paul e sabine acoustics and architecture doc

Itis withthe needs of this rather largegroup ofpossible one can bemoreconsciousthan is the authorofthelackof the treatment ofsimple harmonic motion andthe develop-ment of the wave equat

03

OSMANIA UNIVERSITY LIBRARY

ACOUSTICS AND ARCHITECTURE

McGRAW-HILL BOOK COMPANY, INC.

NEW YORK AND L.ONDON

1932

McGRAW-HiLL BOOK COMPANY,INC PRINTED INTHEUNITED STATES OFAMERICA

parts thereof,maynot be reproduced

inany formwithout permission of

the publishers.

THEMAPLE PRESSCOMPANY, YORK, PA.

The last fifteen years have sen a rapidly growing

interest, both scientific and popular, in the subject ofacoustics The discovery of the thermionic effect and the

d?vicefor the quantitativestudy ofacoustical phenomena.

reproduction of sound These have led to a demand for

by the necessity of minimizing thenoise resultingfromtheeverincreasing mechanization ofall our activities.

claim the attention ofa largegroup of engineers and

tech-nicians Manyof these have had to pickup most of their

most collegesandtechnical schools give onlyscant

instruc-tion in the subject. Further, the fundamental work of

any auditorium which he may design Some knowledge

ofthebehaviorof soundinroomshas thusbecomea

neces-sarypart of the architect's equipment

Itis withthe needs of this rather largegroup ofpossible

one can bemoreconsciousthan is the authorofthelackof

the treatment ofsimple harmonic motion andthe

develop-ment of the wave equation in Chap. II would be much

vi PREFACE

use of the differential equation of the motion of a particleunder the action of an elastic force The only excuse for

the treatment given is the hope that it mayhelp the mathematical reader tovisualize moreclearly the dynamic

inherent difficulties of a strictly logical approach*to tho

dimensions are not great in comparison with the wave

length Thus, inChap. Ill, conditionsinthe steady state

areconsideredfromthe wavepoint ofview;while inChap.

Thetheoryof reverberationisbasedupon certain ing assumptions An understanding of these assumptions

simplify-and the degree towhich theyare realized in practical cases

shouldlead toamoreadequateappreciationofthe 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 fieldin

promi-nence seemsto be given tothe results of work done in thiscountryand particularly to that of the RiverbankLabora-

his real motive in writing a book has been to give

per-manent form to those portions of his researches which in

preserving

Grateful recognitionismadeofthekindnessofnumerous

authors in supplying reprints of their papers It is also a

Miss Cora Jensen and Mr. C A Anderson of the staff of

anddrawings for the text

PREFACE vii

andunfailinginterest inthe solution of acousticalproblems

have made the writingof those pages possible

RIVEBBANKLABORATORIES,

GENEVA,ILLINOIS,

July, 1932.

CHAPTER X MEASUREMENT AND CONTROLOFNOISEINBUILDINGS . 204

Ideas of the nature of heat, light, and electricity have

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 clearlydiscerned fromthe very

musical interval says: "I assertthat the ratio of a musical

that is, by the number of pulses of air waves which strike

thetympanumofthe ear causingitalsoto vibratewiththe

same frequency." In the "Principia," Newton states:

"

Whenpulses arepropagated through afluid,everyparticleoscillates with a very small motion and is accelerated and

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 toparticle as the wave advances On the theoretical side,

the study of sound considered as the physical cause of the

study of the mechanics of solids andfluids.

2 ACOUSTICS AND ARCHITECTURE

On the physical side, acoustics naturally divides itselfinto three parts: (1) thestudyof vibrating bodies including

solids and partially inclosed fluids; (2) the propagation of

of the mechanism of the organ of perception bj; means of

induce nerve stimuli There is still another branch ofacoustics, which involves not only the purely physicalproperties of sound but also the physiological and psycho-

logical aspects ofthe subject as well as the studyof sound

inits relation tomusic and speech.

Of the three divisions of purely physical acoustics, thestudy ofthe laws of vibrating bodies has, up until the last

twenty-five years, received byfar the greatest attention ofphysicists The problems of vibrating strings, of thin

theattentionofthe bestmathematical minds Alistoftheoutstanding names in the field would include those of

side of the subject. Galileo, Chladni, Savart, Lissajous,

Melde, Kundt, Tyndall, and Koenig are some who have

of strings, bars, thin membranes, plates, and air columns

com-pleteness,andthetheoretical solutions, in part,

the agreement between the theory and experiment is lessexactthaninanyotherbranchof physics Thisisdue

partly to thefact thatin many casesit is impossible to set

upexperimentalconditionsinkeeping withtheassumptions

theoretical solution of a general problem may be obtained

in mathematical expressions whose numerical values can

be arrived at

INTRODUCTION 3

Velocity ofSound

changes taking place in the medium through which thisenergyispropagated Thefirstproblemistodeterminethe

with the assumption that the motion of the individual

particle ofairisone of purevibration and thatthismotion

is transmitted with a definite velocity from particle to

particle, he deduced the law that the speed of travel of a

numerically equal to the square root of the ratio of the

dilata-tion Suppose, for example, that we have a given volume

dV will result The ratio of the change of pressure tothe

change of volume perunit volume gives us themeasure of

the elasticity,theso-called "

coefficientof elasticity" oftheair

namely, that if the temperature of a fixed mass of gas

remains constant, thevolumewillbeinverselyproportional

to thepressure. Thisisthelawofthe isothermalexpansion

andcontraction of gases Itiseasytoshowthatundertheisothermal condition, the elasticity ofa gas atanypressure

is numerically equal to that pressure; so that Newton's

lawforthe velocitycofpropagationofsoundinairbecomes

> density >p

4 ACOUSTICS AND ARCHITECTURE

76 cm of mercury is 0.001293 g. per cubic centimeter A

formulathe value ofcshould be

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

The experimentally determined value of c is about 18

Newton formula This disagreement between theoTy and

experimentwasexplainedin 1816, byLaplace, whopointed

that are set up by the vibrations of sound It is a matter

of common experience that if a volume of gas be suddenlycompressed, itstemperaturerises. Thisriseoftemperature

place slowly, allowing time for the heat of compression to

be conductedaway bythe walls of the containing vessel or

to other parts ofthe gas. Inother words, the elasticity of

air for the rapid variations of pressure in a sound wave is

rapid changes withnoheat transfer (adiabatic compression

and rarefaction) is 7 times the isothermal elasticity where

7is the ratioofthe specificheatof themedium atconstant

experimentally determined value of this quantityfor air is

1.40, so that theLaplace correctionofthe Newtonformula

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

Table I gives the results of some of the better known

Other determinations have been made, all in closeagreement with thevaluesshowninTableI, so thatitmay

known with afairlyhigh degree of accuracy The weight

of all the experimental evidence is to the effect that thisvelocityisindependentofthepitch, quality,andintensityof

onlyupontheratio oftheelasticityanddensityofthe

is independent of the pressure, since a change in pressure

produces a proportionalchangeindensity, leaving theratio

the densityof air isinversely proportional to the

willincreasewithrisingtemperature. Thevelocityofsound

c t at the centigrade temperature t is given by the formula

6 ACOUSTICS AND ARCHITECTURE

c t = 331.2 + 0.60*

As an illustration of the application of thefundamental

equation for the velocity of sound in a liquid medium, we

compressibilityof wateris defined as the change of volume

per unitvolumeforaunitchangeof pressure Forwaterat

per c.c. per

dyne per sq cm The coefficient of elasticity as defined

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 ofsound in

sea water ranging from 1,445 to more than 1,500 m. per

secondattemperatures rangingfrom to22 C fordepths

asgreat as100m. Here,asinthe caseofair, thedifference

tends to make the computed less than the measured

theoretical value of the velocity.1

Thevelocityofsoundinwateristhus approximatelyfour

times as great as the velocity in air, although water has a

densityalmosteightytimes thatofair. Thisis duetothe

much greaterelasticity of water

Propagation ofSound in OpenAir

homo-geneousmediumissimple, yet the applicationof thistheory

1 It isimportanttohavea clear idea of themeaningof theterm ity" as denned above In popular thinking, there is frequently encoun-

example, ifwe assumeasourceofsoundofsmall areasetup

in theopenairaway fromall reflecting surfaces, weshouldexpect the energy to spread in spherical waves with the

source, the total energy from the source passes through

the surface of asphere ofradiusr,atotalareaof4?rr2 IfE

is the energy generated persecond at the source, then the

energy passing through a unit surface of the sphere would

be E/^Trr*; that is, the intensity, defined as theenergyper

second through a unit area of the wave front, decreases as

the square of the distance r increases This is the

well-known inverse-square law of variation of intensity with

distance fromthe source, statedinall theelementarybooks on the subject. As a matter of fact, search of the

text-literature fails to reveal any experimental verification ofthis law so frequentlyinvoked inacoustical measurements

The difficulty comes in realizing experimentally the ditions of "no reflection" and a "homogeneous medium."

con-Out of doors, reflectionfrom the ground disturbs the ideal

8 ACOUSTICS AND ARCHITECTURE

of the room result in a distribution of intensity in which

usually thereis littleornocorrelationbetweenthe intensity

and the distance fromthe source

Figure 1 istakenfromareport ofaninvestigationonthepropagationofsoundin freeatmospheremade byProfessor

Louis V King at Father Point, Quebec, in 1913!l A

source.

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

pre-vailing at the time ofthese measurements

Figure 2 gives the results of measurements made in a large

havebeen on the assumption of an intensity decreasing as

the square of the distance increases The measured

does not fall off with increasing distance from the sourceTram

INTRODUCTION 9

nearly so rapidly as wouldbe the case if the intensityweresimplythatofatrain ofspherical wavesproceedingfroma

source;and wenotethat the intensitymayactually increase

as we go away from the source

AcousticPropertiesofInclosures

behavior of sound within an inclosure cannot, in general,

be profitably dealtwith from the standpoint ofprogressive

the subject matter of the first part of "Architectural

rooms One may draw the obvious inference from Fig

2 that, within an inclosed space bounded bying surfaces, the intensity at any point is the sum oftwo

sound-reflect-distinct components: (1) that due to the sound coming

decreasewithincreasing distancefromthesource according

to the inverse-square law; and (2) that which results from

inclosure From the practical point of view, theproblem

of auditorium acoustics is to provide conditions such that

that the study of the subject of the acoustic properties of

upon the audibility andintelligibilityofthe direct portion.Search of the literature reveals that practically no

systematic scientific study of this problem was made prior

one entire volume of which is devoted to acoustics, only

to the subject, beginning with the classic work on

10 ACOUSTICS AND ARCHITECTURE

references, we find, for the most part, only opinion, based

on more orless superficial observation Nowhere is there

evidence either ofa thoroughgoing analysis ofthe problem

or ofany attempt at its scientific solution

series of articles by Wallace C Sabine at that time an

analysis ofthe conditions necessary to secure goodhearing

by quoting an introductory paragraph from the first of

these papers.1

No one can appreciate the condition of architectural acoustics the science ofsoundas applied to buildings whohasnotwitha pressing case in hand sought through the scattered literature for some safe

money compels a careful considerationandemphasizesthe meagerness

andoften-repeatedstatementsaresuchas the following : that the

suggestions is the ratios of the harmonic intervals in music, but the

connection is untraced and remote. Moreover, such advice is rather

front of the galleries, to the back or front of the stage recess? Fewrooms have a flat roof where should the heightbe measured? One

auditoriums beelliptical. SandersTheaterisbyfar the bestauditorium

in Cambridge and is semicircular in general shape but with a recess that makes it almost anything; and, on the other hand, the lecture

is inwoodandtheFogglectureroomis plasterontile;oneseizesonthis

only to be immediately reminded that Sayles Hall in Providence is

largely linedwithwoodandis bad. Curiously enough, eachsuggestion

isadvancedas if it aloneweresufficient. Asexamplesofremediesmay

becited the placing of vasesaboutthe roomfor thesakeof resonance,

wrongly supposedtohave beenthe object of the vases inGreektheaters,and the stretching of wires, evennowafrequent thoughuseless device.

1

SABINE, WALLACE C., "Collected Papers on Acoustics," Harvard

In a succeeding paragraph, Sabinestates verysuccinctlythe necessary and sufficient conditions for securing good

hearing conditionsin anyroom He says:

Inorder that hearingmaybe goodinanyauditorium, it is necessary that the sound should be sufficiently loud; that the simultaneous

components of a complex sound should maintain theirproper relative

eachotherandfromextraneous noises. Thesethree are the necessary,

architectural problem is, correspondingly, threefold, and in this

intro-ductory paper anattemptwillbemadeto sketchanddefinebrieflythe subject on this basis of classification. Within the three fields thus

defined is comprised withoutexception thewholeof acoustics.

Very clearly, Sabine puts the problem of securing good

acoustical difficultiesrather thanas one ofimproving

gainedbyquantitative observationsandexperiment during

confirms the correctness of thispoint of view It is to be

said thatduringthe twenty-five yearswhichSabinehimself

devoted to this subject, his investigations were directed

alongthelineshere suggested Most ofthe workofothers

since his time has been guided byhis pioneer work in this

in the special field.

CHAPTER II

NATURE AND PROPERTIES OF SOUND

We may define sound either as the sensation produced

it as the physical cause ofthat stimulus For the present

undulatory movement of the air or of any other elastic

An undulatoryorwave motion ofamediumconsists oftherapid to-and-fro movement of the individualparticles, this

motion being transmitted atadefinitespeedwhichis

medium.

SimpleHarmonic Motion

For a clear understanding of the origin andpropagation

elemen-taryway, theidealsimplecaseof

thetransfer ofa simpleharmonic

circularmotion upona diameter

of the circular path

Thus ifthe particleP (Fig 3)

is movingwith a constant speed

upon the circumference of a

is moving upon a horizontal

diameter so that the vertical line throughP always passes

through P',thenthemotionofP'issimpleharmonicmotion

NATURE AND PROPERTIES OF SOUND 13

If the angular velocity of P is co radians per second, and

the displacement of Pf

is given bythe equation

ThepositionofP' aswellasitsdirection of motionis given

phase angle and is a measure of the phase of the motion

ofPf

. Thus when the phase angle is 90 deg., 7r/2 radians,

Pf

When the phase angle is 180 deg., TT radians, Pf

undisplacedposition, movingto the left. When the phase

is 360 deg., or 2?r radians, P' is again in its undisplaced

ofPf

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

phase in advance ofthe displacement

uniform speed s, on the circumference of a circle of radius

A, is s2/A = (Aco)2

/A = Aco2

. Since the tangentialspeed

namely, Aco2 sin co. Let m be the mass of the particle

P'y its acceleration, and Fx the force which produces its

motion Then bythe second law of motion

Fx = m = mAu2

sinut = mAw2

sin (o)t +TT) (5)

is directlyproportional to the displacement butofopposite

sign Thus if P' is displaced toward the right, it is acted

upon by a force toward the left, which increases as the

increases The force F always acts to

14 ACOUSTICS AND ARCHITECTURE

elastic body is subjected to strain are of just this type

action of an elastic restoring force will perform a simple

free movement of any body under the action of elasticforces only can be expressed as the resultant of a series of

PARTICLE

FIG 4 Simple harmonic motion projected on a uniformlymovingfilm.

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

acceleration, the traces of these magnitudes onthe moving

The maximum excursion on one side of the undisplacedposition, the distance A, is the amplitude of the vibration

The number of complete to-and-fro excursions per second

is thefrequency/of vibration Sinceone complete

to-and-fro movement ofP1

NATURE AND PROPERTIES OF SOUND 15corresponding to2?r radians ofangular motion, then

2irf

the kinetic energy of the particle Pr

is a maximum at 0,

andthe kinetic energy is

(6)

At the maximum excursion, when the particle is

momen-tarily stationary, the kinetic energy is zero The totalenergy 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 throughoutthe cycleis one-halfthe

maxi-mum or j^wwM.2 or Trra/M.2

.

1 The total energy, kinetic

and potential, of a vibrating particle equals J^mco2A2

or

Wave Motion

to particle as a plane wave, that is, a wave in which all

particles of the medium in anygiven plane at right angles

the wave be generated by the rapid to-and-fro movement

simple harmonic motion by the "disk-pin-and-slot"

of the uniform circular motion of the disk is transmitted

vertical slot in the piston head as the disk revolves We

1Themathematicalproofof this statement is not difficult. Theinterested reader withanelementaryknowledgeof the calculusmayeasilyderive the

16 ACOUSTICS AND ARCHITECTURE

laterally by having the propagating medium confined

ina tubeso long thatweneed notconsiderwhat happens atthe open end Represent the undisturbed condition of

a complete vibration of asingle particle.1

Fia 5a Compreseional planewave movingto the right.

Fia 56 Compressional planewavereflected to the left.

called the wave length of the wave motion and is denoted

at the instant that P > the first particle, having made acomplete 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 thermalagitation, andthe pressurewhich the gas

NATURE AND PROPERTIES OF SOUND 17

moving to the right. The first member of the family

thirty-nine-fortieths ofa vibration is, at this instant,displacedslightly

The phase difference between the motions of two adjacent

particles is 27T/40 radians, or 9 deg P20 is 180 deg. inphase behind P and is in its undisplaced position and

moving to the left. P40 is 2?r radians or 360 deg. behind

P0; andthemotionsof thetwoparticles coincide,P havingperformed one more complete vibration thanP40 -

particles *, 3^, %> and 1 period respectively later than

It will be noted that the motion of the particles is in

the line of propagation of the disturbance This type of

a waveconsists of alternate condensationsand rarefactions

be propagated through agas Insolids, theparticlemotion

may be at right angles to the direction of travel, and the

spoken of as a transverse wave As a matter of fact, in

compressional and transverse waves result and the motion

the body of a liquid is of the compressional type At the

free surface ofa liquid, waves occur in which the particles

andsurface tension

Equation ofWaveMotion

18 ACOUSTICS AND ARCHITECTURE

amplitude and frequency of vibration The equation

undis-placedpositionoftheparticle. Inthe caseofa plane wave,since all the particles in a given plane perpendicular tothedirection of propagationhave the same phase, the distance

of this plane from the origin is sufficient to fix the phase

oftheparticle'smotion relativetothatofaparticlelocated

at the origin. Call this distancex.

InFig 5a, considerthemotion ofa particleata distance

x from the origin P . Let c be the velocity with which

thedisturbancetravels, orthe velocityofsound Thenthe

distancexis x/c The particle at xwillrepeat the motion

theparticleatx,referredtothetimewhen P isinitsneutral

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

whatsoever, and, in an elastic medium, this motion would

NATURE AND PROPERTIES OF SOUND 19

frequency of vibration

vibration of a single particle, for example, the distance P

to P4 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 importantrelationshipmakesitpossible to compute

thefrequency of vibration, thewave length,orthe velocity

the velocity of sound in air at any temperature, is known,

fre-quency Table I of Appendix A gives the frequencies and

octave of the tempered andphysical scales.

The frequencies and wave lengths given are for thefirst

octave above middle C (C3 ). To obtain the frequencies

multiply the frequencies given in the table by 2. In the

octave above this weshould multiplyby 4, and in the nextoctave by 8, etc. For the octaves below middle C we

should divide by 2, 4, 8, etc.

Density andPressure Changesin a CompressionalWave.

In the preceding paragraphs, we have followed the gressive change in the motion of the individual particles.

pro-Figure 5a also indicates thechanges that occur in the

F , with a corresponding separation at P2 o- One-quarter

20 ACOUSTICS AND ARCHITECTURE.

period, thereisagain a condensationatF - A wavelength

therefore asdefined above includesone complete

condensa-tionand one rarefaction ofthe 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

fromthe fact that the displacement of eachparticle at any

instantisslightly differentfromthatofanadjacentparticle.

particles were displaced by equal amounts at the same

particles, that is, no variation in the density. It can beeasilyshownthat ina plane wavethecondensation isequal

dis-placement varies from particle to particle. Expressedmathematically,

We thus arrive at the interesting relationthat the

particlevelocity to the wave velocity. Further, it appears

7T/2 radians in advance of the particle displacement.

At constant temperature, the density ofa gas is directly

I, for the rapid alternations of pressure in a sound wave,

the temperature rises in and the

NATURE AND PROPERTIES OF SOUND 21

increases more rapidly than the density, so that the

frac-tional changein density Whence we have

- *

The maximum pressure increment, which may be called

the pressure amplitude, is therefore 2iryPAf/c

We haveseen that thetotalenergy, kineticandpotential,

of a particle of mass m vibrating with a frequency/ and

amplitudeA is 2ir2mA2

f2 If there areNofthese particles

percubic centimeter, the totalenergyinacubic centimeter

f2. The product Nm is the weight per cubic

centimeter of the medium, or the density. The totalenergy percubiccentimeteris, therefore,2w2

pf2A2

y ofwhich,

on the average, halfis potential and half kinetic

The term"intensityofsound" maybe usedin twoways:

either as the energy per unit volume of the medium or asthe energy transmitted per second through a unit section

perpendiculartothe direction ofpropagation The former

and the latter as the "energy flux." We shall denote the

energy densityby symbol/ andthe energy fluxby symbol

/ If the energy is being transmitted with a velocity of c

cm per second,then theenergypassingin 1 sec. through 1

J = cl = 27r2

pA2

amplitude given above may be combined to give a simple

22 ACOUSTICS AND ARCHITECTURE

Nowitcan be shownthat theaverage valueof the square

of dP over one complete period is one-half the square

ofitsmaximumvalue Hence, ifwedenotethesquareroot

of the mean square value of the pressure increment by p,

Eq (12) maybe putin the very simple form

J = 2! = P.' =J^ = Pl Q3)

in which

The expression \/cp hasbeen calledthe "acoustic

values of c andrfor various media

reasonfor callingthe expressionr, the acoustic resistanceof

in a circuit whose electrical resistance is R is given by theexpression

mediumE, the e.m.f corresponds to the effective pressureincrement, and the electrical resistance to r, the "acoustic

resistance" ofthe medium The analogue of theelectrical

particle velocity

The mathematical treatmentofacousticalproblemsfrom

thestandpointofthe analogouselectricalcaseislargelydue

who introduced the term

"acousticalimpedance" toinclude boththe resistance and

NATURE AND PROPERTIES OF SOUND 23

of Crandall's "

and to the recently published "Acoustics" by Stewart and

Lindsay

only It will be noted that it does not involve the

preferred to those giving the amplitude

density changes in the airtake placeadiabatically, thatis,without transfer ofheatfromoneportionof themediumto

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 therarefaction The

relation between the temperature and the pressure in an

adiabatic change is givenby the relation

P+dP

Now 60/0 will in any case be a very small quantity,

*yand the numerical value of

^

is 3.44 Expandingthesecond member of (14) by the binomial theorem and

24 ACOUSTICS AND ARCHITECTURE

Obviously the temperature fluctuationsin a sound wave

are extremely small too small, in fact, to be measurable;but the fact of thermal changes is of importance when we

porous bodies

phe-nomena that constitute a sound wave having been'dealt

with,it isnextof interesttoconsider theorderofmagnitude

purposeweshall startwiththe pressure changes inasound

bars (dynes per square centimeter), approximately

intensityJ for thispressure isfound tobe p* -f- 41.5 = 0.6

ergper second per squarecentimeter or 0.06 microwattper

square centimeter From the relation that J = HH;2

max.

is 27T/A, and the amplitude of vibration is therefore

0.000053cm A moment'sconsiderationof theminuteness

inthedirectexperimental determinationofthese quantities

and why precise direct acoustical measurements are so

since the development of the vacuum tube as a means ofamplifyingveryminuteelectricalcurrentsthat quantitative

ComplexSounds

In the precedingsections we havedealt with the case of

motion The tone produced bysuch a source is known as