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Prediction of sound transmission loss of single leaf walls due to
an improvement of the sea method
KUROKI, Soichiro
鹿児島大学工学部研究報告, 28: 117-124
1986-11-01
http://hdl.handle.net/10232/11380
http://ir.kagoshima-u.ac.jp
PREDICTION OF SOUND TRANSMISSION LOSS
OF SINGLE LEAF WALLS
DUE TO AN IMPROVEMENT OF THE SEA METHOD
Soichiro KUROKI
(Received May 31, 1986)
ABSTRACT
The SEA (statistical energy analysis) method can accurately predict the sound transmission loss
of a finite wall. However, there is a large discrepancy between the calculated value and the measured
value at lower frequencies, since the random incidence mass law deduced for the infinite panel is part
ly applied in this method.
In this paper, to improve this point, the radiation factor derived by investigating the directivity
characteristic of transmitted waves through a finite wall was introduced into the coupling loss factor
between source and receiving rooms. The experimental and theoretical values agreed well over a wide
frequency range.
1. Introduction
It is a well known fact that the measured sound transmission losses of single leaf walls by use of
the two-room method are generally greater than the calculated values by using the random incidence
mass law, when the area of wall becomes smaller. The mass law of sound transmission through an infi
nite panel by A. London
is popular as the classical method.
On the other hand, M. J. Crocker and A. J. Price2 introduced the Statistical Energy Analysis
(SEA) Method to predict the sound insulation of the finite walls. The SEA method is characterized
mainly in that the total sound transmission energy through a finite partition is the sum of the sound
radiation energy on resonant mode and on nonresonant mode. Further, the transmission suite (source
and receiving rooms) is also analyzed. The theoretical values agree well with the experimental ones in
higher frequencies around the coincidence effect region. However, there is a large difference at low
frequencies. Since these authors applied the 80° random incidence mass law of the infinite panel as
the coupling loss factor (713) due to nonresonant transmission, this point remains to be improved.
There are some recent developments in the sound transmission theories for a finite wall. R. J.
Donato3) has deduced a correction factor at low frequencies by using a wavenumber approach. T.
Kawai4) has analyzed numerically the transmission mechanism of a normal incident plane wave through
a circular plate. It was indicated that the plate vibrates in such a way that a piston motion given as
the mass law and a flexural motion damped by the internal loss are superposed, further the later one
is influenced by the edge conditions.
H. Sato5 has studied the directivity characteristic of transmitted wave of a rectangular wall vib
rating with an infinite baffle. It becomes non—directional in order the higher value of incident angle
when the wavenumber of incident sound or the dimensions of wall diminishes from infinity. Therefore,
118
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S28-SI (1986)
the window whose area is small does not show characteristic of infinte wall in the lower frequency re
gion. The above results are considered to correspond with the 80° random incidence mass law by
Crocker and Price which agrees well with the measured value. The author ascertained that the calcu
lated values by using the 80° random incidence mass law agreed with our own experimental values6.
While, a later paper by A. Elmallawany7) demonstrated to improve the coupling loss factor of
nonresonant transmission at low frequencies using the correction factor deduced by Donato. Here the
calculated values are in good agreement with the measured values at the low frequencies.
The author has attempted to compensate the SEA method for the finite size of the single leaf
wall. The radiation factor deduced by Sato was introduced into the coupling loss factor of nonresonant
transmission. The measured and theoretical values of sound transmission loss are made to reach good
agreement over wide frequency range by using our method.
2. Improvement of the Coupling Loss Factor 713 for Nonresonant Transmission
Crocker and Price obtained the coupling loss factor 713 due to nonresonant mass law transmis
sion between the source room and the receiving room as the following relationship1,
101og7,3=-TLm+101og(T^r)
:
(1)
where S is the wall area; c is the speed of sound in air; V\ is the volume of source room and w is
angular frequency. TLm is the random incidence mass law TL value. This value is given by the TL of
80° random incidence mass law which is determined by comparing the theoretical value with the ex
perimental one.
Using the correction factor according to Donato , Elmallawany obtained a least square equation
and then improved the coupling loss factor as follows 7) ;
101og7i3=-TL,;+101og(Tp^r)
TU=TLm+5(^-)-°-72 '
(2)
(3)
where TLm is the transmission loss for a diffused sound field from the mass law, k is wave number,
and a is half the smallest dimension of the panel based on the radius of the largest circle which can
just fit into the rectangular area, respectively. Since the value in the parentheses of second term of
Eq. (3) is limited within 1 to 6.5, this method can be applied to the relatively small walls or at the low
frequency region.
Sato has deduced the modified sound transmission loss TL$ of a finite wall for a random inci
dence sound field, baced on the theory of wave equation, as Eq. (4) and (5);
TLs = TL0-101ogQ-3
(4)
f2>r
Q=]0 X{ka)smd dd
(5)
where TLo is the normal incidence mass law and Q represents spatial average of radiation factor
A(ka) for a spatial direction of incident plane wave (incident—angle 6 ) in receiving room, and a is half
a dimension of wall. The Q value of the square wall has been calculated numerically. In the case that
the wall is not square, the derivation of Q is within ±1 to 2 dB.
The author has tried to introduce this radiation factor into the coupling loss factor due to nonre
sonant transmission of the SEA method. To facilitate computer calculation, the values of Q calculated
by Sato in Reference 5 are converted in the following procedure:
AR=-101ogQ+5
(6)
Prediction of Sound Transmission Loss of Single Leaf Walls due to an Improvement of the SEA Method 119
and Eq. (7) represents the mathematical form.
AR = AUa)B
(7)
The two constants, A and B, can be determined by the least square method. The errors remain less
than ±0.5dB. Finally, the coupling loss factor is obtained as follows:
10 log T^-TLs+lOlogf^^)
(8)
TLs = TL0+ 9.2(A;ar0-51- 8
(9)
where ka is limitied within 0.5 to 64.
3. Verification of the Improved SEA Method
The experimental work is done in the Kagoshima University transmission suite laboratory in con
formity to the JIS Standards "Method for Laboratory Measurement of Sound Transmission Loss"
(JIS A 1416). The volume of source and receiving rooms are 207m3 and 102m3, respectively. The
panels were measured 2,468 by 2,377mm and divided vertically into two parts by a wooden stud (50X
100 mm section). The panel was nailed on to both the wooden frame and the stud. The distance of ev
ery nail is about 15cm. Table 1 shows the physical properties of the panels.
To calculate the sound transmission loss of the wall using the SEA theory1,8), some parameters
must be known beforehand; for example, the volume of the source and receiving rooms and the rever
beration time of the receiving room when the object panel is mounted. Since the panel is divided by a
rigid stud, the rectangular wall were calculated on the half area (1,234X2,377 mm2) as a square wall
having the same area. The internal loss factor rj[nt used in the prediction was 0.01. The sound speed in
air was calculated by measuring the temperature of air in rooms. The panel edge conditions were in
tended to be fixed.
Figure 1 and 2 show the comparisons between the measured TL of single wall of aluminium panel
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gure as a dashed—dotted line.
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Prediction of Sound Transmission Loss of Single Leaf Walls due to an Improvement of the SEA Method
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O
; experimental data.
; theoretical random incidence mass law,
; theoretical according to Elmallawany, Eq. (3),
; theoretical according to our method, Eq. (9),
value were in good agreement at high frequencies above 250Hz (Figure 1). However, the discrepancy
becomes greater at lower frequency below 250Hz. On the other hand, the computed value using Eq. (2)
and (3) from Elmallawany agreed very well with the measured one at low frequencies below 315Hz. But
the deviation in the range of 400 to 2.5kHz as wellknown as the mass control region cannot be neg
lected. The above results can be considered to be caused by the applicable range of the correction
factor according to Eq. (3). This range is limited below 500Hz for the measured wall.
The value calculated by using Eq. (8) and (9), due to the present improved coupling loss factor,
predicts the transmission loss with high accuracy as shown in Figure 2- (a). Comparing the results
shown in Figure 2- (b), in which the dimensions of wall are 2,468mm and 2,377mm in computing, we
found that our method can also estimate the effect of a stud. The applicable range of frequency using
Eq. (9) is lower than 4kHz for the measured wall.
Prediction of Sound Transmission Loss of Single Leaf Walls due to an Improvement of the SEA Method 123
Figure 3 and 4 show the results for 1.2mm thick aluminium panel and 1.5mm thick steel panel, in
order. The solid and broken lines show the calculated values by using our method and Elmallawany's
method, respectively. The applicable range of each method is equal to the 3.2mm thick aluminium
panel.
As shown in Figures 1-4, the calculated values by using Eq. (8) and(9), our method, agree accur
ately with the measured values except a few low frequencies below 250Hz. However, it has to be men
tioned that the Elmallawany's method agrees much better below 250Hz.
Kuga9 has carried out the measurement for many materials. As the system of measurement was
described in detail in Reference 9, our method was applied to the five kinds of glasses which the
edges has been clamped by the wood frame with the soft rubber 5mm thick on the both sides. The
effective dimensions for sound transmission are 1,470mm and 860mm except the area of wood frame.
In computing based on the SEA method, the volumes of both source and receiving rooms were the
same value (125m3), and the reverberation time of receiving room took the value in Table 1 of Refer
ence 9, when the standardized test concrete panel (using mortared concrete block) was mounted, for
every glasses commonly. The sound velocity in air took the value at normal temperature and the edge
condition was assumed to be fixed. The two values of the internal loss factor of glasses in calculating
employed 0.01 and 0.03.
In Figure 5, the comparisons of the theoretical values and the experimental values are shown.
The three kinds of lines for the calculated values indicated the random incidence mass law (—•—),
Elmallawany's method (
, applicable range^800Hz), and our method (
, applicable
ranged 6kHz).
It is found that the calculated values are varied greatly by the internal loss factor used. It is
mentioned that the theoretical value using Elmallawany's method is nearly equal to the value using our
method at low frequencies < 800Hz. Above the critical coincidence frequency, the theoretical value
using the internal loss factor r/int = 0.03 is in good agreement with the TL of each glass. Below the
critical coincidence frequency, while the computed values of the 3mm thick glass using rjint = 0.01
agree well, but the used r/[nt value increased in proportion to the thickness of glass, finally 7int = 0.03
can predict well the TL of the 10mm thick glass. It can be considered that the increasing of the sur
face density causes to lose the more energy at the edge by a relation between the surface density and
the edge condition.
4. Conclusions
It has been shown that the improved SEA method using the present coupling loss factor can pre
dict the sound transmission loss with high accuracy and can estimate also the effect of a stud. Howev
er, an attention has to be paid to the applicable range of frequency and to the value of the internal
loss factor 7int. The deviation between computed and measured values was significantly influenced by
the used rjint value, and the deviation below the critical coincidence frequency fc is more sensitive than
that above jc. It can be presumed that the rjmi value changes due to both the surface density and the
edge condition.
Therefore, in order to obtain the extended coupling loss factor applicable for the wide frequency
range, it will be necessary to study the frequency characteristic of the internal loss factor.
References
1) A. London, "Transmission of Reverberant Sound Through Single Walls," Research Paper RP1908,
124
SKft*^X^SPef5E*^
H 28 -§- (1986)
J. Res. Nat. Bur. Stand. , 42, 605, 1949.
2) M. J. Crocker and A. J. Price, "Sound transmission using statistical energy analysis," J. Sound
Vib.% 469, 1969.
3) R. J. Donato, "Sound Transmission through a Double-Leaf Wall," J. Acoust. Soc. Am. , 51, 807,
1972.
4) T. Kawai, "Sound Transmission through a Single Partition-Normal Incidence on a Circular
Plate-," J. Acoust. Soc. Jpn. , 29, 186, 1973. (in Japanese)
5) H. Sato, "On the Mechanism of Outdoor Noise Transmission through Walls and Windows -A mod
ification of infinite wall theory with respect to radiation of transmitted wave—," J. Acoust. Soc.
Jpn. , 29, 509, 1973. (in Japanese)
6) S. Kuroki, "Noise reduction under the control of sound incident—angle to single-leaf wall," Sum
maries of Technical Papers of Annual Meeting, A. I. J. , Part D, 195, 1985. (in Japanese)
7) A. Elmallawany, "Improvement of the Method of Statistical Energy Analysis for the Calculation of
Sound Insulation at Low Frequencies," Applied Acoustics, 15, 341—345, 1982.
8) M. J. Crocker and F. M. Kessler, Noise and Noise Control Vol. II (CRC Press, Florida, 1982), p.
74-.
9) S. Kuga, "ON THE SOUND TRANSMISSION LOSS OF SINGLE GLASSES, BI-GLASSES,
PAIR GLASSES, SINGLE WINDOWS AND DOUBLE WINDOWS," Trans, of A. I. J. , 96, 36,
1964. (in Japanese)