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Sound transmission paths through a statistical energy analysis model
Applied Acoustics 30 (1990) 45-55
Sound Transmission Paths through a Statistical Energy Analysis Model
Robert J. M. Craik Department of Building, Heriot-Watt University, Riccarton, Edinburgh EHI4 4AS, UK (Received 25 July 1989; revised version received 2 February 1990; accepted 9 February 1990)
ABSTRACT The most common method of determining the performance o f a system using statistical energy analysis is to solve a series of simultaneous equations. However, there can be many advantages in determining the performance on a path by path basis. The sound transmitted by a path is derived and the valid paths defined. The advantages of this form of analysis are discussed.
INTRODUCTION
Statistical Energy Analysis (SEA) is a useful method of determining the performance of complex systems. However, the usual method of determining performance is to solve a series of simultaneous equations and this requires the use of computers for all but the most simple systems. In this paper it is shown that the performance can be determined by studying individual paths through the system and then summing the contribution of each path. The two forms of analysis give identical results but the path analysis gives greater insight into the dominant mechanisms of sound transmission. 45 Applied Acoustics 0003-682X/90/$03"50 © 1990 Elsevier Science Publishers Ltd, England. Printed in Great Britain
Robert J. M. Craik
46
Room
Fig. I.
II
J
L
J 1
3
Plan of part of a building comprising two rooms, a common wall and an external wall.
In order to see how the path by path analysis is carried out, the system shown in Fig. 1 is used as an example. It shows part of a building with two rooms, a party wall and an external wall. Figure 2 shows the corresponding SEA model assuming that there is a noise source in room 1. For simplicity the non-resonant path 1-3 is not included. In order to calculate the performance of a particular system the coupling loss factors and total loss factors must be determined. These are usually simple algebraic expressions. Expressions for the coupling loss factors for the system shown in Figs I and 2 are given in Ref. I together with the relationships between energy and pressure or velocity. In this paper subscripts 1, 2, 3, 4 and 5 refer to the system shown in Figs 1 and 2. Subscripts a, b, c, d, ..., z refer to general systems.
Wall
4
Room I
/
Wall 5 Wall 2 Room 3
Fig. 2. Statistical energy analysis model of the system shown in Fig. 1.
47
Sound transmission paths through an SEA model
MATRIX SOLUTION To determine the performance of the system by the matrix method the power balance equations for all subsystems must be determined. For the example in Figs 1 and 2 this gives - E t r / t + E2r/21 + E4r/4t = -
Wl/e~
Et~/I.~ - E2q2 + E3r/32 + E j / , , + E5~/52 = 0 E2r]23 -- E a q 3 + f s q 5 3 = 0
(1)
E t q t , t + E2r]2,, -- E,t.r/~ + E s q s , * = 0 E2t725 + E3~35 + E,trl.,t 5 - E5~ 5 = 0
where q~b is the C L F from a to b, qa is the total loss factor of subsystem a and Wa is the power input into subsystem a. This can then be rewritten in matrix form which in its general form becomes 2 -eo ~ab
- - ?~b
tl~
tlb~
l~cb
- - tlc
-
"'"
?]:b
Eb
--
...
q:~
Ec
--
WJ /co (2)
-
Given the power input to each subsystem and the coupling and total loss factors the energy in each subsystem can be found• This can then be related to the pressure or velocity as required• The advantage of this method of calculating performance is that the energy in each subsystem is determined for all external sources and for all forms of coupling at the same time. The disadvantage is that a computer is invariably required to carry out the calculations. Furthermore the answers give no information about the importance of the individual paths through the system. In the example in Fig. 1 there would be no indication as to the relative importance of the direct and flanking paths in transmission from room 1 to room 3. If there are several noise sources then the importance of the individual sources can be found by repeating the calculations with one source at a time.
P A T H BY P A T H S O L U T I O N An alternative method of determining the performance of the system is to look at the individual transmission paths through the system• Calculations
48
Robert J. M. Craik
can be carried out to give either the level difference or the energy for a given power input. Usually the level difference is more useful in building acoustics.
Path description--level difference The general equation for the noise reduction due to transmission along a specific path can be found by considering the flanking path 1 4 5 3 for the case when there is a source in room 1. Some of the energy in room 1, E t, will go to wall 4. Considering room I as the only source of power gives the power balance equation of subsystem 4 as E l vl 14 = EJI.~
(3)
Again q~4 is the coupling loss factor (from 1 to 4) and q,, is the total loss factor (of subsystem 4). Some of the energy in wall 4 will go to wall 5. The power balance equation for subsystem 5, considering only power flow from subsystem 4, is then E~q,~ s = Es~I._.
(4)
Substituting in eqn (3) for E,~ gives E5=
E l q l ,~q.t s
q4q5
(5)
Some of the energy in subsystem 5 will go to room 3. Considering only the power from wall 5 gives the power balance for room 3 as Esq53
= E3q 3
(6)
Substituting for E5 gives the ratio of energies as E__ 2 = q14q45q53 E1 r/J/st/3
(7)
due to transmission along the path 1 4 5 3. This analysis can be extended to transmission along any path. In general for a path a b c d . . . z the energy ratio of the last to first subsystems will be E z = 17ab~bc?]cd" " " ~yz
Ea
(8)
qbqcY/d..,r/:
The energy in a room is given by ~ p2 V E = Pc 2
(9)
where P is the sound pressure (N/m2), V is the room volume (m3), p is the density of air (kg/m 3) and c the wavespeed in air (m/s). If the source and
Sound transmission paths through an SEA model
49
receiving subsystems a and z are both rooms the sound pressure level difference can be written as
La -- L . = 1 0 1 o g ~ = 1 0 -
p~
logV r/br/~'"r/:V" 1 L ~~--~-~b~••.--~.~.-Va J
(lO)
where L is the sound pressure level (dB re 2 x 10-5 N/m2).
Path description energy arising from power input If subsystem a is excited by a power input, W., then the power balance of subsystem a due to direct excitation will be
W. = E.corl.
(11)
Some o f t h e energy in a will go to b and then c and so on. Following the same procedure as for the level difference the energy in subsystem z due to the direct excitation of a followed by transmission along the path a b c... z will be E~ = W. r/.br/bcr/¢a.., r/y~ co r/j/br/¢.., r/.
(12)
If subsystem z is a room then the sound pressure level in room : due to transmission along the path a b c ... z can be given in terms of the power input (dB re 10-12W) as
L~ = L w . + 10 log F'r/"br/bc-------"""r/~z-1 - lOlog I f V:] + 17.4 k ~/.~/b... rl: j
(13)
where Lw is the sound power level of the noise source (dB re 10-12W). The constant 17.4 is obtained by converting energy to pressure, by substituting eqn (9) into eqn (12), and then expressing all terms in dB. Together eqns (8), (10), (12) and (13) give the energy or sound pressure level in the receiving room relative to the source level or power input. Clearly the total energy in any subsystem must be the sum of the contributions of all the individual paths. The energy from the sum of the paths must also be the same as that obtained from the matrix method. In the next section the paths which give this equivalence are defined.
Path definition--level difference The possible paths can be found by considering the paths from room 1 to room 3 of the system shown in Fig. 1. The power balance equations for
50
Robert J. M. Craik
subsystems 1-5 given by eqn (1)can be rewritten as E~ =
~1
+
E272t
E~q~t + - -
71 E2
Elqt2 + E3~32 + E~q~z + E5752 02 72 72 q2
E2q23 E3 ----+ ~3
E5q53 ~3
E , = Etqt* ~ + q~
E2~24 ..... + E575" - ~
(14)
74
E 5 = Ezq25 + E3735 + _E~745 _ ~5 75 ~5
Starting with the power balance for subsystem 3 and substituting for and E5 gives
E2
E 3 = Et~t2~2~3- k E3732~2~3+ E4~¢202~3+ E575272__.____~ 3 ~2~3 721?3 7273 0273 + E2 725753 + E3~3575~3+ Ea ~45~53 7573 75~3 7573
(15)
From the definition of a path from eqn (8) the first term is the sound transmission along path 1 2 3. Substituting for E~, E 3, E4 and E5 in all other terms gives 712723 E 3 = El--+ q273
E1
714742723 ~4q273
+ El 0t2~25753 + E, 7t474s75a + 7205~3 747573
(16)
This gives the energy in room 3 as the direct path and the first 3 flanking paths (1423, 1253 and 1453) plus other terms (not given). Substituting again for E 2 - E 5 into all the other terms gives terms corresponding to paths 12323, 12423, 12523, 14523, 14253, 12353 and 12453 plus extra terms. This process of substituting for E 2 - E 5 can be continued indefinitely generating more and more terms corresponding to longer and longer paths. In general, except for systems with only two subsystems, the number of paths is infinite as eqns (14) are rewritten as an infinite series. The series solution is exact and the only approximation that occurs is when a finite rather than an infinite number of paths are computed.
Sound transmission paths through an SEA model
51
TABLE ! Sound Transmission Paths from Room 1 to Room 3 for Level DifferenceCalculations for the System Shown in Figs 1 and 2 Number of subsystems in path
Number of paths
2 3 4 5
0 1 3 7
6
19
7 8 9 10
47 123 311 803
Paths
123 1253, 1423, 1453 12323, 12353, 12423 12453, 12523, 14253, 14523 123253, 123523, 124253, 124523,125253,125323, 125353, 125423, 125453, 142323, 142353, 142423, 142453, 142523, 145253, 145323, 145353, 145423, 145453
The total energy in a receiving subsystem (in this case 3) is the sum o f the contributions from each path. A path is any possible route through the system which starts with the source subsystem and ends with the receiving subsystem. It m a y 'pass through' any subsystem any n u m b e r o f times except the source subsystem. The path m a y also 'pass along' any connection any n u m b e r o f times in any direction except connections to the source subsystem. For the system in Fig. 1 the paths would be as given in Table 1. Path definitior,
caergy arising from power input
If the problem is to find the energy in a subsystem due to an external power source then the contribution from individual paths will be as given in eqn (13) and the total energy will again be the sum of the paths. The same calculation process can be followed to give the paths through the system. In this case substitution is also m a d e for E r The paths will be the same except that the source subsystem can be 'passed
52
Robert J. M. Craik
TABLE 2 Sound Transmission Paths to Room 3 for a power Input into Room I for the System Shown in Figs 1 and 2 Number of subsystems in path
Paths
Number of paths
3 4 5
0 I 3 9
6
27
7 8 9 10
78 232 675 1993
123 1253, 1423, 1453 12123, 12323, 12353, 12423, 12453, 12523, 14123, 14253, 14523 121253, 121423, 121453, 123253, 123523, 124123. 124253, 124523, 125253, 125323,125353, 125423. 125453, 141253, 141423. 141453, 142123, 142323, 142353, 142423, 142453, 142523, 145253, 145323, 145353, 145423, 145453
through' an unlimited number of times and connections into the source subsystem may be 'passed along'. The paths from an external source in subsystem 1 to subsystem 3 are given in Table 2. Consistency relationship For coupling between two subsystems the consistency relationship (17)
r l a q a b = rlbtTb a
relates the coupling in opposite directions where n is the modal density. If a 'path loss factor', q', is defined as being the fraction of energy flowing from the source subsystem to the receiving subsystem along a particular path in one radian cycle then , ++++...:
17°bPIb++q+a
• • • qv.+
=
rlbrl+rla.., fly
(18)
Sound transmission paths through an SEA model
53
Applying the consistency relationship to each term in eqn (18) gives naq'~...z = n~q'z...~
(19)
which is simply the consistency relationship with path loss factors substituted for coupling loss factors. Thus the consistency relationship holds not only for coupling loss factors but also for entire paths. In a similar manner it can also be shown to apply to the sum of several path loss factors. The path loss factor is a useful concept as the power balance for a subsystem z for transmission along a path a b c... z will then be Ear/'.... : = E:.:
(20)
Thus the path loss factor can be used in exactly the same way as a coupling loss factor.
DISCUSSION In the first part of the paper a definition of a path was given in terms of the coupling and total loss factors. In the next part the number of paths through the system necessary to give the same overall solution as the matrix method was described. It is necessary that the sum of all paths be the same as the matrix method. However, the 'path' solution given in this paper is not unique. Other equations can be ascribed to paths and the number of paths necessary to give the matrix solution ascertained. For example it is possible to define a meaning for paths such that the number of paths is finite. This would have the advantage that all the paths could be computed. However the expression for the individual paths would be more complicated. The expression given in this paper for the path level difference is extremely simple and this is the advantage of this method of calculation. However, it is a consequence of this that the number of paths be infinite since a path can recross the same 'route' any number of times. When the sound transmission by individual paths is known the relative importance of each path can be determined. If remedial action is to be taken to reduce the overall transmission of sound then the most effective treatment will be to reduce the dominant path. An example of such a path analysis is given in Ref. 3. Although there are many long paths the rate at which the number of paths increase is less than the rate at which the contribution of those paths decreases. This must be true as the infinite sum of the contributions of all the paths is finite (and can be found from the matrix method). The result is that
54
Robert J. M. Craik
long paths can be ignored though what constitutes a long path will depend on the system being considered. For sound transmission in buildings, as in the example of Figs i and 2, the longer paths tend to be much less important than short paths. Ira building is only made of solid walls then on average each structural element will be connected to about 8-10 other elements. Since all structural coupling loss factors are usually roughly similar the total loss factor being the sum of the coupling loss factors will be 8-10 times higher than an average CLF. Each time a path crosses a structural joint the sound transmitted is reduced by a factor which is the ratio of the coupling to total loss factor. This is about ~ to ~0 or approximately a reduction of 10dB. Thus typically path 1 4 2 5 crosses one joint more than 123 and will be 10dB less important. Similarly path 1 2 4 5 3 will be 10dB less than path 1253. Thus, in the example in Fig. 1, the dominant path will be the direct path. The three paths that cross one joint will together be about 3 t h s or 5 dB less important than the direct path. The 7 paths that cross 2 joints will be about ~--~oths or 11 dB less important than the direct path and the 19 paths that cross 3 joints will be about T6-ff6ths ~9 or 17 dB less important than the direct path. Of course the actual importance of a path will depend on the details of the construction. The noise reduction of a particular path can be measured approximately in a straightforward manner. The level difference can be rewritten as
L ~ - L . = 101ogqb + 101ogr/¢+ 10log q___L ~ab
~bc
~cd
• . . + log~/: r/~.: + 10log ~V-
(21)
The first term on the right is the energy level difference between subsystems a and b when a is excited if the model has only 2 subsystems (a and b). In practice a and b will only be part of the system but the energy ratio is usually a good approximation for the ratio of the C L F to the TLF. In the same way the second term is approximately the energy level difference between b and c when b is excited. The third term is approximately the energy level difference between c and d when c is excited and so on. Thus by measuring the attenuation at each step of the path the overall reduction of the path can be obtained. It is assumed that when subsystem a is excited that the sound picked up at b is due to transmission along the connection a b. If most of the sound goes from a to b along another route, say a d b then there is little point in having a connection in the model between a and b and little
55
Sound transmission paths through an SEA model TABLE 3
Example of Data Measured Source
SPL room 1
Acceleration Wall 4
Room 1 Wall 4 Wall 5
100
85 70
SPL room 3
Difference (dB)
80
15 10 20
Wall 5
60 100
45dB point in measuring it. Typically the error introduced by using the energy ratio as the ratio o f loss factors is of the order of 1 dB. In practice it is pressure or acceleration that is measured not energy. H o w e v e r since the intermediate subsystems are always measured twice, once as a source subsystem and once as a receiving subsystem, the terms which convert pressure or acceleration to energy cancel and the actual measurements o f pressure or acceleration can be used. F o r example, for the path 1453, a noise source is first placed in room 1 and the SPL in r o o m 1 and the acceleration of wall 4 is measured. Wall 4 is then excited and the acceleration o f wall 4 and wall 5 are measured. Finally wall 5 is excited and the acceleration o f wall 5 and the SPL in r o o m 3 are measured. If the data that are measured are as in Table 3, then the path level difference will be 45 dB. The units of the difference are those o f the first and last measurements and as both are SPLs so the difference is a sound pressure level difference. An example o f this is given in Ref. 3.
REFERENCES 1. Craik, R. J. M., The noise reduction of flanking paths. Applied Acoustics, 22 (1987) 163-75. 2. Lyon, R. H., Statistical Energy Analysis of Dynamical Systems: Theory and Applications. MIT Press, Cambridge, MA, 1975. 3. Craik, R. J. M., The noise reduction of the acoustic paths between two rooms interconnected by a ventilation duct. Applied Acoustics, 12 (1979) 161-79.
Sound Transmission Paths through a Statistical Energy Analysis Model
Robert J. M. Craik Department of Building, Heriot-Watt University, Riccarton, Edinburgh EHI4 4AS, UK (Received 25 July 1989; revised version received 2 February 1990; accepted 9 February 1990)
ABSTRACT The most common method of determining the performance o f a system using statistical energy analysis is to solve a series of simultaneous equations. However, there can be many advantages in determining the performance on a path by path basis. The sound transmitted by a path is derived and the valid paths defined. The advantages of this form of analysis are discussed.
INTRODUCTION
Statistical Energy Analysis (SEA) is a useful method of determining the performance of complex systems. However, the usual method of determining performance is to solve a series of simultaneous equations and this requires the use of computers for all but the most simple systems. In this paper it is shown that the performance can be determined by studying individual paths through the system and then summing the contribution of each path. The two forms of analysis give identical results but the path analysis gives greater insight into the dominant mechanisms of sound transmission. 45 Applied Acoustics 0003-682X/90/$03"50 © 1990 Elsevier Science Publishers Ltd, England. Printed in Great Britain
Robert J. M. Craik
46
Room
Fig. I.
II
J
L
J 1
3
Plan of part of a building comprising two rooms, a common wall and an external wall.
In order to see how the path by path analysis is carried out, the system shown in Fig. 1 is used as an example. It shows part of a building with two rooms, a party wall and an external wall. Figure 2 shows the corresponding SEA model assuming that there is a noise source in room 1. For simplicity the non-resonant path 1-3 is not included. In order to calculate the performance of a particular system the coupling loss factors and total loss factors must be determined. These are usually simple algebraic expressions. Expressions for the coupling loss factors for the system shown in Figs I and 2 are given in Ref. I together with the relationships between energy and pressure or velocity. In this paper subscripts 1, 2, 3, 4 and 5 refer to the system shown in Figs 1 and 2. Subscripts a, b, c, d, ..., z refer to general systems.
Wall
4
Room I
/
Wall 5 Wall 2 Room 3
Fig. 2. Statistical energy analysis model of the system shown in Fig. 1.
47
Sound transmission paths through an SEA model
MATRIX SOLUTION To determine the performance of the system by the matrix method the power balance equations for all subsystems must be determined. For the example in Figs 1 and 2 this gives - E t r / t + E2r/21 + E4r/4t = -
Wl/e~
Et~/I.~ - E2q2 + E3r/32 + E j / , , + E5~/52 = 0 E2r]23 -- E a q 3 + f s q 5 3 = 0
(1)
E t q t , t + E2r]2,, -- E,t.r/~ + E s q s , * = 0 E2t725 + E3~35 + E,trl.,t 5 - E5~ 5 = 0
where q~b is the C L F from a to b, qa is the total loss factor of subsystem a and Wa is the power input into subsystem a. This can then be rewritten in matrix form which in its general form becomes 2 -eo ~ab
- - ?~b
tl~
tlb~
l~cb
- - tlc
-
"'"
?]:b
Eb
--
...
q:~
Ec
--
WJ /co (2)
-
Given the power input to each subsystem and the coupling and total loss factors the energy in each subsystem can be found• This can then be related to the pressure or velocity as required• The advantage of this method of calculating performance is that the energy in each subsystem is determined for all external sources and for all forms of coupling at the same time. The disadvantage is that a computer is invariably required to carry out the calculations. Furthermore the answers give no information about the importance of the individual paths through the system. In the example in Fig. 1 there would be no indication as to the relative importance of the direct and flanking paths in transmission from room 1 to room 3. If there are several noise sources then the importance of the individual sources can be found by repeating the calculations with one source at a time.
P A T H BY P A T H S O L U T I O N An alternative method of determining the performance of the system is to look at the individual transmission paths through the system• Calculations
48
Robert J. M. Craik
can be carried out to give either the level difference or the energy for a given power input. Usually the level difference is more useful in building acoustics.
Path description--level difference The general equation for the noise reduction due to transmission along a specific path can be found by considering the flanking path 1 4 5 3 for the case when there is a source in room 1. Some of the energy in room 1, E t, will go to wall 4. Considering room I as the only source of power gives the power balance equation of subsystem 4 as E l vl 14 = EJI.~
(3)
Again q~4 is the coupling loss factor (from 1 to 4) and q,, is the total loss factor (of subsystem 4). Some of the energy in wall 4 will go to wall 5. The power balance equation for subsystem 5, considering only power flow from subsystem 4, is then E~q,~ s = Es~I._.
(4)
Substituting in eqn (3) for E,~ gives E5=
E l q l ,~q.t s
q4q5
(5)
Some of the energy in subsystem 5 will go to room 3. Considering only the power from wall 5 gives the power balance for room 3 as Esq53
= E3q 3
(6)
Substituting for E5 gives the ratio of energies as E__ 2 = q14q45q53 E1 r/J/st/3
(7)
due to transmission along the path 1 4 5 3. This analysis can be extended to transmission along any path. In general for a path a b c d . . . z the energy ratio of the last to first subsystems will be E z = 17ab~bc?]cd" " " ~yz
Ea
(8)
qbqcY/d..,r/:
The energy in a room is given by ~ p2 V E = Pc 2
(9)
where P is the sound pressure (N/m2), V is the room volume (m3), p is the density of air (kg/m 3) and c the wavespeed in air (m/s). If the source and
Sound transmission paths through an SEA model
49
receiving subsystems a and z are both rooms the sound pressure level difference can be written as
La -- L . = 1 0 1 o g ~ = 1 0 -
p~
logV r/br/~'"r/:V" 1 L ~~--~-~b~••.--~.~.-Va J
(lO)
where L is the sound pressure level (dB re 2 x 10-5 N/m2).
Path description energy arising from power input If subsystem a is excited by a power input, W., then the power balance of subsystem a due to direct excitation will be
W. = E.corl.
(11)
Some o f t h e energy in a will go to b and then c and so on. Following the same procedure as for the level difference the energy in subsystem z due to the direct excitation of a followed by transmission along the path a b c... z will be E~ = W. r/.br/bcr/¢a.., r/y~ co r/j/br/¢.., r/.
(12)
If subsystem z is a room then the sound pressure level in room : due to transmission along the path a b c ... z can be given in terms of the power input (dB re 10-12W) as
L~ = L w . + 10 log F'r/"br/bc-------"""r/~z-1 - lOlog I f V:] + 17.4 k ~/.~/b... rl: j
(13)
where Lw is the sound power level of the noise source (dB re 10-12W). The constant 17.4 is obtained by converting energy to pressure, by substituting eqn (9) into eqn (12), and then expressing all terms in dB. Together eqns (8), (10), (12) and (13) give the energy or sound pressure level in the receiving room relative to the source level or power input. Clearly the total energy in any subsystem must be the sum of the contributions of all the individual paths. The energy from the sum of the paths must also be the same as that obtained from the matrix method. In the next section the paths which give this equivalence are defined.
Path definition--level difference The possible paths can be found by considering the paths from room 1 to room 3 of the system shown in Fig. 1. The power balance equations for
50
Robert J. M. Craik
subsystems 1-5 given by eqn (1)can be rewritten as E~ =
~1
+
E272t
E~q~t + - -
71 E2
Elqt2 + E3~32 + E~q~z + E5752 02 72 72 q2
E2q23 E3 ----+ ~3
E5q53 ~3
E , = Etqt* ~ + q~
E2~24 ..... + E575" - ~
(14)
74
E 5 = Ezq25 + E3735 + _E~745 _ ~5 75 ~5
Starting with the power balance for subsystem 3 and substituting for and E5 gives
E2
E 3 = Et~t2~2~3- k E3732~2~3+ E4~¢202~3+ E575272__.____~ 3 ~2~3 721?3 7273 0273 + E2 725753 + E3~3575~3+ Ea ~45~53 7573 75~3 7573
(15)
From the definition of a path from eqn (8) the first term is the sound transmission along path 1 2 3. Substituting for E~, E 3, E4 and E5 in all other terms gives 712723 E 3 = El--+ q273
E1
714742723 ~4q273
+ El 0t2~25753 + E, 7t474s75a + 7205~3 747573
(16)
This gives the energy in room 3 as the direct path and the first 3 flanking paths (1423, 1253 and 1453) plus other terms (not given). Substituting again for E 2 - E 5 into all the other terms gives terms corresponding to paths 12323, 12423, 12523, 14523, 14253, 12353 and 12453 plus extra terms. This process of substituting for E 2 - E 5 can be continued indefinitely generating more and more terms corresponding to longer and longer paths. In general, except for systems with only two subsystems, the number of paths is infinite as eqns (14) are rewritten as an infinite series. The series solution is exact and the only approximation that occurs is when a finite rather than an infinite number of paths are computed.
Sound transmission paths through an SEA model
51
TABLE ! Sound Transmission Paths from Room 1 to Room 3 for Level DifferenceCalculations for the System Shown in Figs 1 and 2 Number of subsystems in path
Number of paths
2 3 4 5
0 1 3 7
6
19
7 8 9 10
47 123 311 803
Paths
123 1253, 1423, 1453 12323, 12353, 12423 12453, 12523, 14253, 14523 123253, 123523, 124253, 124523,125253,125323, 125353, 125423, 125453, 142323, 142353, 142423, 142453, 142523, 145253, 145323, 145353, 145423, 145453
The total energy in a receiving subsystem (in this case 3) is the sum o f the contributions from each path. A path is any possible route through the system which starts with the source subsystem and ends with the receiving subsystem. It m a y 'pass through' any subsystem any n u m b e r o f times except the source subsystem. The path m a y also 'pass along' any connection any n u m b e r o f times in any direction except connections to the source subsystem. For the system in Fig. 1 the paths would be as given in Table 1. Path definitior,
caergy arising from power input
If the problem is to find the energy in a subsystem due to an external power source then the contribution from individual paths will be as given in eqn (13) and the total energy will again be the sum of the paths. The same calculation process can be followed to give the paths through the system. In this case substitution is also m a d e for E r The paths will be the same except that the source subsystem can be 'passed
52
Robert J. M. Craik
TABLE 2 Sound Transmission Paths to Room 3 for a power Input into Room I for the System Shown in Figs 1 and 2 Number of subsystems in path
Paths
Number of paths
3 4 5
0 I 3 9
6
27
7 8 9 10
78 232 675 1993
123 1253, 1423, 1453 12123, 12323, 12353, 12423, 12453, 12523, 14123, 14253, 14523 121253, 121423, 121453, 123253, 123523, 124123. 124253, 124523, 125253, 125323,125353, 125423. 125453, 141253, 141423. 141453, 142123, 142323, 142353, 142423, 142453, 142523, 145253, 145323, 145353, 145423, 145453
through' an unlimited number of times and connections into the source subsystem may be 'passed along'. The paths from an external source in subsystem 1 to subsystem 3 are given in Table 2. Consistency relationship For coupling between two subsystems the consistency relationship (17)
r l a q a b = rlbtTb a
relates the coupling in opposite directions where n is the modal density. If a 'path loss factor', q', is defined as being the fraction of energy flowing from the source subsystem to the receiving subsystem along a particular path in one radian cycle then , ++++...:
17°bPIb++q+a
• • • qv.+
=
rlbrl+rla.., fly
(18)
Sound transmission paths through an SEA model
53
Applying the consistency relationship to each term in eqn (18) gives naq'~...z = n~q'z...~
(19)
which is simply the consistency relationship with path loss factors substituted for coupling loss factors. Thus the consistency relationship holds not only for coupling loss factors but also for entire paths. In a similar manner it can also be shown to apply to the sum of several path loss factors. The path loss factor is a useful concept as the power balance for a subsystem z for transmission along a path a b c... z will then be Ear/'.... : = E:.:
(20)
Thus the path loss factor can be used in exactly the same way as a coupling loss factor.
DISCUSSION In the first part of the paper a definition of a path was given in terms of the coupling and total loss factors. In the next part the number of paths through the system necessary to give the same overall solution as the matrix method was described. It is necessary that the sum of all paths be the same as the matrix method. However, the 'path' solution given in this paper is not unique. Other equations can be ascribed to paths and the number of paths necessary to give the matrix solution ascertained. For example it is possible to define a meaning for paths such that the number of paths is finite. This would have the advantage that all the paths could be computed. However the expression for the individual paths would be more complicated. The expression given in this paper for the path level difference is extremely simple and this is the advantage of this method of calculation. However, it is a consequence of this that the number of paths be infinite since a path can recross the same 'route' any number of times. When the sound transmission by individual paths is known the relative importance of each path can be determined. If remedial action is to be taken to reduce the overall transmission of sound then the most effective treatment will be to reduce the dominant path. An example of such a path analysis is given in Ref. 3. Although there are many long paths the rate at which the number of paths increase is less than the rate at which the contribution of those paths decreases. This must be true as the infinite sum of the contributions of all the paths is finite (and can be found from the matrix method). The result is that
54
Robert J. M. Craik
long paths can be ignored though what constitutes a long path will depend on the system being considered. For sound transmission in buildings, as in the example of Figs i and 2, the longer paths tend to be much less important than short paths. Ira building is only made of solid walls then on average each structural element will be connected to about 8-10 other elements. Since all structural coupling loss factors are usually roughly similar the total loss factor being the sum of the coupling loss factors will be 8-10 times higher than an average CLF. Each time a path crosses a structural joint the sound transmitted is reduced by a factor which is the ratio of the coupling to total loss factor. This is about ~ to ~0 or approximately a reduction of 10dB. Thus typically path 1 4 2 5 crosses one joint more than 123 and will be 10dB less important. Similarly path 1 2 4 5 3 will be 10dB less than path 1253. Thus, in the example in Fig. 1, the dominant path will be the direct path. The three paths that cross one joint will together be about 3 t h s or 5 dB less important than the direct path. The 7 paths that cross 2 joints will be about ~--~oths or 11 dB less important than the direct path and the 19 paths that cross 3 joints will be about T6-ff6ths ~9 or 17 dB less important than the direct path. Of course the actual importance of a path will depend on the details of the construction. The noise reduction of a particular path can be measured approximately in a straightforward manner. The level difference can be rewritten as
L ~ - L . = 101ogqb + 101ogr/¢+ 10log q___L ~ab
~bc
~cd
• . . + log~/: r/~.: + 10log ~V-
(21)
The first term on the right is the energy level difference between subsystems a and b when a is excited if the model has only 2 subsystems (a and b). In practice a and b will only be part of the system but the energy ratio is usually a good approximation for the ratio of the C L F to the TLF. In the same way the second term is approximately the energy level difference between b and c when b is excited. The third term is approximately the energy level difference between c and d when c is excited and so on. Thus by measuring the attenuation at each step of the path the overall reduction of the path can be obtained. It is assumed that when subsystem a is excited that the sound picked up at b is due to transmission along the connection a b. If most of the sound goes from a to b along another route, say a d b then there is little point in having a connection in the model between a and b and little
55
Sound transmission paths through an SEA model TABLE 3
Example of Data Measured Source
SPL room 1
Acceleration Wall 4
Room 1 Wall 4 Wall 5
100
85 70
SPL room 3
Difference (dB)
80
15 10 20
Wall 5
60 100
45dB point in measuring it. Typically the error introduced by using the energy ratio as the ratio o f loss factors is of the order of 1 dB. In practice it is pressure or acceleration that is measured not energy. H o w e v e r since the intermediate subsystems are always measured twice, once as a source subsystem and once as a receiving subsystem, the terms which convert pressure or acceleration to energy cancel and the actual measurements o f pressure or acceleration can be used. F o r example, for the path 1453, a noise source is first placed in room 1 and the SPL in r o o m 1 and the acceleration of wall 4 is measured. Wall 4 is then excited and the acceleration o f wall 4 and wall 5 are measured. Finally wall 5 is excited and the acceleration o f wall 5 and the SPL in r o o m 3 are measured. If the data that are measured are as in Table 3, then the path level difference will be 45 dB. The units of the difference are those o f the first and last measurements and as both are SPLs so the difference is a sound pressure level difference. An example o f this is given in Ref. 3.
REFERENCES 1. Craik, R. J. M., The noise reduction of flanking paths. Applied Acoustics, 22 (1987) 163-75. 2. Lyon, R. H., Statistical Energy Analysis of Dynamical Systems: Theory and Applications. MIT Press, Cambridge, MA, 1975. 3. Craik, R. J. M., The noise reduction of the acoustic paths between two rooms interconnected by a ventilation duct. Applied Acoustics, 12 (1979) 161-79.