Prediction Methods for the Sound Transmission of Building Elements* Existing techniques for predicting sound transmission loss generally provide unrealistic values for all but the simplest type of building elements. Ben I-I. $harpf describes an i m p r o v e d prediction procedure a n d gives e x a m p l e s of its application.** A few years ago, the US Department of Housing and Urban Development (HUD) sponsored a program designed to study techniques of increasing the sound insulation of building elements. The theory available at the time indicated that the majority of existing constructions were capable of providing significantly greater values of transmission loss than those measured in the laboratory. While this was undoubtedly true, there was also a concern that the theory was inadequate for obtaining realistic estimates of the transmission loss for all but the simplest structures. Therefore, it was necessary to examine and modify the theory to improve the accuracy of the prediction methods. The results of the study indicated that it was possible to improve the performance of existing structures and, more importantly, that it was possible to predict the performance of fairly complex structures.1 These prediction techniques were used to design low-cost structures with values of transmission loss required for buildings, as well as structures exhibiting extremely high values of transmission loss suitable for special applications. The techniques and their development are described here in the hope that others may find them useful for design purposes. The principles involved in the prediction method are discussed briefly in the belief that a basic understanding of the mechanisms involved in sound transmission is almost as important as the prediction equations themselves in the optimum design of structures. *Received 6 January 1977; revised 3 July 1978 fWyle Laboratories/Wyle Research, 2361 Jefferson Davis Highway, Suite 404, Arlington, Virginia 22202 **This p a p e r is a partial tultillment ot the INCE requirements tor membership. It is based primarily on work that was sponsored by the US Department of Housing a n d Urban Development. 1 V o l u m e 11 / N u m b e r 2 Thin Panel Theow The simplest type of structure under consideration is the single panel whose thickness is small compared with the associated airbome and structure-borne waves. If the panel dimensions are much greater than the wavelength of bending waves, it can be shown by classical methods that at frequencies less than the critical frequency, the transmission coefficient ~'0 for sound waves incident at single angle ~ is given by the expression 2 r 0 = 10 log [1 + (~mcos 0/2pc)2], dB, (1) where ~ is the radial frequency (2 ~f), m is the mass of the panel per unit area, and pc is the characteristic impedance of the air. The corresponding transmission loss TL 0 of the panel at this angle of incidence is given as TL0 = 10 log (11%). To determine the transmission coefficient for excitation by a reverberant sound field, it is generally assumed that all angles of incidence are equally probable and that the average value of the coefficient is given by integrating %, multiplied by an appropriate weighting factor, over all angles in the range 0 to ~-/2. However, the transmission loss value obtained in this way is found to be about 3 dB lower than the measured values. The agreement between the calculated and measured results can be improved by arbitrarily limiting the integration range from 0 to 0z (with 0t < ~/2), where 0z is chosen simply for the agreement to be good. It is found that different laboratories require different values of 0t in order for the calculated results to agree with those measured in the laboratory. The values of 01 used by various workers range from 78 ° to 85 °. The explanation that is usually given to warrant this empirical correction is that the sound field in a reverberation chamber is not totally diffuse and little sound 53 energy is incident to the panel at grazing angles of incidence. However, there appears to be no experimental justification for this assumption. The simple transmission loss theory does, of course, assume that the panels are of infinite lateral dimensions. At low frequencies, the majority of panels tested in transmission loss facilities are not very large compared with the bending wavelengths. In this case, the resonant frequencies or modes of the panel and the coupling of the incidence sound waves to these modes must be taken into consideration. The transmission of sound through a finite single panel has been treated in the published literature. 3.4 In Ref. 3, a classical approach is adopted by considering a plane wave incident to a panel in a baffle; the solution is obtained in matrix form. In Ref. 4, the panel is taken to the common wall between two reverberation chambers. The solution is found by evaluating t h e coupling between the sound fields in both rooms and the panel. An approximation in this solution is that the sound pressure on the incident side of the panel is much greater than that on the receiving side. Presumably, the solution is valid only for panels of high transmission loss, although how high has yet to be determined. At frequencies below the critical frequency, both methods give similar results, and it is found that the major portion of sound energy is transmitted by forced vibration of the pane] rather than by resonance vibration. It also seems that the major transmission is from sound energy that is incident at small angles to the normal of the panel. The expression for the transmission loss given in Ref. 4 is TL,. = 20 log (c0 m/2pc) (2) - 10 log [1.5 + In (2f/Af)], dB, the panel. In this frequency range, the transmission loss is given by the expression 3 TL = 20 log (oJrn/2pc) + 10 log (2"0c0/~-~0c), where T/is the loss factor for the panel material and coc is the critical radial frequency (2 ~-fc). This expression is identical to that derived by Cremer. = The expressions in Eqs. 4 and 5 give values of transmission loss that agree with measured values (see Fig. 1). Eq. 4 is valid only at frequencies less than approximately one-half the critical frequency (1/2 f~). At frequencies between 1/2 f~ andre, resonance transmission assumes a greater importance in determining the transmission loss and analytical expressions do not seem to give good agreement with the measured results. Until more accurate expressions are available, an approximate method that can be used to predict the transmission loss in this frequency range is to describe a straight line between the value of transmission loss at the frequency 1/2f~ (Eq. 4) and the value at the frequency fc (Eq. 5). The behavior of panels in the frequency range where the structural wavelength is comparable to the panel thickness is much more complex. ~ It is found that in the case of thick panels, such as concrete or masonry, the critical frequency occurs at very low frequencies (that is, 600 Hz or less), so the mass law expression is valid only up to about 300 to 400 Hz. At frequencies greater than the critical frequency, the transmission loss tends to be about 6 dB less than that indicated by an extension of the mass law expression. For design purposes, this characteristic can be accounted for by assureing that the "effective" mass of a thick panel is one-half (corresponding to a 6-dB reduction in transmission loss) that of its actual mass. where Af is the bandwidth of the sound signal used for testing. If one-third octave bands of noise are utilized, then Eq. 2 becomes TL., = 20 log (~m/3.8pc), dB (3) TL,, = 20 log (mr) - 48, dB, (4) (5) Laminated Panels To a great extent, the transmission loss of a single panel is determined by the mass of the panel; the greater the mass or or 60 I ' ' MASS I ' ' J ' ' CONTROLLED 1 ' ~ I ' ~ I ~1----"- ' ' I ' ~ t COINCIDENCE CONTROL where m is the panel mass in kg/m 2. This is the familiar mass law. At higher frequencies, the coincidence effect is exhibited (where the impedance terms representing the panel mass and bending stiffness are equal in magnitude and opposite in phase). The lowest frequency at which this effect occurs is termed the critical frequency fc, given by fc = (c2/2~h) (m/B)~, where h is the thickness of the panel and B is the bending stiffness. At frequencies greater than the critical frequency, the transmission loss is quite dependent on the internal losses in 541 Oj 40 CALCULATED oZ~ EQUATION Z / 4 y ~ C A L ~ E ~ " "MEASURED e::: ('rl = O.Ot2) I- 0 L, 63 , I ,, 125 I, 250 , 1 ~. , I , 500 1000 FREQUENCY , I , 2000 , I,, 4000 1 8000 , Hz Figure I ~ Measured and calculated values of the transmission loss of 1.59-crn gypsum board. The panel size is 2.44 by 3.05 m for all presented data. NOISE CONTROL ENGINEERING / September-October 1978 the thicker the panel for a given material, the greater the transmission loss - - except at frequencies near the critical frequency. Since the value of the critical frequency is inversely proportional to the thickness of the panel, any attempt to increase the transmission loss of the panel by increasing its thickness automatically lowers the critical frequency, perhaps into a frequency region of major importance. As a result, the two most desirable properties for any single panel are high density and low stiffness - - properties that are normally incompatible in a single material. In practice, building elements are required to exhibit a high stiffness at low or zero frequencies in order to withstand lateral loads. Thus, the ideal panel would exhibit a stiffness that was high at low frequencies, reducing to a low value at high frequencies. This characteristic can be obtained with laminated panels in which the adhesive layer is designed to shear and provide a panel impedance lower than the bending impedance of the combination. At low frequencies, the two panels behave as though they were rigidly connected, exhibiting a bending stiffness eight times that of either panel alone (the panels are assumed to be identical). At high frequencies, the shearing effect of the adhesive layer reduces the bending stiffness of the combination to that of each of the individual panels. As a result, the critical frequency of the combination can be increased by a factor of two without affecting the low frequency stiffness, provided that shearing of the adhesive occurs at a frequency less than the critical frequency of the combination. The characteristics of such a multilayer panel are largely determined by the properties and thickness of the adhesive layer. It is possible to remove this dependency by the technique of spot laminating, whereby the adhesive is applied in small discrete amounts to a square lattice over the panels' surface. The general characteristics of such a multilayer panel are the same as those described above, except that the two panels decouple and move more or less independently at a frequency determined mainly by the relative spacing of the adhesive spots. It is therefore possible to design the decoupling frequency by making the correct choice of adhesive lattice spacing.. For most types of panel material, it is found that a spacing of 0.3 to 0.6 m is quite adequate. The effect of panel decoupling is demonstrated in Fig. 2, where the measured values of transmission loss are given for two spot-laminated sheets of 1.27-cm gypsum board and for a single sheet of 1.27-cm gypsum board. No reduction in the critical frequency from its value of approximately 3000 Hz is noted, yet the transmission loss at low frequencies has been increased by 6 dB. analysis involving modal coupling and previous attempts have been largely unsuccessful in obtaining good agreement with measured results. Fortunately, it is possible to attain good agreement for double panels by taking the classical expression for the transmission coefficient (as developed by London) and making a few, simple approximations. 5 In this" way, the transmission loss of a double panel (with no interconnections between the panels) can be determined for three principal frequency regions by means of the following expressions (Ref. 1): f< fo TLM TL = TLm I + TLm2 + 20 log TLmI + TLm~ + 6 One method of obtaining transmission loss values higher than that available from a single panel is by the introduction of one or more additional panels with intervening air spaces. The multiple panel construction formed in this way is naturally more complex to analyze than the corresponding case for a single panel, since the transmission loss is dependent on a greater number of construction parameters. Methods of V o l u m e 11 / N u m b e r 2 f> h where TL:~, TLml, and TLm2 are the values of the mass law transmission loss calculated from Eq. 4 for the total construction (M = ml + m2), panel 1 (=rn~), and panel 2 (=m2), respectively. The quantity f0 is given by 113/m,/-m-~ded,where me is equal to 2m i m J(ml + m 2), and represents the frequency at which the fundamental mass-spring-mass resonance of the panel masses and the cavity air stiffness occurs. The quantity d is the separation of the two panels in metres, andh is equal to (55/d) Hz. The transmission loss calculated by these expressions is in good agreement with measured values at frequencies less than the critical frequency of either panel, as shown in Fig. 3. With the aid of the previous discussion and the approximate expressions that have been derived, it is now possible to examine the effects of coincidence on the transmission loss of a double panel. The values of the transmission loss of each of 80 ; ' I ~ ] I ' ' I ' I ' ; ' I 60 MASS LAW - 1.27 cm GYPSUM --X BOARD ' I ' ' I \ \ MEASURED 1.27 + 1.27 cm Z o ; MASS LAW - 2.54 cm GYPSUM _ _ BOARD "o 0.J I SPOT ~ LAMINATED-~ 4° "GYPSUM j - ~ ' \ J BOARD ~ S " " =E 03 y ~ ' ~ Z Double Panel Theory - 29 fo< f< f~ (6) (fd) ° ° o l-- k_MEASURED 1.27 cm GYPSUM BOARD I,, 63 I 125 ,, I,, I, 250 500 , I, 1000 ,I 2000 =, I, 4000 , I 8000 FREQUENCY , Hz Figure 2 - - The meesured values of transmission loss for a single sheet and two 1.27-cm spot-laminated sheets of gypsum b o a r d 55 the individual panels will, of course, deviate from that calculated according to the mass law at frequencies in the vicinity of and greater than their critical frequencies. As a result, the (second) panel that is not exposed directly to the source of sound will experience an increase in the level at the critical frequency of the first panel. Similarly, this second panel will readily transmit energy at its critical frequency. The increases in energy transmitted by the two panels at their critical frequencies are contained implicitly in their respective values of transmission loss. Eq. 6 indicates that the two panels act independently in providing the overall transmission loss. Therefore, to a first approximation, the effect of coincidence inthe double panel construction can be taken into account by taking the sum of the effects of coincidence in the transmission loss of each of the individual panels. As a result, it is possible to use Eq. 6, with the values of TL,, I and TL,,= taken as the measured or calculated values of the transmission loss for the individual panels including the effects of coincidence. This prediction method using the approximate expressions is fairly accurate even for the case where the two panels are ~dentical (see Fig. 4). In this example, mechanical connections between the two panels were minimized by locating the panels in the separate, isolated rooms of the transmission loss facility. It was necessary to seal the perimeter of the construction, and it is felt that this is the reason for the deviations between measured and predicted values in the region of 1000 Hz. The predicted values were obtained by inserting measured values of transmission loss for the individual panels into Eq. 6. It will be noted that the measured data shown in Figs. 3 and 4 include the effect of absorption in the cavity between the two panels. The importance of this absorption and its effect on the transmission loss in different frequency regions will be discussed. The Effect of Absorption in Double-Wall Cavities The basic acoustic theory for double-wall constructions assumes that the air contained in the cavity separating the walls acts as a stiffness element at low frequencies. This implies that the air is unable to escape from the cavity and that the sound pressure is constant over the entire cavity volume. The lateral dimensions of practical double-wall constructions, however, are sufficiently large compared with a wavelength for standing acoustic waves, or modes, to be set up in the cavity. Clearly, the cavity can no longer be represented as a simple stiffness element in the frequency range containing such standing waves. It is therefore natural to expect that the measured values of transmission loss will differ from the values predicted using the simple theory - - that is, unless the lateral modes are adequately damped. In a conventional 5.1 cm by 10.2 cm wood frame construction, 2.44 m high, with studs 0.41 or 0.61 m on center, the lowest acoustic mode occurs at approximately 70 Hz, or well below the lowest frequency that is usually of interest for sound insulation in residential buildings. In this case, the stiffness assumption is incorrect over the complete frequency range. This is only part of the problem, however. If there is little or no acoustic absorption in the cavity, the standing waves may be of large amplitude and may transmit considerable energy to the walls. In fact, at the pressure antinodes in the cavity, the high values of sound pressure will produce an effect similar to that of direct mechanical connections. It would therefore be expected that the resulting strong acoustical coupling between the panels at the natural frequencies of the cavity would significantly reduce the transmission loss of a doublepanel construction. Furthermore, it is expected that the addi8O 80 I m '~ ' ' I ' ' I ' ' I ' ' I ' ~ I ~ o 6 ' ' I ' • ' I ' ~ I ' MEASURED ' I ~ ' I T ~ I ' VALUES ' I • • ' ' I' o MEASURED V A L U E S / / 60 ~ PREDICTED . 4o 50 . O3 (/') • o, Z O o I "~m - 40 • ' o3 m_ m i1~ 20 o..._Jl L o.0 I, 63 , I , 125 , I , 250 , I , 500 , I = 1000 FREQUENCY HARDBOARD , I , 2000 , I , 4000 ~ I 8000 , Hz Figure 3 - - Measured values of the transmission loss of a double panel compared with values calculated by the approximate method. Fiberglass batts were used in the cavity for the measured data. 56 ,e::.GYPSUM BOARD 0 I= 63 ; I ] 25 ,, I 250 j, I ,., 500 FREQUENCY I, 1000 ,I 2000 ,, I 4000 L , I 8000 , Hz Figure 4 - - Measured and calculated values of the transmission loss of 1.59-crn gypsum board. Fiberglass batts were used in the cavity for the measured data. The boldfaced line represents data predicted from Eq. 6 and measured values of TL for individual panels. NOISE CONTROL ENGINEERING / September-October 1978 tion of acoustical absorption to the cavity would reduce the amplitude of standing waves and result in an increase in the transmission loss. 80 . ' / .j rn "Io Experimental evidence to demonstrate modal coupling between the panels has been obtained by measuring the transmission loss of a double wall (2.44 m by 3.05 m), in which the individual walls were completely isolated. In the experiments, one wall of the construction was placed in the source room, the other in the receiving room of a transmission loss measurement facility in which the two rooms were mechanically isolated. The edges of the cavity consisted of wood studs that were slightly smaller than the space between the walls, connected to one of the two walls. The walls were of 0.32-cm and 0.64-cm hardboard, chosen so that the effects of coincidence were removed from the frequency range of interest. The space between the two walls was 16 cm. The results of the experiments are shown in Fig. 5. In the absence of absorption, curve c of this figure shows that the strong acoustic coupling between the walls results in almost a single-wall performance at frequencies less than the first cavity resonance perpendicular to the plane of the walls (that is, 1100 Hz). At higher frequencies, the phase of the sound pressure varies over the thickness of the cavity and the acoustic coupling is weaker. In this frequency range, the transmission loss is seen to increase and behave more like that expected of a double wall, although the predicted values (calculated from Eq. 6) are not attained. The introduction of a 5.1-cm layer of fiberglass insulation board (density 48 kg/m ~) across the entire cavity width produces a remarkable improvement in the transmission loss (see curve b), resulting in good agreement between theory and experiment. With a 10.2-cm layer of fiberglass in the cavity, the mass of the absorption material is comparable to the mass of the walls, which explains the additional increase in transmission loss over and above that predicted by the simple theory (see curve a). One of the interesting features of Fig. 5 is that the transmission loss of the construction at low frequencies is apparently quite dependent on the presence of absorption material in the cavity. This result is contrary to common opinion. In the context of meeting certain STC (Standard Transmission Class) requirements, it is an important result, since the STC of double-wall constructions is often determined by the values of transmission loss at these low frequencies. It is common and less costly to use standard foil-backed fiberglass batts in wall cavities, rather than the fiberglass insulation board. Since the density of the batts is lower than that of the board, their effectiveness in damping the cavity modes is lower. Measured results of the transmission loss of the double hardboard wall construction are given in Fig. 6 for the two types of absorption material in the cavity. At low frequencies, the values are essentially the same within experimental error, but a reduction on the order of 4 to 5 dB is noted at frequencies in excess of 500 Hz. It can be concluded that both types of material are equally effective in damping the low-frequency lateral cavity modes, but that the batts are V o l u m e 11 / Numb er 2 " 60 03 m O -J z 40 O 03 03 ,~ z~ ~0 ,~ nl0 • O A, PREDICTED " o/ FROM -~. ,;,/ EQUAT ION. V " • • & A , & & A . & & SS LAW I,, 63 I , 125 ~ I, 250 I I,, 500 I, 1000 , I , 2000 , I t 4000 , I 8000 FREQUENCY , Hz. • 10.2 cm full layer (a) 0 5. I cm full layer (b) /~ None (c) Figure 5 - - Measured values of the transmissian loss of an isolated double-wall construction with and without full-layer cavity absorption. The construction consists of 0.64-cm and 0.32-cm hardboard with a spacing of 16 cm. 80 ' o 5., c., F,8 ~ 60 ss +OARD/ • 8.9 cm FIBERGLASS BATTS ~. . • I "" o/• " u~ o, Z _o 40 U~z <~ 20 n~" f 63 L_ MASS LAW 125 250 500 1000 2000 4000 8000 FREQUENCY , Hz Figure 6 - Transmission loss values for an ideal double wall with a full-layer fiberglass insulation board and fiberglass botts less effective than the board in the frequency range where the higher order cavity modes occur (that is, those perpendicular and oblique to the surface of the walls) because of the lower density and flow resistance. If the modal coupling theory is correct, it should be possible to provide acoustic absorption solely at the periphery of the cavity. This should, in fact, be the optimum position for the material. Fig. 7 shows the result of introducing layers of 57 80 I ' ' I ' ~ I 15.3cm e--e z~ --,~ zJ ' ~ I ' ' I [ ' I t THICKNESS OF / ~' PERIMETER / ) ABSORPTION // 5.1 cm ' ' 1 ' ' l , 4000 , .,,_,.... CALCULATED -x~.S~/ 0 I, 63 ~ I J I I I 125 250 I I, 500 I 1 / I I I000 , I i 2000 i I 8000 FREQUENCY,Hz Figure 7 - - Measured values of transmission loss of an isolated double-wall construction with perimeter absorption. The construction consists of 0.64-cm and 0.32-cm hardboard with a spacing of 16 Cm. 8O I T , i ' T I ' ' I 5 , I ' ' I ~ ' I ' ' I • 0.61 m x 0.61m LATTICE / m o "1o NO ABSORPTION/ / 60 PREDICTED FROM ~ - ~ / ~ 4o _m ~ / ..i /-/P ~ if) Z rr 20 }0 / SS LAW I, 63 , t ,, ] 25 t, 250 ~ I, 500 ~ I , 1000 , I ,, 2000 I ~, I 4000 8000 FREQUENCY,Hz Figure 8 - - Measured values of transmission loss of an isolated double-wall construction with a 0.61 m by 0.61 m lattice in the cavity. The construction consists of 0.64-cm and 0.32-cm hardboard with a spacing of 16 cm. fiberglass (density 48 kg/m2), 5.1 cm and 15.3 cm thick, around the periphery of the cavity for the construction described. Several points can be noted: The transmission loss at low frequencies increases as the thickness of the absorbent material at the periphery is increased. The predicted values for the construction are not attained, but it is reasonable to assume that they would be approached more closely with thicker layers of material. 58 The slight dip in the curves at 1000 Hz corresponds to the first cavity resonance perpendicular to the plane of the walls. This will be evident since the damping at the periphery of the cavity will not be fully effective in damping this mode. At frequencies greater than the first cavity resonance, the presence of higher order cavity modes (perpendicular and oblique to the plane of the walls) reduces the overall value of transmission loss. However, the individual resonances are not noticeable. At the critical frequency of the 0.64-cm sheet of hardboard (5000 Hz), there is a marked reduction in the measured values. Obviously, perimeter absorption has little effect on the transmission loss at the critical frequency. The principles of modal coupling provide an interesting method by which the transmission loss of double walls can be increased without the use of absorption. If the cavity is divided into a large number of smaller cavities by means of a lattice network, the entrapped air will behave as a stiffness element up to high frequencies (UP to the lateral modal frequencies of the individual elements in the lattice). This is demonstrated in the measured results of Fig. 8, where the lattice was 0.61 m square and consisted of wood studs mounted in the cavity with no contact with the walls of the construction. At low frequencies, the measured results follow the predicted curve closely. The strong coupling effect of the first and second lateral modes of the lattice (in the 315 Hz and 630 Hz one-third octave bands) is evident. Note that the space between the modes is sufficiently large for the transmission loss to increase well above the value predicted by the mass law at frequencies between the modes. In contrast, the space between successive modes for the construction shown in Fig. 5 is only 70 Hz (corresponding to the 2.44 m dimension of the construction), thus explaining the low values of transmission loss in the entire low-frequency region. The lattice has very little effect at high frequencies. If the lattice dimensions were 0.15 m rather than 0.61 m, it is anticipated that the predicted results would be approached at all frequencies up to 1000 Hz without the use of absorption material. The conclusion that can be drawn is that the modal coupling theory appears to be valid. The use of peripheral absorption alone apparently is not sufficient to attain the possible high values of transmission loss at the higher frequencies. Dividing the cavity into smaller individual cavities, while providing good results at low frequencies, again has similar limitations at high frequencies. In practice, of course, the transmission loss of double-wall constructions at any given frequency has an upper limit that is determined by the type and number of mechanical connections between the two walls. As a result, the usefulness of absorption material in single stud walls is debatable, and should be considered on a case-by-case basis, using available methods of prediction. However, the upper limit introduced by connections usually occurs at a frequency which is a few one-third octaves greater than the fundamental massspring-mass resonant frequency of the double wall. The addition of absorption material can apparently increase the NOISE CONTROL ENGINEERING / September-October 1978 transmission loss at these frequencies, and hence, should be considered useful. In other forms of double-wall construction that incorporate framing consisting of staggered studs, split studs, or double studs, or where the walls are resiliently mounted to the studs, the addition of absorption material in the cavity can result in a significant increase in transmission loss - - on the order of 3 to 8 dB, depending largely on the wall construction. In the case of double windows, it is only possible to place absorption material around the perimeter of the cavity between the glass panels, and it is important that this should be done. However, the full potential of the sound insulation is difficult to achieve. Keeping in mind that standing waves will be excited in the cavity, it is advisable to design a double window such that the frequency of the standing waves does not coincide with that of the fundamental mass-spring-mass resonance of the construction. In addition, square windows should be avoided. WOODEN J : I !I , ~ - - " SUPPORTED One of the major assumptions in the previous analysis of double-panel structures is that the two individual panels are completely isolated from one another. This means that the only path of energy transfer between the two panels is an airborne path. In practice, it is necessary to have some form of connection between the panels to provide the added stiffness for the construction to withstand lateral loads. These connections usually take the form of wooden or metal studs in building structures and metal ribs and stringers in aerospace structures. Their effect is to provide an additional transmission path in parallel to the airborne path previously considered, with the result that acoustic radiation from the structure is increased and the transmission loss correspondingly reduced. It is not usually possible to eliminate these interpanel connections, or sound bridges as they are called; thus, in the design of multiple panel structures it is necessary to be able to determine the effect that they have on the transmission loss. There are two basic'types of interpanel connections. One of these, the line connection, is commonly found in building constructions in the form of wooden or metal studs, in which the two panels are connected along a line or a series of lines. The other (which is not so common) is the point connection, consisting of a connection, or a number of connections, having small cross-sectional area that approximates to a point. An illustration of one type of point connection is given in Fig. 9. The method that will be used to determine the reduction in transmission loss of a double panel caused by the insertion of a number of such sound bridges is to add together the acoustic power radiated by the action of the bridges WB and that radiated by the ideal isolated panel Wp. The result will then be compared with the power radiated in the absence of sound bridges (Wp). The reduction in transmission loss TLB due to the presence of sound bridges is given by the expression Volume 11 / N u m b e r 2 PANEL Figure 9 - - Method of prouiding a point con nection to one panel in (] double-panel construction TLB = 10 log [1 + Double Panels with Sound Bridges STUD (WB/Wp)]. (7) The overall transmission loss TL of the double panel is TL = TLI - TLB (8) where TLI is the transmission loss of the construction .in the absence of sound bridges, as given by Eq. 6. Consider a double-panel construction that is subject to acoustic excitation from an unidentified noise source (see Fig. 10). The panel not directly exposed to the noise source will be exposed to the sound field created in the cavity between the two panels. If the resultant rms velocity of this second panel is v2, then the sound power Wp radiated because of the forced response of the panel at frequencies less than the critical frequency is given by the expression Wp ~ pcSv2 2, (9) where S is the area of the panel. This expression also holds for frequencies greater than the critical frequency for both free and forced wave radiation. To the power Wp must be added the power radiated by the action of the sound bridges which are assumed to connect the two panels. It has been shown by Heckl 6 that the sound power WB radiated by a panel at frequencies less than the critical frequency, when excited by a mechanical force such as that provided by the action of the sound bridges, is given by the expression WB = pcKv ~, (10) where v is the rms velocity of the area over which the force is acting, and K is given by K = { (8/~"3) ~c2 for a point force, (2/~-)I~c for a line force, (11) 59 PANEL PANEL I 2 point impedance: Z line impedance: (17) Z = 2 (1 + j)mc(fffc) ½per unit length. = Therefore, the value of the ratio (v/v i) can be determined for any combination of line and point connections to either of the two panels. The ratio (vffv2) can be obtained by analyzing an analog electrical circuit corresponding to a double panel in the absence of sound bridges. It can be shown that V SOUND EXCITATION F VffV2cc{ff 2 Vi where £c = the critical wavelength of the panel (C/fc), I = the length of the line over which the force acts. Under conditions where the second term in the brackets of Eq. 7 is much greater than unity, the rate of increase of TLB (the detraction in transmission loss) with frequency is 12 dB per octave forf<ft and 6 dB per octave forf>fz (since TL oc 20 log [v~/v2]). The transmission loss of an ideal double panel increases at 18 dB per octave and 12 dB per octave in the two frequency ranges, respectively, as shown in Fig. 11. Thus, the transmission loss of a double panel with sound bridges will increase at a rate of only 6 dB per octave over the entire frequency range where the transmission loss is governed by the bridges. The curve will thus be parallel to the mass law line, and is given by the expression Therefore, the ratio of sound power radiated by the sound bridges to that radiated by the panel is given by (12) where n is the number of line or point bridges. The ratio (v/v 5) can be determined by dividing it into two components: V/V 2 = (V/Vl) (VffV2). (13) Examination of Fig. 10 indicates that the force F induced in the sound bridge from the panel exposed to the sound excitation can be written as F = (V 1 -- v ) Z I , TL = TLM + ATLM. (19) At lower frequencies, when the value of the second term in Eq. 7 is less than or comparable to unity, the slope of the curve will vary between the limits of 18 dB and 6 dB per octave. Thus, the general form of the transmission loss for a bridged double panel is as illustrated in Fig. 11. The value of ATLM is determined by incorporating Eqs. 11, 16, and 18, with the necessary constants of proportionality, into Eq. 7, and hence, into Eq. 8. The results are point connections: ATLM = 20 log (e fc) + K - 45, dB, (14) where Zx is the impedance of the first panel as seen by the sound bridge. If the sound bridge is assumed to be rigid and massless, then it follows that F = Z2v, (18) ---V 2 Figure 10-- Nornenclature for calculatingthe effect of sound bridges (WB/Wp) = (n K/S)(v/v~) ~, fo<f<ft.f~ft. (20) line connections: ATLM = 10 log (b fc) + K - 18, dB, (21) (15) where whereZ2 is the impedance of the second panel as seen by the sound bridge. Thus, the ratio v/v i is given by the expression v/vl = Zff(Z1 + Z2). (16) It has been shown by Cremer that expressions for the panel impedance Z are as follows: 7 60 K = 20 log [ml (Z1 + Z2)/Z~ (ml + m2)l, (22) e = spacing of point connections (assumed to be regular) in metres, b = spacing of line connections (studs) in metres. NOISE CONTROL ENGINEERING / September-October 1978 80 12 dB PER ' ' I ' ' I ' • MEASURED ' I ' ' I DATA I ' ' I ' ' I 60 u3 u3 O ..J Z O3 0 ' PREDICTED FROM EQUATION 24 m fD ' PER OCTAVE ', ..J OCTAAVE 4O i O3 ~0 Z 0 Z 2o Or) Z <1fo F- < I-- i~__. I _ MASS LAW 6 dB PER OCTAVE 0 , 63 i 125 i , I 250 , I I 500 l . I i 1000 , I . 2000 , I 4000 ~ . I 8000 FREQUENCY, Hz t t I fo (LOG) f£ FREQUENCY, Hz Figure 11 General forrn forthe transmission loss of a double panel with sound bridges - - Eqs. 20 and 21 have been developed and presented so that the reader can incorporate any combination of line and point connections to the two panels of a double-panel constmction. However, in practice, it is found that the following approximations to these expressions are acceptable: Figure 12 - - Measured and predicted transmission loss for a double-panel construction of 1.59-cm and 0.95-cm gypsum board with wooden line studs 0.61 m on center and fiberglass batts in the cavity Design Methods (24) The expressions given in the preceding sections are sufficient for the design of a construction that must satisfy a specific transmission loss requirement. In many cases where the requirement is not severe, a simple single panel may suffice, provided, of course, that the mass required to achieve the transmission loss is not too high. If a practical single panel does not provide sufficient transmission loss, a brief review of the HUD noise control guide will show whether there is an existing construction that will satisfy the requirement. 8 If both of these approaches fail to come up with a desirable constmction or if the requirement itself is for a construction with low cost and/or high transmission loss, then it may be necessary to design a construction by means of the expressions in the preceding section. wherefc is the highest critical frequency of the two panels, and ml is the mass of the other panel. Eqs. 23 and 24 may upset some purists because the principal of reciprocity appears to have been violated - - it is possible to get different values of ATLMby interchanging the panels[ However, if the instructions are carefully followed, quite accurate results can be obtained, as indicated in Figs. 12 and 13. If in doubt, the more complex expressions given in Eqs. 20 and 21 can be used. In the design of a structure, the required value of transmission loss is known, and the structural parameters must be selected so that the quantity TLM + ATLM is equal to this value. In other words, the design equations presented earlier must be used in reverse. To simplify matters in the first stage of design, it is convenient to assume that the panels in a double-panel construction are identical. Once the overall characteristics of the construction are determined, the individual panels can be modified for optimum transmission loss or lowest cost. Taking the expressions for TLMand ATLMand solving for the structural parameters gives the following resuits. point connections (to one panel only): ATLM = 20 log (efc) (23) + 20 log [m~/(m~ + m2)] - 45, dB, wherefc is the critical frequency of the panel supported by the point connections, and ml is the mass of the other panel. line connections (to both panels): ATLM = 10 log (bfc) + 20 log [ml/(m, + m2)] -18, dB, Volume 11 / Number 2 61 80 ' I ' ' I ' ' I ' ~ • MEASURED DATA l ' i I ' ' I i , PREDICTED FROM I 60 18 dB / O C T A V E / ,8 dB PER E Q U ~ g 40 I- J// 40 / 20 r/i / i. : CHARACTERISTIC ', I fo ~. 72 HZ fo , .... /~ I t 63 k/ 63 I , 125 , ! , 250 ~ 1 , 500 I I I I I J i I . 1000 2000 4000 . I 125 250 500 1000 2000 4000 8000 FREQUENCY, Hz 80O FREQUENCY, Hz Figure 13 Measured and predicted transmission loss for a double-panel construction with 1.59-cm gypsum board on both sides of wooden studs 0.61 cm on center, and fiberglass batts in the cavity. One panel of gypsum board is mounted on point connections 0.61 cm on center. - - point connections: M e f~ = (l/f) antilog [(TLs + 99)/20], (25) line connections: M2b fc = (I/f)2antilog [(TLs + 72)/I0], (26) where M is the total mass of the construction ( = m I + m 2) and TLs is the specified or required transmission loss. As an example, suppose the transmission loss requirement shown in Fig. 14 is required for an internal loadbearing wall construction. To define a construction that will satisfy this requirement, the following steps in the calculation must be performed. 1) Draw a straight line with a slope of 6 dB per octave tangential to the required transmission loss characteristic (see Fig. 14). This establishes on upper bound to the curve that must be satisfied by a single panel or a bridged double panel. 2) Note the value of the transmission loss given by this line at a certain frequency-- say, 1000 Hz for convenience and insert the value into Eq. 4 to determine the mass of the single panel that would provide the straight line characteristic. In this case, TLM at 1000 Hz is equal to 58 dB; hence, a mass of 199 kg/m 2 is required. 3) Determine the feasibility of using a single panel of mass 199 kg/m ~to satisfy the requirement. Such a high mass can be obtained only by using concrete or masonry walls, which invariably exhibit low values for the critical frequency. For example, a 7.6-cm concrete panel of mass 176 kg/m 2 has a critical frequency of approximately 400 Hz. At frequencies 62 Figure 14 example - - Required transmission loss characteristic for the design greater than 400 Hz, the concrete panel will provide a transmission loss approximately 6 dB less than that calculated according to the mass law (Eq. 4). That is, its effective mass is one-half of its actual mass. Thus, a 15.2-cm concrete panel of mass 398 kg/m 2 is required to satisfy the transmission loss requirement shown in Fig. 14. Examination of the measured values of transmission loss for a 15.2-cm concrete panel shows that this panel would in fact satisfy the requirement. If the 15.2-cm panel is too massive or undesirable for other reasons, it is necessary to consider a double-panel construction. 4) Consider the possibility of a double panel with line connections (that is, a common wooden or metal stud wall). Insert the value of the required transmission loss (58 dB) at a given f r e q u e n c y (1000 Hz) in Eq. 26 to determine the required value of the quantity M2bfc. In this case, the minimum requirement is given by M2bfc = 1.0 × 10 7 (kg2/m/s). If the stud spacing b is taken as 0.61 m, the requirement becomes M2fc = 1.6 × 107 (kg2/m2/s). Using gypsum board, it is possible to obtain values of the critical frequency in the range from 2500 Hz for a 1.59 cm thickness to 6000 Hz for a 0.64 cm thickness. Taking a median value of 4000 Hz (0.95-cm gypsum board), the minimum requirement for the total mass of the construction, excluding the studs, is then NOISE CONTROL ENGINEERING / September-October 1978 M ~ 64 kg/m 2. 5) Consider the possibility of a double panel with one of the panels mounted on point connections. Repeating the general method described in Step 4, but this time using Eq. 25, shows that for a panel with a critical frequency of 4000 Hz, mounted on points with a lattice spacing of 0.61 m, the minimum requirement for the total mass of the construction, excluding the studs, is M ~ 29 kg/m 2. This is significantly less than the 64 kg/m ~ required with the same panel mounted directly on the studs. The remainder of this example therefore assumes the presence of point connections for the panel with a critical frequency of 4000 Hz, although the method for the case of line connections is exactly the same. 6) Calculate the transmission loss for the total mass of 29 kg/m 2according to Eq. 4 and insert the mass law line onto the diagram (see Fig. 14). 7) Draw a straight line with a slope of 18 dB per octave tangential to the required transmission loss curve as shown in Fig. 14. This represents the transmission loss of a double panel without sound bridges in the frequency region from fo to fz (see Eq. 6). 8) Determine the frequencyfo at which the mass law line intersects the line at 18 dB/octave. In this case, f0 ~ 72 Hz. 9) Determine the spacing d of the panels in a doublepanel construction with each panel of mass 1/2 × 29 kg/m 2 (the optimum condition) for the frequency fo to be 72 Hz, using the expression given earlier. In this case, d = 0.17 m. ent thickness or different materials while maintaining the required structural parameters as determined in the above design process. It is also possible to obtain a significant increase in transmission loss in the region of the critical frequency by using small squares of resilient material, such as PVC from tape, as the point connections to a wooden stud, through which the panel can be nailed. References I. B.H. Sharp, "A Study of Techniques to Increase the Sound Insulationof Building Elements," US Department of Housing and Urban Development, NTIS PB 222 829/4, 2. L. Cremer, Akustische Z., 7, 81 (1942). 3. E. C. 5ewell, "Transmission of Reverberant Sound Through a SingleLeaf PartitionSurrounded by an InfiniteRigidBaffle," d. Sound Vib., 12, I, 21-32 (1970). 4. R. Josse and C. Lamore, "TransmissionDu Son Par Une Paroi Simple," Acustica, 14, 267-280 (1964). 5. A. London, "Transmission of Reverberant Sound Through Double Walls," Bureau of Standards Journal, 44, Research Paper RP 2058 (January 1950). 6. M. Heckl, Acustica, 9, 378 (1959). 7. L. Cremer and M. Heckl, Korperschall (Springer-Verlag,Berlin, 1967). 8. "A Guide to Airborne, Impact and Structurebome Noise Con~ol in Multi-FamilyDwellings," US Department of Housing and Urban Development (Washington, DC). It would appear from this result that the requirement would be satisfied by 20.3-cm wooden studs (actual dimensions, 19 cm) with 1.59-cm gypsum board (m = 12.6 kg/m 2) mounted on both sides. However, the critical frequency of 1.59-cm gypsum board is 2500 Hz, which is well below the required value of 4000 Hz. The critical frequency can be raised by using 0.95-cm gypsum board (fc = 4000 Hz); however, since the mass of this material is only 7.3 kg/m 2, it is necessary to use two laminated panels. Checking back through the calculations shows that this combination of materials with a spacing of 14 cm in place of 19 cm would provide a value of 80 Hz for fo, which is close to that required. Thus, the final construction is as follows (stated in English units for convenience): 2 in. by 6 in. wooden studs, 24 in. on center; %-in. gypsum board nailed to one side; on the other side, two laminated panels of %-in. gypsum board mounted on point connections 24 in. on center. Fiberglass batts (3V2 in.) to be included in the cavity. It is interesting to compare the total mass of this construction (29 kg/m 2 excluding studs) with that of the single panel with equivalent performance (398 kg/m~). The design method described is based on the simplified expressions given earlier, without considering the effect of coincidence on the individual panels. The magnitude of this effect can, of course, be reduced by selecting panels of differVolume 11 / N u m b e r 2 63