12

The Forty-fifth Thomas Hawksley Lecture THE TRANSMISSION AND RADIATION OF ACOUSTIC WAVES BY STRUCTURES By Leo L. Beranek, B.A., M.S., S.D., D.Sc.-Hon.* Of major interest to architects, designers of vehicles, and acousticians is the control of structure-borne sound. Walls and panels are set into vibration by airborne waves or by vibrating mechanisms. The panel so excited will radiate sound and it may carry the vibrations to other panels or bodies. In this lecture, the author will treat the problem of acoustic transmission through walls and panels in the audible frequency range. At low frequencies the panel vibrates as a plate or a stretched membrane. At higher frequencies the panel may behave as a quasiinfinite sheet. Above a particular 'critical' frequency, the wavelength of the bending waves in the panel will be longer than the compressional waves in air at the same frequency. The two wavelengths may be brought into coincidence provided the airborne wave impinges on the panel at an angle 8 determined by cos 8 = c,jcb where ca and cb are the speeds of sound in the air and panel, respectively. At the coincidence angle, an airborne wave striking one side of a panel will set it into a level of vibration such that the magnitude of the airborne wave radiated from the other side may be only a few decibels below that of the incident wave. This effect will be discussed both for airborne waves impinging at individual angles of incidence and for waves at many angles of incidence simultaneously. Measurements on plates, concrete sheets, and masonry walls will be presented and analysed. Analysis of the different types of response to airborne wave excitation will be made. Rules for the selection of simple and complex structures for buildings and vehicles will be suggested. Means for reducing thc response of structures to airborne wave and mechanical excitation include the use of damping materials in or on the structure, the introduction of discontinuities and the use of sound-absorbing blankets in the structure. Recent data on damping materials and means for utilizing them to produce maximum reduction of flexural waves will be presented. It will be shown that by proper utilization of such materials a structure-borne wave may be attenuated in a given distance by a factor of 10 or more than when utilized in a conventional manner.

INTRODUCTION

SOUNDIS DEFINED as a time-varying alteration in pressure, stress, particle displacement, or shear, occurring in an elastic medium. The medium in which the sound exists is indicated by an appropriate adjective before the word 'sound' such as airborne, water-borne, or 'structure-borne. Contrary to popular usage, not all sound waves evoke an The MS. of this lecture was received a t the Institution on 10th November 1958. For a yeport of the meeting, in London, a t which this lecture was given, see p . 35. * Lecturer, Massachusetts Institute of Techiiology, and President, Bolt Beranek and Newman Inc., Cambridge, Massachusetts. Proc Instn Mech Engrs

auditory sensation. Sound is of importance to engineers because, if intense enough, it may affect human beings, may cause malfunction of electronic or mechanical equipments, and may produce structural failures. Two kinds of sound waves are found in elastic media, namely, compressional and shear waves. In gases, sound waves are compressional. Only in second order, because of viscosity, do shear waves exist. In perfect gases, the air particles would always move back and forth in a direction parallel to the direction in which a sound wave is travelling. Vol 172 1959

THE TRANSiMISSION AND RADIATION OF ACOUSTIC WAVES BY STRUCTURES

In liquids, compressional waves are also of primary importance. Shear waves occur in second order, but ore of greater importance than in gases. In structures, both compressional and shear waves are of primary importance. When a bar is excited at its end by motion parallel to its axis, a compressional wave travels along the bar. So little airborne sound is radiated by a compressional wave in a bar that the ear is not likely to hear it. The wave usually may only be perceived by touch or by an electromechanicaltransducer. Distortional (torsional) waves in a circular rod are a form of shear wave, but, owing to the shape of the rod, produce very little sound in a surrounding medium. The waves of most interest in structures are bending (flexural) waves. They have associated with them large transverse displacements that may readily couple to compressional waves in surrounding fluids. Bending waves are easily excited in a bar or plate by airborne or waterborne sound waves. In turn, bending waves readily radiate sound energy into fluid media. Thus, a vibrating engine in a ship excites bending waves in the ship’s hull. These waves radiate sound into the surrounding water. Some of the ways in which sound may travel from a source in one room to an observer in a second room are shown in Fig. 1. Airborne sound waves, 1, may impinge, either

ROOM

I

ROOM 2

Fig. 1. Diagrammatic Representatioiz of Some of the Ways in Which a Machine (Source of Sound) May Create Sound in an Adjoining Room

directly or after reflection, on the common wall between rooms 1 and 2 and excite bending waves that cause airborne sound waves 1A and B to be radiated into room 2. Airborne sound waves, 2, may impinge on other walls of room 1 and the resulting bending waves may be transmitted by the walls to room 2 and radiated as airborne sound waves 2A and B. The mechanical motion of the machine may induce bending waves in the floor (or walls) that may travel through the floor and walls to produce airborne waves 3A and B. As we shall see later, the intensity of the transmitted sound depends on many things such as the frequency, intensity and angles of incidence (on the walls) of the incident waves, the wall dimensions and materials, the manner in which the walls join each other and the existence of damping materials. In airplanes or missiles, the fuselage may be set into vibration by (1) sound waves produced by the engines or propellers, (2) direct excitation from the vibration of the engines, or (3) by travelling turbulence vortices over the Proc Instn Mech Engrs

13

exterior surfaces due to the forward motion of the body through the air. Structure-borne waves in a fuselage radiate airborne waves to the passengers’ ears, or cause vibrations of equipment. In automobiles, vibrations of the engine or road surfaces are transmitted through the structure to the sides and roof of the car. Airborne sound is then radiated to the interior. Air turbulence and engine intake and exhaust noise also create structure-borne sound waves. Much of the work on structure-borne sound in buildings has occurred since the 1939-45 war. The stimulus for h s activity has been, in part, the need for large-scale reconstruction of war-damaged buildings. It is not surprising, therefore, that three of the leading countries in this area of investigation have been England, Germany, and Holland. The literature on structure-borne sound has largely appeared in Acustica and, in particular, in the Akustische Beihefte, a German supplement to Acustica. Much practical information and data have appeared in publications of the British Building Research Station ((I)*; Parkin and Humphreys (2)) ; The Netherlands Institute for Public Health Engineering, TNO, The Hague (3); in the United States Air Force research reports (4); and in the Proceedings of the Goettingen Symposium (5). Papers from time to time have appeared in the Journal of the Acoustical Society of America and in Noise Control. Much of the American effort has been on the quieting of aircraft and ships and has not yet been published. Some American information on sound transmission in buildings is summarized by Cook and Chrzanowski (6). Other information will be published soon by Beranek (7). Papers related to structure-borne sound also appear in the Soviet ‘Acoustical Journal’. (Published in the United States in English by the American Institute of Physics under the name Soviet Physics: Acoustics.) Extensive work has been done by Cremer and his students in Germany (8) and recently in the United States by Watrers (9). In the material which follows we shall discuss some properties of elastic waves; define acoustic transmission factor and transmission loss; present the phenomenon of wave coincidence; tabulate some physical properties of common structural materials; give formulae and charts for determining the transmission loss of single and double walls; give formulae and charts for estimating the transmission of sound through discontinuities in structures ; discuss coupling between panels of a double wall; mention transmission of impact sound through floors; and, finally, treat the reduction of transmitted sounds by vibration damping and sound-absorbing materials. Many of the computational charts and procedures have not before appeared in the public press. S O M E P R O P E R T I E S O F E L A S T I C WAVES IN P L A T E S

It is beyond the scope of this lecture to cover thoroughly the various types of elastic waves. Excellent papers have been presented recently which review experimental and

* A numerical list of references is given in the Appendix. Vol 173 1959

LEO L. BERANEK

14

theoretical work in this field (Davies (10);Schmidt (11); Exner, Gueth, and Immer (12); Cremer (8);Kurtze (I 3); Junger (14);Deresiewicz (IS); Watters (9)). The structure that we are most concerned with is the vibrating plate. Several of the formulae needed in the latter part of this manuscript are presented here. Longitudinal Wave Velocities In a thin bar, a longitudinal wave can be propagated along its axis whose low-frequency phase velocity is the 'longitudinal bar velocity' c, = (E/pp)l/2 ft/sec (mjsec) . . (1) Where E is Young's modulus in 1bfW (or newtonlm2) and pp is the density of the plate in slug/fV (or kg/m3)*.

In a plate, owing to the constraint on the sides, the lowfrequency phase velocity is slightly different. It is known as the 'longitudinal plate velocity', and is given by CLI

= Ja(-l3 .

E

= . CL

-

*

(2)

In calculations we generally make the approximation that is given by equation (1). The quantity u is the Poisson's ratio, equal to 0.3 in most cases. c,'

The Wave Equation The classical equation of motion for a bending wave on a thin plate is

where 7 is the displacement of the plate perpendicular to its surface in ft (or m) and h is the thickness of the plate in ft (or m). Bending Wave Velocity and Wavelength From a solution to equation (3) (Morse 15a) we find that the velocity of propagation of the bending wave is cB = (1*8hfc,')1/2 ft/sec (or mlsec) . (4) where h is the thickness of plate in ft (or m) and f is the frequency of the wave in cycles per sec. The wavelength of the bending wave in the plate is given by -,

We see immediateiy that there is an essential diEerznce between a longitudinal wave in a thin bar (or in air) and a bending wave in a plate. In the case of a longitudinal wave, the wavelength is inversely proportional to frequency because the speed of travel is constant. In the case of a bending wave, the wavelength is inversely proportional to the square root of the frequency. Because bending waves of different frequencies travel at different velocities, the wave form of a complex wave is not preserved and the medium when excited into bending waves is said to be dispersive. * To obtaiii sluglft3 IbIft3 is divided by 32.2 frlsecz. Proc Instn Mech Engrs

Supported Rectangular Plate For any plate of finite size there is a galaxy of frequencies at which resonance occurs. For each resonance there is a different arrangement of nodal lines and of maximum lines of vibration over the surface of the plate. These vibrating regions radiate airborne sound waves. The strength of the radiation depends on how hard the plate is driven and on the particular configuration of the pattern of vibrations in the plate. Assume that the wall between a room and out of doors is made from a solid, single sheet of metal supported at the edges. This sheet of metal has mass and stiffness distributed throughout its surface. If we were to strike the centre of this wall with a hammer, it would ring or resonate at a number of frequencies, and it would radiate sound. The lowest of these resonance frequencies would occur when the panel moves everywhere in phase. For example, if the centre of the panel were displaced and then released the panel would vibrate back and forth perpendicular to the plane of the sheet. The largest amplitude of vibration would be at the centre, and the amplitude would decrease gradually to zero at the corners. The manner in which the panel vibrates at the higher resonance frequencies may be visualized by assuming that bending waves, excited by the hammer blow, travel out to the boundaries and are reflected backward across the panel. Whenever the wavelength of the sound in the panel is such that reflected waves reinforce other travelling waves of the same wavelength, a resonance condition is created and the panel vibrates vigorously at some parts and exhibits lines of zero motion at others (Fig. 2). After the initial hammer blow, the vibration persists for a length of time that depends on the internal and external damping of the panel.

U

b

Fix. 2. Vibratioiz Patterns of Flat Plates Mounted Horizontally on a Single Support at the Centre with No Clamping at the Edges The plates were set into vibration with a violin bow and sand was sprinkled on top. The sand collected at the lines of zero motion as shown above. The two different patterns shown are produced by pulling the bow across the edge at two different places, thereby exciting two different resonant conditions.

A rectangular thin plate, supported (but not clamped in frames) at the four edges has resonance frequencies (called normal frequencies) as computed by

where cL is the longitudinal wave velocity in ft (or m), h is Vol173 1959

THE TRANSMISSION AND RADIATION OF ACOUSTIC WAVES BY STRUCTURES

the plate thickness in ft (or m), and I,, l, are lateral dimensions in fi (or m). By a ‘supported plate’ we mean that at the boundaries, no transverse motion is possible but that the slope of the plate is not constrained. In equation (6) the quantities YZ, and n,, can have any integral values, independent of each other, 1,2,3. . . The lowest normal frequency is f l , with n, = 1 and ny = 1. If the panel were rigidly clamped at the boundaries, thereby restricting both the motion and slope to zero, the lowest resonance frequency would be twice as high and all others would be somewhat higher than for the ‘supported’ case. Below the lowest resonance frequency, stiffness alone controls the movement of the panel. That is to say, mass and damping are unimportant. We must not conclude that for all frequencies above this lowest resonance frequency, the panel is mass controlled. Equation (6)shows that hundreds of resonances are possible which require the existence of both mass and stiffness. Indeed, for undamped systems like a bell, a multitude of resonances can be perceived. However, if there are significant energy losses either in the plate or at the boundaries, the higher resonances are less pronounced and the surface of the plate on the average (averaged both in space and over a band of frequencies) moves nearly as though it were mass controlled, that is to say, as if it were made of a large number of little masses free to slide by each other without intermass constraining forces. The mass-controlled region may extend from two or three times the lowest resonance frequency up to the ‘critical frequency’. By critical frequency, we mean the frequency at which the bending wavelength AB in the panel equals the wavelength of the radiated wave h in the second medium. As we shall see shortly, above the critical frequency the stiffness of the panel again may not be neglected.

.

Summary In this section it was stated that stiffness in a panel is important in determining the amount of radiated sound (1) below the lowest resonance frequency and (2) above the critical frequency. In between (1) and (2), on the average, practical panels often may be treated as though they were mass controlled. Formulae for wave velocities and wavelengths and for the resonance frequencies of supported rectangular plates were also given. T R A N S M I S S I O N COEFFICIENTS A N D TRANSMISSION LOSSES OF DOUBLE A N D S I N G L E WALLS

Throughout this paper we shall be concerned with the type of wall” that is used as a barrier between a region in which there is a source of airborne sound and a region in which there is an observer of airborne sound. For example, one application of the results is the party wall between two flats in a multi-family dwelling. The terminology and formulae

* We shall speak

15

in common use for specifying how well a wall transmits sound are now described.

Definitions The decibel (dB) is basically a unit implying a given ratio between two powers. I n acoustics, the difference in level of one power with respect to another is commonly expressed in decibels. The formula for calculation of acoustical and electrical power level differences is electrical}power level difference = 10 logloW2 dB {sound W1 Thus a ratio of 10 in power corresponds to a level difference of 10 dB. A ratio of 100 in power corresponds to a level difference of 20 dB. Similarly a power ratio of 0.1 corresponds to a level dif€erenceof - 10 dB, etc. I n acoustics it is customary to choose a value for Wl above which the levels of all other powers W2are rated. This value of Wl is called a ‘reference power’. The most common reference power in America is 10-13 watt, so that sound power level = 130+ 10 log W2dB re 10-13 watt Obviously, W2must also be expressed in watts. Because acoustic power is proportional to the meansquare sound pressure measured at a point in a freetravelling sound wave, we often substitute p12 for Wl and p22 for W2and call the result sound pressure level difference. PI2 sound pressure level difference = 10 log -

PZ2 Pl

= 20 log - dB P2

where p 1 and p2 are r.m.s. sound pressures. A common reference sound pressure is 0.0002 dyn/cm2 (called 0.0002 microbar and equal to 4 . 2 10-7 ~ lbift2). So sound pressure level = 74+20 logp, dB re 0.0002 microbar where pl is expressed in microbars. The ‘sound transmission coefficient’ T of a wall is the fraction of the sound power in the incident airborne wave that appears in the transmitted airborne wave on the secondary side of the wall. The ‘sound transmission loss’ TL of a wall in decibels is

TL = 10loglo

(t)

dB

. . .

(7)

A situation of common interest, namely, a wall with two parallel panels is shown in Fig. 3. Here are illustrated (1) the incident and reflected sound waves on the source side of a double wall; (2) the standing sound wave between the two panels of a double wall; and (3) the wave transmitted through the double wall. For a single wall, item (2) above vanishes. Various authors (Schoch (16); London (~7); Beranek and Work (18); Cremer (19);Schoch and Feher ( 2 0 ) ) have derived for both double and single walls the desired ratio of the acoustic intensities of the incident and transmitted waves for air medium on both sides defined as

of a wall as being made up of one or more panels separated by air spaces and containing, in the general case, soundabsorbing niaterials.

Proc Instn Mech Engrs

Vol 173 1959

LEO L. BERANEK

16

The vertical bars indicate that the r.m.s. of the ratio of the sound pressures is desired. A single panel is one whose two sides move together. Hence, if v, is the velocity of the panel surface on the source side taken perpendicular to the surface and v2 is the same on the opposite side, then v1 = v 2 = v,~.For a homogeneous panel, t h s relation is generally satisfied provided its thickness is less than one-sixth of the wavelength of the longitudinal wave A, in the panel at the frequency being considered. Secondly, let us define the specific transmission impedance of a panel as the complex ratio of the pressure difference on the two sides of the wall to the velocity of the wall perpendicular to its surface. Thus, Z, = Ap/v,, . . . * (9) where 2, is a complex number with the units of lb.sec/ft3 (or newton sec/m3);d p has the units of lb/ft? (or newton/m2); and v, has the units of ft/sec (or mjsec). Both d p and v,,are complex quantities. The complex aspects of these quantities indicate the time-phases relative to a reference time. These quantities are point quantities and are equal in magnitude at all positions on an infinite plate driven by a wave of infinite extent. Finite plates are handled in practice as discussed later.

Double Wall Assume that both panels in the double wall of Fig. 3 are identical. Then, from the above references, the complex

and pc is the characteristic resistance of air (witha density p and a velocity of propagation c) in lb.sec/fF (or newton sec/m3), w = 2rf, where f is the frequency in cycles per second, j is the operator 4- 1, and d is the separation of the panels in feet (or in metres). Under the special case that the specific transmission impedance of each panel in the wall is reactive, that is to say, the real part is near-zero, we have Z,=_R,+jX,=O+jX, . . (12) The quantity X,is called the ‘specific transmission reactance’. We may now, with the help of equations (7) and (8), write for the transmission loss of a double wall,

{ + ($1

TL = 10 log,, 1

x 2

[cosp-+($)

+

cosz x

cos+ sin/3I2}

. . .

(13)

T o enable conversion from metric to English to mixed English systems of units, Table 1 has been given. The specific transmission impedance 2, of a wall is nearreactive and equal approximately to jX, only when the frequency of the incident wave is below the ‘critical frequency’. Above the critical frequency, any damping of the bending wave, either internal or applied to the surface, introduces a significant dissipation term R,.

Single Wall For a single wall of specific transmission impedance Z,, equation (lo), with d = 0, reduces to Yt

The factor of 2 appears in the denominator because (Fig. 3) not only the separation d has been reduced to zero, but one of the walls of impedance 2, has been eliminated. That is to say, Z, for a single wall is taken to be half that for the walls in Fig. 3. If a purely reactive specific transmission impedance, equation (12), is assumed, the transmission loss for a single wall, using equations (7) and (8),becomes

F e . 3. Geometrical Situation Illustrating the Transmission of Sound Through a Double Wall with Parallel Panels The incident sound wave p i impinges on the first panel at angle 4. Part of the sound energy is reflected (wave f i r ) and part passes into the air space between the walls. I n the inter-wall air space, a standing wave is set up with components travelling to the right p + and components travelling to the left p - . Part of the sound power passes through the double wall and produces the transmitted sound wave p r a t an anglc 4.

ratio of the incident and transmitted sound pressures for a double wall (London (21))is

Reverberant Source Room Frequently a wall is located between two rooms, with the source room being very reverberant. In a fairly reverberant room, the sound impinges on a wall from all angles below a limiting angle,,,,4, that is usually not much greater than 80”. T o obtain the ‘average transmission coefficient’ 7 we need to perform the operation given by

-- - f

LIMIT

r cos 4 sin +d+

J n

where, by definition,

p E (wd cos r$yc = (27rd cos +)/A Proc Instn Mech Engrs

.

(11)

’-

[m,lM1r

. .

(16)

cos 4 sin +d+

J o

Vol 173 19-59

T H E TRANSMISSION AND RADIATION OF ACOUSTIC WAVES BY STRUCTURES

17

Table 1. Units Used in Connection with TL Formulae

System of

units

MS

d

mkS

newton m2

m -

kg m2

metre

1

cgs

cm2

dyn

cm sec

gm cm2

cm

1

Foot-slug-sec

Ib

@

ft -

slug -

ft

1

-_

sec

sec

ft2

Approximate PO at sea level

PC

ku.

(see equation (37))

410

105

n-sec/mJ 41

106

d-sec/cm-’

2.12 x 103

b Mixed English system of units. I

I

I

I

Approximate Po at sea level

in.lb.sec

1

lb

I

in.

1

lb

1

in.

1

386

0.584

lb/(inz-sec)

I

14.7

M J p c in Table l a exactly equals w / ( p c ) i n Table lb. to neuTton/m2 multiply by 47.88. T o convert from lb/ftt To convert from lb/ft2 to kg/m2 multiply by 4.882. T o convert from Ib/ftz to lb/in2 multiply by 0.006 95. T o convert from slug/ft2 to kg/m2 multiply by 0.151. To convert from newton/mz to dyn/cm2 multiply by 10. T o convert from kg/m2 to gm/cm2 multiply by 0.1.

Summary In this section formulae were presented for the transmission losses of double and single walls at any angle of wave incidence in terms of the specific transmission impedance 2, of the wall. Also presented was the equation for determining the average transmission coefficient for a wall with the source side exposed to a highly reverberant sound field in which many angles of incidence below a limiting angle exist simultaneously. These formulae are of particular importance in determining the transmission losses of double and single walls in the frequency region where the component panels are mass controlled.

condition of equal wavelengths in the panel and in the air is designated ‘wave coincidence’. Similarly, if an airborne sound wave is incident on a panel at such a frequency and angle that wave coincidence occurs, the panel is set into motion readily, just as though it were at resonance. A condition of wave coincidence is illustrated in Fig. 4. The incident wave with wavelength h travels from the lower DIRECTION OF BENDING WAVE

W A V E COINCIDENCE

It is not surprising that an intense wave is radiated by a panel when it is driven at a frequency equal to one of its resonance frequencies. Less expected is the fact that for every frequency above a certain critical frequency there is a particular angle do at which even an infinitely large vibrating panel radiates sound as though it were at resonance. The reason for this increased radiation of sound is that above this critical frequency the wavelength of the bending wave in the panel A, is able to become equal to the wavelength in air X projected on the panel, and a high degree of coupling between the panel and the air is achieved. This Proc Insrn Mech Engrs

V I B R ~ T ~ N GPANEL

Fig. 4. Wave Coincidence The wavelength of the bending wave in the panel is AS. A sound wave in air, whose wavelength is A, impinges on a plate at the angle 40.When A/sin 40 is equal to AS, the intensity of the transmitted wave approaches the intensity of the incident wave. Vol 173 1959

LEO L. BERANEK

18

left and impinges on the panel, which is set into vibration. The resulting vibration appears as a bending wave with wavelength A, that travels as shown (vertical on the page). If X is less than A,, then, at a particular angle of incidence +o, it is possible for the projccted wavelength (;\/sin do)to equal the bending wavelength A,, as shown. Under this condition, the panel vibrates at an amplitude almost equal to the amplitude of the air particles in the incident wave. In turn, the panel radiates a transmitted wave with almost this same amplitude also at the coincidence angle $o. In other words, the panel radiates a wave (the transmitted wave) that is almost as intense as the exciting wave (the incident wave). Hence, the transmission loss at that frequency and angle is very small. The condition for wave coincidence is sin+o = h/A, . . . . (17) Obviously, if the wavelength of the sound in air is greater than the wavelength of the sound in the plate, no wave coincidence can occur, because the sine cannot be greater than 1.0. Foi a given frequency f, c = Af and cB = A,f, so that

.

-

siii(b0 = C/C, . . (18) where c,, is the velocity of propagations of the bending wave in the plate and c is that in air. T o develop the terminology further, when a fixed frequency is assumed, the angle at which wave coincidence takes place is defined as the ‘coincidenceangle’. When a fixed angle is assumed, the I-requency at which wave coincidence takes place is defined as the ‘coincidence frequency’. The author does not know who first discovered the phenomenon of wave coincidence. G. W. Pierce described it in a United States patent* that was filed for on 2nd August 1933. In the field of ultrasonics, Sanders (22) also described the effect. Cremer (8) of Germany presented the concept in 1942 specifically in relation to the transmission of sound through walls, and deserves credit for explaining the difference between the simple wall theory (assuming panel resonances only) and actual panel measurements that involve radiations of sound at angles oblique to the panel. Critical Frequency The critical frequency fc is defined as the lowest frequency at which wave coincidence occurs and is that frequency for which A, = A, or c, = c. I n other words, the critical frequency is the lowest possible coincidence frequency and occurs for grazing incidence sound, that is, do = 90”. From equation (5) and setting c, = c, we get, in a consistent set of units,

In mixed English units, we have

where h is in inches, c and cLf in ftlsec, pp in lb/ft3, and E in lb/ft2. Equations (17) to (20) are strictly valid only for cases where the wavelength of the flexural wave is greater than about six times the thickness h of the panel (A > 6A). Coincidence Frequency Another way of looking at the phenomenon of wave coincidence is to say that the airborne wave divides the panel into segments, each h/(2 sin 4) in length. Each segment behaves like a supported b x . Its effective mass varies in direct proportion to its length and its thickness and its effective stiffness varies in inverse proportion to the cube of the length and in direct proportion to the cube of the thickness. The bar is naturally expected to vibrate very easily when the mass reactance equals the stiffness reactance. Because the stiffness reactance of the segmented length of panel varies directly 3s the square of the frequency and because the mass reactance is independent of frequency (due to the fact that the segmented length is inversely proportional to frequency), it is found that the critical freqiiency has the form shown in equation (19). I n addition we observe by this logic that the coincidence frequency varies in inverse proportion to sin‘+ We can also show this relation by combining equations (5) and (17) to yield

SOME PHYSICAL PROPERTIES OF C OMiW 0 N M A T E R I A L S

It is intended that this lecture may be used for numerical calculations. Accordingly the properties of several common structural materials are presented in this section. Surface Density and Mass Surface densities, weights, and masses in mks, English, and mixed English units for a number of materials are given in Table 2. These quantities are used in the calculation of transmission loss at frequencies below the critical frequency.

M , f c Froduct and 7 Two quantities are needed for the calculation of transmission loss at frequencies above the critical frequency. These are the internal damping factor 17 and the product of the surface density M, (or, in mixed English units, the surface weight w) and the critical frequency f,. The nunrerical values of these quantities for a number of common building materials are given in Table 3. Critical Frequencies A chart of critical frequencies for several common structural materials as a function of plate thickness is given in Fig. 5.

-

TRANSMISSION LOSS OF SINGLE PANELS

*

PIERCE,G. W. 1933 U.S. Patent No. 2 063 945 filed 2nd August 1933, ‘Transmission and Reception of Soiind Waves’.

Proc Instti hfech Etzgrs

Mass-controlled L i m p Panels Laboratory measurements of the transmission loss of panels that are either damped internally or at the boundaries reveal V d 173 1959

THE TRANSMISSION AND RADIATION OF ACOUSTIC WAVES BY STRUCTURES

Table 2a. Surface Densities and Wights of Common Building Materials per Unit Thickness

19

Table 2b. Surface Densities and Weights of Building BZocks ~~

Material

w/h, lb/ft2 per in. of thickness

MJh slug/ftZ per ft of thickness

14 9 10-12

5.2 3.4 3.7-4.5

2700 1700 1900-2300

12 8 5 13 59

4.5 3.0 1.9 4.8 22.0

2300 1500 1000 2500 11 000

5 9 6

1.9 3.4 2.2

1000 1700 1150

Aluminium Asbestos board iTrankte) Brick Concrete dense cinder . cinder fill . Glass Lead . Plaster light-weight aggregate (Perlite or Vermiculite binder) . sand aggregate Plexiglas or Lucite . Sand dryloose . dry packed wet Steel , Wood timber. fir plywood

:

. . .

.

.

Material

. .

7-8 9-10 10

40

2.6-3 34-3.7 3.7 15.0

.

3-5 3

1.1-1.86 1.1

Weight/ft2

Mass/mz

lb/ft2

Masslft-2 Ms, slug/ftZ

25

0.78

120

35

1.09

170

35

1.09

170

54

1.68

260

37

1.15

180

W,

1300-1 500 1700-1900 2000 7700 580-1000 580

Hollow cinder block, 6 in. thick* . I-Iollow cinder block, 6 in. thick, # in. plaster on each side Hollow dense concrete block, 6 in. thiclc . Hollow dense concrete*block, 6 in. thick, sand filled Solid dense concrete block, 4 i n . thick .

.

.

.

.

& kglm2

* No@nal thickness. Approximately $ in. smaller to allow for mortar joint. neglected in predicting the TL, the panel is described as ‘limp’ in this frequency region. Then, the specific transmission impedance of equation (12) is simply a mass reactance, Zs=jwM, . . (22) where M , is the mass per unit area in slug/fiz (or kg/m2). Substitution of equation (22) into equations (14) and (8) yields

.

that at some frequencies such panels behave nearly as though they were a lot of little masses free to slide on each other. This frequency range lies below the critical frequency and above about twice the frequency of the first panel resonance. Because the stiffness of the panel can be

1 7

*

See Table 1 for values of pc and units for M3. Note that if pc is taken as 84, the quantity w in lblft2 may be substituted for MS.

Table 3. Internal Damping Factors and Products of Surfate-density and Critical-frequarcyfm Common Building Materials Product of surface-density and critical frequency

Material

(lb/ft2 x c/s) wfc

1 2 3 4

Aluminium. . Brick. Concrete, dense poured . Concrete (clinker) slab plastered on both side’s 2 ii. thick

5

.

Masonry block hollow cinder (nominal 6 in. thick) hollow cinder, # in. sand plaster each side (nominal 6 in. thick) hollow dense concrete (nominal 6 in. thick) hollow dense concrete, sand-filled voids (nominal 6 in. thick) solid dense concrete (nominal 4 in. thick) Fir timber . Glass. Lead chemical or tellurium

.

.

6 7 8

9 10 11 12 13 14 ~~

7000-12 OOO ‘Ogo00 o0

1

I

10 000

.

.

. .

Antimonial (hard) Plaster, solid, on metal or gypsum lath Plexiglas or Lucite Steel . . . Plasterboard (+ in. to 2 in.) Plywood (+ in. to la in.) Wood waste materials bonded with plastic, 5 lb/ftz

.

. .

.

(kg/m2 x c/s>

Wfc

?*

MJC

217 217-373 279

34 700 34 700-58 600 43 900

10-4 0.01 om2

48 800

0.005

23 200

0.005

162 147 269

25 500 23 OOO 42 200

OW5 0.007

345 31 242

54 100 4880 38 800

Varies with frequency (Fig. 12) 0.012 0.04 0.002

3850

605 000

0.015

3240 404 225 621 217 81 466

508 000 63 400 35 400 97 500 34 200 12 700 73 200

0.002 0.005 0.002 10-4 0.03 0.01

310

4750

. .

(slug/ftz x c/s)

11 100 7800 ‘Oo0 124 000 (approx.) 104 000 13 000 7250 20 000 7000 2600 15 000

1

Internal damping factor at 1000 c/s

-

* These values for I ) are approximate and in most cases are based on very limited data. Proc Znstn Mech Engrs 3

Vo1173 I959

LEO 1.BERANEK

20

I:. jc-

CYCLES PER SECOND

Fig. 5. Critical Frequency f, Plotted as a Function of Plate Thickness h This is the lowest frequency at which the coincidence effect is possible. At this frequency, the T L is quite small.

Limp-wall Transmission Losses If the incident sound wave impinges on the limp, masscontrolled panel at normal incidence so that = 0 (Fig. 3) we obtain, using equation (7), the so-called 'normalincidence, limp-wall mass law',

+

[TLIo = 10loglo

[l+rS)2] . dB

(24)

Further, insertion of equation (23) in equation (16) and integration up to +,,, = 90" and then inserting the result in equation (7) yields the so-called 'random-incidence, limp-wall mass law', [TLIrandom = [TL]O- 10 log,, (0'23[TL]o) dB (25) Equation (25), as written, is an approximation valid for [ TLl0 greater than about 15 dB (see Fig. 6 for values below 15 dB). Equation (25) says that if the sound on the primary side comes from a highly reverberant room where all angles of incidence from waves of equal energy impinge on the wall, the transmission loss is significantly reduced compared to its normal incidence value. Equations (24) and (25) are plotted as the upper and lower lines, respectively, in Fig. 6. These are idealized curves because they assume that all resonances are suppressed. For a finite-sized panel that is not sufficiently damped, the transmission loss curve will vary above and below these curves as the panel goes through anti-resonances and resonances. Measurements made in the field in buildings on walls between flats and offices reveal that transmission losses generally lie about 5 dB below the normal incidence curve. We thereby draw such a curve in Fig. 6 and call it the field incidence curve. A transmission-loss curve of [ TLIO--5 dB approximates a diffuse sound field with a limiting upper angle of about 78" (equation (16)).

.

proc Instn Mech Engrs

=FREQUENCY L SURFACE WEIGHT-+

x lbift'

Fig. 6. Theoretical Transmission Loss Curves for Masscontrolled Limp Panels Field data indicate that in practical situations the field incidence curve gives satisfactory predictions below the coincidence frequency and above about twice the lowest panel resonance frequency. (After Watters (9) and London (17))

Solid Panel Above Critical Frequency In the region near and above the critical frequency f,, the mass-law curves in Fig. 6 are no longer useful because the transmission loss is decreased due to the coincidence effects. I n other words, because wave motion in the wall is involved, the bending stiffness of the panel must be considered. Cremer (23) shows that the wall impedance is given by cL2M/1W sin4 qcL2M$*w3 sin4 4 2, = 12c4 +j[wM,-12c4

+

. . -

1

(26) where q is the damping factor from the complex Young's modulus given by E ' = E(l+jT) . . (27) The first term in equation (26) is the damping term because it contains 17 and because it is the real part of the impedance. It also is affected by the panel stiffness because cL2= E/pp, as we saw from equation (1). The second term is determined only by the surface mass. The third term is the one that contains the principal information about the stiffness because it would persist even if there were no damping and because cL2 contains the real part of the Young's modulus E. Note that although the panel density n/i, appears in the third term it is actually cancelled out by pp in cL2. Substitution of equation (26) in equation (10) and solving for I/. from equation (8) yields (Feshbach (24))

. .

We see that if 4 is set equal to zero, which is the case for a Vol I73 I959

THE TRANSMISSION AND RADIATION OF ACOUSTIC WAVES BY STRUCTURES

wave incident on the panel normally, we obtain for 1 / ~ exactly the same expression as we had in equation (23). I n other words, when the wave is normally incident and is plane, it cannot set the bending wave into existence. In the practical case we must know how to determine the transmission loss for near-random incidence sound. Without going into the mathematicalramifications, the theoretical results are shown in Fig. 7 (Feshbach (24)). T o use this figure the values of w or M,, f,, and q for common materials are needed. These values are found in Table 3.

:t/c

-CIS

X Ib/frl

A -DECIBELS

--

..

11,

I. - JI

x

kg/m'

J/Jc

Fig. 7. Transmission Loss for Randomly Incident Sound on a Single Homogeneous Panel of Wezght w lb/ft2 (or Mass in kg/m2) and Damping Constant q The frequency range near and above the critical frequency fc only is considered. The curves are not continued below 40 dB in the f/fcregion between 1 and 2 because there they become a function of both wf, and 7.

Practical Design Chart for a Solid Panel The theoretical chart in Fig. 7 reveals several interesting facts: first, below about 0.6 of the critical frequency the transmission loss is independent of damping and depends almost entirely on A . Secondly, we note that at f/fc = 0.3, A is related to wf (or M,f) according to the 'field incidence' curve in Fig. 6. As an example, assume wf, = 5000 c/s.lb/ft2 and f/jc= 0.3. Then A = 10 dB and ( T L + A ) - A = T L = 40-10 = 30 dB. From Fig. 6 for wf = 5000 ~ 0 . 3 = 1500 c/s.lb/ft2, we obtain from the field incidence curve, T L = 30 dB. Thirdly, all of the lines above f/fc= 1 are nearly parallel to each other, and increase at the rate of about 10 dB per doubling of frequency. Finally, from field observation with panels as mounted in buildings, the dip in the transmission loss between f/fc= 1 and f/fc= 2, Proc Instn Mech Engrs

21

generally does not drop much below (TL+ A ) f 36 or above 44 dB. Taking into account the observations of the previous paragraph, one can, without much loss in generality, make up a single design curve for a wide variety of panels. The desired curve is given in Fig. 8. To estimate the transmission loss curve for a single solid panel (above the lowest panel resonance frequency) the following procedure is followed: (1) Select a piece of graph paper like that used to draw Fig. 8. Label the abscissa 'frequency in c/s'. The abscissa should be a logarithmicscale so that each octave (doubling) of frequency has the same extent along the abscissa. Label the ordinate 'transmission loss in dB'. (2) From the data in the table on Fig. 8 and from the panel thickness determine the surface density of the panel. For example, suppose the panel is a 4-in. thick slab of dense concrete. The surface density (weight) is 4~ 12 = 48 lb/ft2. (3) From the field incidence curve in Fig. 6, determine the TL at one frequency. For example, suppose, for the panel of (2), that we want the transmission loss at 100 c/s. The TL from Fig. 6 for wf = 4800 c/s.lb/fV is 41 dB. Plot this point on the graph paper and draw a line through it with a slope of 6 dB/octave (Fig. 8). (4) Determine the plateau height in dB from the table in Fig. 8 and draw a horizontal line at that value. This line intersects the field incidence T L line at point A. In our example, the plateau height is 36 dB. The lOO-c/s point that we determined in (3) was 41 dB. A difference of 5 dB is a frequency change of 5/6 octave. Hence, point A will fall 5/6 octaves below 100 CIS,i.e. at 55 c/s. (5) Starting from point A, mark off the length of the line AB according to the plateau breadth shown in the table under Fig. 8. I n our example, the plateau breadth for dense concrete is 3.3 octaves, or a ratio of 9.8 in frequency. Hence, point B falls at 55 x 9.8 = 540 c/s. (6) Extend the line above point B at a rate of about 10 dB/octave for the first octave, gradually decreasing the slope to about 6 dB/octave (Fig. 8).

Sandwich (Honeycomb) Panels Today, the demand in building is for stronger, lighter weight materials. One such structure, shown in Fig. 9, is a 'sandwich' panel. It has surface sheets, such as aluminium, bonded to both sides of a light-weight honeycomb core. The sandwich panel structure combines high stiffness with low weight, with the result that bending waves are propagated with higher velocities than in a heavier, solid panel of equal thickness. If we assume that the stiffness of the panel is concentrated in the surface sheets, with the core merely serving to hold these sheets apart, we find the following approximate equation for the critical frequency:

Vol 173 19S9

LEO L. BERANEK

22

I

I

I

I

I

I

I

-

v)

J W

DEPENDS ON SIZE OF PANEL,EDGE DAMPING AND INTERNAL DAMPING

m W u

0

I v)

s * LOW DAMPING M N E R I A L S HAVE THE

z 0 Lo I?

$a TlDdB

THESE VERY NUMBERS ARE FOR A TYPICAL PANEL IN PLACE +S*HOLOW BLOCK THE VALUES WERE DETERMINED FOR A 6 IN PLASTERED BLOCK

PLATEAU HEIGHT IN DECIBELS

1-

-

I

I

I

4I

I

I

I

I I

I

OCTAVE

- CYCLES PER SECOND Fig. 8. Practical Design Chart for Single Limp Panels FREQUENCY

A reverberant sound field on the source side of the wall is assumed. The part of the curve left of A is determined from Fig. 6. The plateau height and the length of the line from A to B is determined from the Table. The part above B is determined by extrapolation. This chart is fairly accurate for large panels. For example, masonry panels must be greater than 6 ft x 8 ft; thin panels, like those in airplanes, must be greater than 2 ft x 3 ft; etc. (After Watters (9)J

As we have indicated, it is important to use a consistent system of units throughout. Appropriate constants must be introduced when units other than those specified are used. Once the coincidence frequency and the surface mass are known, the honeycomb panel is treated as though it were a homogeneous isotropic panel.

*J SURFACE SHEETS

LABORATORY T E S T S FOR WALL CONSTANTS

In order to utilize the formulae and design charts for the sound transmission loss of panels, one must have available the relevant physical constants of any particular wall. These constants are the surface mass (or surface weight), the critical frequency and the damping factor. The test procedures described here are from Cremer (23) and Watters (9).

SECTION A - A

Fig. 9. Construction of a Sandwich (Honeyconzb)Panel

where M, is the total surface mass of the panel, including the core (kg/m2 in m k s units or slug/fi2 in English units), d the centre-to-centre spacing of the surface sheets (m or ft) (Fig. 9), t the thickness of one surface sheet (same units as d ) (Fig. 9), E Young’s modulus for the surface-sheet material (newtonlm2 in m k s units or Ib/ft2 in English units), and c the speed of sound in air (m/sec or ftlsec). Proc Insm Mech Engrs

The mass and critical frequency may sometimes be determined by static methods. The bar or strip is weighed to yield the mass. The critical frequency is determined by supporting a thin bar or strip horizontally on knife-edge supports located at the ends of the bar. The distance between the supports l and the sag of the bar at the mid-point tmax. are measured. The critical frequency of a homogeneous, isotropic bar or plate is given by

where.,,[

is in inches and I is in feet. ?lo1 1’3 1959

THE TRANSMISSION AND RADIATION OF ACOUSTIC WAVES BY STRUCTURES

Very stiff materials such as masonry and honeycomb panels are too stifF to give an easily observable deflection in such a test. Also, many materials would fail in tension if supported in such a manner. A simple dynamic test, which gives both the velocity of propagation of bending waves in the material and the internal damping of these waves, is as follows: First mount the bar so that it can freely vibrate (free-free end conditions) and then measure the resonance frequencies and the Q values (band width) of the resonances. The width of the bar must be small compared to the shortest bending wavelength. Care must be taken to avoid torsional modes of vibration by symmetrically locating the exciting force and the mounting supports about the longitudinal axis of the bar. Standing waves will be set up in this bar, and at resonance the vibrations may be intense. At the first resonance, the standing wave will consist of one-half bending wavelength plus an end correction at each end of about Q wavelength (Morse 15a). The bending velocity of propagation is to a close approximation C s f

-

If0

+

n = 1, 2, 3 . . .

. (31)

( 42 %), where I is the actual length of bar, fo the frequency of resonance, and n the number of standing-half waves in the bar aside from the end corrections. The damping factor q may be determined from either a measurement of the band width of the resonance peaks or of the decay time of the vibration amplitude at the resonance frequency. Thus,

4-in. thick concrete block wall, a strip one block-width wide and 8 ft high is built on a 5 in. x 14 in. board hung from &-in. diameter rods that are 13 in. apart. The loading effect of the swing and the vibration shaker appear to be negligible above about 100 c/s. In conducting the experiment, the vibration shaker is tightly connected to the bottom of the vertical strip. A lightweight accelerometer is mounted near the top of the strip. The shaker is excited by an audio-oscillator whose frequency is slowly swept so as to search out the frequency and band width of the various resonances. The nature of the resonances of a typical strip is shown in Fig. 10. The results of the measurements are given in Figs. 11 and 12. Finally, in Fig. 13, there is shown a comparison between calculated data (using Figs. 6, 7, and 8) and measured field data. The transmission loss at higher

7-

I0 dB

3

4

6

?l=Tf,’

*

*

*

(33)

*

where Af, is the band width in c/s between the frequencies on either side of the resonance curve where the amplitude of vibration has dropped to 0.707 of its maximum value, f o the frequency of resonance in c/s, and T the time in sec required for the amplitude of free bending vibrations on the bar to drop to 10-3 of their initial value. Equation (32) is valid for r ) < 0-1, while equation (33) is exact. An alternative method for measuring the bending wavelength (and thus cB) is to scan the standing wave pattern in the resonant bar with a movable vibration pick-up. This technique is recommended if the bar is relatively thick and if there is any doubt as to the mode of vibration of the bar. To support a bar (or strip) so that its motion is not significantly altered by the mounts, it may be placed in a horizontal position and be continuously supported on soft sponges or, if of masonry, seated on a ‘swing’. A swing consists of two thin rods in tension supported from the ceiling with a wooden board spanning the bottom ends. The board is slightly larger than the strip of masonry wall that sits on it, and the steel rods are far enough apart to avoid touching the sides of the strip. For a 12-in. wide, Proc Instn Mech Engrs

3

4

I00

FREQUENCY

2.2

2

6

20

or

23

- C Y C L E S PER SECOND

6

6

I OCO

Fig. 10. Acceleration at End of a 6-in. Thick Plastered (Both Sides) Cinder Block Strip, 9.9 f t High and 16 in. Wide The driving force was held constant. (After Watters (9).)

..

FREOUENCY

- CYCLES PER SECOND

Fig. 11. Velocity of Propagation of Bending Waves C, in Plastered 6-in. Cinder Block (After Watters (9).) Vol 173 I959

LEO L. BERANEK

24

,$WITHOUT

2

20

100 FREQUENCY

SAND

1

r

- CYCLES

e

I

1

6

8

1000 PER SECOND

10 000

Fig. 12. Damping Factor 7 for 6-in. Hollow, Dense Concrete Block

Fig. 13. Measured and Calculated Transmission Loss for 6-in. Hollow Cinder Block, Plastered

(After Watters (9).)

(After Watters (9).)

frequencies is lower than that calculated. This may be due to the nature of the damping at the edge of the panel and to the size of panel. The effect of adding sand in the cavities of a masonry wall has been reported by Kuhl and Kaiser (25) and is shown by the data of Watters in Fig. 12. DISCONTINUITIES IN S T R U C T U R E S

Bending waves are attenuated when they travel around a bend (comer) (Cremer (26)). Kurtze, Tamm, and Vogel(27) determined experimentally over a frequency range of 2002000 c/s the transmission of bending waves around a 90”

bend made by joining two 1.2 in. x 1.2 in. Plexiglas bars. Qualitatively they show that at a corner the bending waves behave as shown in Fig. 14. The solid and dashed lines in Fig. 14 illustrate the displacements (greatly exaggerated) of the bars at two instants of time separated by t period. The source was located about 6 fi from the comer on bar No. 1. It is seen that when the bending stiffness of the comer is greatly reduced, either by inserting a rubber layer or by a ‘rolling’ joint, the transmitted bending wave is greatly attenuated.

r2

RUBBER

UBAR I

EAR I

R I G I D JOINT

FR*

ROLLING

ELASTIC JOINT

JOINT

t \

\ \



DIRECTION OF MOTION

\

iIOCM

BAR 1

BAR 1 BAR 1

Fig. 14. Qualitative Representation of the Transmission of Bending Waves Around a YO” Corner for T h e e Types of Corner Construction (After Kurtze, Tamm, and Vogel (27).) Proc Insrn Mech Epigrs

t’ol I73 1959

THE TRANSMISSION AND RADIATION OF ACOUSTIC WAVES BY STRUCTURES

In travelling around a rigid corner, the bending wave was attenuated by 3-5 dB. With 0.08 in. of rubber, installed as shown in Fig. 14, the attenuation was about 20 dB; with 0.2 of rubber, about 25 dB, and with 1.2 in. of rubber about 30 dB. With the ‘rolling’ joint the attenuation was about 12 dB. The importance of these findings is that sound from a vibrating machine may be produced in an adjoining room by a common wall not only due to airborne excitation of that wall, but also due to structure-borne excitation of that wall (and other walls). In practice, joints in the form of T’s or +’s, are important. For a rigid T the attenuation below 1000 c/s is 6-7 dB between the vertical member and each of the members at right angles. For a rigid the attenuation is 9-10 dB, Westphal (28). These attenuations are exceeded in walls of rooms at frequencies above 1000 c/s. In double walls, of the type shown in Fig. 3, a ‘sound bridge’ is often intentionally or unintentionally built in. The ‘bridge’, for example, might be a rod conducting compressional waves, which excites bending waves in the radiating wall. Cremer (29) has analysed the effect of such a tie. He considers the particular case of a 1.5-in.-thick concrete sheet ffoated 0-4 in. over a 4.5-in.-thick concrete slab. Cremer finds that a sound bridge in the form of a rod with a cross-sectional area of between 0-2 and 0.6 in2 decreases the transmission loss over no sound bridge by 1+30 dB. The exciting impact on the ffoated slab was applied 6 ft from the sound bridge. The deleterious effect is less pronounced when the point of excitation is removed a greater distance from the bridge. Masonry walls are frequently held together for structural reasons by resilient wire clips. Their deleterious effects are not great if the walls are very heavy. However, between two flexible leaves, they may reduce the transmission loss of a double wall by a significant amount. In the case of a light-weight wall, the coupling will be less for a massive bridge than for a light-weight bridge. Two additional cases of interest have been studied recently by Kurtze (30). One of these cases is shown in Fig. 150. A bending wave is established in member A. The transmitted bending wave is measured in B. If the thicknesses hl and h2 are equal, the effect of adding the L-shaped member on top, with a vertical length of leg L, is to cause a transmission loss of 4 dB. This transmission loss will reduce to about 2 dB at values of L, = A/2, A, 3A/2,2AYetc. It will increase to 7 dB at L, = A/4, 3A/4, 5x14, etc. We remember that for a simple ‘T’, that is, with L, infinitely long, the TL from A to B is 4 dB and from A to C is 6.5 dB. Now, if (Fig. 1%) the thickness h2 is made equal to 1.75hl, the transmission loss averages 11 dB, with maxima and minima of 16 and 6 dB respectively at the same values of L,lA as given above. In Fig. 15b, a case which embodies a double slot is shown. The material in member B is reduced to 0.45h, as shown. The TL from A to B averages 12 dB with maxima and minima at 16 and 8 dB respectively and at the same values of Lz/Aas previously. Finally, if both a double slot and the

+

Proc Instn Mech Engrs

25

b ~

HIA

!

--

T

045h1

t

B

Fig. 15. Structures for Which Bending Wave Transmissioii Has Bee8 Computed (After Kurtze (30).)

value of h, = 1.75hl are chosen, the TL from A to B averages 19 dB with maxima and minima of 25 and 13 dB respectively. COUPLING OF RADIATING S U R F A C E TO A MEDIUM

Below the critical frequency, a solid panel does not couple well to an air medium. At the critical frequency, a panel radiates a sound wave that is nearly equal in particle displacement to the surface displacement of the panel itself This is also true at each frequency above the critical frequency-the airborne sound wave being radiated at the coincidence angle. The radiation from panels above and below the critical frequency has been discussed by Westphal (28) and Goesele (31). Simply summarized, they show that below the critical frequency f, the radiated sound may be up to 20 dB less than above the critical frequency for the same strength of bending wave in the plate. Two of their graphs are shown in Figs. 16 and 17. The value of 0 dB on the ordinates corresponds to the radiation of sound that would occur if the panel were vibrating as an infinitely large rigid plate, back and forth perpendicular to its surface, with a peak velocity v equal to the peak velocity of the travelling bending wave in the plate. The first graph, Fig. 16a, shows the effect of damping (expressed as D = number of dB the bending wave is attenuated in travelling a distance in the plate equal to a wavelength of the bending wave) on the density of the radiated wave. In Fig. 17, they show the effect on the intensity of the radiated wave of finite plate size, blh; (where b is the length of a side of a square plate and A, is the bending wavelength given by equation (5)). It is seen that, below coincidence, for a given intensity of bending wave in the plate, increased damping or making the plate small increases the intensity of the radiated wave. Fig. 16 does not indicate, however, that for a given excitation 1701173 1959

LEO L. BERANEK

26

SOLID DOUBLE WALLS

a

J/J,

b

Fig. 16. Relative Radiation of a Free-travelling Plane Bending Wave from an Infinitely Large Plate T h e parameter D is the damping of the bending wave (in dB/wavelength). (After Westphal (28).)

c2

0.3

3

0-4

I/ I ,

Fig. 17. Eflect of Finite Plate Size (Width b) on the Radiation EfJiency of Plates Note both the upper and lower ordinates. D (Fig. 16) is equal to about 0.01. (After Goesele

(31).)

of the panel, damping usually decreases the intensity of the bending wave set up in the panel. The practical sigmficance of these findings lies in their application to the transmission of noise (vibrations) via the structure from one room to another or in flanking a double wall. For example, a sound wave from room 1 may arrive in room 2 as wave 1A or 2A or 2B. If there were no loss in travel of wave 2 through the structure, the waves 2A and 2B combined would be as strong as wave 1A. In other words, each would be 3 dB weaker than wave 1A. However, the presence of the common wall reduces wave 2A by 6.5 dB and 2B to 4 dB. Damping in the structure or a discontinuity in the path of the wave will cause greater reductions. To get the most noise reduction out of a wall, light-weight stiffmaterials should not be used. Heavy, non-stifFmaterials, for example, a thin sheet of metal loaded with ceramic tiles such as are used in bathrooms, would be better. Proc Instn Mech Engrs

A glance at Fig. 6 reveals that, at a given frequency, the field transmission loss for each doubling in mass of a single solid panel increases by only 6 dB. In modern building structures where light-weight movable partitions are becoming common practice, and in aircraft and light-weight trains, it is not possible to achieve desired TL values simply by increasing panel thickness. Emphasis, therefore, must be placed on making the critical frequency high, on multipleleaved walls and on the introduction of porous sound absorbing materials between the leaves. In this section we shall treat sound transmission through solid double walls. Two cases will be considered : (1) those in which the two panels of the double wall are vibration isolated at the edges and are not bound together at any point in between, and (2) those in which ties of some sort exist either between the two panels or at the edges. We shall call the former a double wall with isolated panels and the latter a double wall with bridging (or flanking paths).

Double Walls with Isolated Panels The acoustic behaviour of a double wall with isolated panels depends on the mass of the panels, on the depth of the air space between the two panels, on the critical frequency of each panel, on the mass-air-mass resonance fres. of the overall structure, and on the angles of incidence in the exciting sound field. London (17)has solved the case of sound transmission through isolated double walls under the following restrictions : (1) The panels are identical. (2) They are excited at frequencies below their critical frequency. (3) They are mass-controlled so that panel resonances need not be considered. With these assumptions and equation (13) and Fig. 3 we may write the reciprocal of the transmission factor as 1

2

. . .

(34) where /3 = (wd cos +)/c = (2nd cos +)/A (see equation (1I)), w = 2rf = angular frequency in rad/sec, M,is the mass of each panel in slug/ft2 (or kg/m2), pc the characteristic resistance of air in suitable units (see Table l), 4 the angle of incidence of sound wave (see Fig. 3), d the spacing of the panels in ft (or m) (see Fig. 3), c the speed of sound in ft/sec (or mlsec), p the density of air in slug/fV (kg/m3), and X the wavelength of sound in air in ft (or m). Perfect transmission occurs, that is, T = 1, when the bracketed portion of the equation vanishes. That is to say when

Vol173 1959

THE TRANSMISSION A N D RADIATION OF ACOUSTIC WAVES BY STRUCTURES

21

In the special but frequently encountered case that 2nd/h is small, we can replace the cotangent by the reciprocal of its argument, so that

We call this the 'mass-air-mass resonance condition' because the masses of the walls and the depth of the air space are involved. If we define a basic resonance wo as that mass-air-mass resonance existing for a wave at normal incidence (4= 0),then,

where k, is the correction factor introduced if non-consistent systems of units for Po and M , are used (see Table l), and Po the atmospheric pressure in lb/ft2 (or newton/m2). Note that from the wave equation for gases, yPo always equals pc2. For air, y = 1.4. And w,,,, = W ~ / C O S . (38) Finally, we substitute equation (37) into equation (34) and in turn into equation (7) to obtain the transmission loss in decibels.

+

TI, = 1010g,o{l+($)2(F) [cos /3-f(:)

WOMs

r9) +

.

.

The mass-air-mass resonance is f,,the driving frequency is f and M , is the mass of each panel. The angle of incidence 4 = 60'. (See Table 1 for units.)

cos*+x

cos sin /3]

'} dB

.

(39)

where

. .

(40) wo woMr 8=2(")(9f_)cos+ We have in equations (39) and (40), three dimensionless variables, (w/wo), ( w & f r / p c )and , +. For four values of (w,,Ms/pc),namely 3, 10, 30, and 100, the transmission loss in decibels is plotted in Fig. 18 for = 60". For two cases, namely (woMr/pc) = 10 and 100 the transmission loss is plotted in Figs. 19 and 20 for 4 = o", 30", 45", 60",and 80". No assumptions except the three of London given above are incorporated in the graphs. We note from the graphs and from equation (38)that the frequency of the mass-air-mass resonance shifis upward as the angle of incidence increases, in proportion to l/cos 4. It appears from equation (39) that when the incident sound wave impinges on the wall at grazing incidence, the transmission loss approaches zero. In practice, zero transmission loss does not occur for at least three reasons. First, grazing incidence sound waves cannot exist ifthe wall yields. Secondly, absorption of sound in the faces or at the edges of the panels tends to dampen the wave that is travelling in the air space parallel to the panels. Thirdly, zero transmission loss can only occur at grazing incidence for infinitely broad waves travelling along infinitely large walls. Let us now consider the region of the graphs (Figs. 18-20) above aboutflf, = 8.Assume also that the wall spacing d is less than & the wavelength in air. In this region, the

+

Proc Instn Mech Engrs

1/10

Fig. 19. Theoretical Transmission Loss for Double Wall with Completely Isolated Panels The mass-air-mass resonance is fo, the driving frequency is f, and the angle of incidence is 4. The value of woMr/pc = 10, where M I is the mass of each panel. (See Table 1 for units.)

reciprocal of the transmission factor (I/.) varies nearly as cos64. Using equations (16) and (7) we find that if the reverberant incident sound field is composed of waves impinging on the wall from all angles below a limiting angle +L, the average transmission loss is given by

where [ T L ] ,is the transmission loss at normal wave incidence Vol I73 1959

LEO L. BERANEK

28

Fig.21. Theoretical Transmission Loss for an Isolated Double

+

Fig. 20. Theoretical Transmission Loss fur Double Wall

Wall for = 0" and with No Restriction on the Separation Between the Walls

with Completely Isolated P a d s The mass-air-mass resonance is fo, the driving frequency is f, and the angle of incidence is 4. The value of w&fS/pc = 100, where Mr is the mass of each panel. (See Table 1 for units.)

(4 = 0). For example, if we say that the sound field is diffuse, but contains no angles of incidence greater than 80°, then T L = [TL]o-28 dB (42) I n other words, the transmission loss actually obtained is highly dependent on the composition of the incident sound field. For the frequency region above fHo equal to about 12, resonances due to standing waves between the panels occur whenever the equality of equation (35) exists. For 4 = 0 and wMS/2pclarge, these resonances occur when 2ad . p = - =h x . . . . (43)

. .

or

d=A/2 . . . * (44) The behaviour of the transmission loss curve for 4 = 0 and for frequencies up to 100fo is illustrated in Fig. 21. Because the resonance dips are quite narrow, it is customary to assume for simplicity an average transmission loss above f = 1% that increases at a rate of about 6 dB per doubling of frequency. For other angles of incidence and for (wM,cos 4/2pc) large, the inter-panel resonances occur at

. .

d= h/(2 cos+) . * (45) The addition of sound absorbing materials in the air space, will remove these resonances entirely, and the average transmission loss will rise at a more rapid rate above f = 10fo than 6 dB/octave.

Double Wall with Bridging Examples of the relative acoustic behaviour of some parProc Instn Mech Engrs

ticular single and double brick walls with and without bridging are shown in Fig. 22. The bottom curve in Fig. 22 shows the transmission loss as a function of frequency for a single brick wall as measured in a number of British dwellings with a reverberant sound field on the source side of the wall. The coincidence frequency is expected to be at about 220 c/s (see Table 3). The TL in the horizontal region is expected, from Fig. 8, to extend from 55 to 300 c/s with a height of about 33 dB. The average value in Fig. 22 is about this amount. The TL above 300 c/s behaves about as predicted from Fig. 8, namely, it rises 10 dB/octave and then gradually tapers off to 6 dB/ octave. When two 5-in.-thick plastered brick walls are combined to produce an isolated double wall with a 12-in.deep air space between the panels, the results at low frequencies may be predicted from equations (39) and (41). For such a wall, the mass-air-mass resonance frequency fo is about 10 c/s and (woMs/pc) is 35. At f = 100 c/s and 4 = 0" a transmission loss of about 82 dB is calculated. For a diffuse sound field with a limiting angle of +L = 80" it is found from equation (42) that = [TLlO-28 dB = 82-28 = 54 dB. Above 100 c/s, one would expect the TL to increase first at a rate of 12 dB/octave and later at only 6 dB/octave. One example of a completely isolated double wall is found in the literature (Moeller (32)). That double wall is formed by the two facing walls of two adjacent well-isolated broadcast studios (see Fig. 22). As shown by curve 1 in Fig. 22, the transmission loss at 100 c/s, with a reverberant sound field was measured as 57 dB. For the first 2 octaves above f = 100 c/s, the TL rises at a rate of only about 7 dB/octave. This lower rate than the predicted 12 dB is due to the effects of wave coincidence. In the next Octave the TL should rise at a rate of 6-10 dB/octave, but the data are incomplete and do not show this frequency region. A double wall of similar construction, but with flanking is now to be compared with the preceding wall (see the V d 1-2' 1959

THE TRANSMISSION AND RADIATION OF ACOUSTIC WAVES BY STRUCTURES

middle curve in Fig. 22). With flanking, the transmission loss obtained should be considerably lower than the predictions of Figs. 18-21. The measured TL values are reduced by 15-30 dB below those of the upper curve over the entire audible frequency range. The presence of the UTcm.-wide foundation beneath the double wall reduced the flanking transmission because the foundation has a high moment of inertia to bending waves. If the 16-cm. slab carried straight through with no 30-cm.-wide foundation beneath, the improvement of the double wall over a single wall would be only 2-4 dB. In several examples taken from the British Building Research Station tests, almost no improvement over singlewall performance was obtained by use of double masonry walls with butterfly wire ties between. IMPACT S O U N D TRANSMISSION

Noise in a structure can be produced by the impact of a rigid body against one of its surfaces. The chief difference between the excitation of vibration in a panel by airborne sound waves and that by impact is the extent of the area over which the driving force is applied. Theoretically, one can handle this problem by assuming this area of contact to be a point. Cremer (23) has made a theoretical study of ‘the reaction of a driven continuum to a point source’. He derives the Proc Iturn Mech Engrs

29

mechanical point impedance (ratio of an alternating driving force to the resultant alternating velocity) for plates of steel, aluminium, asphalt, and plywood as a function of their thickness. He then derives the spectrum of the impact sound as a result of dropping a mass on the iniinite plate, assuming several values of an added elastic layer at the surface of the plate. An added elastic layer could, in practice, be the result of adding a resilient covering to the surface of the plate. Cremer next discusses theoretically the case of adding a floating floor over a floor-ceiling. As indicated earlier in this paper, Cremer also considers the case of ‘sound bridges’, that is, solid ties, between the floating floor and the ceding below. Because of the lack of space in this text, the reader is referred to Cremer (23) for further study. Measured data in comparison with predictions of the sound insulation of floating floors with sound bridges have been published by Heckl(33). He reports good agreement. DAMPING OF P A N E L S

It is apparent by now that above the critical frequency of a panel, bending waves are readily excited by airborne or water-borne waves and, having been excited, they readily produce sound waves on the other side of the panel in an air or water medium. Furthermore, bending waves travel easily along a structure and around corners. Undesired v001173 1959

LEO L. BERANEK

30

sound (noise) is produced by vibrating structures in buildings, aircraft, ships, automobiles, and so forth. If the induced bending waves are intense enough, the material of the structure may fail, or electronic or mechanical equipment attached to it may malfunction. Failures of these types are often found in jet aircraft and in missiles. To alleviate these undesired phenomena, methods for controlling or reducing the amplitudes of the bending waves must be found. In structures where high tensile strength at low weight is a primary factor, such as in aircraft, it is not possible to interrupt the continuity of the structure to insert elastic layers. Furthermore, the introduction of heavy strips, with high moments of rotational inertia perpendicular to the direction of travel of the bending wave, may not be possible owing to weight restrictions. Therefore, highly efficient, light-weight damping materials are needed to control the excitation and transmission of bending waves in structures. Historically, damping has often been treated as a somewhat mysterious phenomenon, best approached by trialand-error methods. In recent years, Oberst (34, Oberst and Becker ( 3 9 , and Liinard (36)have successfully analysed the damping produced by single homogeneous layers of visco-elastic damping materials. They have found the damping mechanism there to be associated with the stretching of the damping layer. Kerwin (37) and Ross, Kerwin, and Dyer (38) have shown that the damping of a constrained layer is caused by the shear motion of the damping layer between the plate being damped and the outer constraining

Fig. 23. Four Types of Damped Plates a Single plate with single layer of damping material.

foil. Following ROSS,Kerwin, and Dyer, four types of damped structure are shown in Fig. 23. As the caption to the figure indicates, single or double plates are damped by one or two non-constrained or constrained layers of damping material.

b Single plate with equal damping layers on both sides. C Single plate with thin damping layer and thin constraining sheet. d Two plates with single damping layer between.

bending wave in a distance equal to 1 wavelength, Dh, D, = 13.67 dB/wavelength (47)

.

Two common non-constrained damping structures for application to a single plate are shown in Fig. 23a and b. Both structures are assumed to have the same weight. The following symbols are adopted for the basic parameters of the structures : El is the real part of the Young’s modulus of the plate, H I the thickness of the plate, E2 the real part of the Young’s modulus of the damping layer, H2the thickness of the damping layer, K2 = E2H2the real part of the extensional stiffness of a unit length of the damping layer, v2 the damping factor of the damping layer, equal to the ratio of the imaginary part to the real part of Young’s modulus, and 7 the damping factor of the overall structure. The damping factor 7 is a dimensionless quantity which can be used in several ways to measure the degree to which a composite structure is damped. Some examples are : (1) T o compute the time rate of decrease of squared amplitude of free bending vibrations on a plate, D,,

-

~

~

-~-- -- -

~

--

where fo = resonance frequency in cjs. Proc Insrn Mech Engrs

I

D1= 13.67/A dB/unit length

.

(48)

side of the resonance frequency, ~

- ~

where fo = resonance frequency in cjs. It is assumed now, as Oberst does, that the product of the damping factor v2 times the extensional stiffness of the damping layer K2 is less than & the extensional stiffness of the original bar K1. The relative damping factor for a bar with a single homogeneous layer is a function of the relative elastic (Young’s) modulus (E2/E1)as well as the relative thickness of the damping layer (H2/H1).In Fig. 24, the damping is plotted as a function of the relative thickness with the relative Young’s modulus as the parameter. The curves Vol I 7 1 1959

THE TRANSMISSION AND RADIATION OF ACOUSTIC WAVES BY STRUCTURES

001 '

"005

1-

I

I

I 0002

, R A T I O OF THICKNESS OF DAMPING LAYER TO THICKNESS CF P L A T E

Fig. 24. Relative Damping Factor 7 / v 2for Homogeneous Damping Layer (After Ross, Kerwin, and Dyer (3%)

show an approximate square-law relation for the damping as a function of thickness. They also show that for thick damping layers the damping factor for the overall structure becomes about 0.8 that for the damping layer itself. To illustrate the best performance to expect from a nonconstrained damping layer, calculations were made for the best one of the damping materials reported by Oberst and Becker (35). The properties of the materials and the results of the calculations are given in Fig. 25. The chart is a plot of qmm.(for best type of damping material) as a function of the ratio of the treatment weight to the plate weight. The damping is seen to rise as the square of the weight ratio. Ross, Kerwin, and Dyer find that dividing the damping layer into two layers (Fig. 23b) reduces the resulting damping by about one half. This result is to be expected because the effectiveness of a damping layer is proportional to the square of the thickness of the layer.

Constrained Damping Layers When the visco-elastic layer is constrained by a stif€ foil (Fig. 23c), the mechanism responsible for the major component of the damping is the shear motion of the damping layer. A relatively thin layer may be quite effective and the requirements for optimum damping materials are quite different from those for the single layer. The symbols of the preceding paragraphs are used here, augmented as follows: GIis the real part of the complex Proc Instn Meclt Engrs

I

31

BEST OF OQERST119541 OAHPINC MATERIAL ASSUME0 AS FOLLOWS

qPE2

--

-

El YI

-92 2 -

4 a 10' dyn/crnl

Y,-Wl/UNIl

W E - 38.5 Ib/lrl

2 I 10" d y n / m z

-

WTlUNlT VOUIME OF R A T E 4 480 Ib/ftl

-

-

32

LEO L. BERANEK

Fig. 26. Graph of the Ratio of the Composite Structure Damping Factur 7 to the Damping Factor /3 of the Damping Material Plotted as a Function of the Modified Shear Parameter I’ The graph is based on calculationsassuming a constraineddamping layer with & the thickness of the plate. (After Ross, Kenvin, and Dyer (38).)

as the layer becomes ‘too stiff‘ and the foil stretches ap-

preciably as the plate bends, the damping decreases. The shear parameter is a complicated function of frequency and temperature, depending as it does on the shear modulus of the damping material, which is a function of frequency and temperature. Also, the shear parameter is a function of the thickness of the damping layer. For a given operating temperature and frequency it is usually possible to select a damping material or layer thickness, or both, that will yield close-to-maximum damping. When this is done, the damping achieved is primarily a function of the relative stiffness of the constraining layer, and of the loss factor, 6. Assume that a proper choice of damping material has been made to give optimum damping. In order to achieve optimum damping the value of r will lie somewhere between 0.02 and 0.06. For this case, the optimum damping factor rlopt. is plotted in Fig. 27 as a function of the ratio of the treatment weight to the plate weight. The range of values of (0.5, 1, and 1.5) cover the range from ‘easy to achieve’ to ‘possible to achieve’.

Comparison of Constrained an d Non-Constrained Damping Layers The damping achievable by non-constrained damping layers depends roughly on the square of the weight of the material, while that for constrained layers depends approximately linearly on weight. At this writing, it appears that if we choose the best type of damping mzterial and optimize it for temperature and frequency, the two types are likely to give about equal values of damping for weights between 10 and 20 per cent of the base plate weight. Proc Iiistn Mech Engrs

Fig. 27. Calculated Optimum Composite Structure Damping Factor of a Plate with a Constrained Damping Layer for Three Values of the Damping Factor p of the Damping (After Kerwin (37).) Material

Below 10 per cent, the constrained layer damping is likely to be more effective. Above 20 per cent the non-constrained layer damping should be more effective. Sandwich Plate A sandwich plate (Fig. 23d) is a special type of constrained damping layer consisting of a thin damping layer between Vol 173 1959

THE TRANSMISSION AND RADIATION OF ACOUSTIC WAVES BY STRUCTURES

two identical stiff plates. The maximum damping is obtainable with a value of modified shear parameter, I‘, equal to about 0.2. The result of calculations assuming this value of r is shown in Fig. 28 (Ross, Kerwin, and Dyer (38)).

Fig. 28. Damping of Sandwich Structures Embodying a Thin Damping Layer (After Ross, Kerwin, and Dyer (38).)

For low values of tlie material loss factor, the maximum loss factor of the composite sandwich is 3 that of the damping material, while at higher values of 8, this ratio decreases.

S O U N D ABSORBENT S T R U C T U R E S

In those cases where weight is of primary importance and where the reduction of noise is more important above 300 c/s than below, sound absorbing structures embodying dissipative blankets are used. Such an application is noise reduction in aircraft. Generally, an outside ‘skin’ for the fuselage cabin is chosen of such thickness that it provides the necessary low-frequency noise reduction. Bending wave transmission from one part of the fuselage to another is reduced by damping ‘tapes’. These tapes are of the constrained-layer type. To reduce the high frequencies the entire interior of the cabin is lined with a 1-4 in. thickness of very fine-fibre acoustic blanket. Such a blanket does little good at low frequencies, but becomes very effective above 200-400 c/s depending on its thickness. An example of the transmissionloss to be expected through a metal panel with a surface density of 0.28 lb/fi* combined with a 2-in.-thick homogeneous isotropic fibrous acoustic blanket spaced 2 in. away from the panel is shown in Fig. 29. The parameter for the curves is the specific flow resistance in rayl/cm of blanket thickness. Specific flow resistance can be measured by maintaining a steady flow of air, ZJ (cmlsec), through the blanket of interest and measuring the pressure difference,dp (dyn/cm2), across the blanket necessary to maintain that flow. The specific flow resistance R1= dp/(vd) rayl/cm (dyn-sec/cm“), where d is the blanket thickness in cm. A specific flow resistance of 100 rayl/cm is, for example, measured on glass fibre blankets with a density about 35 times that of air and with fibres that are about 0.8p in diameter. It is seen that the addition of the blanket does little good below 400 c/s. Values of 90 dB transmission loss are never obtained in practical installations because of flanking paths. Transmission losses for many varieties of sound absorbing materials and structures have been discussed elsewhere (Beranek and Work (18)).

2IN

FREQUENCY

- CYCLES

33

00

- II -- 2 IN -

PER SECOND

F2. 29. Aircraft Structure with Homogeneous, Isotropic Acoustic Blanket.

Sozrnd Source

was on the Metal Panel Side The ‘sound attenuation’ is the ratio of the sound pressures on the two sides of the structure for normal wave incidence. T h e ‘transmission loss’ would be about 6 dB less above 100 c/s. (After Beranek and Work (IS).) Proc Insrn Mech Engrs

L’ol 173 1959

LEO L. BERANEK

34

CONCLUSION

This lecture presents formulae, charts, and tables for determining the transmission loss of single and double walls. It gives data on the transmission of bending waves around comers and through structural bridges. It presents information on the use of damping materials in damping bending waves in structures. It briefly discusses the transmission loss of dissipative structures. The author hopes that the information presented will be useful in architectural, aircrafi, ship, and automobile design. The author wishes to express his appreciation to Dr. I. Dyer, Dr. E. Kenvin, Jun., Dr. D. ROSS,and Dr. P. Franken for their assistance in editing the paper. He also wishes to express his deepest thanks to the Institution of Mechanical Engineers of Great Britain for the opportunity to present the 1958 Thomas Hawksley Lecture. APPENDIX REFERENCES

(I) BUILDING RESEARCH STATION 1948-56 Summaries of part

of the Building Research Station work have appeared in the following places : 1948, ‘Symposium on Noise and Sound Transmission’, published as a report by The Physical Society of London, 1949 (subjects included: ‘Party Walls with Improved Sound Reduction’, W. A. Allen; ‘Sound Insulation Measurements on Windows and on Cavity Brick Walls’, G. H. Aston; ‘Studies of Sound Insulation by Discontinuous Structures’, W. A. Allen; ‘Floating Floors’, H. R. Humphreys; ‘A Study of Domestic Noise’, W. A. Allen; ‘Sound Insulation Between Flats’, P. H. Parkin and H. R. Humphreys). 1954, ‘Recent Research on Sound Insulation in Houses and Flats’, P. H. Parkin and E. F. Stacey, 3. Roy. Inst. Brit. Archit., July 1954. 1956, Build. Res. Sta. Dig. Nos. 88 and 89, ‘Sound Insulation of Dwellings’, Building Research Station, Garston, Watford, Herts, England. (2) PARKIN,P. H. and HUMPHREYS, H. R. 1958 ‘Acoustics of Buildings’ (Faber and Faber, London). (3) DUTCH Results of Dutch studies summarized in the proceedings of two symposia: 1952 ‘Symposium on Noise Nuisance and Sound Insulation in House Building’ published in De Zngenieur, Nos. 33,34,and 36: ‘Inquiry About the Noise Nuisance in Flats’, C. Bitter; ‘The Former and Present Method of Measuring the Airborne Sound Insulation in the Field’, M. L. Kasteleyn and J. van den Eijk; ‘Sound Transmission Through Windows’, J. van den Eijk and M. L. Kasteleyn; ‘The Reduction of Noise in Staircase Halls’, J. van den Eijk, M. L. Kasteleyn, C. Bitter, and A. H. M. Basart. 1954 ‘Symposium on Noise Nuisance and Sound Isolation’, published in De Zngenieur, Nos. 38,41,44, and 47: ‘Design of Floors and Sound Insulation’, M. L. Kasteleyn; ‘Lightweight Double Walls Having a Large Transmission Loss’, C. W. Kosten; ‘Sound Insulation in Floors and its Bearing Upon the Problem of Sound Nuisance’, C. Bitter and P. van Weeren; ‘The Tentative Dutch Standard V-1070: Sound Insulation in Dwellings’, W. P. van Leening. 1955 ‘Sound Nuisance and Sound Insulation of Dwellings 1’, C. Bitter and P. van Weeren, Report No. 24, Res. Inst. Publ. Hlth Engng, TNO, The Hague (September 1955). (4) USAF 1952-55 ‘Handbook of Acoustic Noise Control’, Phys. A C O ~ Svol. Z . , 1 and Supplement 1, Wright Air Proc Instn Mech Engrs

Development Center, Wright-Patterson Air Force Base, Ohio. WADC Tech. Rep. No. 52-204 (and Supplement), prepared by the Staff of Bolt Beranek and Newman Inc. ( 5 ) GOETTINGEN SYMPOSIUM1955-56 ‘Symposium on Sound and Vibrations in Solid Bodies, Goettingen, Germany, 19 to 22 April 1955’, Papers appear in Acustica (Akust. Beih.), vol. 6, p. 49 (1956). (In German, French, and English.) (6) COOK,R. K. and CARZANOWSKI, P. 1957 ‘Handbook of Noise Control’, edited by C. M. Harris, Chapter 20, ‘Transmission of Air-borne Noise Through Walls and Floors’ (McGraw-Hill Book Co., New York). (7) BERANEK, L. L. ‘Noise Reduction’ (in Press). Text to be published by McGraw-Hill Book Co. with a number of contributors. (8) CREMER, L. 1942 Akust. Z., vol. 7, p. 81. (9) WATTERS, B. G. ‘The Transmission Loss of Some Masonry Walls’ (in the Press). (10) DAVIES,R. M. 1953 Rev. appl. Mech., vol. 6, p. 1, ‘Stress Waves in Solids’. (XI) SCHMIDT, H. 1954 Acustica, vol. 4, p. 639, ‘Die Schallausbreitung in Koernigen Substanzen’. (12) EXNER,M. L., GUETH,W. and IMMER, F. 1954 Acuctica, vol. 4, p. 350, ‘Untersuchung des akustichen Verhaltens koerniger Substanzen bei Anregung zu Schubschwingungen’. (13) KURTZE,G. 1956 Acustica (Akusr. Beih.), vol. 6, p. 154, ‘Koerperschalldaempfung durch koernige Medien’. (14) JUNGER, M. C. 1958 ‘Structure-borne Noise’, review paper presented at August 1958, First Symposium on Naval Structural Engineering, sponsored by Office of Naval Research, Washington, D.C., and Stanford University. T o be published in the proceedings of that symposium. (15) DERESIEWICZ,H. 1958 Appl. Mech. Rm., vol. 11, p. 259, ‘A Review of Some Recent Studies of the Mechmcal Behavior of Granular Media’. (Iga) MORSE,P. M. 1948 ‘Vibration and Sound’, second edition (McGraw-Hill Book Co., New York). SCHOCH, A. 1937 ‘Die Physikalischen und Technischen Grundlagen der Schalldammung im Bauwesen’ (S. Hinel, Leipzig). LONDON, A. 1949 Bur. StandardsJ., Wash., D.C., Research RP1998, vol. 42, p. 605, ‘Transmission of Reverberant Sound Through Single Walls’. BERANEK, L. L. and WORK,G. A. 1949 3.acoust. SOC.Amer., vol. 21, p. 419, ‘Sound Transmission Through Multiple Structures’. CREMER,L. 1950 ‘Die Wissenschaftlichen Gmndlagen der Raumakustik‘ (S. Hinel Verlag, Leipzig). Undated Dept. Sci. Industr. Res., Lond., Sponsored Research (Germany) Report No. 1, Series B, ‘The Propagation of Structure-borne Sound‘. 1953 Acustica, vol. 3, p. 317, ‘Calculation of Sound Propagation in Structures’. K. 1952 Acustica, vol. 2, p. 289, SCHOCH,A. and FEHER, ‘The Mechanism of Sound Transmission Through Single Leaf Partitions’. LONDON,A. 1950 3.acoust. SOC.Amer., vol. 22, p. 270, ‘Transmission of Reverberant Sound Through Double Walls’. F. H. 1939 Canad.3. of Res., vol. 1. SANDERS, CREMER,L. 1955 ‘Insulation of Air-borne Sound by Rigid Partitions’ and ‘Insulation of Impact Sound’, Sections 11.2and 11.3 respectively of WADC Tech. Rep. 52-204, vol. 1 and Supplement 1,prepared by Bolt Beranek and Newman Inc. for Wright-Patterson Air Force Base, Ohio. FESHBACH, H. 1953 ‘Transmission Loss of Infinite Single Plates for Random Incidence’. Unpublished report prepared for Bolt Beranek and Newman (see Cremer (23)). Vol 173 I959

THE TRANSMISSION AND RADIATION OF ACOUSTIC WAVES BY STRUCTURES

(25) I K w , W. and KAISER,H. 1952 Acustica, vol. 2, p. 179, ‘Absorption of Structure-borne Sound in Building Materials Without and With Sand-filled Cavities’. (26) CREMER, L. 1956 Acustica, vol. 6,No. 1,p. 59,‘Berechnung von Korperschallvorgangen’. (27) KURTZE, G.,TAMM, K. ~ ~ ~ V O GS.E L 1955 , Acustica, vol. 5, p. 223, ‘Model Investigation of Bending Wave Damping at Corners’. (28) WESTPHAL, W. 1957 Acusrica (Akusr. Beih.), vol. 7, p. 335, ‘Ausbreitung von Koerperschall in Gebaeuden’. (29) CRE~ER, L. 1954 Afutrica, vol. 4,p. 273, ‘Berechnung der Wirkung von Schallbruecken’. (30) KURTZE, G. 1958 ‘Noise Transmission by Boundary Layer Turbulence’. Quart. Progr. Rep., April-June 1958, submitted to Office of Naval Research, 30th July 1958, Bolt Beranek and Newman Inc. (31) GOESELE, K. 1956 Acustica (Akust. Beih., No. l), vol. 6, p. 94, ‘Radiation Behavior of Plates’. (32) MOELLER, F. 1946 Teknisk. Ukeblad, No. 45 (Norway), ‘Sound Isolation with Structural Engineering’. (33) HECKL,M. 1956 Acustica (AKutr. Beih.), vol. 1, p. 91, ‘Messungen an Schallbruecken Zwischen Estrich und Rohdecke’. (34) OBERST,H. 1952 Acusrica, vol. 2 (Akust. Beih., vol. 4), p. 181, ‘ h e r die D h p f u n g der Biegeschwingungen darner Bleche durch fest haftende BeGge’.

Proc Instn Mcch Engrs

4

35

(34) OBERST,H.1956 Anrsrica, vol. 6 (Akust. Beih., vol. l), p. 144, ‘Wekstoffe mit extrem hoher innerer Dhpfung’. 1956 KututstofJe, vol. 46-5, p. 190,‘Akustische Anwendung von Schaumstoffen’. (35) OBERST,H. and BECKER,G. W. 1954 Acustica, (Akusr. Beih., vol. l), p. 433, ‘Uber die Dampfung der Biegeschwingungen donner Bleche durch fest haftende Belige 11’. (36) LIBNARD, P. 1951 Rech. aero., vol. 20, p. 11, ‘Etude d’une Mtthode de Mesure du Frottement IntCrieur de Revhements Plastiques Travaillant en Flexion’. 1957 Ann. Tdlkomm., vol. 12-10, p. 359, ‘Les Mesures d’hortissement dans les Mattriaux Plastiques ou Fibreux’. (37) KERWIN,E. M. 1958a ‘Vibration Damping by a Constrained Damping Layer’, Bolt Beranek and Newman Inc. Report No. 547, 29 April 1958, submitted to Convair, San Diego, California. 1958b ‘Vibration Damping by a Stiffened Damping Layer’, Paper U-8, 55th Meeting of the Acoustical Society of America, 10 m y 1958. (38) ROSS, D., KERWIN, E. M. and DYER, 1. 1958 ‘ F l e d Vibration Damping of Multi-layer Plates’, Report No. 564, prepared for the Office of Naval Research, Washington, D.C., by Bolt Beranek and Newman Inc., 26 June 1958.

Vo1173 1959