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The transmission loss of finite sized double panels in a random incidence sound field
J. Somd Vib. (1968) 8 (l), 126-133
THE TRANSMISSION
LOSS OF FINITE SIZED DOUBLE
IN A RANDOM
INCIDENCE
SOUND
PANELS
FIELD
A. CUMMINGS AND K. A. MULHOLLAND Department of Building Science, The University, Liverpool, England (Received 7 December 1967)
The extension of the multiple-reflection theory to take account of absorption at the edges of the cavity between the septa of a double panel is described, and an expression is obtained for the random incidence transmission loss of a double panel with edge absorption, given values of the panel dimensions and edge absorption coefficient. A curve is given of the measured random incidence transmission loss of a thin double aluminium panel with glass fibre absorbent around the edge of the cavity and theoretical curves given for the same panel with different values of the edge absorption coefficient. From the curves it is inferred that using a value of the edge absorption coefficient of unity in the theory, gives the best agreement between theory and experiment. A curve of the measured transmission loss of a thin polythene panel with glass fibre absorbent around the cavity edges and a theoretical curve with the absorption coefficient equal to unity are given. Good agreement between experiment and theory is obtained without the somewhat unrealistic assumptions of our previous paper. The effect of truncation of multiple reflections between the septa of a double panel upon the “mass-spring-mass” resonance is considered, and the finite panel theory is compared to modal theory.
1. INTRODUCTION
Existing theories for the transmission loss of double panels [l-4] do not allow for the absorption of obliquely incident sound waves at the edge of the air-space between the panels. This fact leads to considerable disagreement between theoretical and measured transmission loss curves for double panels, particularly in the region of the mass-spring-mass resonance frequency. In our previous paper [l] this was overcome by choosing values of panel face absorption coefficient so as to obtain a fit with measured data. In this paper it will be shown that by modifying the multiple reflection theory it is possible to allow for the absorption of sound waves at the boundary of the air-space. This produces agreement between theoretical and measured transmission loss curves, without the unrealistic assumption of panel face absorption coefficient that was involved in our previous paper. 2. THEORY
In the multiple-reflection theory as originally derived [l], it was assumed that each septum of the double panel is infinite in extent. In this derivation a geometric series of terms (see Figure l), each representing a sound wave emerging from the panel, is summed to infinity. This treatment is valid for a finite panel provided that the boundary of the air-space is totally reflective. A totally reflective boundary would produce a mirror-image between the panels of the multiple reflections which would occur beyond the boundary if the boundary were not present. This would produce an infinite series of multiple reflections between the panels, which would then behave as infinite panels with an infinite airspace between them. 126
TRANSMISSION
LOSS OF FINITE SIZED PANELS
127
For a cavity with a sound absorbing boundary, however, the sound waves reflected from the boundary are only partially reflected back into the cavity.
x,x2(1.-x,)(1-x2)exp (-3jkdax e)
~;x,x2(i-x,)2B-x2)Zep (-5fidcc6 e) . +d--+l I I I
I I
Figure 1. Sound transmission through an infinite double panel. Figure 2 shows two panels with sound waves arriving at the face of one panel at different points. Each incident wave sets up its own series of internal reflections in the cavity and these, on reaching the boundary, are partially reflected back into the cavity as shown. This series is partially reflected back from the opposite edge of the cavity and this process will continue indefinitely.
Figure 2. Sound transmission through a finite double panel with an absorbent cavity boundary.
128
A. CUMMINGS AND K. A. MULHOLLAND
In this model, the incident waves are split up into n bands (see Figure 3). The first band of incident waves produces n emergent bands on its first traverse of the cavity, by internal reflection. On its second and all subsequent traverses it produces n emergent bands. The second incident band produces n - 1 emergent bands on its first traverse, n emergent bands on its second traverse, n on its third, and so on.
nth
Figure 3. Bands of sound waves incident upon a finite double panel. The nth incident band produces only one emergent band on its first traverse, and n on its subsequent traverses. It is seen from Figure 3 that n may be found approximately from the formula n = [S/2d tan 01. (1) This formula ignores certain effects near the edges of the double panel, but the authors feel that in the light of subsequent approximations used to obtain a formula for a three-dimensional panel this approximation is satisfactory. Summing the emergent waves from each incident band gives, for the first incident band, C#Jt,= a $ ay + . . . + q-1
-t
+ a(uyn + uy*+1 + . . . + uy*“-1) + + a*(uy*” + uy*“+’+ . . . UJJ~“-~)+ . . . to infinity;
for the second incident band, +t* = a f uy + . . . + uyn-2 + -I-a(uy”-’ + uy” + . . . + uy*n-2) + + o~*(uy~“-~ + uy*” + . . . + uy3ne2) + . . . to infinity;
TRANSMISSION LOSSOFFINITESIZED PANELS
129
for the 12thincident band, &,=a+ + a(ay + ay2 + . . . + ay”) + + a2(ay”+’ + ay”+2 + . . . + ay2”) + . . . to infinity. Here a = $i X1x2 where x = 1 +jwm cos 8/(2pc), y = (1 - x1) (1 - x2) exp(- 2jkdcos 19)and Q = the fraction of the sound pressure reflected from the cavity boundary. a is related to the energy absorption coefficient of the boundary by the equation absorption coefficient = 1 - ~c.c~~. If we define
,=#I
r=n-I
S2= IX w’,
S, = I: Y',
r=l
r=O
and
then r=n
,2;,A, = S, + as2 S3S4.
(2)
Hence
z
t=‘=l=__ nk
A,
X,X,Y(l
XIX2
1-Y
41
and
-Y”> - Y12
1 _4 [
-Y”)
(1 - CCJ)“) I
(3)
7= jt12.
We can find the random incidence transmission factor of the panel from 81 7sin8cosedB / - b r= 81 sin t3cos e d0 i 0
and hence the transmission loss (TL) from TL = lOlog(l/?). This is a two-dimensional treatment, but it can be extended to apply to a three-dimensional system by taking the value of S as (a + b)/2, where a and b are the width and breadth of the panels. This approximation is valid provided that a 2: b. Theoretical transmission loss curves were computed using this formula, for different values of CL.Each value for transmission loss was found by averaging ten energy transmission factors (F) distributed over one-third octave frequency bands. This was done so that the theoretical results could be compared with measurements made using white noise sampled in thirdoctave bands. The KDF9computer used needed 40 min to obtain a transmission loss curve of seventeen results using KIDSGROVE ALGOL 60. 9
130
A. CUMMINGS AND K. A. MULHOLLAND
3. EXPERIMENTAL
RESULTS
Figure 4 shows a curve of measured transmission loss as a function of frequency for a thin double aluminium panel with thick glass fibre absorbent around the edges of the air space between the panels. Theoretical transmission loss curves for the panels are also shown. These were computed using the parameters relevant to the measured panels, and various values of CL. A value of the limiting angle 8,, of 80” was used. This had previously been found to give the best agreement between experiment and theory for the measuring facility used. The theoretical curve for cc= 1 shows a mass-spring-mass resonance “trough” and a standing-wave dip. This is identical to a curve for an infinite panel, and the “trough” and standing-wave dip are characteristic of such curves. However, the measured transmission loss curve does not exhibit the mass-spring-mass resonance trough except for a dip near the
01
100
II
I/f
1
Illlll
1
I
1000
u/f
I C/2d
I
4( )O
Frequency (Hz 1
Figure 4. Random g/cm2, d==7.11 cm.
incidence transmission
loss of a double aluminium
panel ml = m2 = 0.239
normal incidence mass-spring-mass resonance (“lower London frequency”) and the standingwave dip is much less pronounced than in the curve for a = 1. Theoretical transmission loss curves for a = O-75, CL= 0.5 and a = 0 are also shown. The curve for a = 0 agrees most closely with the measured curve and lies within 5 dB of it above 300 Hz. This was to be expected as the glass fibre absorbent used around the edge of the air-gap was several feet thick and would have a high absorption coefficient at all frequencies. The curves for a = 03 and a = O-75 lie between the curves for a = 0 and a = 1. The curve for a = 0 shows the same general shape as the measured curve, with a dip at the “lower London frequency”, and does not exhibit the extreme behaviour of the curve for a = 1. It was assumed that a value of a = 0 for the glass fibre absorbent used would give the best agreement between theory and experimental results for all panels. Figure 5 shows a curve of measured transmission loss as a function of frequency for a thin polythene double panel, and theoretical curves for a = 0 and a = 1. Again the curve for a = 0 agrees well with the measured curve. The curve for a = 1 shows a mass-spring-mass resonance trough which does not appear in the measured curve. It is seen that the oblique mass-spring-mass resonance is reduced in magnitude when a is reduced from 1 to 0. When a = 0, the number of internal reflections for obliquely incident
131
TRANSMISSION LOSS OF FINITE SIZED PANELS
sound waves becomes finite. Therefore, the magnitude of the mass-spring-mass resonance must depend upon the number of internal reflections of the sound waves between the panels. This was investigated. Figure 6 shows a curve of the theoretical transmission loss of a typical double panel as a function of the number of emergent waves taken into account in the
25-
20iii D 4
15-
100
llf
1000
4000
u/f
Frequency (Hz)
Figure 5. Random g/cm2, d = 7.1 I cm.
incidence transmission
I
IO
loss of a double polythene
I
20
I
30 Number of emergent wws
panel m, = m3 = 0.047
I
40
Figure 6. Theoretical transmission loss of a double panel at mass-spring-mass g/cm2, d = 10 cm,f= 86 HZ, 0 = 0”.
I
50
resonance frequency
ml = m2 = 1
summation from which the total emergent wave amplitude is derived. The frequency was chosen as the normal incident mass-spring-mass resonance frequency for the panel, and the angle of incidence as zero. The graph therefore shows how the mass-spring-mass resonance depends upon the number of internal reflections. For one emergent wave only the sound insulation is twice that of a single panel. As more terms are added the sound insulation falls, tending asymptotically to zero. It is seen, however, that even with fifty multiple reflections taken into account, the insulation is still 3 dB.
132
A. CUMMINGS AND K. A. MULHOLLAND
Thus, truncation of the multiple reflection series in the manner described in this paper will markedly reduce the transparency of the panel due to oblique mass-spring-mass resonance. The result is that the random incidence curve shows a normal incidence mass-spring-mass resonance (where the truncation is least) and above this frequency a sharp rise in insulation. London [2], in deriving his infinite panel transmission loss theory, used a limiting angle of 90” initially, and explained the discrepancy between his predictions and the experimental results by assuming that there was a resistive term in the panel impedance. He chose values of this term to obtain good agreement between his theoretical and experimental results. Mulholland, Parbrook and Cummings [l] obtain good agreement between theoretical and experimental transmission loss curves by attributing an absorption coefficient to the inner surfaces of the panels. Values of this were chosen to give good agreement at all frequencies but it was not shown by independent means that the values chosen were realistic. Both of these treatments are now considered by the authors to be inadequate. There is no experimental evidence to suggest the existence of a resistive element in the wall impedance as postulated by London. Our previous treatment assumes that all the absorption takes place on the surfaces of the panels, whereas absorption may be taking place both on the panel surfaces and at the boundary of the cavity. In this paper, the other extreme is being considered where all the absorption takes place at the boundary of the cavity. We consider that this latter condition is more satisfactory in that it more closely resembles the experimental conditions under which the sound insulation measurements were made. Also, it is possible to predict the transmission loss of any double panel constructed of septa whose individual sound insulation can be predicted from the mass law and coincidence theory provided that the absorbing properties of the cavity boundary are known. 4. COMPARISON
WITH MODAL THEORY
This theory contrasts with the more commonly used modal theory. A method of solving this problem using modal theory would be to attempt to write down standing wave solutions within the cavity. This method, because of the restrictions on cavity wave number and directions and also on wave impedance at both ends of the cavity, leads to the conclusion that only certain wave modes are allowed in the cavity. It is thus necessary to work out mode coupling integrals through the panels and air-spaces. This has been attempted by White and Powell [3] and it is very complicated. We are of the opinion that since a mode cannot be excited instantaneously by transient waves such as are present in reverberant sound fields our theory is legitimate in that it traces the course of the wave as it attempts to set up its own (probably self-cancelling) mode and thus radiates sound energy. In order to substantiate this claim we are carrying out a series of tests with double panels having highly reflecting edges to see if the insulation differences predicted by our theory appear. The present results do show that for cavities with highly absorbing edges this theory gives good agreement with measured insulation values. 5. CONCLUSION It is seen that the theory given for the transmission loss of finite sized double panels gives reasonable agreement with measured transmission loss curves. No unrealistic assumptions are made, and the theory as given is considered to be in a usable form. REFERENCES 1. K. A. MULHOLLAND, H. D. PARBR~~K and A. CUMMINGS 1967 J. Sound Vib. 6,324. Transmission
loss of double panels.
TRANSMISSION LOSS OF FINITE SIZED PANELS
2. A.
1313
LONDON 1950 J. acoust. Sot. Am. 20,270. Transmission of reverberant sound through double walls. 3. P. H. WHITE and A. POWELL1965 J. acoust. Sot. Am. 40,82 I. Transmission of random sound and vibration through a rectangular double wall. 4. L. L. BERANEK and G. A. WORK 1949 J. acoust. Sot. Am. 21, 419. Sound transmission through multiple structures containing flexible blankets.
APPENDIX n s
: NOMENCLATURE
number of sound waves emergent from panels linear dimension of panel (for two-dimensional panel) d panel separation 0 angle of incidence of sound 4 scalar velocity potential a fraction of sound pressure reflected from absorbent x pressure transmission coefficient of single panel w angular frequency m surface mass density of panel p density of air c velocity of sound in air j 2/-l k wave number in air t pressure transmission coefficient of double panel 7 energy transmission coefficient e1 limiting angle of incidence TL transmission loss a linear panel dimension b linear panel dimension l,f mass-spring-mass resonance frequency at normal incidence ulf mass-spring-mass resonance frequency at limiting angle of incidence
THE TRANSMISSION
LOSS OF FINITE SIZED DOUBLE
IN A RANDOM
INCIDENCE
SOUND
PANELS
FIELD
A. CUMMINGS AND K. A. MULHOLLAND Department of Building Science, The University, Liverpool, England (Received 7 December 1967)
The extension of the multiple-reflection theory to take account of absorption at the edges of the cavity between the septa of a double panel is described, and an expression is obtained for the random incidence transmission loss of a double panel with edge absorption, given values of the panel dimensions and edge absorption coefficient. A curve is given of the measured random incidence transmission loss of a thin double aluminium panel with glass fibre absorbent around the edge of the cavity and theoretical curves given for the same panel with different values of the edge absorption coefficient. From the curves it is inferred that using a value of the edge absorption coefficient of unity in the theory, gives the best agreement between theory and experiment. A curve of the measured transmission loss of a thin polythene panel with glass fibre absorbent around the cavity edges and a theoretical curve with the absorption coefficient equal to unity are given. Good agreement between experiment and theory is obtained without the somewhat unrealistic assumptions of our previous paper. The effect of truncation of multiple reflections between the septa of a double panel upon the “mass-spring-mass” resonance is considered, and the finite panel theory is compared to modal theory.
1. INTRODUCTION
Existing theories for the transmission loss of double panels [l-4] do not allow for the absorption of obliquely incident sound waves at the edge of the air-space between the panels. This fact leads to considerable disagreement between theoretical and measured transmission loss curves for double panels, particularly in the region of the mass-spring-mass resonance frequency. In our previous paper [l] this was overcome by choosing values of panel face absorption coefficient so as to obtain a fit with measured data. In this paper it will be shown that by modifying the multiple reflection theory it is possible to allow for the absorption of sound waves at the boundary of the air-space. This produces agreement between theoretical and measured transmission loss curves, without the unrealistic assumption of panel face absorption coefficient that was involved in our previous paper. 2. THEORY
In the multiple-reflection theory as originally derived [l], it was assumed that each septum of the double panel is infinite in extent. In this derivation a geometric series of terms (see Figure l), each representing a sound wave emerging from the panel, is summed to infinity. This treatment is valid for a finite panel provided that the boundary of the air-space is totally reflective. A totally reflective boundary would produce a mirror-image between the panels of the multiple reflections which would occur beyond the boundary if the boundary were not present. This would produce an infinite series of multiple reflections between the panels, which would then behave as infinite panels with an infinite airspace between them. 126
TRANSMISSION
LOSS OF FINITE SIZED PANELS
127
For a cavity with a sound absorbing boundary, however, the sound waves reflected from the boundary are only partially reflected back into the cavity.
x,x2(1.-x,)(1-x2)exp (-3jkdax e)
~;x,x2(i-x,)2B-x2)Zep (-5fidcc6 e) . +d--+l I I I
I I
Figure 1. Sound transmission through an infinite double panel. Figure 2 shows two panels with sound waves arriving at the face of one panel at different points. Each incident wave sets up its own series of internal reflections in the cavity and these, on reaching the boundary, are partially reflected back into the cavity as shown. This series is partially reflected back from the opposite edge of the cavity and this process will continue indefinitely.
Figure 2. Sound transmission through a finite double panel with an absorbent cavity boundary.
128
A. CUMMINGS AND K. A. MULHOLLAND
In this model, the incident waves are split up into n bands (see Figure 3). The first band of incident waves produces n emergent bands on its first traverse of the cavity, by internal reflection. On its second and all subsequent traverses it produces n emergent bands. The second incident band produces n - 1 emergent bands on its first traverse, n emergent bands on its second traverse, n on its third, and so on.
nth
Figure 3. Bands of sound waves incident upon a finite double panel. The nth incident band produces only one emergent band on its first traverse, and n on its subsequent traverses. It is seen from Figure 3 that n may be found approximately from the formula n = [S/2d tan 01. (1) This formula ignores certain effects near the edges of the double panel, but the authors feel that in the light of subsequent approximations used to obtain a formula for a three-dimensional panel this approximation is satisfactory. Summing the emergent waves from each incident band gives, for the first incident band, C#Jt,= a $ ay + . . . + q-1
-t
+ a(uyn + uy*+1 + . . . + uy*“-1) + + a*(uy*” + uy*“+’+ . . . UJJ~“-~)+ . . . to infinity;
for the second incident band, +t* = a f uy + . . . + uyn-2 + -I-a(uy”-’ + uy” + . . . + uy*n-2) + + o~*(uy~“-~ + uy*” + . . . + uy3ne2) + . . . to infinity;
TRANSMISSION LOSSOFFINITESIZED PANELS
129
for the 12thincident band, &,=a+ + a(ay + ay2 + . . . + ay”) + + a2(ay”+’ + ay”+2 + . . . + ay2”) + . . . to infinity. Here a = $i X1x2 where x = 1 +jwm cos 8/(2pc), y = (1 - x1) (1 - x2) exp(- 2jkdcos 19)and Q = the fraction of the sound pressure reflected from the cavity boundary. a is related to the energy absorption coefficient of the boundary by the equation absorption coefficient = 1 - ~c.c~~. If we define
,=#I
r=n-I
S2= IX w’,
S, = I: Y',
r=l
r=O
and
then r=n
,2;,A, = S, + as2 S3S4.
(2)
Hence
z
t=‘=l=__ nk
A,
X,X,Y(l
XIX2
1-Y
41
and
-Y”> - Y12
1 _4 [
-Y”)
(1 - CCJ)“) I
(3)
7= jt12.
We can find the random incidence transmission factor of the panel from 81 7sin8cosedB / - b r= 81 sin t3cos e d0 i 0
and hence the transmission loss (TL) from TL = lOlog(l/?). This is a two-dimensional treatment, but it can be extended to apply to a three-dimensional system by taking the value of S as (a + b)/2, where a and b are the width and breadth of the panels. This approximation is valid provided that a 2: b. Theoretical transmission loss curves were computed using this formula, for different values of CL.Each value for transmission loss was found by averaging ten energy transmission factors (F) distributed over one-third octave frequency bands. This was done so that the theoretical results could be compared with measurements made using white noise sampled in thirdoctave bands. The KDF9computer used needed 40 min to obtain a transmission loss curve of seventeen results using KIDSGROVE ALGOL 60. 9
130
A. CUMMINGS AND K. A. MULHOLLAND
3. EXPERIMENTAL
RESULTS
Figure 4 shows a curve of measured transmission loss as a function of frequency for a thin double aluminium panel with thick glass fibre absorbent around the edges of the air space between the panels. Theoretical transmission loss curves for the panels are also shown. These were computed using the parameters relevant to the measured panels, and various values of CL. A value of the limiting angle 8,, of 80” was used. This had previously been found to give the best agreement between experiment and theory for the measuring facility used. The theoretical curve for cc= 1 shows a mass-spring-mass resonance “trough” and a standing-wave dip. This is identical to a curve for an infinite panel, and the “trough” and standing-wave dip are characteristic of such curves. However, the measured transmission loss curve does not exhibit the mass-spring-mass resonance trough except for a dip near the
01
100
II
I/f
1
Illlll
1
I
1000
u/f
I C/2d
I
4( )O
Frequency (Hz 1
Figure 4. Random g/cm2, d==7.11 cm.
incidence transmission
loss of a double aluminium
panel ml = m2 = 0.239
normal incidence mass-spring-mass resonance (“lower London frequency”) and the standingwave dip is much less pronounced than in the curve for a = 1. Theoretical transmission loss curves for a = O-75, CL= 0.5 and a = 0 are also shown. The curve for a = 0 agrees most closely with the measured curve and lies within 5 dB of it above 300 Hz. This was to be expected as the glass fibre absorbent used around the edge of the air-gap was several feet thick and would have a high absorption coefficient at all frequencies. The curves for a = 03 and a = O-75 lie between the curves for a = 0 and a = 1. The curve for a = 0 shows the same general shape as the measured curve, with a dip at the “lower London frequency”, and does not exhibit the extreme behaviour of the curve for a = 1. It was assumed that a value of a = 0 for the glass fibre absorbent used would give the best agreement between theory and experimental results for all panels. Figure 5 shows a curve of measured transmission loss as a function of frequency for a thin polythene double panel, and theoretical curves for a = 0 and a = 1. Again the curve for a = 0 agrees well with the measured curve. The curve for a = 1 shows a mass-spring-mass resonance trough which does not appear in the measured curve. It is seen that the oblique mass-spring-mass resonance is reduced in magnitude when a is reduced from 1 to 0. When a = 0, the number of internal reflections for obliquely incident
131
TRANSMISSION LOSS OF FINITE SIZED PANELS
sound waves becomes finite. Therefore, the magnitude of the mass-spring-mass resonance must depend upon the number of internal reflections of the sound waves between the panels. This was investigated. Figure 6 shows a curve of the theoretical transmission loss of a typical double panel as a function of the number of emergent waves taken into account in the
25-
20iii D 4
15-
100
llf
1000
4000
u/f
Frequency (Hz)
Figure 5. Random g/cm2, d = 7.1 I cm.
incidence transmission
I
IO
loss of a double polythene
I
20
I
30 Number of emergent wws
panel m, = m3 = 0.047
I
40
Figure 6. Theoretical transmission loss of a double panel at mass-spring-mass g/cm2, d = 10 cm,f= 86 HZ, 0 = 0”.
I
50
resonance frequency
ml = m2 = 1
summation from which the total emergent wave amplitude is derived. The frequency was chosen as the normal incident mass-spring-mass resonance frequency for the panel, and the angle of incidence as zero. The graph therefore shows how the mass-spring-mass resonance depends upon the number of internal reflections. For one emergent wave only the sound insulation is twice that of a single panel. As more terms are added the sound insulation falls, tending asymptotically to zero. It is seen, however, that even with fifty multiple reflections taken into account, the insulation is still 3 dB.
132
A. CUMMINGS AND K. A. MULHOLLAND
Thus, truncation of the multiple reflection series in the manner described in this paper will markedly reduce the transparency of the panel due to oblique mass-spring-mass resonance. The result is that the random incidence curve shows a normal incidence mass-spring-mass resonance (where the truncation is least) and above this frequency a sharp rise in insulation. London [2], in deriving his infinite panel transmission loss theory, used a limiting angle of 90” initially, and explained the discrepancy between his predictions and the experimental results by assuming that there was a resistive term in the panel impedance. He chose values of this term to obtain good agreement between his theoretical and experimental results. Mulholland, Parbrook and Cummings [l] obtain good agreement between theoretical and experimental transmission loss curves by attributing an absorption coefficient to the inner surfaces of the panels. Values of this were chosen to give good agreement at all frequencies but it was not shown by independent means that the values chosen were realistic. Both of these treatments are now considered by the authors to be inadequate. There is no experimental evidence to suggest the existence of a resistive element in the wall impedance as postulated by London. Our previous treatment assumes that all the absorption takes place on the surfaces of the panels, whereas absorption may be taking place both on the panel surfaces and at the boundary of the cavity. In this paper, the other extreme is being considered where all the absorption takes place at the boundary of the cavity. We consider that this latter condition is more satisfactory in that it more closely resembles the experimental conditions under which the sound insulation measurements were made. Also, it is possible to predict the transmission loss of any double panel constructed of septa whose individual sound insulation can be predicted from the mass law and coincidence theory provided that the absorbing properties of the cavity boundary are known. 4. COMPARISON
WITH MODAL THEORY
This theory contrasts with the more commonly used modal theory. A method of solving this problem using modal theory would be to attempt to write down standing wave solutions within the cavity. This method, because of the restrictions on cavity wave number and directions and also on wave impedance at both ends of the cavity, leads to the conclusion that only certain wave modes are allowed in the cavity. It is thus necessary to work out mode coupling integrals through the panels and air-spaces. This has been attempted by White and Powell [3] and it is very complicated. We are of the opinion that since a mode cannot be excited instantaneously by transient waves such as are present in reverberant sound fields our theory is legitimate in that it traces the course of the wave as it attempts to set up its own (probably self-cancelling) mode and thus radiates sound energy. In order to substantiate this claim we are carrying out a series of tests with double panels having highly reflecting edges to see if the insulation differences predicted by our theory appear. The present results do show that for cavities with highly absorbing edges this theory gives good agreement with measured insulation values. 5. CONCLUSION It is seen that the theory given for the transmission loss of finite sized double panels gives reasonable agreement with measured transmission loss curves. No unrealistic assumptions are made, and the theory as given is considered to be in a usable form. REFERENCES 1. K. A. MULHOLLAND, H. D. PARBR~~K and A. CUMMINGS 1967 J. Sound Vib. 6,324. Transmission
loss of double panels.
TRANSMISSION LOSS OF FINITE SIZED PANELS
2. A.
1313
LONDON 1950 J. acoust. Sot. Am. 20,270. Transmission of reverberant sound through double walls. 3. P. H. WHITE and A. POWELL1965 J. acoust. Sot. Am. 40,82 I. Transmission of random sound and vibration through a rectangular double wall. 4. L. L. BERANEK and G. A. WORK 1949 J. acoust. Sot. Am. 21, 419. Sound transmission through multiple structures containing flexible blankets.
APPENDIX n s
: NOMENCLATURE
number of sound waves emergent from panels linear dimension of panel (for two-dimensional panel) d panel separation 0 angle of incidence of sound 4 scalar velocity potential a fraction of sound pressure reflected from absorbent x pressure transmission coefficient of single panel w angular frequency m surface mass density of panel p density of air c velocity of sound in air j 2/-l k wave number in air t pressure transmission coefficient of double panel 7 energy transmission coefficient e1 limiting angle of incidence TL transmission loss a linear panel dimension b linear panel dimension l,f mass-spring-mass resonance frequency at normal incidence ulf mass-spring-mass resonance frequency at limiting angle of incidence