- 


Weight 


Average 


Material 


ness, 


per square 
foot 


reduction, 




inches 


pounds 


decibels 


Wood 


2 5 <6 


6 85 


30 2 


Wood 


2% 


7 00 


30 7 


Wood 


3 


7.65 


31 5 


Wood, steel sheathed .... 


3 


9.6 


33.0* 


Wood, steel sheets, inclosed. 


3 


11.6 


35.6 


Wood, lead sheets, inclosed 


3 


15.3 


37.3 











* These data are published with the kind permission of Mr Irving Hamlin of Evanston, 
Illinois, and of the Compound and Pyrono Door Company of St. Joseph, Michigan, for 
whom these tests were made. 

It is of interest to compute the reduction in the example 
given at the end of Chap. XI. When a door giving a 
reduction of 35 db. is placed in the brick wall, in place of the 
oak door with a reduction of 25 db., we find, upon calcu- 
lation, that with the 35-db. door the over-all reduction is 
41 db. only 6 db. lower than for the solid wall. With the 
25-db. door the over-all reduction is 31.7 db. The com- 
parison is instructive, showing that in any job of sound 
insulation, improvement is best gained by improvement 
in the least insulating element. 

Porous Materials. 

In contrast to impervious septa of glass, wood, and 
steel, porous materials allow the direct transmission of the 
pressure changes in the sound waves by way of the pore 
channels. We should thus expect the sound-transmitting 
properties of porous materials to differ from those of solid 
impervious materials. Experiment shows this to be the 
case. With porous materials, the reduction in transmission 
increases uniformly with the frequency of the sound. Fur- 
ther, for a given frequency the reduction in decibels 
increases linearly with the thickness of the material. 1 



1 For a full account of the study of the transmission of sound by porous 
materials, see Amcr. Architect, Sept. 28, Oct. 12, 1921. 



TRANSMISSION OF SOUND BY WALLS 257 

Experiments on six different materials showed that, in 
general, the reduction of sound by porous materials can 
be expressed by an equation of the form 

R = 10 log k = 10 log (r + qf) (80) 

where r and q are empirical constants for a given material, 
and t is the thickness. 

Both r and q are functions of the frequency of the sound, 
which with few exceptions in the materials tested increase 
as the frequency is raised. In general, the denser the 
material of the character here considered the greater the 
value of q. From the practical point of view, the most 
interesting result of these tests is the fact that each addi- 
tional unit of thickness gives the same increment in insu- 
lating value to a partition composed wholly of a porous 
material. 

Davis and Littler, 1 working at the National Physical 
Laboratory at Teddington, England, by the method 
described in Chap. XI made similar tests on a somewhat 
heavier felt than that tested at the Riverbank Laboratories. 
For frequencies above 500 vibs./sec. they found a linear 
relation between the reduction and the number of layers 
of felt. For frequencies between 250 and 500 vibs./sec. 
their results indicate that the increase in reduction with 
increasing thickness is slightly less than is given by Eq. (80). 
The difference in method and the slightly different defini- 
tions of the term " reduction" are such as to explain this 
difference in results. 

Equation (80) suggests that the reduction in trans- 
mission by porous materials results from the absorption of 
acoustic energy in its passage through the porous layers. 
Each layer absorbs a constant fraction and transmits a 
constant fraction of the energy which comes to it. Let 
/i be the average intensity throughout the sound chamber, 
and Ii/r be the intensity in the sound chamber, directly 
in front of the panel. Let l/q be the fraction of the energy 

1 Phil. Mag., vol. 3, p. 177, 1927. 



258 



ACOUSTICS AND ARCHITECTURE 



transmitted by a unit thickness of the material; then 
(1/qY is the fraction transmitted by a thickness t. Hence 

7 2 , the intensity on the farther side, will be -A> 
whence 



and 



log Y = log (r + 0) 

* 2 

R = 10 log (r + qt) 



The altered character of the phenomenon when layers 
of impervious material are interposed is shown by reference 

to the curves of Fig. 97. The 
straight line gives reduction 
at 512 for thicknesses from 
1 to 4 in. of standard hair felt. 
The upper curve gives the 
reduction of alternate layers of 
felt and heavy building paper. 
We note a marked increase 
in reduction. Experiments 
showed that this increase is 
considerably greater than the 
measured value of the reduc- 









^ 








y 








/ 


/ 








/ 




s/ 






/ 


y 


^ 








\/ 


1 1 inch hair felt 
e Hair felt and 
building paper 




/ 



































2 3 

Number of Layers 

Fia. 97. Reduction by (1) hair . 
felt, (2) by alternate layers of hair tlOn afforded by the paper 
felt and building paper. 



additional unit of paper and felt does not produce an equal 
increment in the reduction. In similar experiments with 
thin sheet iron and felt, Professor Sabine obtained similar 
results. Commenting on this difference, he states: "The 
process [in the composite structure] must be regarded not 
as a sequence of independent steps or a progress of an 
independent action but as that of a structure which must 
be considered dynamically as a whole/' 

Subsequent work disclosed the fact that the average 
reduction produced by masonry walls over the entire 
frequency range is almost wholly determined by the mass 
per unit area of the wall. This suggests the possibility 



TRANSMISSION OF SOUND BY WALLS 



259 



that this may also hold true for the composite structures 
just considered. Accordingly, in Fig. 98 we have plotted 
the reduction against the logarithm of the number of 
layers for the paper and felt, for the sheet iron and air 
space, and for the sheet iron, felt, and air space. (In 
Sabine's experiments, >^-in. felt was placed in a 1-in. 
space between the metal sheets.) We note, in each case, 
a linear relation between the reduction and the logarithm 
of the number of layers, which, in any one type of con- 



40 

30 
g 
1 
1 

20 
10 
















y 


1 Building paper and I" felt 
~ 2 Sheet steel with fair space 
. 3Sheetskc/,'/2Ye/t 
'/z'a/r space 

\_freq 'uency-512 '/sec 1 
/ 





'7* 


/_ 


-~^ 













/ 


/* 














/ 








^^ 






/ 






^- 


^""^^ 








/ 
^^ 


-^ 










/ 

^ 










/ 






^^ 














^^ 












^ 


-T 












^^ 


""" 












^^ 






5 


4 








01 OZ Q 


5 06 07 Ofl 



Logarithm of Number of Layers 

FIQ. 98. Intensity reduction in transmission by composite partitions of steel 

and felt. 

struction, is proportional to the weight. As far as these 
experiments go, the results lead to the conclusion that 
the reduction afforded by a wall of this type wherein 
non-porous layers alternate with porous materials may be 
expressed by an equation of the form 



k 



Y- = aw b 



where w is the weight per unit area of the composite 
structure, and a and b are empirical constants whose 
values are functions of the frequency of the sound and also 
of the materials and arrangement of the elements of which 
the wall is built. The values of 6 for the single frequency 
512 vibs./sec. for the three walls are as follows: 



260 



ACOUSTICS AND ARCHITECTURE 



Paper and felt 1 .67 

Steel with air space 1.47 

Steel, felt, air space 3.1 

We shall have occasion to refer to a similar law when 
we come to consider the reduction given by masonry walls. 
It is to be borne in mind that under the conditions of the 
experiment, we are not dealing with structurally isolated 
units. The clamping at the edges necessarily causes the 
entire structure to act as a unit. 

The figures given in Table XVIII show the average 
reductions given by porous materials. As will subse- 
quently appear, one cannot draw conclusions from the 
results of tests conducted on porous materials alone as 
to how these will behave when incorporated into an other- 
wise rigid construction. In such cases, sound reduction 
is dependent upon the mechanical properties of the struc- 
ture as a whole, rather than upon the insulating properties 
of its components. 

TABLE XVIII. REDUCTION OF SOUND BY POROUS PARTITIONS 



Num- 
ber 


Material 


Average 
reduction, 
decibels 


17 


Hair felt 1 in. thick 


7 1 


18 


Hair felt 2 in. thick 


10 5 


19 


Hair felt 3 in. thick . 


13 4 


20 


Hair felt 4 in. thick 


16 7 


21 


3 layers 1-in. felt, alternated with 4 layers building 
paper . . . . 


31.7 


22 


4 layers /^-in. Cabot quilt 


22 


23 


4 layers *2~Hi Flaxlinum 


29 6 









Continuous Masonry. 

By " continuous masonry " we shall mean single walls, 
as contrasted with double walls, of clay or gypsum tile, 
either hollow or solid, of solid plaster laid on metal lath 
and channels, and of brick or concrete. These include 
most of the common types of all-masonry partitions. 
Figure 99 shows the general similarity of the curves 



TRANSMISSION OF SOUND BY WALLS 



261 



obtained by plotting the reduction as a function of the 

frequency. This similarity in shape justifies our taking 

the average reduction as a measure of the relative sound- 

insulating merits of walls in general. The graph for the 

4 in. of hair felt is instructive, as 

showing the much smaller sound- 

insulating value of a porous material 

and the essentially different charac- 

ter of the phenomena involved. In 

Table XIX, the average reduction 

for 16 partitions of various masonry 

materials is given, together with 

the weight per square foot of each 

finished construction. 

In the column headed "Relative 




256 



512 1024 2048 4096: 
Frequency 



stiffness" are given the steady pres- for homogeneous partitions. 
sure in pounds per square foot over the entire face of the 
walls that produce a yielding of 0.01 in. at the middle 
point of the wall. In each case, the test partition was 6 by 
8 ft. built solidly into the opening. Gypsum plaster was 
used throughout this series of tests. 

TABLE XIX. CONTINUOUS MASONRY WALLS 



Num- 
ber 


Construction 


Weight 
per square 
foot, 
pounds 


Relative 
stiff- 
ness 


Average 
reduc- 
tion 


Log 
weight 


24 


2-in. gypsum tile, unplaatered 


10.4 




25 9 


02 


25 
26 
27 
28 
29 
30 
31 
32 


3-in. hollow gypsum tile, unplastered. . 
l^-in. plaster on metal lath 
3-in. solid gypsum tile, unplastered . . . 
2-in. solid gypsum tile, K~ in - plaster . . 
4-in. clay tile, unplastered 
2-in. solid gypsum tile, }$-in. plaster . . 
2-in. solid gypsum tile, IJ^-in. plaster 
4-in clay tile J^j-in plaster 


11.1 
13.9 
14.2 
15.0 
17.0 
19.6 
21.4 
22 


5.8 
122.0 


27.2 
29 6 
31 4 
31 
33 2 
34 2 
35 2 
36 3 


.05 
.14 
15 
.18 
23 
29 
33 
36 


33 


2^-in. plaster on metal lath 


23 2 


24 6 


37 9 


38 


34 
35 


3-in. solid gypsum tile, l^-i". plaster. . 
4-in. clay tile, 1-in. plaster 


25.4 
27.0 


187.0 


39 
39 8 


.41 
43 


36 


4-in clay tile 1^-in plaster 


28 6 


173 


40 5 


46 


37 


3V-in plaster on metal lath 


32 5 


111 


41 


51 


38 


4^$ in plaster on metal lath. . 


41 8 




45 4 


62 


39 


8-in. brick, 1-in. plaster 


88.0 




53.8 


95 















262 



ACOUSTICS AND ARCHITECTURE 



One notes in the table a close correspondence between 
the weight and the reduction. Figure 100, in which the 
average reduction is plotted against the logarithm of the 
weight per square foot of continuous masonry partitions, 
brings out this correspondence in a striking manner. The 



55 



40 




/ - Continuous masonry plaster, file and 'brick 
2- Cinder, slag and haydife concrete 




80 90 100 



30 30 40 

Weight per Sq. Ft 

FIG. 100. Riverbank measurements on (1) plaster, tile, and brick; (2) cinder, 
slag, and haydite concrete. 

equation of the straight line, along which the experimental 
points lie, is 

R = 10 log k = 10(3 log w - 0.4) 



This may be thrown into the form 
k = QAw 3 



(81) 



It should be pointed out and clearly borne in mind that 
Eq. (81) is a purely empirical equation formulating the 
results of the Riverbank tests on masonry walls. The 
values of the two constants involved will depend upon the 
range and distribution of the test tones employed. Thus it 
is clear from Fig. 99 that if relatively greater weight in 
averaging were given to the higher frequencies, the average 
reductions would be greater. Equation (81) does, however, 
give us a means of estimating the reduction to be expected 
from any masonry wall of the materials specified in Table 
XIX and a basis of comparison as to the insulating merits 



TRANSMISSION OF SOUND BY WALLS 263 

of other constructions and other materials. For example, 
other things being equal, a special construction weighing 
20 Ib. per square foot giving an average reduction of 40 
db. would have the practical advantage of smaller building 
load over a 4-in. clay tile partition giving the same reduc- 
tion but weighing 27 Ib. per square foot. Further, having 
determined the reduction for a particular type of con- 
struction, Eq. (81) enables us to state its sound-insulating 
equivalent in inches of any particular type of solid masonry 
brick, let us say. Thus a staggered- wood stud and metal- 
lath partition weighing 20 Ib. per square foot showed a 
reduction of 44 db., which by Eq. (81) is the same as that 
for a continuous masonry partition weighing 41 Ib. per 
square foot. Now a brick wall weighs about 120 Ib. per 
cubic foot, so that the staggered-stud wall is the equivalent 
of 4.1 in. of brick. The use of the staggered stud would 
thus effect a reduction of 21 Ib. per square foot in building 
load over the brick wall. On the other hand, the over-all 
'thickness of the staggered-stud wall is 7>^ in., so that the 
advantage in decreased weight is paid for in loss of available 
floor space due to increased thickness of partitions. Obvi- 
ously, the sound-insulating properties of partition-wall 
constructions should be considered in connection with 
other structural advantages or disadvantages of these 
constructions. The data of Table XIX and the empirical 
formula derived therefrom make quantitative evaluation 
of sound insulation possible. 

Relative Effects of Stiffness and Mass. 

Noting the values of the relative stiffness of the walls 
listed in Table XIX, we see no apparent correspondence 
between these values and the sound reductions. This does 
not necessarily imply that the stiffness plays no part in 
determining the sound transmitted by walls. At any 
given frequency, the transmitted sound does undoubtedly 
depend upon both mass and stiffness. The data presented 
only show that under the conditions and subject to the 
limitations of these tests the mass per unit area plays the 



264 ACOUSTICS AND ARCHITECTURE 

predominating r61e in determining the average reduction 
over the entire range of frequencies. Dynamically con- 
sidered, the problem of the transmission of sound by an 
impervious septum is that of the forced vibration of a 
thick plate clamped at its edges. The alternating pressure 
of the incident sound supplies the driving force, and the 
response of the partition for a given value of the pressure 
amplitude at a given frequency will depend upon the mass, 
stiffness (elastic restoring force), the damping due* to 
internal friction, and also the dimensions of the partition. 
In the tests considered, the three factors of mass, stiffness, 
and damping vary from wall to wall, and it is not possible 
to segregate the effect due to any one of them. The ratio 
of stiffness to mass determines the series of natural fre- 
quencies of the wall, and the transmission for any given 
frequency will depend upon its proximity to one of these 
natural frequencies; hence the irregularities to be noted 
in the graphs of Fig. 99. 

It is easy to show mathematically that if we have a 
partition of a given mass free from elastic restraint and 
from internal friction, driven by a given alternating 
pressure uniformly distributed over its face, the energy of 
vibration of the partition would vary inversely as the 
square of the frequency and that the ratio of the intensity 
of the incident sound to the intensity of the transmitted 
sound would vary directly as the square of the frequency. 
At any given frequency, the ratio of the intensities would 
be proportional to the square of the mass per unit area. 
If we were dealing with the ideal case of a massive wall 
free from elastic and frictional restraints, the graphs of 
Fig. 99 would be parallel straight lines, and the exponent of 
w, in Eq. (81), would be 2 instead of 3. For such an ideal 
case, the increase with frequency in the reduction by a wall 
of given weight would be uniformly 6 db. per octave. 

The departure of the measured results from the theo- 
retical results, based on the assumption that both the 
elasticity and the internal friction are negligible in com- 
parison with the effects of inertia, indicates that a 



TRANSMISSION OF SOUND BY WALLS 



265 



complete theoretical solution of the problem of sound trans- 
mission must take account of these two other factors. 
With a homogeneous wall, both the stiffness and the fric- 
tional damping will vary as the thickness and, conse- 
quently, the weight per unit area are varied. The empirical 
equation (81) simply expresses the over-all effect of varia- 
tion in all three factors in the case of continuous masonry 
walls. There is no obvious theoretical reason for expecting 
that it would hold in the case of materials such, for example, 
as glass, wood, or steel, in which the elasticity and damping 
for a given weight are materially different from those of 
masonry. 

Exceptions to Weight Law for Masonry. 

As bearing on this point we may cite the results for 
walls built from concrete blocks in which relatively light 
aggregates were employed. Reference is made to the line 2, 
in Fig. 100. The description of the walls for the three 
points there shown, taken in the order of their weights, is 
as follows: 





Total 








thick- 


Weight per 


Reduc- 


Material 




square foot, 


tion, 




inches 


pounds 


decibels 


4-in. hayditc block, 1-in. plaster . ... 


5 


23.2 


43 


4-in. cinder concrete block, 1-in. plaster 


5 


32 3 


47.0 


8-in. blast-furnace slag blocks, 1-in. 








plaster 


9 


56 


52.6 











For this series we note a fairly linear variation of the 
reduction with logarithm of the weight per square foot. 
But the reduction is considerably greater than for tile, 
plaster, and brick walls of equal weight. The three walls 
here considered are similar in the fact that the blocks are 
all made of a coarse angular aggregate bonded with 
Portland cement. The fact that in insulating value they 
depart radically from what would be expected from the 
tests on tile, plaster, and brick serves to emphasize the 



266 



ACOUSTICS AND ARCHITECTURE 



point already made, that there is not a single numerical 
relation between weight and sound reduction that will 
hold for all materials, regardless of their other mechanical 
properties. 

Bureau of Standards Results. 

The fact of the predominating part played by the weight 
per unit area in the reduction of sound produced by walls 
of masonry material was first stated by the writer in 1923. x 
No generalization beyond the actual facts of experiment 
was made. Since that time, work in other laboratories 
has led to the same general conclusion in the case of other 



bO 
50 
40 
30 
20 
10 

-2 














<Ma$c 
3/ 


nry> 

J 










<Lead,q> 


iss,steel> 


''>' 

"' - i& 












G 
ff4 

6 


A 

'S 


'/ 




Paper, fabric, alum 


num 


G 


/ 




tit 


ood, fib 


er boar 


te. 






^ 


\ - Bureau of Standards measurements 
1-Rtverbank measurements 
"5-Heim burgers measurements 


/ 


/ 


Logar 


thm of 


Weight 


'Area 






-1 

! 


5 -1 

1 1 111! 


-0 

1 


500 

1 1 Mill 1 


5 10 1 

MM!!! 1 


5 2 

1 1 III 



001 002 004 01 02 0.4 0.6 10 20 40 60 100 20 40 60 100 
Weight/Area 

Fio. 101. Reduction as a function of weight per unit area. Results from 
different laboratories compared. 

homogeneous materials. The straight line 1 (Fig. 101) 
shows the results of measurements made at the Bureau of 
Standards 2 on partitions ranging from a single sheet of 
wrapping paper weighing 0.016 to walls of brick weighing 
more than 100 Ib. per square foot. For comparison, the 
Riverbank results on masonry varying from 10 to 88 Ib. 

1 Amer. Architect, July 4, 1923. 

2 CHRISLER, V. L. f and W. F. SNYDBR, Bur. Standards Res. Paper 48, 
March, 1929. 



TRANSMISSION OF SOUND BY WALLS 267 

per square foot and also results for glass, wood, and steel 
partitions are shown. 

We note that from the Bureau of Standards tests the 
points for the extremely light materials of paper, fabric, 
aluminum, and fiber board show a linear relation between 
the average reduction and the logarithm of the weight per 
unit area. The heavier, stiffer partitions of lead, glass, 
and steel as well as the massive masonry constructions 
tested by the Bureau show, in general, lower reductions 
than are called for by the extrapolation of the straight line 
for the very light materials. 

Taken by themselves, the Bureau of Standards figures 
for heavy masonry constructions, ranging in weight from 
30 to 100 Ib. per square foot, do not show any very definite 
correlation between sound reduction and weight. On the 
whole, the points fall closer to the Riverbank line for 
continuous masonry than to the extension of the Bureau 
of Standards line for light materials. 

Figure 101 gives also the results on homogeneous struc- 
tures ranging in weight from 10 to 50 Ib. per square foot, 
as reported by Heimburger. 1 As was indicated in Chap. 
XI, in Kreuger's tests, a loud-speaker horn was placed 
close to the panel and surrounded by a box, and the 
intensities with and without the test panels intervening 
were measured. This method will give much higher 
values of the reduction than would be obtained if the whole 
face of the panel were exposed to the action of the 
sound. It is worth noting, however, that the slope of the 
line representing Kreuger's data is very nearly the same 
as that for the Riverbank measurements. 

Here, again, the data presented in Fig. 101 lead one to 
doubt whether there is a single numerical formula connect- 
ing the weight and the sound reduction for homogeneous 
structures that will cover all sorts of materials. Equation 
(81) gives an approximate statement of the relation 
between weight and reduction for all-masonry construc- 
tions. The Bureau of Standards findings on light septa 

1 Amer. Architect, vol. 133, pp. 125-128, Jan. 20, 1928 



268 



ACOUSTICS AND ARCHITECTURE 



(less than 1 Ib. per square foot) can be expressed by a 
similar equation but with different constants, namely: 

k = SGOw 1 - 43 (82) 

It should be borne in mind that both Eqs. (81) and (82) 
are simply approximate generalizations of the results 
of experiment and apply to particular test conditions. 

The data obtained for the sound-proof doors bear inter- 
estingly upon this point. Plotting the sound reduction 
against the weight per square foot, we find again a very 
close approximation to a straight line. From the equation 
of this line one gets the relation 



as the empirical relation between weight per square foot 
and sound reduction for these constructions. Here the 
exponent 2.1 is close to the theoretical value 2.0, derived 
on the assumption that the mass per unit area is the only 
variable. In this series of experiments, this condition was 
very nearly met. The thickness varied only slightly, 
2% to 3 in., and the metal was incor- 
porated in the heavier doors in such a 
way as not materially to increase their 
structural stiffness, while adding to 
their weight. 

Lacking any general theoretical 
formulation, the graphs of Fig. 101 
furnish practical information on the 
degree of sound insulation by homo- 
geneous structures covering a wide 
variation in physical properties. 




Fio. 102. Double 
partitions completely 
separated. 



Double Walls, Completely Separated. 

In practice, it is seldom possible to 
build two walls entirely separated. 
They will of necessity be tied together at the edges. 
The construction of the Riverbank sound chamber and 
the test chambers is such, however, as to allow two 
walls to be run up with no structural connection what- 



TRANSMISSION OF SOUND BY WALLS 



269 



soever (Fig. 102). This arrangement makes it possible 
to study the ideal case of complete structural separation 
and also the effect of various degrees of bridging or tying 
as well as that produced by various kinds of lagging fill 
between the walls. Figure 103 gives the detailed results 
of tests on a single wall of 2-in. solid gypsum tile and of two 
such walls completely separated, with intervening air 
spaces of 2 in. and 4 in., respectively. One notes that the 
increased separation increases the insulation for tones up 
to 1,600 vibs./sec. At higher frequencies, the 2-in. 



/u 
60 
50 
40 
30 
20 
10 








^x"V''' 


7 




, 


/3~~ 


'/ N 


/ 


,j 


.^"^ 

/ 


/ 






f. 


/ 


I 


-X / 






/ 
* / 


^r 


1 - Single wall 
?-2 -walls, ?" separation 
J 2worlls t <t" separation 


/^ 





1E8 



E50 



512 IOZ4 2048 4096 
Frequency 

FIG. 103. Effect of width of air space between structurally isolated partitions. 

separation is better. Another series of experiments showed 
that still further increase in the separation shifts the dip 
in the curve at 2,048 vibs./sec. to a lower frequency. This 
fact finds its explanation in the phenomenon of resonance 
of the enclosed air, so that there is obviously a limit to the 
increased insulation to be secured by increasing the 
separation. 

Figure 104 shows the effect (a) of bridging the air gap 
with a wood strip running lengthwise in the air space and 
in contact with both walls and (&) of filling the inter-wall 
space with sawdust. One notes that the unbridged, 
unfilled space gives the greatest sound reduction and, 
further, that any damping effect of the fill is more than 
offset by its bridging effect. Experiments with felt and 



270 



ACOUSTICS AND ARCHITECTURE 



granulated blast-furnace slag showed the same effect, so 
that one arrives at the conclusion that if complete structural 
separation were possible, an unfilled air space would be the 
most effective means of securing sound insulation by means 
of double partitions. As will appear in a later section, 




256 



2046 4096 



512 1024 

Frequency 

FIG. 104. Effect of bridging and filling the air spaoe between structurally iso- 

lated partitions. 

this conclusion does not include cases in which there is a 
considerable degree of structural tying between the two 
members of the double construction. 

Table XX gives the summarized results of the tests on 
double walls with complete structural isolation. 

TABLE XX . DOUBLE WALLS COMPLETELY SEPARATED 



Num- 
ber 


Specifications 


Weight per 
square foot, 
pounds 


Reduc- 
tion, 
decibels 


Equivalent 
masonry, 
inches 


40 


Double 2-in. solid-gypsum tile, 
unplastercd, unbridged, 2-in. 
separation 


20 4 


56 


10 


41 
42 
43 
44 
45 


The same, bridged at middle.. . . 
The same, filled with sawdust . 
The same, filled with slag. . . 
The same, filled with felt 
The same as No. 40 but with 4-in. 
separation 


20.4 
23.0 
30 9 
22.3 

20.4 


48 
47.7 
48.6 
55.1 

59 


5 4 
5.4 
5 7 
9 3 

12 5 


46 


The same as No. 45 but bridged 
top and bottom . . 


20 4 


53 3 


8 3 


47 


The same as No. 45 but with inner 
faces lined with 1-in. felt. . . 


22 3 


65 





TRANSMISSION OF SOUND BY WALLS 



271 



Double Partitions, Partially Connected. 

Under this head are included types of double walls in 
which the two members are tied to about the same degree 
as would be necessary in ordinary building practice. In 
this connection, data showing the effect of the width of the 
air space may be shown. This series of tests was conducted 
with two single-pane K-in. plate glass windows 82 by 34 in. 
set in one of the sound-chamber openings. Spacing 
frames of 1-in. poplar to which 2^-in. saddler's felt was 
cemented were used to separate the two windows. 

The separation between the windows was increased by 
increasing the number of spacing frames. It is evident 
that the experiments did not show the effect of increased 
air space alone, since a part of the transfer of sound energy 
is by way of the connection at the edges. However, the 
results presented in Table XXI show that the spatial 
separation between double walls does produce a very 
appreciable effect in increasing sound insulation. 

TABLE XXI, DOUBLE WINDOWS, K-IN. PLATE GLASS 



Num- 
ber 


Description 


Reduction, 
decibels 


Equivalent 
masonry, 








inches 


48 


Sashes in contact . . 


33.2 


1.7 


49 


13^-in. separation 


38.6 


2 5 


50 


4^-in. separation 


40 1 


2 9 


51 


7^-in. separation . 


44.2 


4 


52 


9^-in. separation 


46 3 


4 7 


53 


133^-iri. separation 


48.2 


5 5 


54 


16-in. separation. . 


48.8 


5 8 



Experiments in which a solid wood spacer replaced the 
alternate layers of felt and wood showed practically the 
same reduction, as shown by the alternate wood and felt, 
indicating that the increasing insulation with increasing 
separation is to be ascribed largely to the lower transmission 
across the wider air space rather than to improved insula- 
tion at the edges. 



272 



ACOUSTICS AND ARCHITECTURE 




gypsum 



"*4" batten \- Rock lath r < - 
cr/es- 7' apart 



24" 



fe 



: 57 



3 /4"C-/2"O.C. &$* /Vo30 flat expJath 

Efcfc 



^ 






^t Gypsum plaster 
Ce/ofex (Loose) 




3-5$ 



^ P/asfer on tile 
FIG. 105. Double walls with normal amount of bridging. 



/y 
60 
50 

30 
20 
























^ 


/"- 


""^iv 


/ 




/^x 


/^ 


\ 


V 


& 


AerM^ 


j4/4flw/ 

rtn/ifer 


TDCvut-r? 


X^wte>7 
^2-0(7. 




2 - 7A $ame,batfitn plate 



126 256 512 1024 2048 4096 
Frequency 

FIQ. 106. Double walls: (1) loosely tied; (2) closely tied. 



TRANSMISSION OF SOUND BY WALLS 



273 



Figure 105 shows a number of types of double wall con- 
struction that have been tested, with results shown in 
Table XXII. 

TABLE XXII. DOUBLE WALLS, CONNECTED AT THE EDGES 



Num- 

u__ 


Description 


Weight 
per 
square 


Reduc- 
tion, 


Thick- 
ness, 


Equiv- 
alent 


Der 




foot. 


decibels 


inches 


masonry, 






pounds 








55 


Staggered 2 X 4-in wood studs: 












a. Metal lath, \4-\n. gypsum plaster 


19.8 


44.4 


7H 


4 




b. %-in. Celotex, >a-in. gypsum plaster 


13.0 


41.6 


7>* 


3.3 


56 


2 X 2 in. wood studs, set on 6-in. plate. 


) 










K-in. gypsum plaster on M-m. Celotex 


>12.2 


52.2 


8 


7 5 




H-in. Celotex stood loosely between . . . 


) 








57 


K-in steel studs, rock lath, K-in. gypsum 












plaster: 












o. Batten plates between angles 7 ft. O.C. 


12.0 


45.3 


5H 


4.3 




6. Batten plates 2 ft. O.C .. 


12 


35.9 


5% 


2.1 


58 


Double metal lath on %-in. channels, ?*- 












in. gypsum plaster: 












o. Without cross-bracing clips. 


18 


49.5 


4 


6.0 




6. With cross-bracing clips 


18.0 


39.7 


4 


2.9 


' 59 


Double 3-in. hollow gypsum tile: 












a. Unplastered, 3-in air space 


22 


42 6 


9 


3.6 




6. Unplastered 2H-in. air space, ^j-in 












felt 


22 6 


46 


9 


4.5 




c Same as 6, with 1-in. plaster. 


31 8 


48.0 


10 


5.4 




d. Plastered, M-in. Celotex in 2-in. air 












space 


32 


47 8 


9 


5 4 















Figure 106 shows in a striking way the effect of the 
bridging by the batten plates tying together the two >-in. 
steel angles forming the steel stud of No. 57. Table XXII 
indicates the limitation imposed by excessive thickness upon 
sound insulation by double wall construction. In only one 
case that of the undipped double metal-lath construction 
is the double construction thinner than the equivalent 
masonry. The moral is that, generally speaking, with 
structural materials one has to pay for sound insulation 
either in increased thickness using double construction or 
by increased weight using single construction. 

Wood-stud Partitions. 

The standard wood-stud construction consists of 2 by 
4-in. studs nailed bottom and top to 2 by 4-in. plate and 



274 



ACOUSTICS AND ARCHITECTURE 



header. One is interested to know the effect on sound 
insulation of the character of the plaster whether lime or 
gypsum the character of the plaster base wood lath, 
metal lath, or fiber boards of various sorts and finally the 
effect of filling of different kinds between the studs. 
Table XXIII gives some information on these points. 
The plaster was intended to be standard scratch and 
brown coats, J- to ^g-in. total thickness. The weight in 
each case was determined by weighing samples taken from 
the wall after the tests were completed. Figure 107 shows 
the effect of the sawdust fill in the Celotex wall both with 
and without plaster. The contrast with the earlier case, 

TABLE XXIII. WOOD-STUD WALLB 









.Weight 


R educ- 


Equiva- 


Num- 
ber 


Plaster material 


Plaster base 


per square 
foot, 
pounds 


tion, 
decibels 


lent 
masonry, 
inches 


60 


Gypsum . 


Metal lath 


17 4 


33 2 


1 8 


61 


No. 60 filled with granulated slag 


Metal lath 


27.4 


41.6 


3 4 


62 


Gypsum . . 


Wood lath 


18.0 


33.4 


1.8 


63 


Lime.. 


Wood lath 


17.4 


43 2 


3.8 


64 


None. ... 


?-in. Celotex 


3 


26.2 


1 


65 


No. 64 filled with sawdust 


>2~m. Celotex 


6.6 


34 


1 8 


66 


Gypsum 


J-2-in. Celotex 


12 


37 


2 3 


67 


No 66 filled with sawdust 


M-in. Celotex 


15 6 


39 8 


2 9 


68 


Gypsum, % in. thick 


H-in. masomte 


16 


43 


3 8 


69 


Gypsum 


H-in. felt, H- 












in. furring 












metal lath 


18 


45 4 


4.6 



where there was no structural tie between the two members 
and in which the sawdust filling actually decreased the 
insulation, is instructive. In the wood-stud construction, 
the two faces are already completely bridged by the studs, 
so that the addition of the sawdust affords no added bridging 
effect. Its effect is therefore to add weight and possibly 
to produce a damping of the structure as a whole. This 
suggests that the answer to the question as to whether a 
lagging material will improve insulation depends upon the 
structural conditions under which the lagging is applied. 
In Fig. 107, it is interesting to note the general similarity 
in shape of the four curves and also the fact that the 



TRANSMISSION OF SOUND BY WALLS 



275 



addition of the sawdust mak'es a greater improvement in 
the light unplastered wall than in the heavier partition 
after plastering. 



50 
40 
30 
20 

to 
















4- , 


>^ 






^J 


'X" 


.--" 


^ 


/? 


s /-- 
." 

-"'' / 


-x' 




V 


x I 


1 Ce/oi 
~2 Celot 
3 CeM 
A- Ce/oi 


Vf,unp/as 
vx,unplast 
'ex, pi as fe 
ex,p/aste 


fared, uni 
wecLsam 
red, unfit 
red.satvo 


Wed 
titsffi/r 
ted 
y,s//?// 



128 256 512 1024 2048 4096 
Frequency 

FIG. 107. Effect of filling wood stud, Celotcx, and plaster walls. 

60r 




8 10 20 30 40 50 60 60 100 

Weight per Sq Ft. 

FIG. 108. Various constructions compared with masonry walls of equal weight. 

General Conclusions. 

Figure 108 presents graphically what general conclusions 
seem to be warranted by the investigation so far. In the 



276 ACOUSTICS AND ARCHITECTURE 

figure, the vertical scale gives the reduction in decibels; and 
the horizontal scale, the logarithm of the weight per square 
foot of the partitions. The numbered points correspond 
to various partitions described in the preceding text. 

1. For continuous masonry of clay and gypsum tile, 
plaster, and brick the reductions will fall very close to the 
straight line plotted. Wood-stud construction with gyp- 
sum plaster falls on this line (No. 62). Lime plaster on 
wood studs and gypsum plaster on fiber-board plaster 
bases on wood studs give somewhat greater reductions than 
continuous masonry of equal weight (Nos. 63, 68, 66). 
Glass and steel show greater reductions than masonry of 
equal weights (Nos. 1 and 10). The superiority of lime 
over gypsum plaster seems to be confined to wood-stud 
constructions. The Bureau of Standards reports the results 
of tests in which lime and gypsum plasters were applied to 
identical masonry walls of clay tile, gypsum tile, and 
brick. In each case, two test panels were built as nearly 
alike as possible, one being finished with lime plaster. In 
each case, the panel finished with the gypsum plaster 
showed slightly greater reduction than a similar panel 
finished with lime plaster. The difference, however, was 
not sufficiently great to be of any practical importance. 1 

These facts bring out the fact referred to earlier, that the 
sound insulation afforded by partitions is a matter of struc- 
tural properties, rather than of the properties of the 
materials comprising the structure. 

2. The reduction afforded by double construction is 
a matter of the structural and spatial separation of the two 
units of the double construction (cf. Nos. 45, 40, 42, also 
58a and 586). In double constructions with only slight 
structural tying, lagging fills completely filling the air 
space are not advantageous. In hollow construction, where 
filling appreciably increases the weight of the structure, 
filling gives increased insulation (compare No. 64 with No. 
65, and No. 66 with No. 67). In each case, the increased 
reduction due to the filling is about what would be expected 

1 See Bur. Standards Sci. Papers 526 and 552. 



TRANSMISSION OF SOUND BY WALLS 277 

from the increase in weight. It is fairly easy to see that since 
the filling material is incorporated in the wall, it can have 
only slight damping effect upon the vibration of the structure 
as a whole. Following this line of reasoning, the increase 
in reduction due to filling should be proportional to the 
logarithm of the ratio of the weight of the filled to the 
unfilled wall. This relationship is approximately verified 
in the instances cited. However, the slag filling of the metal 
lath and plaster wall (Nos. 60 and 61) produces a somewhat 
greater reduction than can be accounted for by the increased 
weight, so that the character of the fill may be of some slight 
importance. 

Meyer's Measurements on Simple Partitions. 

Since the foregoing was written, an important paper by 
E. Meyer 1 on sound insulation by simple walls has come 
to hand. 

This paper not only reports the results on sound trans- 
mission but also gives measurements of the elasticity and 
damping of 12 different simple partitions ranging in weight 
from 0.4 to 93 Ib. per square foot. The frequency range 
covered was from 50 to 4,000 vibs./sec. Actual measure- 
ments of the amplitude of vibration of the walls under 
the action of sound waves were also made. For this 
purpose, a metal disk, attached to the wall, served as one 
plate of an air condenser. This condenser constituted a 
part of the capacity of a high-frequency vacuum-tube 
oscillator circuit. The vibration of the walls produced a 
periodic variation in the capacity of this wall plate-fixed 
plate condenser, a variation which impressed an audio- 
frequency modulation upon the high-frequency current 
of the oscillator. These modulated high-frequency cur- 
rents were rectified as in the ordinary radio receiving set, 
and the audiofrequency voltage was amplified and measured 
by means of a vacuum-tube voltmeter. The readings of 
the latter were translated into amplitudes of wall vibrations 

1 Fundamental Measurements on Sound Insulation by Simple Partition 
Walls, Sitzungber. Preuss. Akad. Wis. t Phys.-Math. Klasse, vol. 9, 1931. 



278 



ACOUSTICS AND ARCHITECTURE 



by allowing the wall to remain stationary. The fixed 

plate of the measuring condenser was moved by means of a 

TABLE XXIV. MEYEU'S DATA ON VIBRATION OF WALLS 



Kind of wall 


Thick- 
ness, 
inches 


Weight 
per 
square 
foot, 
pounds 


Natural 
frequency 


Damp- 
ing, 
db. per 
sec. 


Elasticity 
modulus, 
kg./cm.2 


Trans- 
mission 
loss, 
decibels 


Calcu- 
lated 


Meas- 
ured 


Wood plate 


0.2 

08 
1.0 
3.5 

4 75 
4 3 
( 3.5 
< 6 
( 10.5 


0.45 

3.2 
6.3 
13 

25.0 
26 7 
31.2 
46 5 
93.0 


7.8 

10.5 
16.0 
24 

32.0 
31.0 
27 5 
45 
75 


7.0 

12.0 
17.0 
30.0 

38.0 
28.0 
29.0 
48.0 
51 


9.5 

5 4 
11.1 
41.0 

54 
16.0 
87.0 
69.0 
48 


110,000 

18,600,000 
42,000 
4,300 

5,600 
8,100 
13,000 
12,000 
11,000 


18 5* 

34.0 
33.0 
38 

39.0 
41.0 
42.0 
43 
49.0 


Sheet iron, stiffened in 
middle 




Pressed straw, plastered. . 

Schwrrnmbteinwand, plas- 
tered 


Pumieo concrete 


Brick wall, plastered 



micrometer screw. The tube voltmeter reading was thus 
calibrated in terms of wall movement. The author of the 
paper claims that amplitudes as small as 10~ 8 cm. can be 
measured in this way. 

The same device was also used to measure the deflection 
of the walls under constant pressure. From these measure- 
ments the moduli of elasticity were computed. Finally, 
by substituting an oscillograph for the vacuum-tube 
oscillator, records of the actual vibration of the walls 
when struck by a hammer were made. From these 
oscillograms the lowest natural frequency and the damping 
were obtained. 

In Table XXIV, data given by Meyer are tabulated. 
We note the low natural frequency of these walls, for the 
most part well below the lower limit of the frequency range 
of measurements. We note also their relatively high 
damping, with the exception of the steel and wood. This 
high damping would account for the fact that the changes 
in sound reduction with changes in frequency are no more 
abrupt than experiments prove them to be. It is further 
to be noted that, excepting the first three walls listed, 
the moduli of elasticity are not widely different for the 



TRANSMISSION OF SOUND BY WALLS 



279 



different walls. The steel, however, has very much 
greater elasticity than any of the other walls, and it is to 
be noted that the steel shows a much greater transmission 
loss for its weight than do the other materials. This fact 
has already been observed in the Riverbank tests. We 
should expect this high elasticity of steel to show an even 
greater effect than it does were it not for the fact that the 
great elasticity is accompanied by a relatively low damping. 
Meyer's work throws light on the "why" of the facts 
that the Riverbank and the Bureau of Standards researches 
have brought out. For example, the fact that the relation 
between mass and reduction is so definitely shown by the 



60 

50 

g40 
| 

230 

20 
10 


i 


1 1 1 
Meyers 
Knudsen 
PESabme 
a Paper, Fabr 
b We, Plaster, 








-- 












s 

*^j 


__ 


ic, efc [ 
Brick // 


9ur fffS 
^/verbal 

<? 


land] 

7A7 

!b 






. 


8&* 
^9 
b 


4- 


?. 

^^: 


"-"""^ 


$&- 

* 

r 


s' '10 






-c 


a . 
4 - 


. 


_^ 


^ 








)2 ( 


) C 


2 ( 


Loq of 
40 


the We 
6 


r /A ?o , 


2 \ 


I 1 


> i 


\.. 2, 



II I I I 1 1 1 



~04 0506 0810 



I I I I 1.1 1 II 

30 40 5060 ao 10 



20 



_L 



20 30 40 5060 80100 



Weight/Area 
FIG. 109. Comparison of results obtained at different laboratories. 

walls listed in Table XIX finds an explanation in the fact 
that the modulus of elasticity is probably fairly constant 
throughout the series of walls and that the internal fric- 
tioiial resistance is nearly constant, hence the damping 
increases according to a definite law with increasing 
massiveness of the walls. These facts, together with the 
fact that the fundamental natural frequency is low in all 
cases, would leave the predominating role in determining 
the response to forced vibrations to be played by the mass 
alone. 

Further, Meyer's work would seem to show that in the 
cases of wide variation in the damping and elasticity, this 
effect of mass alone may be masked by these other factors. 
Sheet iron is a case in point. The fact that sheet iron only 



280 



ACOUSTICS AND ARCHITECTURE 



0.08 in. thick stiffened at the middle shows so high a reduc- 
tion is quite in agreement with the Riverbank tests on 
windows, which showed that the cross bracing of the sash 
effected an appreciable increase in the insulating power. 

In Table XXV, Meyer's results are given together with 
figures on comparable constructions as obtained at the 
Riverbank Laboratories. These values are plotted in 
Fig. 109. On the same graph, values given by Knudsen 1 
are shown for a number of walls, the characters of which 
are not specified. For further comparison, the Bureau of 
Standards line for light septa of paper, fabric, aluminum, 
and wood as well as the Riverbank line for tile, plaster, 
and brick are shown. 

TABLE XXV 



Num- 
ber 


Wall 


Weight per 
square foot, 
pounds 


Trans- 
mission 
loss, 
decibels 


Author- 
ity 


1 


Wood 0.2 in. thick 


0.45 


18.5 


Meyer 


2 


Plate glass, 0.11 to 0.16 in. thick 


2.4 


28 


Meyer 


3 


Plate glass, 0.27 to 0.31 in. thick 


3 2 


29 


Meyer 


3a 


Plate glass, 0.25 in. thick 


3.2 


27.0 


Sabine 


36 


Plate glass, 0.25 in. thick (cross 










braced) 


3 2 


30.0 


Sabine 


4 


Rabitzwand 1 in. thick. . . . 


6 1 


33 


Meyer 


5 


Sheet iron stiffened 


3.2 


34.0 


Meyer 


5a 


Sheet steel, no bracing. 


10.0 


34.0 


Sabine 


6a 


Celotcx plastered 


12.0 


37.0 


Sabine 


66 


Pressed straw plastered 


14.0 


38 


Meyer 


7 


Schwemmsteinwand plastered . . 


25.0 


39.0 


Meyer 


8 


Pumice concrete, cork interior . 


27 


41 


Meyer 


9 


3?--in. wall plastered 


31.2 


42.0 


Meyer 


10 


Brick wall unplastered . . 


22.0 


33 


Meyer 


11 


6-in. brick wall plastered. . 


46 


43.0 


Meyer 


12 


10}^-in. brick wall plastered 


93 


49.0 


Meyer 



One notes very fair agreement between Meyer's figures 
and the Riverbank figures for glass (c/. Nos. 3, 3a, 36). 
Moreover, the transmission loss shown by the plastered 
compressed straw measured by Meyer is quite comparable 

1 Jour. Acous. Soc. Amer., vol. 2, No. 1, p. 133, July, 1930. 



TRANSMISSION OF SOUND BY WALLS 281 

to the Riverbank figures for plastered Celotex board of 
nearly the same weight (cf. Nos. 6a, 66) . With the excep- 
tion of the 10>^-in. brick wall, Meyer's values for brick 
do not depart very widely from the line for plaster, tile, 
and brick, shown by the Riverbank measurements. The 
unplastered brick (No. 10) and the Schwemmsteinwand 
(No. 7) both fall below Meyer's line. On the other hand, 
the pumice concrete, which falls directly on Meyer's line, 
is similar in structure to those materials, such as cinder and 
slag concrete, which according to the Riverbank tests 
showed higher transmission losses than walls of ordinary 
masonry of equal weight. 

On the whole, Meyer's results taken alone might be 
thought to point to a single relation between the mass and 
the sound insulation by simple partitions. Viewed criti- 
cally and in comparison with the results of other researches 
on the problem, however, they still leave a question as to 
the complete generality of any single relation. Obviously 
there are important exceptions, and for the present at 
least the answer must await still further investigation. 
The measurements of the elasticity and damping of struc- 
tures in connection with sound-transmission measurements 
is a distinct advantage in the study of the problem. 



CHAPTER XIII 
MACHINE ISOLATION 

In every large modern building, there is usually a certain 
amount of machine installation. The operation of venti- 
lating, heating, and refrigerating systems and of elevators 
calls for sources of mechanical power. In many cases, 
both manufacturing and merchandising activities are 
carried on under a single roof. It therefore becomes 
important to confine the noise of machinery to those por- 
tions of a building in which it may originate. There are 
two distinct ways in which machine noise may be propa- 
gated to distant parts of a building. 

The first is by direct transmission through the air. The 
second is by the transmission of mechanical vibration 
through the building structure itself. These vibrations 
originate either from a lack of perfect mechanical balance 
in machines or, in the case of electric motors, from the 
periodic character of the torque on the armature. These 
vibrations are transmitted through the machine supports to 
the walls or floor, whence, through structural members, 
they are conducted to distant parts of the building. The 
thoroughly unified character of a modern steel structure 
facilitates, to a marked degree, this transfer of mechanical 
vibration. In the very nature of the case, good construc- 
tion provides good conditions for the transfer of vibrations. 
The solution of the problem therefore lies, first, in the design 
of quietly operating machines and, second, in providing 
means of preventing the transfer of vibrations from the 
machines to the supporting structure. 

The reduction of vibration by features of machine design 
is a purely mechanical problem. 1 

1 For a full theoretical treatment the reader should consult a recent text 
on the subject: "Vibration Problems in Engineering," by Professor S. 

282 



MACHINE ISOLATION 283 

Related to this problem but differing from it in certain 
respects is the problem of floor insulation. In hotels 
and apartment houses, the impact of footfalls and the sound 
of radios and pianos are frequently transmitted to an 
annoying degree to the rooms below. Experience shows 
that the insulation of such noise is most effectively accom- 
plished by modifications of the floor construction. The 
difficulty arises in reconciling the necessities of good con- 
struction with sound-insulation requirements. 

Natural Frequency of a Vibrating System. 

The usual method of vibration insulation is to mount 
the machine or other source of vibration upon steel springs, 
pads of cork, felt, rubber, or other yielding material. The 
common conception of just how such a mounting reduces 
the transmission of vibration to the supporting structure is 
usually put in the statement that the "pad damps the vibra- 
tion of the machine/' 

As a matter of fact, the damping action is only a part 
of the story, and, as will appear later, a resilient mounting 
may under certain conditions actually increase the trans- 
mission of vibrational energy. It is only within recent 
years that an intelligent attack has been made upon the 
problem in the light of our knowledge of the mechanics of 
the free and forced vibration of elastically controlled 
systems. 

A complete mathematical treatment of the problem is 
beyond our present purpose. 1 We shall try only to present 
as clear a picture as possible of the various mechanical 
factors involved and a formulation without proof of the 
relations between these factors. For this purpose, we 
shall consider the ideally simple case of the motion of a 

Timoshenko, D. Van Nostrand Company, New York, 1928. A selected 
bibliography is to be found in " Noise and Vibration Engineering," by 
S. E. Slocum, D. Van Nostrand Company, New York, 1931. 

1 For such a mathematical treatment, the reader should consult any 
text on the theory of vibration, e.g., Wood, "Textbook of Sound," p. 36 
et scq., G. Bell & Sons, London, 1930. Crandall, "Theory of Vibrating 
Systems and Sound," p. 40, D. Van Nostrand Company, New York, 1926. 



284 



ACOUSTICS AND ARCHITECTURE 



system free to move in only one direction, displaced from 
its equilibrium position, and moving thereafter under the 
action of the elastic stress set up by the displacement and 
the frictional forces generated by the motion. Figure 110 
represents such an ideal simple system. The mass m is 
supported by a spring, and its motion is damped by 
the frictional resistance in a dashpot. Assume that T y the 
force of compression of the spring, is proportional to the 





Fia. 110. The mass m moves under the action of the elastic restoring force of the 
spring and the frictional resistance in the dash pot. 

displacement from the equilibrium position (Hooke's 
law). 



where s is the force in absolute units that will produce a 
unit extension or compression of the spring. We shall call s 
the " spring factor." 

Assume further that the frictional resistance at any time 
due to the motion of m is proportional to the velocity at 
that time and that it opposes the motion. The frictional 
force called into play by a velocity is r. The force due 
to the inertia of the mass m moving with an acceleration 
is m. If there is no external applied force, then the force 
equation for the system is 



= 



(83) 



MACHINE ISOLATION 285 

In mathematical language, this is a " homogeneous linear 
differential equation of the second order/' and its solutions 
are well known. 1 

For the present purpose, the most useful form for the 
solution of Eq. (83) is 

(84) 



sn 

where k, called the " damping coefficient/' is defined by 
the equation k = r/2m, and coi is defined by the equation 



(85) 

<p is an arbitrary constant whose value depends upon the 
displacement at the moment from which we elect to measure 
times, /i is the frequency of the damped system. If the 
system is displaced from its equilibrium position and then 
allowed to move freely, its motion is given by Eq. (84). 
There are two possible cases. If r 2 /4w 2 > s/m, i.e., if the 

(s r 2 \ 
4 2) i g negative, 

and its square root is imaginary. The physical inter- 
pretation of this is that in such a case the motion is not 
periodic, and the system will return slowly to its equilibrium 
position under the action of the damping force. Under 
the other possibility, r 2 /4m 2 < s/m, the system in coming 
to rest will perform damped oscillations with a frequency of 
1 



-- .__ B j n the usual dashpot damping, the resist- 



ance term is large, thus preventing oscillations. The 
automobile snubber is designed to increase the frictional 

1 WOOD, "Textbook of Sound," p. 34. MELLOR, "Higher Mathematics 
for Students of Chemistry and Physics," p. 404, Longmans, Green & Co., 
London, 1919. 

For the solution in the analogous electric case of the discharge of a 
condenser through a circuit containing inductance and resistance, see 
PIEBCE, "Electric Oscillations and Electric Waves," p. 13, McGraw-Hill 
Book Company, Inc., New York, 1920. 



286 



ACOUSTICS AND ARCHITECTURE 



resistance and thus reduce the oscillations that would other- 
wise result from the elastic action of the spring. 

In any practical case of machine isolation in buildings, 
the free movement of the machine on a resilient mounting 
will be represented by the second case, r*/4ra 2 < s/m. 
The motion is called " damped sinusoidal oscillation" and 
is shown graphically in Fig. 111. The dotted lines show 
the decrease of amplitude with time due to the action of 
the damping force. 




1.0 Sees, 



. Graph of damped sinusoidal oscillation. Ao/Ai Ai/A? 



Forced Damped Oscillations. 

In the foregoing, we have considered the movement 
when no external force is applied. If impacts are delivered 
at irregular intervals, the motion following each impact is 
that described. If, however, the system be subjected to a 
periodically varying force, then in the steady state it will 
vibrate with the frequency of the driving force. Thus in 
alternating-current generators and motors there is a periodic 
torque of twice the frequency of the alternating current. 
Rotating parts which are not in perfect balance give rise 
to periodic forces whose frequency is that of the rotation. 

For mathematical treatment, suppose that the impressed 
force is sinusoidal with a frequency / = co/2w and that it 
has a maximum value F . Then the motion of the system 



MACHINE ISOLATION 287 

shown in Fig. 110, under the action of such a force, is 
given by the equation 

ml + r + s = F Q sin otf (86) 

The solution of this equation is well known, 1 and the 
form of the expression for the displacement in the steady 
state in terms of the constants of Eq. (86) will depend upon 
the relative magnitudes of m, r, and s. There are three 
cases : 

w2 o 

Case I |^- 2 < (small damping) 

r 2 s 
Case II -2 = (critical damping) 



T 2 S 

Case III 4^-2 > (large damping) 

Practical problems of machine isolation come under 
case I, and the particular solution under this condition is 
given by the equation 

^o / - i - 

L*. = Ao = ^ \4fc^F W i _ W 2 (87) 



Here o> = s/m, k = r/2m, and o> = 2?r times the frequency 
of the impressed force. It can be easily shown that the 
natural frequency of the undamped system is given by the 

relation __ 

i / 

* = 27T = 2w\m 

From Eq. (87) it is apparent that the amplitude of the 
forced vibration for a given value of the impressed force is 
a maximum when w = w , i.e., when the frequency of the 
impressed force coincides with the natural frequency of 
the vibrating system. In this case, Eq. (87) reduces to 

T~ Fo 



^ ~~ m \4A; 2 co 2 ~ r 
This coincidence of the driving frequency with the 

1 WOOD, A. B., "A Textbook of Sound," p. 37. 



288 ACOUSTICS AND ARCHITECTURE 

natural frequency of the vibrating system is the familiar 
phenomenon of resonance, and Eq. (88) tells us that the 
amplitude at resonance in the steady state is directly 
proportional to the amplitude of the driving force and 
inversely proportional to the coefficient of frictional 
resistance. It is apparent that when the impressed fre- 
quency is close to the resonance frequency, the frictional 
resistance plays the preponderant r61e in determining the 
amplitude of vibration set up. If we could set up a system 
in which there were no frictional damping whatsoever, 
then any periodic force no matter how small operating at 
the resonance frequency would in time set up vibrations of 
infinite amplitude. This is the scientific basis for the often 
repeated popular statement that the proper tone played 
on a violin would shatter the most massive building. 
Fortunately for the permanence of our buildings, move- 
ments of material bodies always call frictional forces into 
play. The vibrations set up in the body of an automobile 
for certain critical motor speeds is a familiar example of 
the effect of resonance. It frequently occurs that machines 
which are mounted on the floor slab in steel and concrete 
construction set up extreme vibrations of the floor. This 
can be frequently traced to the close proximity of the oper- 
ating speed of the machine to a natural frequency of the 
floor supporting it. The tuning of a radio set to the 
incoming electromagnetic frequency of the sending station 
is an application of the principle of resonance. 

Inspection of Eq. (87) shows that while the frictional 
damping is most effective in reducing vibrations at or near 
resonance, yet increasing k decreases the amplitude for 
all values 'of the frequency of the impressed force. In 
general, it may be said therefore that if we are concerned 
only with reducing the vibration of the machine on its 
support, the more frictional resistance we can introduce 
into the machine mounting the better. As we shall see, 
however, this general statement does not hold true, if we 
concern ourselves with the transmission of vibration to the 
supporting floor. 



MACHINE ISOLATION 289 

Inertia Damping. 

We consider now the question of the effect of mass 
upon the amplitude of vibration of a system under the 
action of an impressed periodic force. In the expression 
for the amplitude given in Eq. (87), it would appear that 
increasing m will always decrease A , since m appears in 
the denominator of the right-hand member of the equation. 
It must be remembered, however, that m is involved in 

both k( r/2m) and co (= ^s/m). Putting in these 
values, Eq. (87) may be thrown into the form 



(s - <mo> 2 ) 2 



(88) 



It is clear that the effect produced on A by increasing m 
will depend upon whether this increases or lowers the 
absolute value of the expression (s wco 2 ) 2 . If co 2 = 
sjm < co 2 , then increasing m decreases (s mco 2 ) 2 , decreas- 
ing the denominator of the fraction and hence increasing 
the value of A Q . In other words, if the driving frequency 
is below the natural frequency, increasing the mass and 
thus lowering the natural frequency brings us nearer to 
resonance and increases the vibration. If, on the other 
hand, the driving frequency is above the natural frequency, 
the reverse effect ensues. Added mass is thus effective in 
reducing vibration only in case the driving frequency is 
above the natural frequency of the vibrating system. By 
similar reasoning it follows that under this latter condition, 
decreased vibration is effected by decreasing the spring 
factor, i.e., by weakening the supporting spring. 

Graphical Representation. 

The foregoing discussion will perhaps be clarified by 
reference to Fig. 112. Here are shown the relative ampli- 
tudes of vibration of a machine weighing 1,000 Ib. for a 
fixed value of the amplitude and frequency of the impressed 
force. These are the familiar resonance curves plotted so 



290 



ACOUSTICS AND ARCHITECTURE 



as to show their application to the practical problem in 
hand. Referring to the lower abscissae, we note that 
starting with a low value of the spring factor, the amplitude 
increases as s increases up to a certain value and then 
decreases. The peak value occurs when the relation 
between the spring factor and the mass of the machine is 
such that s/m = co 2 . We note further that increased 
damping decreases the vibration under all conditions but 



0.1 02 0.4 




FIG. 112. Amplitudes of forced vibrations with a fixed driving frequency and 
varying values of the stiffness of the resilient mounting. The upper abscissae 
are the ratios of the natural frequencies to the fixed driving frequency 
5 = Trr/VSw = fc//o = Ao/Ai, in Fig. 111. 



that this effect is most marked when the natural frequency 
is in the neighborhood of the impressed frequency. Finally, 
it is to be observed that if we are concerned only with 
reducing the vibration of the machine itself, this can best 
be done by using a very stiff mounting, making the natural 
frequency high in comparison with the impressed frequency, 
i.e., with a rigid mounting. As we shall see subsequently, 
however, this is the condition which makes for increased 
transfer of vibrations to the supporting structure. 



MACHINE ISOLATION 291 

Transmission of Vibrations. 

Let us suppose that the mass m of Fig. 110 is a machine 
which due to unbalance or some other cause exerts a 
periodic force on its mounting. Suppose that the maximum 
value of the force exerted by the machine on the support 
is FI and that this transmits a maximum force F 2 to the 
floor. The transmissibility r of the support is defined as 
Fi/Fz. Soderberg 1 has worked out an expression for the 
value of T in terms of the mass of the machine and the 
spring factor and damping of the support. 

A convenient form for the transmissibility of the support 
is given by the equation 



r = (89) 

\4A 2 co 2 + (co 2 - co 2 ) 2 

where co = 27r/ and co = 2?r/, / , and / being the natural 
frequency and the impressed frequency respectively. 

In most cases of design of resilient machine mounting, 
the effect of frictional damping is small. Neglecting 
the term 4& 2 co 2 , Eq. (89) reduces to the simple form 



_ _ __ 

CO 2 - C0 2 CO 2 _ l R 2 - I 

<0 * 

where R is the ratio of the impressed to the natural fre- 
quency. We note that for all values of R between and 
\/2, r is greater than unity. In the neighborhood of 
resonance, therefore, the resilient mounting does not 
reduce but increases the vibratory force on the floor. 
Figure 113 gives the values of the transmissibility for 
varying values of the spring factor and for four different 
values of the logarithmic decrement d = Trr/Vsra. We 
note that the transmissibility is less than unity only for 
low values of the natural frequency w = s/m. By 
inspection of Eq. (89) it is seen that, when w /a> = J^ \/2^ 
the value of r is unity regardless of the damping. This is 
shown in the common point for all the curves of Fig. 113. 

1 SODERBERG, C. R., Elec. Jour., vol. 21, pp. 160-165, January, 1924. 



292 



ACOUSTICS AND ARCHITECTURE 



This means that for any resilient mounting to be effective 
in reducing the transmission of vibration to the supporting 
structure, the ratio of s/m must be such that the natural 
frequency is less than 0.7 times the driving frequency. 
We have already noted that if it is a question of simply 
reducing the vibration of the machine itself, placing the 
natural frequency well above the impressed frequency will 
be effective. This, however, increases the transmission 
of vibration to the floor, as shown by the curves of Fig. 113. 

08 10 15 



28 
2.4 

20 


: s 16 

i 

12 


1.0 

08 
04 

s 






y detriment 


>>- -h|f 


i 


\-Detfnph 










t# 














\ 


1 rf - 

2 <y * / 

3 t? 31 - 














\ 


4 <f 62 










/ 


/ 


/ 

X 4 


^ 














^ 




^^ 


__ 








~~<f 
















// 
















__ -"' 


/ 






I 


i i 


i 






-^ 


.0 9.5 10.0 10.5 HO 115 12.0 125 13 



Log.S 

FIG. 113. Transmissibility of resilient mounting. The upper abscissae are the 
ratios of the natural frequencies to the driving frequency. 

In other words, isolation of the machine has to be secured 
at the price of increased vibration in the machine itself. 
The limit of the degree of isolation that can be secured by 
resilient mounting is thus fixed by the extent to which 
vibration of the machine on its mounting can be tolerated. 
If the floor itself were perfectly rigid, then a rigid mounting 
would be best from the point of view both of reduced 
machine vibration and of reduced transmission to other 
parts of the building. In general, resilient mounting will 
be effective in reducing general building vibration only 



MACHINE ISOLATION 293 

provided the natural frequency of the machine on the 
resilient mounting is farther below the driving frequency 
than is the natural frequency of the floor, when loaded with 
the machine. Obviously, a complete solution of the 
problem calls for a knowledge of the vibration character- 
istics of floor constructions. Up to the present, these 
facts are lacking, so that the effectiveness of a resilient 
mounting in reducing building vibrations in any particular 
case is a matter of some uncertainty, even when the 
machine base is properly designed to produce low values of 
the transmissibility to the supporting floor. 

We shall assume in the following discussion that the 
natural frequency of the machine on the resilient base is 
farther below the driving frequency than is the natural 
frequency of the floor. In such case, the proper procedure 
for efficient isolation is to provide a mounting of such com- 
pliance that the natural frequency is of the order of one- 
fifth the impressed frequency. Inspection of the curves of 
Fig. 113 shows that decreasing the natural frequency below 
this gives a negligible added reduction in the transmission. 

Effect of Damping on Transmission. 

Inspection of Fig. 112 shows that damping in the support 
reduces the vibration of the spring mounted body at all 
frequencies, while Fig. 113 shows that only in the frequency 
range where the transmissibility is greater than unity does 
it reduce transmission. It follows, therefore, that when 
conditions are such as to reduce transmission, damping 
action is detrimental rather than beneficial. For machines 
that are operated at constant speed, which is always 
greater than the resonance speed, the less damping the 
better. On the other hand, when the operating speed 
comes near the resonance speed, damping is desirable to 
prevent excessive vibrations of the mounted machine. 

Practical Application. Numerical Examples. 

There are two distinct cases of machine isolation that 
may be considered. The first is that in which the motion 



294 ACOUSTICS AND ARCHITECTURE 

of the machine produces non-periodic impacts upon the 
structure on which it is mounted. The impacts of drop 
hammers and of paper-folding and paper-cutting machines, 
in which very great forces are suddenly applied, are cases 
in point. Under these conditions, a massive machine base 
mounted upon a resilient pad with a low spring factor and 
high damping constitutes a system in which the energy of 
the impact tends to be confined to the mounted machine. 
In extreme cases, even these measures may be insufficient, 
so that, in general, massive machinery of this type should 
be set up only where it is possible to provide separate 
foundations which are not carried on the structural mem- 
bers of the building. 

The more frequent, and hence more important, problem 
is that of isolating machines in which periodic forces result 
from the rotation of the moving parts. It appears from the 
foregoing that the isolation of the vibrations thus set up 
can be best effected by resilient mounting so designed that 
the natural frequency of the mounted machine is well 
below the frequency of the impressed force. 

i /^ 

In the practical use of the equation / = 9~~A/ , s and m 

must be expressed in absolute units. In the metric system, 
s is the force in dynes that produces a deflection of one 
centimeter, and m is the mass in grams. Engineering 
data on the compressibility of materials are usually given 
in graphs on which the force in pounds per square foot is 
plotted against the deformation in inches. For a given 
material, the spring factor is roughly proportional to the 
area of the load-bearing surface and inversely proportional 
to the thickness. 1 If a force of L Ib. per square foot 
produces a decrease in thickness of d in. in a resilient 
mounting whose area, A sq. ft., carries a total mass m Ib. 
then 

1 This statement is not even approximately true for a material like rubber, 
which is practically incompressible when confined. When compressed, 
the change in thickness results from an increase in area, so that Young's 
modulus for rubber increases almost linearly with the area of the test sample. 



MACHINE ISOLATION 295 

J .- ~\l U\J*S -I 

2ir \ md 

As an example, let us take the case of a motor-driven 
fan weighing 8,000 lb., with a base 3 by 4 ft. Suppose 
that a light-density cork is to be used as the isolating 
medium. For 2-in. thickness, a material of this sort 
shows a compression of about 0.2 in. under a loading of 
5,000 lb. per square foot. If the cork is applied under 
the entire base of the machine, then we shall have the 
natural frequency of the machine so mounted 

f = 3 13 /^OQQ"^"^ = 19 l 

If the vibration to be isolated is that due to the varying 
torque on the armature produced by a 60-cycle current, 
the applied frequency is 120 vibs./sec., and the mounting 
will be effective. If, on the other hand, the motor operates 
at a speed of 1,800 r.p.m., then the vibration due to any 
unbalance in the motor will have a frequency of 30 per 
second. Effectively to isolate this lower-frequency vibra- 
tion, we should need to lower the natural frequency to 
about 6 per second. This can be done either by decreasing 
the load-bearing area of the cork or by increasing its 
thickness. We can compute the area needed by Eq. (90), 
assuming that the natural frequency is to be 6 vibs./sec. 



, ft t 5,000 X A 

/ = 6 = ~ 



/ 8,000 X 0.2 

from which we find A 1.17 sq. ft. This gives a loading 
of about 6,800 lb. per square foot. A larger area of material 
is perhaps desirable. We can keep the same spring factor 
by using a larger area of a thicker material. Thus if we 
double the thickness, we should need twice the area, giving 
a loading of only 3,400 lb. per square foot and the same 
transmission. 

It is apparent from the foregoing that the successful 
use of a material like cork or rubber involves a knowledge 
of the stress-strain characteristics of the material and the 



296 



ACOUSTICS AND ARCHITECTURE 



frequencies for which isolation is desired. Figure 114 
shows the deformation of three qualities of cork used for 
machine isolation. 1 The lines AB and CD indicate the 




0.10 0.20 030 0.40 050 0.60 0.70 0.80 0.90 1.00 
Deformation in Inches 

Fio. 114. Compressibility of three grades of machinery cork. (Courtesy of 
Armstrong Cork Co.) 



11,000 
10,000 

+. 9,000 
u_ 

<?f 8,000 
1.7.000 
1 6.000 

^ 5l0 
is 4.000 

- 1 3,000 
2,000 
1,000 


1 






NATURAL FREQUENCY VS. _ 
LOAD PER SQUARE FOOT 
MACHINERY MOUNTED ON - 
2" LIGHT DENSITY CORK 
























































\ 














\ 














\ 














\ 


i 














\v 














^<> 





- 






.. . 



30 



70 



Natural Frequency 

FIG. 115. Natural frequencies for various loadings on the light density cork of 

Fig. 114. 

limits of loading recommended by the manufacturers. 
We note that the increase in deformation is not a linear 
function of loading, indicating that these materials do not 

1 Acknowledgment is made to the Armstrong Cork Company for per- 
mission to use those data. 



MACHINE ISOLATION 



297 



follow Hooke's law for elastic compression. For any 
particular loading, therefore, we must take the slope of 
the line at that loading in computing the load per unit 
deflection. 

In Fig. 115, the natural frequencies of machines mounted 
on the light-density cork are given for values of the load 
per square foot carried by the cork. The compressibility 
of cork is known to vary widely with the conditions of 
manufacture, so that the curves should be taken only as 
typical. Similar curves for any material may be plotted 



zu 
.16 

JC 

JT 

.12 

1 

* 8 
1 
4 






AREA OF RUBBER PAD 
VS. 
LOAD PER PAD, FOR 
SQUARE PADS-T'THIC 


K 

^ 


















_^^ 


**M 


**~\ 

fcyc 


/es 


per 


sec 
















,*-- 


** 


^^ 










^ 


*** 


























^ 


**"" 








































^ 


^ 
















- 


SS* 


m " 


*tes- 

fC 


*W 


mi 


m <~ 














X 


x"" 








\ 







- 








*/es 


per 


set 








y 


/ 








*** 


*** 


^ 
































/ 


x 


^ 


+** 


+** 








































/ 

X 


^ 












































X 
















































100 200 300 400 500 600 700 800 900 1,000 1,100 1.2C 



load per Pc*d , Lb. 

FIG. 116. Effect of area of rubber pads upon the natural frequency for various 
loadings. (Hull and Stewart.) 

using Eq. (90) giving the deformation under varying load- 
ings. For the heavier-density corks shown in Fig. 114, 
the loading necessary to produce any desired natural 
frequency would be considerably greater. For 1-in. cork 
the loading necessary to produce any desired natural 
frequency would theoretically have to be twice as great 
as those shown, while for 4-in. material the loading would 
need to be only half as great. 

As has been indicated, when rubber is confined so as 
not to flow laterally, its stiffness increases. For this 
reason, rubber in large sheets is much less compressible 
than when used in smaller units. The same thing is true 
to a certain degree of cork when bound laterally. To 
produce a given natural frequency, cork confined laterally 
by metal bands will require greater loading than when free. 



298 ACOUSTICS AND ARCHITECTURE 

The curves of Fig. 116 are taken from a paper by Hull 
and Stewart. 1 They show the size and loading of square 
rubber pads 1 in. thick that must be used to give natural 
frequencies of 10 and 14 vibs./sec. These pads are of a 
high-quality rubber containing 90 per cent pure rubber. 
From the curves we see that the 8,000-pound machine 
mounted on 10 of these pads each approximately 10 sq. 
in. will have a natural frequency of 10 vibs./sec.; while 
if it is mounted on 10 pads each 17 sq. in., the natifral 
frequency is 14 vibs./sec. 

Natural Frequency of Spring Mountings. 

The computation of the natural frequency of a machine 
mounted on metal springs is essentially the same as that 
when the weight is distributed over an area. Suppose it 
were required to isolate the 8,000-lb. machine of the 
previous example using springs for which a safe loading 
is 500 Ib. We should thus need to use for the purpose 
8,000 -r- 500 = 16 springs, designed so that each spring 
with a load of 500 Ib. will have a natural frequency of 6 
per second. Then we may compute the necessary deflec- 
tion for this load by the formula 

r 500 



6- 3.^ 50()d 

from which 

d = 0.27 in. 

From the known properties of steel, springs may be 
designed having any desired characteristics over a fairly 
wide range. From the standpoint of predictability of 
performance and the control of spring factor to meet 
any desired condition, spring mounting is advantageous. 
Because of the relatively low damping, in comparison with 
organic materials, the amplitude of vibration of the machine 
and the transmission to the floor are large when the machine 
is operating at or near the resonance speed. The trans- 

1 Elastic; Supports for Isolating Rotating Machinery, Trans. A. I. E. E., 
vol. 50, pp. 1063-1068, September, 1931. 



MACHINE ISOLATION 



299 



mission, however, is less when the machine is operating 
at speeds considerably above resonance. 

Results of Experiment. 

Precise experimental verification of the principles de- 
duced in the foregoing is difficult, due to the uncertainty 
pertaining to the vibration characteristics of floor con- 
struction. We have assumed that the floor on which the 
machine is mounted is considerably stiff er, i.e., has a higher 
natural frequency than that of the mounted machine. 



(a) 



(c) (d) 

FIG. 117. (a) Vibration of machine with solid mounting, (b) Vibration of 
floor with solid mounting, (c) Vibration of machine mounted on U. S. G. 500 Ib. 
clip, (d) Vibration of floor machine mounted on U. S. G. 500 Ib. clip. 

The oscillograms of machine and floor vibrations in Figs. 
117 and 118 were kindly supplied by the Building Research 
Laboratory of the United States Gypsum Co. They were 
obtained by direct electrical recording of the vibration 
conditions on and under a moderately heavy machine with 
a certain amount of unbalance, carried by a typical clay- 
tile arch concrete floor. The operating speed was 1,500 



300 



ACOUSTICS AND ARCHITECTURE 



r.p.m. In Fig. 117 are shown the vibration of the machine 
and floor, first, when the machine is solidly mounted on 
the floor and then with the machine mounted on springs 



(c) 



(d) 



"""* 



FIG. 118. (a) Vibration of machine with solid mounting, (fe) Vibration of 
floor with solid mounting, (c) Vibration of machine on 1-in. high-density cork; 
loading 1,000 Ib. per square foot, (d) Vibration of floor, machine mounted on 
1-in. high-density cork; loading 1,000 Ib. per square foot, (c) Vibration of 
machine on 2-in. low-density cork; loading 3,000 Ib. per square foot. (/) Vibra- 
tion of floor, machine mounted on 2-in. low-density cork; loading 3,000 Ib. per 
square foot. 

each of which was designed to carry a load of 500 Ib. and 
to have, so loaded, a natural frequency of 7.5 vibs./sec. 
Figure 118 shows the necessity of proper loading in order 



MACHINE ISOLATION 301 

to secure efficient isolation by the use of cork. The middle 
curves show increased vibration both of the machine and 
of the floor when 1 in. of heavy-density material loaded 
to 1,000 Ib. per square foot is used. This is probably 
explained by the fact that on this mounting the natural 
frequency of the machine approximates that of the floor 
slab. In the lower curves, the loading on the cork is much 
nearer what it should be for efficient isolation. From the 
graph of Fig. 115, we see that the natural frequency of 
the 2-in. light-density cork loaded 3,000 Ib. to the square 
foot is about 9 per second. This is a trifle more than one- 
third the driving frequency and is in the region of efficient 
isolation. These curves show in a strikingly convincing 
manner the importance of knowing the mechanical proper- 
ties of the cushioning material and of adjusting the loading 
and spring factor so as to yield the proper natural fre- 
quency. In general, this should be well below the lowest 
frequency to be isolated, in which case higher frequencies 
will take care of themselves. 

Tests on Floor Vibrations under Newspaper Presses. 

In June, 1931, the writer was commissioned to make 
a study of the vibrations of the large presses and of the 
floors underneath them and in adjacent parts of the 
building of the Chicago Tribune. For the purpose of this 
study, a vibration meter was devised consisting of a light 
telephonic pick-up, associated with a heavy mass the 
inertia of which held it relatively stationary, while the 
member placed in contact with the vibrating surface moved. 
The electrical currents set up by the vibration, after being 
amplified and rectified, were measured on a sensitive 
meter. The readings of the meter were standardized by 
checking the apparatus with sources of known vibra- 
tions and were roughly proportional to the square of the 
amplitude. 

In the Tribune plant, three different types of press 
mounting had been employed. In every case, the presses 
were ultimately carried on the structural steel underneath 



302 



ACOUSTICS AND ARCHITECTURE 



the reel room at the lower-basement floor level. Still 
a fourth type had been used in the press room of the 
Chicago Daily News, and permission was kindly granted to 
make similar measurements there. The four types of 
mounting are shown in Fig. 119. A shows the press 
columns mounted directly on the structural steel with no 




FIG. 119. Four types of newspaper press mountings studied. 

attempt at cushioning. In JS, the press mounting and 
floor construction are similar, except that layers of alternat- 
ing %-in. steel and ] /4-in. compressed masonite fiber board 
are interposed between the footings of the press columns 
and the structural girders. The loading on these pads 
was about 13,000 Ib. per square foot. G shows the press 



MACHINE ISOLATION 



303 



supports carried on an 18-in. reinforced concrete base, 
floated on a layer of lead and asbestos J in. thick. This 
floated slab is carried on a 7J^-in. reinforced concrete 
bed carried on the girders. D is essentially the same as 
(7, except that a 3-in. continuous layer of Korfund (a steel- 
framed cork mat) is interposed between the floated slab 
and the 12-in. supporting floor. Here the loading was 
approximately 4,000 Ib. per square foot. 

Experiment showed that the vibration varied widely for 
different positions both on the presses themselves and on 
the supporting structure. Accordingly, several hundred 
measurements were made in each case, an attempt being 
made to take measurements at corresponding positions 
about the different presses. Measurements were made 

TABLE XXVI 



Tost 


Insulation 


On press 


On column 
footing or 
floated slab 


On reel- 
room floor 


On floor 
above 
press room 


A 
B. 

C.. .. 
D 


None 
Masonite and steel 
Load and asbestos 
Steel-bound cork 


1,375 
1,450 
1,000 
2,000 


72 
55 
412 


262 
21 
41 
30 


34 
2 G 
9.4 

8 















with the presses running at approximately the same speed, 
namely, 35,000 to 40,000 papers per hour. 



TABLE XXVII 







Ratio of press vibrations to 


Test 


Mounting 


Footings or 
supporting 
slab 


Reel-room 
floor 


Floor above 
press room 


A 


Direct to struc- 










tural steel 




5:1 


40:1 


B 


Masonite and 










steel 


20:1 


69:1 


440:1 


C . .. 


>2-in. asbestos 










and lead 


18:1 


24:1 


102:1 


D 


3-in. cork 


5:1 


67:1 


250:1 



304 ACOUSTICS AND ARCHITECTURE 

The averages of the measurements made are shown in 
Table XXVI. 

Since the vibrations of the presses themselves vary 
rather widely, it will be instructive to find the ratio of the 
press vibrations to the vibration at the other points of 
measurement. These ratios are shown in Table XXVII. 

So many factors besides the single one of insulation 
affect the vibration set up in the building structure that 
it is dangerous to draw any general conclusions from these 
tests. Oscillograph records of the vibrations showed no 
preponderating single frequency of vibration. The weight 
and stiffness of the floor structures varied among the 
different tests, so that it is not safe to ascribe the differences 
found to the differences in the mountings alone. However, 
it is apparent from comparison of the ratios in Table XXVII 
that all of the three attempts at isolation resulted in less 
building vibration than when the presses were mounted 
directly on the steel. Conditions in A and B were nearly 
the same except for the single fact of the masonite and 
steel pads B. The vibrations of the presses themselves 
was about the same in the two cases (1,375 and 1,450), 
so that it would appear that this method of mounting in 
this particular case reduced the building vibration in about 
the ratio of 13:1. The loading was high 13,000 Ib. per 
square foot and the masonite was precompressed so as 
to carry this load. The steel plates served to give a 
uniform loading over the surface of the masonite. 

Study of the figures for C and D discloses some interesting 
facts. In C (Table XXVI), with the lead and asbestos, 
we note that the vibrations of the press and of the floated 
slab are both low and, further, that there is only slight 
reduction in going from the slab to the reel-room floor. 
Comparison with Z), where vibration of both presses and 
floated slab was high, indicates that the cushioning action 
of the % in. of lead and asbestos was negligibly small, in 
comparison with 3 in. of cork. This latter, however, was 
obtained at the expense of increased press vibration. The 
loading on the 3 in. of cork was comparatively low. In 



MACHINE ISOLATION 305 

the light of both theory and experiment, one feels fairly 
safe in saying that considerably better performance with 
the cork would have resulted from a much higher loading. 

General Conclusions. 

It is fairly apparent from what has been presented that 
successful machine isolation is a problem of mechanical 
engineering rather than of acoustics. . Each case calls for a 
solution. Success rests more on the intelligence used in 
analysis of the problem and the adaptation of the proper 
means of securing the desired end than on the merits of 
the materials used. Mathematically, the problem is 
quite analogous to the electrical problem of coupled cir- 
cuits containing resistance, inductance, and capacity. 
The mathematical solutions of the latter are already at 
hand. Their complete application to the case of mechani- 
cal vibrations calls for more quantitative data than are 
at present available. In particular, it is desirable to know 
the vibration characteristics of standard reinforced floor 
constructions and the variation of these with weight, 
thickness, and horizontal dimension. Such data can be 
obtained partly in the laboratory but more practically 
by field tests on existing buildings of known construction. 
Since the problem is of importance to manufacturers of 
machines, to building owners, to architects, and to struc- 
tural engineers, it would seem that a cooperative research 
sponsored by the various groups should be undertaken. 

A more detailed theoretical and mathematical treatment 
of the subject may be found in the following references. 

REFERENCES 

HULL, E. H.: Influence of Damping in the Elastic Mounting of Vibrating 

Machines, Trans. A. S. M. E., vol. 53, No. 15, pp. 155-165. 
and W. C. STEWART: Elastic Supports for Isolating Rotating 

Machinery, Trans. A. I. E. E. t vol. 50, pp. 1063-1068, September, 1931. 
KIMBALL, A. L.: Jour. Acous. Soc. Amer., vol. 2, No. 2, pp. 297-304, October, 

1930. 

NICKEL, C. A.: Trans. A. I. E. E. t p. 1277, 1925. 
ORMONDHOYD, J. : Protecting Machines through Spring Mountings, Machine 

Design, vol. 3, No. 11, pp. '25-29, November, 1931. 
SODERBERG, C. R.i Elfic. Jour., vol. 21, No. 4, pp. 160-165, January, 1924. 



APPENDIX A 

TABLE I. PITCH AND WAVE LENGTH OF MUSICAL TONES 



Tone 


International pitch 
Tempered scale 
A = 435 vib./sec. 


Physical pitch 
Diatonic scale 
C s = 256 vib./sec. 


Fre- 
quency 


Wave length 


Fre- 
quency 


Wave length 


Meters 


Feet 


Meters 


Feet 


Middle C 


258 6 
274.0 
290 3 
307.6 
325 9 
345.3 
365.8 
387 5 
410 G 
435 
460 9 
488.3 
517.3 


1.328 
1 253 
1.183 
1.118 
1.054 
0.995 
0.939 
0.886 
836 
789 
745 
0.703 
0.664 


4 35 
4 11 
3.88 
3 66 
3 46 
3 26 
3.08 
2 91 
2.74 
2 59 
2 44 
2 31 
2.18 


256 

288 

320 
341.3 

384 
426 7 

480 
512 


1.341 
1.191 

1 071 
1.005 

0.894 
806 

714 
0.670 


4.40 
3.91 

3 52 
3.31 

2 93 
2 64 

2 35 
2 20 


G# 


D . . 


D# 


E 


F 


F# . . 


G 


c;# . 


A 

A# 


B 


G 



Velocity of sound at 20 C. = 343.33 m /sec. = 1,126.1 ft./sec. 



307 



308 



ACOUSTICS AND ARCHITECTURE 



TABLE II. COEFFICIENTS OF VOLUME ELASTICITY, DENSITY, VELOCITY 
OF SOUND, AND ACOUSTIC RESISTANCE 



Material 


e 


p 


c 


r 


Steel 
Cast iron . ... 
Brass 


19,600 X 10 8 
9,480 X 10 
6,370 X 10* 


7.8 
7.0 
8.4 


5,010 
3,650 
2,750 


391 X 10 4 
255 X 10 4 
232 X 10 4 


Bronze. 
Load 


3,140 X 10 8 
588 X 10 8 


8.8 
11 4 


1,890 
718 


166 X 10 4 
82 X 10 4 


Wood: 
Teak 


1,570 X 10 8 


86 


4,270 


37 X*10 4 


Fir 
Birch .... 
Water 
Rubber 


880 X 10" 
590 X 10 8 
196 X 10 8 
1 X 10 8 


0.45 
80 
1.0 
1 


4,430 
2,710 
1,400 
100 


20 X 10 4 
22 X 10 4 
14 X 10 4 
1 X 10 4 


Air (0 C.) 


0.014 X 10 


0.00129 


330 


0.0042 X 10 4 



c = coefficient of volume elasticity in bars 

p =s density, grams per cubic centimeter 

c velocity of sound, meters per second 

r acoustic resistance, grams per centimeters"* seconds" 1 



APPENDIX B 

Mean Free Path within an Inclosure. 

Given an inclosed space of volume V and a bounding 
surface S, in which there is a diffuse distribution of sound 
of average energy density 7: To show that p, the mean 
free path between reflections at the boundary, of a small 
portion of a sound wave, is given by the equation 

4F 

v = ~s 

A diffuse distribution is one in which the flow of energy 
through a small section of given cross-sectional area is the 
same independently of the orientation of the area. For the 




FIG. 1. 

purpose of this proof we may consider that the energy is 
concentrated in unit particles of energy, each traveling 
with the velocity of sound and moving independently of all 
the other particles. The number of particles per unit 
volume is /, and the total energy in the inclosure is VI. 

309 



310 ACOUSTICS AND ARCHITECTURE 

In the proof, we shall first derive an expression for the 
energy incident per second on a small element dS of the 
bounding surface and then, by relating this to the mean 
free path of a single particle, arrive at the desired relation. 
In Fig. 1, dV is any small element of volume at a distance 
r from the element of surface dS. We can locate dV on 
the surface of a sphere of radius r by assigning to it a 
colatitude (f> and a longitude 6. We shall express its 
volume in terms of small increments cfy?, dO, and dr of the 
three coordinates. From the figure we have 

dV - r* sin <pdOdr (1) 

The number of unit energy particles in this volume is 

IdV = Ir* sin <f>d<pdOdr (2) 

In view of the diffuse distribution, all the energy con- 
tained in dV will pass through the surface of a sphere of 
radius r. The fraction which will strike dS is given by 
the ratio of the projection of dS on the surface of this 
sphere, which is dS cos <p, to 4?rr 2 , the total surface of the 
sphere. Therefore the energy from dV that strikes dS is 

dS cos <p TJTr IdS . 77/17 /o\ 

-7 j- 21 Id V -j sin (p cos <pd<pdvdr (3) 

Energy leaving dV will reach dS within one second for 
all values of r less than c, the velocity of sound. Hence 
the total energy per second that arrives at dS from all 
directions is given by the summation of the right-hand 
member of Eq. (3) to include all volume elements similar 
to dV within a hemisphere of radius c. This summation is 
given by the definite integral 

IdS fi . . f 2 " r- IcdS (4) 

j I sin (f> cos <pd<f> \ ad \ dr = - A v ' 

47T Jo JO Jo ^ 

The total energy that is incident per second on a unit- 
area is therefore /C/4, and on the entire bounding surface 
S is IcS/4. 

Now we can find an expression for this same quantity 
in terms of the mean free path of our supposed unit energy 



APPENDIX B 311 

particles. If p is the average distance traveled between 
impacts by a single particle, the average number of impacts 
per second of each particle on some portion of the bounding 
surface S is c/p. The total number of particles is VI, so 
that the total number of impacts per second of all the 
particles on the surface S is VIc/p. By the definition of 
the unit particle this is the total energy per second incident 
upon S, so that we have 

TcS Vic _ 

-4- - -y w 

or 

4F 
P =^r (6) 

the relation which was to be shown. 



APPENDIX C 

TABLE 1.- -ABSORBING POWER OF SEATS 



Description 


Absorbing power 


Authority 


128 


256 


512 


1,024 


2,048 


Ash chairs, solid seats, open 














back 


15 


0.16 


17 


0.18 


20 


W. C. Sabine 


Auditorium chairs, solid ve- 
neer seat and back . . 
Cushions, vegetable fiber, can- 


JO. 15 


0.22 
19 


25 
24 


28 
0.39 


0.50 
38 


P. E. Sabine 
F. R. Watson 


vas covered, damask cloth 


75 


1.04 


1 45 


1 59 


1.42 


W. C. Sabine 


Cushions, hair covered with 














canvas and plush. . . 


0.99 


1 13 


1 77 


1.67 


1.37 


W. C. Sabme 


Cushions, hair covered with 














canvas and thin leather . . . 


1.13 


1.27 


1.93 


1.27 


73 


W. C. Sabine 


Cushions, elastic cotton cov- 














ered with canvas and plush 


1.66 


1.88 


2.04 


2.77 


1.95 


W. C. Sabine 


Theater chairs, upholstered 














seat and back, in imitation 














leather 




1.4 


1.6 


1.7 


2.1 


F. 11. Watson 


Theater seats, upholstered in 














mohair (average for 5 types) 




3 1 


3.0 


3 2 


3 4 


F. 11. Watson 


Theater seats, upholstered . 




3 4 


3 


3 2 


3.7 


V. L. Chrisler 


Theater seats upholstered 






2 6 






P. E. Sabine 


Steel chairs wood seats, open 














back . 


10 


15 


15 


0.14 


30 


P. E. Sabine 



312 



APPENDIX C 313 

TABLE II. COEFFICIENTS OF ABSORPTION OF MATEKIALS I 



N-. 




Thick- 


Coefficients 


D'lto 


o 


Material 


ness, 
inches 


128 


256 


512 


1,024 


2,048 


4,096 




1 


Acoustex, excelsior tile 


1 


0.14 


28 


55 


77 


79 


69 


1930 


2 


Acoustex, excelsior tile 


l->6 


20 


41 


69 


86 


79 


65 


193!) 


3 


Akoustohth tile, fine texture. 


7 /6 


09 


27 


29 


50 


62 


39 


1929 


4 


Balsam wool, scrim facing, paper 




















backing 


1 


14 


0.33 


50 


71 


70 


60 


1929 


5 


Balsam wool, "Quiet tile" 


1 


0.12 


40 


60 


72 


77 


62 


1931 


6 


Celotex, Acousti type B . . 


7 /A 




33 


41 


48 


52 


52 


1929 


7 


Celotex, Aciousti type BB . . . 


Ui 




47 


64 


76 


69 


57 


1929 


8 


Cclotex standard on 1 in. furring 


T-io 




19 


22 


21 


20 


19 


1929 


9 


Celotex, standard on 2 X 4-in. 




















studs. ... 


Me 


27 


17 


13 


15 


17 


20 


1929 


10 


Draperies, cotton 10 oz/sq. yd. 




















hung straight in contact with 




















wall 




04 


05 


11 


18 


30 


44 


1929 


11 


Draperies, same but weighing 14 




















oz/sq. yd 




06 


08 


13 


23 


40 


44 


1929 


12 


Draperies, velour weighing 18 




















oz/sq yd. hung 4 in from wall 




09 


33 


45 


52 


50 


44 


1929 


13 


Draperies, same hung 8 in. from 




















wall 




12 


36 


45 


52 


50 


44 


1929 


14 


Draperies, cotton weighing 14 




















oz/sq. yd draped 7 <i 




04 


15 


15 


28 


46 


52 


1929 


15 


Draperies, same draped ' t 




06 


28 


41 


60 


66 


50 


1929 


16 


Draperies, same draped J.(? 




10 


38 


50 


85 


82 


67 


1929 


17 


Felt, standard cattle hair 


1 


13 


41 


56 


69 


65 


49 


1929 


18 


Flaxlinum, semi-stiff fiber board 


W 


13 


18 


35 


60 


64 


59 


1929 


19 


Johns-Manville insulating board, 




















vegetable fiber, on 1-in furring 


H 


16 


24 


16 


18 


20 


26 


1929 


20 


Masonite, standard wood fiber 




















board on 1-in furring. 


H 


21 


36 


34 


34 


37 


46 


1929 


21 


Nashkoto A, asbestos hair felt, 




















painted membrane 


i 


16 


32 


39 


50 


51 


40 


1928 


22 


Nashkote A, membrane perfo- 




















rated in place . 


i 


14 


47 


0.78 


80 


67 


46 


1928 


23 


Nashkote B-316, asbestos hair 




















felt, covered with oilcloth, 




















H 6-in. perforations . . 


i 


13 


40 


71 


79 


72 


56 


1928 


24 


Nashkote B-332, same but with 




















}f2-iii. perforations 


i 


16 


38 


68 


85 


81 


62 


1928 


25 


Plaster, gypsum on wood lath, 




















rough finish 


l /2 


023 


039 


039 


052 


037 


035 


1929 


26 


Plaster, same, smooth finish . 


% 


0.029 


0.026 


032 


041 


048 


035 


1929 


27 


Plaster, gypsum on metal lath, 




















smooth finish . . . 


H 


0.020 


0.022 


032 


0.039 


039 


028 


1929 


28 


Plaster, lime, on wood lath, 




















rough finish 


H> 


039 


056 


061 


089 


054 


070 


1929 


29 


Plaster, lime, on wood lath, 




















smooth finish. . . 


H 


035 


033 


031 


039 


023 


041 


1929 


30 


Plaster, Sabmite on gypsum base 


H 


0.09 


19 


21 


30 


42 


46 


1930 



1 Measurements made by timing duration of audible sound from organ-pipe source. 
Krnpty-room absorbing power measured by variable source methods (loud-speaker). River- 
bank Laboratory tests. 



314 ACOUSTICS AND ARCHITECTURE 

TABLE II. COEFFICIENTS OF ABSORPTION OF MATERIALS. (Continued) 



No. 


Material 


Thick- 
ness, 
inches 


Coefficients 


Date 


128 256 


512 


1,024 


2,048 


4,096 


31 


Plaster improved Sabmite on 




















gypsum base 


H 


0.13 


0.29 


37 


56 


0.60 


58 


1931 


32 


Plaster, Habinite A, on gypsum 




















base, trowel finish 


H 


13 


0.24 


36 


50 


56 


62 


1931 


33 


Plaster, Sabinite A, on gypsum 




















base, float finish 


H 


14 


24 


39 


56 


56 


49 


1931 


34 


Banacoustic tile, l}4-in. rock 
















% 




wool, perforated metal 




20 


47 


84 


90 


95 


74 


1930 


35 


Soundex (excelsior tile) ... 


1 


0.13 


0.36 


57 


0.84 


0.74 


62 


1931 


36 


Soundex (excelsior tile) 


Wi 


21 


49 


80 


84 


74 


77 


1931 


37 


U. S. Gypsum acoustical tile 


Yi 


07 


16 


48 


60 


52 


56 


1931 


38 


U. S. Gypsum acoustical tile 


3 A 


06 


29 


62 


60 


66 


56 


1931 


39 


U. S. Gypsum, acoustical tile 


1 


15 


0.46 


67 


0.60 


64 


0.56 


1931 


40 


U. S. Gypsum, perforated metal, 




* 
















IH-in. mineral wool. 




33 


74 


0.81 


0.81 


0.69 


54 


1931 


41 


Weetfelt, jute ... 


*4 


09 


15 


20 


42 


54 


55 


1930 




Westfelt, jute 


/2 


12 


21 


33 


65 


69 


55 


1930 




Westfelt, jute 


1 


13 


27 


50 


67 


69 


65 


1930 



APPENDIX C 



315 



TABLE III. ABSORPTION COEFFICIENTS OF MATERIALS BY DIFFERENT 

AUTHORITIES, USING REVERBERATION METHODS 
B. S. = Bureau of Standards F. R. W. - F. R. Watson 

B. R. S. = Building Research Station V. 0. K. = V. O. Knudsen 
W. C. S. = W. C. Sabine C. M. S. = C. M. Swan 



No. 


Material 


Thick- 
ness, 
inches 


Coefficients 


Authority 


Date 


128 


256 


512 


1,024 


2,048 


4,096 


1 
2 

3 
4 
5 
6 

7 
8 
9 
10 
11 

12 
13 
14 
15 
16 
17 

18 
19 
20 
21 
22 
23 
24 

25 
26 

27 
28 


Aroustex (excelsior 
tile) 


1 
1H 

l'/2 
1 

2 

1 
18 
18 

94 
Yi 

H 

<K 
IK 
l Ke 

1H 
1M 

m 

1H 
1 
1 
1 


0.11 
0.16 

14 
0.08 
15 
0.21 

0.12 
012 
0.024 

09 
0.11 
0.08 

28 
15 

37 

18 
04 
08 
10 
09 

0.23 
19 

0.12 
0.13 


21 
34 

30 
0.13 
26 
0.48 

24 
013 
025 

0.08 
0.14 

0.18 

0.24 
32 

24 
0.36 
0.50 

33 
03 
0.14 
0.23 
31 

15 
0.26 

25 

20 
0.26 


53 
0.75 

74 
25 
59 
34 

62 
017 
031 
0.20 
21 

0.37 

0.48 

0.47 
0.46 

62 
70 
0.67 

84 
05 
61 
58 
62 
49 
;o 5 

0.28 
03 

0.32 

0.33 
58 


0.81 
0.85 

0.90 
0.54 
74 
0.31 

0.76 
020 
042 


0.81 
0.84 

85 
67 
0.52 
0.41 

76 
0.023 
049 




B. S. 
B. S. 

B. S. 
B. S. 
B. S. 
B. S. 

B. S. 
W. C. S. 
W. C. S. 
W. C. S. 


1931 
1931 

1930 
1930 
1930 
1930 

1931 
1900 
1900 
1900 

1931 

1927 
1927 

1931 
1927 
1927 
1931 

1931 
1900 
1930 
1927 

1926 
1900 

1928 

1929 
1929 


Acoustex (excelsior 
tile) . . . 
Acoustex, with 6 
coats spray paint . 
Akoustolith, grade D 
Akoustolith, grade D 
Arborite, fiber board 
Balsam wool, quiet 
tile 
Brick wall, painted . 
Brick wall, unpainted 
Carpet, lined. .. 
Carpet, on concrete 
Carpet lined with* 
felt 








83 
025 
070 


26 
0.43 

0.63 

49 
0.56 

0.76 
0.76 
0.74 

0.97 
11 
0.56 
72 
77 
61 
) depe 

0.29 
0.36 

0.33 
0.73 


27 
27 

0.75 

60 
0.61 

0.73 
0.76 
0.80 

0.76 
07 
0.64 
0.66 
69 
67 
iding 

0.32 
0.36 

0.28 
0.77 


0.37 
0.27 

0.62 

0.77 

0.02 
65 
0.46 

66 
on ope 

28 
0.71 


B. R. S. 
B. R. S. 

B. S. 

F. R. W. 
V. O. K. 

B. S. 
F. R. W. 
V. 0. K. 

B. S. 
B. R. S. 
B. S. 
W. C. S. 
B. S. 
F R. W. 
ning 

V. 0. K. 
W. C S. 

V. 0. K. 

B. S. 
B. S. 


Celotex, Acousti sin- 
gle B 
Celotex, Acousti type 
B 
Celotex, Acousti dou- 
ble B 
Celotex, Acousti type 
BB 


Celotex, Acoueti type 
BB 


Celotex, Acousti tri- 
ple B 


Cork-tile floor 
Corkoustic C 


Felt, standard hair . . 
Flaxlinum 


Flaxlinum 
Grills, ventilating 


Insulite, standard 
board. 
Linoleum on concrete 
Masonite, standard 
board 


H 

1 A 

1 
1 


Nashkote A, asbestos 
hair felt, painted 
fabric 


Nashkote A, perfo- 
rated after erection 



316 



ACOUSTICS AND ARCHITECTURE 



TABLE III. ABSORPTION COEFFICIENTS OF MATERIALS BY DIFFERENT 
AUTHORITIES, USING REVERBERATION METHODS. (Continued} 



No. 


Material 


Thick- 
ness, 
inches 


Coefficients 


Authority 


Date 


128 


256 


512 


1,024 


2,048 


4,096 


29 


Naahkote B-332 .... 


1 


19 


26 


51 


73 


89 


77 


B. S 


1929 


30 


Nashkote B-332 


*A 


0.12 


21 


40 


63 


0.81 




B. S. 


1929 


31 


Plaster, gypsum on 






















olay tile 




013 


015 


020 


028 


040 


050 


W. C H. 


1900 


32 


Plaster, Acoustichme 


?i 


17 


23 


28 


36 


64 




B S. 


1930 


33 


Plaster, Akoustohth 


M 


13 


21 


19 


23 


33 




B. B. 


1931 


34 


Plaster, Akoustolith 


^ 


21 


24 


29 


33 


37 


42 


C. M. S 




35 


Plaster, Hachmeister- 






















Lind stippled with 






















pins Yi in. deep. . 




16 


19 


25 


36 


44 




B S 


1930 


36 


Plaster, Macoustie 






















perforated with 






















pins K' in. deep. 


Vi 


06 


17 


33 


56 


58 




B. S. 


1931 


37 


Plaster, Reverbolite 






















perforations with 






















large pins 


\<2 


07 


15 


34 


47 


65 




B. S. 


1930 


38 


Sabinite A 


Y* 


19 


20 


37 


60 


61 


46 


B. S 


1932 


39 


Plaster, hydraulic 






















Sabinite 


Yi 


14 


24 


27 


38 


49 




B S 


1931 


40 


Rock wool . . 


1 


35 


49 


63 


80 


83 




V. O. K 


1928 


41 


Rock wool 


2 


44 


59 


68 


82 


84 




V. O. K. 


1928 


42 


Rubber carpet on 






















concrete 


?io 


04 


04 


08 


12 


03 


10 


B. H. S 




43 


Sanacoustic tile 1M- 






















in. rock-wool filler 






















perforated metal. 




17 


41 


82 


94 


85 




B 8 


1930 


44 


Same on ^le-in. 






















furring 




19 


64 


87 


87 


80 




B. S. 


1931 


45 


Soundex (excelsior 






















tile) 


M 


04 


22 


0.45 


72 


75 


65 


B. S. 


1931 


46 


T. M. B. fiber tile 






















(excelsior) . . 


} 2 


07 


15 


28 


51 


71 




B. S. 


1931 


47 


1 he same . . . 


1 


12 


27 


58 


79 


80 




B. S 


1931 


48 


The same 


Itt 


0.17 


36 


78 


85 


85 




B. S. 


1931 


49 


U. S. Gypsum tile 






















(mineral wool) . 


Yz 


00 


20 


48 


64 


66 




B. S. 


1930 


50 


The same 


44 


13 


28 


61 


73 


73 




B. S 


1930 


51 


The same ... 


1 


18 


38 


64 


73 


73 




B. S. 


1930 


52 


Wood paneling 


Yi 


10 


11 


10 


08 


08 


11 


W. C S. 


1900 



APPENDIX D 

TABLE I. NOISE DUE TO SPECIFIC SOURCES 



Source or description of noise 


Mini- 
mum, 
decibels 


Aver- 
age, 
decibels 


Maxi- 
mum, 
decibels 


Distance 
source to 
micro- 
phone, feet 


Hammering on steel plate, 4 blows per second 




113 




2 


Riveter as heard near by 


94 


97 


101 


35 


as heard ordinarily on street 




79 5 




200 


Ulast of explosives, open-cut digging . . 




96 




50 


Subway station underground, noise on platform 










local station: express train passing. 


88 


94 


97 


15 to 25 


local station: local train 


85 


88 5 


91 


6 to 30 


5 turnstiles, rush hour . 


78 


83 


91 


3 to 7 


Steamship whistle (fairly loud), teats near by . 


92 


93 


94 


115 


tests in lOth-floor office, windows open 


59 


61 


65 


1,450 


as heard ordinarily on street 


47 


56 


68 


2,500 


Automobile horn, 34 types directed toward micro- 










phone . . ... 


72 


91 


102 


23 


as heard ordinarily on street 


57 


71 5 


85 


25 to 100 


Klevated tram, on open structure, as heard near by 


85 


89 


91 


15 to 20 


as heard ordinarily on street 


70 


81 5 


91 


15 to 75 


as heard at 750 ft .... 


51 


57 5 


63 


750 


Fire apparatus: siren and bell 




83 




100 






81 




50 


Police whistle, as heard near by 


80 


82 


83 


15 


as heard ordinarily on street . 


64 


74 


83 


15 to 75 


as heard at 185 ft 


55 5 


57 5 


62 


185 


Radio loud-speaker on street 


73 


79 


81 


30 


Motor truck, not muffled . 


70 


77 5 


87 


15 to 50 


changing gears ... 


68 


74 


83 


15 to 50 


as heard ordinarily on street (not including 










horn) . 


55 5 


73 5 


87 


15 to 50 


Electric street car moving fast 


7.3 5 


76 5 


77 5 


10 to 15 


over track crossing 


68 


74 


81 


40 


as heard ordinarily on street 


63 


72 5 


83 


15 to 50 


moving slowly . 


68 5 


69 5 


70.5 


10 to 15 


Motor bus changing gears 


68 


71 


75 


15 to 50 


Automobile squeaking brakes 


62 


71 


76 


15 to 50 


changing gears . . . 


60 


70 5 


83 


15 to 50 


exhaust ... ... 


64 


70 


74 


15 to 50 


as heard ordinarily on street (not including 










horn) ...,.,., . . . 


50 


65 5 


83 


15 to 50 













317 



318 ACOUSTICS AND ARCHITECTURE 

TABLE II. NOISE IN BUILDINGS* 

Level above 
Threshold, 
Location and Source Decibels 

Boiler factory . . 97 

Subway, local station with express passing 95 

Noisy factories 85 

Very loud radio in home 80 

Stenographic room, large office. .. 70 

Average of six factories 68 

Information booth, large railway station . . 57 

Noisy office or department store . . . .57 

Moderate restaurant clatter .... 50 

Average office . . . 47 

Noises measured in residence . . .45 

Very quiet radio in home. . ... 40 

Quiet office 37 

Quietest non-residential location . . . 33 

Average residence. . . .... 32 

Quietest residence measured . 22 

1 Taken from " City Noise," Department of Health, New York City. 



APPENDIX E 

Within the past year, a considerable amount of research has been done 
in an attempt to standardize methods of measurement of absorption coeffici- 
ents. Diverse results obtained by different laboratories on ostensibly the 
same material have led to considerable confusion in commercial applica- 
tion. Early in 1931, a committee was appointed by the Acoustical Society 
of America to make an intensive study of the problem with a view to estab- 
lishing if possible standard procedure in the measurement of absorption 
coefficients. The work of this committee has taken the form of a cooperative 
research to determine first the sources of disagreement before making recom- 
mendations as to standard practice. 

The Bureau of Standards sponsored a program of comparative measure- 
ments using a single method and apparatus in different sound chambers. 
This work was carried on by Mr. V, L. Chrisler and Mr. W. F. Snyder. 
The apparatus developed and used at the Bureau was transported to the 
Riverbank Laboratories, the laboratory at the University of Illinois, and 
the laboratory of the Electrical Research Products, Incorporated, iiiNew 
York. Measurements were made on each of three identical samples in 
each of these laboratories. The Bureau of Standards equipment consisted 
of a moving-coil loud-speaker as a source of sound. The source was rotated 
as described in Chap. VI. The sound was picked up by a Western Electric 
condenser microphone. The microphone current was fed into an attenuator, 
graduated in decibels of current squared, and amplified by a resistance- 
coupled amplifier. By means of a vacuum-tube trigger circuit and a delicate 
relay, the amplified current was made to operate a timing device. 
This device measured automatically the time between the instant of cut-off 
at the source and the moment at which the relay was released. By varying 
the attenuation in the pick-up circuit, one measures the times required for 
the sound in the chamber to decay from the initial steady state to different 
intensity levels and thus may plot the relative intensities as a function of 
the time. It consists essentially of an electrical ear whose threshold can 
be varied in known ratios. 

The purpose of the Bureau's work was to determine the degree to which 
the measured values of coefficients are a function of the room in which the 
measurements are made. 

At the Riverbank Laboratories, a comparison of the results obtained by 
different methods in the same room has been undertaken. The methods may 
be summarized as follows: 

1. Variable pick-up, constant source. 

a. Loud-speaker, fixed current. 

b. Organ pipe, constant pressure. 

2. Variable source (loud-speaker) constant pick-up. 

a. Moving microphone, with electrical timing. 

b. Ear observations (four positions). 

319 



320 



ACOUSTICS AND ARCHITECTURE 



3. Constant source (organ pipe). 

Constant pick-up, ear (calibrated empty room). 

In these measurements, the loud-speaker source was mounted at a fixed 
position in the ceiling of the room. The organ pipes were stationary. In 
all eases, the large steel reflectors already described were in motion during 
all measurements. When the microphone was used, this was mounted 011 
the reflectors. The ear observations were made with the observer inclosed 
in a wooden cabinet placed successively at four different positions in the 
room. A detailed account of these experiments cannot be given here, but 
a summary of the results is presented in Tables I and II. 

TABLE I. ABSORPTION COEFFICIENTS OF THE SAME SAMPLE AS MEASURED 
JN DIFFERENT ROOMS WITH BUREAU OF STANDARDS EQUIPMENT 





Laboratory 


Frequency 


Bureau of 
Standards 


Riverbank 


University 
of Illinois 


K.R.P.I 


128 


18 


14 






18 


256 


0.36 


34 


0.38 


43 


42 


512 


73 


77 


75 


86 


72 


1,024 


89 


88 


87 


95 


83 


2,048 


83 


90 


87 


1 06 


80 


4,096 


69 


67 


74 


74 


56 



TABLE II. ABSORPTION COEFFICIENTS OF THE SAME SAMPLE MEASURED 

IN THE RlVEKBANK SOUND CHAMBER, UsiNG DIFFERENT METHODS 





Method 


Frequency 
















1 


Ib 


"2i 


26 


3 


128 


0.15 


23 


15 


19 


18 


256 


0.36 


0.52 


0.39 


0.48 


45 


512 


71 


69 


73 


0.67 


63 


1,024 


91 


86 


90 


92 


69 


2,048 


85 


74 


82 


74 


71 


4,096 













A full report of the Bureau of Standards investigation has not as yet been 
published, and further study of the sources of disagreement is still in progress 
at the Riverbank Laboratories. The data of the foregoing table serve, 
however, to indicate the degree of congruence in results that is to be expected 
in methods of measurements thus far employed. 

Method la in Table II is the same as the Bureau of Standards method 
except that in the former, we have a moving pick-up with a stationary 
source; while in the latter, we have a stationary pick-up with a moving 



APPENDIX E 321 

source. The agreement is quite as close as can he expected in measurements 
of this sort. In Table II, we note rather wide divergence at certain fre- 
quencies between the results of ear observations and those in which a 
microphone WPS employed. Very recent work at both tho Bureau of Stand- 
ards and the Riverbank Laboratories points to the possibility that in some 
cases, the rate of decay of reverberant sound is not uniform. If this is 
true, then the measured value of the coefficient of absorption will depend 
upon the initial intensity of the sound and the range of intensities over which 
the time of decay is measured. In such a case, uniformity of results can be 
obtained only by adoption by mutual agreement of a standard method and 
carefully specified test conditions. This is tho procedure frequently fol- 
lowed where laboratory data have to be employed in practical engineering 
problems. The measurements of thermal insulation by materials is a caso 
in point. Pending such an establishment of standard methods and specifica- 
tions, the data on absorption coefficients given in Table II (Appendix C) 
may be taken as coefficients measured by the method originally devised and 
used by W. C. Sabiiie. The data given out by the Bureau of Standards for 
1931 and later were obtained by the method outlined in the second para- 
graph of this section. 



INDEX 



Absorbents, commercial, 143, 313 
Absorbing power, of audience, 140 
defined, 58 

empirical formula for, 147 
of seats, 140, 312 
total and logarithmic decrement, 

74 

unit of, 69 

Absorption, due to flexural vibra- 
tion, 130 

due to porosity, 128 
uniform, 169 
Absorption coefficient, definition of, 

86 

effect of area on, 131 
of edges on, 134 
of thickness on, 92, 129 
of mounting on, 137 
of quality of test tone on, 138 
of spacing on, 136 
impedance measurement of, 93 
measurement of, by different 

methods, 119, 319 
of small areas, 135 
stationary-wave measurement of, 

87 

two meanings of, 86 
Absorption coefficients of materials, 

313-316 

Acoustical impedance, 93 
Acoustical materials in hospitals, 228 
Acoustical power of source, 75 
Acoustical resistence, 22, 308 
Air columns, vibrations of, 37 
Amplifying system, 203 
Analysis, harmonic, 26 
Articulation, 156 
Asbestos, 302 



Audiometer, buzzer type, 213 
Auditorium Theater, Chicago, 185 



B 



Bagenal and Wood, 194 
Balcony recesses, allowance for, 151 
Buckingham, E. A., 58, 247 
Bureau of Standards, absorption 
measurements, 90 



Carnegie Hall, New York, 186 
Chrisler, V. L., 89, 122, 142, 237 
Chrisler and Snyder, 115, 266 
Churches, Gothic,, 197 
various types of, 196 
Concert halls, 161 
Condensation, in compressional 

wave, 20 
Cork, for machine isolation, 296, 

300, 302 

Crandall, I. B., 23, 93, 128 
Cruciform plan, reflections in, 199 
Curved shapes, allowable, 176 
defects caused by, 171 

D 

Damping, due to inertia, 289 
effect of, on transmission, 293 

Davis, A. H., 214 

Davis and Kaye, 234 

Davis and Littler, 240, 257 

Davis and Evans, 88, 92 

Decibel scale, 205 

Density, changes in compressional 
wave, 19 

Diffuse distribution, 86 

Doors, sound proof, 255 
transmission by, 254 



323 



324 



ACOUSTICS AND ARCHITECTURE 
E J 



Echoes, focused, 178, 181 
Eckhardt, E. A., 58, 82, 89, 237 
Eisenhour, B. E., 27, 215 
Energy, in coinpressionnl wave, 21 

density, 21 

flux, 21 
Eyring, C., 52, 107, 151, 195 



Fletcher, H., 31, 104 
Fourier series, 25 
Four-organ experiment, 71, 103 
Franklin, W. S., 58 

G 

Gait, R. II., 214, 226 
Griswoid, P. B., 223 

II 

Hales, W. B., 180 

Hearing, frequency and intensity 

range of, 205 
Ileimburger, G., 267 
Hill Memorial, Aim Arbor, 182 
Hospitals, quieting of, 226 
Hull, E. II., 305 
Hull and Stewart, 298, 305 
Humidity, effect of, on absorbing 

power, 105 



Image sources, 52 

Inertia damping, 289 

Intensity, decrease of, in a room, 59 

in a tube, 51, 54 
distribution in rooms, 39, 41 
effect of absorption on, 46 
growth of, in a room, 59 

in a tube, 48 
level in decibels, 206 
logarithmic decrease of, 70 
steady state, 57 



Jaeger, S., 58 



K 



Kimball, A. L., 305 

King, L. V., 7 

Evingsbury, B. A., 211 

Knudsen, V. O., 105, 115, 117, 155, 

161, 208, 280 

Kranz, F. W., 30, 104, 205 
Krueger, H., 242 



L 



Laird, D. A., 224 

La Place, 4 

Larson and Norris, 229 

Lead, for machine isolation, 302 

Leipzig, Gewanclhaus, 194 

Lifschitz, 8., 158 

Loudness, contours of equal, 211 

of noises, 212 

of pure tones, 210 

Loud speaker, for sound-chamber 
calibration, 108 

M 

Machine isolation, 282-305 
Machines, resilient mountings for, 

292, 298, 302 

Masking effect of noise, 213 
Masonite for machine isolation, 302, 

303 
Masonry, sound transmission by, 

260-266 

Maxfield, J. P., 168, 170 
McNair, W. A., 158 
Mean free path, defined, 57 

derivation of formula for, 309 

formula for, 58 

experimental value of, 78 
Meyer, E., 105, 114, 277 
Meyer and Just, 116, 244 
Miller, D. C., 26 
Mormon Tabernacle, 179 



INDEX 



325 



N 

Neergard, C. F., 228 
Newton, Sir I., 1, 3 
Newspaper presses, 301 
Nickel, C. A., 305 
Noise, comparison of, 213 

defined, 30, 204 

measurement and control of, 204- 
231 

physiological effect of, 224 

from ventilating ducts, 229 

very loud, quieting of, 222 
Noise level, as a function of absorp- 
tion, 219 

Noise levels, computation of reduc- 
tion in, 221 

effect of reverberation on, 217 

indoors, 318 

measured reduction in, 219 

outdoors, 317 
Norris, R. F., 52, 229 



O 



Office noises, coefficients for, 225 

Office quieting, effect of, 223 

Open window, absorbing power of, 

69, 133 

Orchestra Hall, Chicago, 172 
Orchestra pit, 195 
Ormondroyd, J., 305 
Oscillations, forced damped, 286 

free damped, 284-286 
Oscillograph, for sound chamber 

calibration, 115 
Osswald, F. W., 163 



Paget, Sir. R., 31 
Paris, E. T., 88, 92, 128 
Parkinson, J. S., 135 
Parkinson and Ham, 209 
Phase angle, 13 
Pierce, G. W., 285 
Pitch of musical tones, 307 
Porous materials, absorption by, 128 
transmission by, 256 



Pressure, changes in compressions! 

wave, 19 

related to intensity level, 210 
Propagation of sound in open air, 6 

in rooms, 7 

R 

Radio studios, 167 

Rayleigh, Lord, 128 

Rayloigh disk, 111 

Reduction factor, compared with 

transmission loss, 249 
definition of, 235, 239, 242 
Reflection from rear wall, 174 
Resonance, defined, 40 

effects in sound transmission, 244 
in rooms, 40 
Reverberation, and character of 

seats, 190 

coefficient, defined, 97 
constant, 60, 77 
effect of, in design, 187 
on hearing, 154 
on music, 157 
on speech articulation, 155 
relation to absorbing power, 67 
to volume, 68 
to volume and seating capacity, 

188 

in a room, 59 
in a tube, 47 
Reverberation equation, absorption 

continuous, 60 
absorption discontinuous, 61 
applies to other phenomena, 83 
complete, 79 
in decibels, 209 

Reverberation meter, Meyer, 114 
Meyer and Just, 116 
Wente and Bedell, 112 
Reverberation theory, assumptions 

of, 80 

Reverberation time, acceptable, 158 
of acoustically good rooms, 157 
for amplified sound, 166 
calculation of, 144, 150 
defined, 51 
experimental determination of, 63 



326 



ACOUSTICS AND ARCHITECTURE 



Reverberation time, optimum, 159 
for speech and music, 160 
with standard source, 164 
variable, 163 

with varying audience, 189 
Riesz, R. R., 208 
Riverbank Laboratories, 99, 236 
Riverside Church, New York, 201 
Rubber, for machine isolation, 297 



S 



Sabine, W. C., 41, 66, 121, 132, 142, 

154, 174, 234, 258 
Schlenker, V. A., 83 
Schuster and Waetzmann, 52 
Seating capacity, related to volume, 

188 

Sensation level, 207 
Shapes, cruciform, 198 
ellipsoidal, 179 
paraboloidai, 182 
spherical, 178 
Simple harmonic motion, definition 

of, 12 

energy of, 15 
Sinusoidal motion, 14 
Slocum, S. E., 283 
Soderberg, C. R., 291 
Sound absorbents, choice of, 192 
location of, 192 
properties of, 127 
Sound chamber, 98 

calibration, 102, 108, 111 
methods, 101 
Sound insulation, computation of, 

250-252 

Sound recording rooms, 167 
Sound source, reaction of room on, 

121 
Sound transmission, coefficient of, 

246 

by continuous masonry, 260-263 
effects of stiffness and mass on, 

263-265, 277 

by doors and windows, 254 
by double walls, 268-270 
by ducts, 229 
by porous materials, 256-260 



Sound transmission, loss, 246 

measured by reverberation 
method, 234 

measurement at Bureau of Stand- 
ards, 237 

mechanics of, 232 

resonance effects in, 244 

theory and measurement of, 232- 

252 
Sounds, musical, 30 

speech, 31 

Spring mountings, experiments with, 
299 

for machine isolation, 298 
Stationary waves, 33 

equations of, 36 

in a tube with absorbent ends, 38 
Stewart and Lindsay, 23 
Strutt, M. J. O., 83 
Swan, C. M., 202 
Synthesis, harmonic, 26 
Synthesizer, harmonic, 27 



Tallant, H., 182 
Taylor, H. O., 39, 90 
Timoshenko, S., 282 
Tone, complex, 25 

pure, defined, 25 

Transmission, effect of damping on, 
293 

of vibrations, 291 
Tuma, J., 90 

Tuning fork, for noise comparison, 
214 

U 

University of Chicago Chapel, 200 
V 

Velocity of sound, in air, 3 

effect of elasticity and density 

on, 3 

of temperature on, 5 
in solids, 308 
in water, 6 



INDEX 



327 



Ventilating ducts, 229 

Vibrating system, natural frequency 

of, 283 
Vibrations, of air columns, 37 

damped, free, 284 

of floors, 301-305 

forced, damped, 286 

transmission of, 291 
Volume, related to seating capacity, 
188 

W 

Walls, double, completely separated, 

268-271 

partially bridged, 271, 272 
Waterfall, W., 243 
Watson, F. R., Ill, 158, 193, 240 
Wave length, relation of, to fre- 
quency, 19 
table of, 307 



Wave motion, 15 

definition of, 12 

equation of, 17 

reflection of, 16 
Waves, equations of progressive, 36 

temperature changes in, 23 

types of, 17 

Weber-Fechner Law, 207 
Webster, A. G., 22 
Weight, effect of, on sound insula- 
tion, 262, 280 
Weight law, exceptions to, 265 

in sound transmission, 262, 264 
Weisbach, F., 90 
Wente, E. C., 93 
Wente and Bedell, 112 
Whispering gallery, 177, 179 
Windows, sound transmission by, 

254 

Wolf, S. K., 164 
Wood, absorption coefficients of, 131 

as an acoustical material, 194 

stud partitions, 272