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Absorption of sound by porous material I
P h y s i c a V I I I , no 2
ABSORPTION
F e b r u a r i 1941
OF SOUND BY POROUS MATERIAL
I
by J. VAN DEN E I J K and C. Z W I K K E R Laboratorium voor Technisehe Physiea. Technische Hoogeschool, Delft
Zusammenfassung Von einer Holzfaserplatte und yon einer P l a t t e aus a k u s t i s c h e m P u t z s i n d die P o r o s i t t i t u n d L u f t w i d e r s t a n d g e m e s s e n . Mit H i l f e d i e s e r Messw e r t e s i n d u n t e r B e n u t z u n g t t l t e r e r T h e o r i e n die zu e r w a r t e n d e a k u s t i s c h e Schluckzahlen berechnet. A n d e r e r s e i t s s i n d diese S c h l u c k z a h l e n d i r e k t g e m e s s e n ; ftir s e n k r e c h t e n E i n f a l l u n t e r B e n u t z u n g e i n e r I n t e r f e r e n z m e t h o d e u n d ffir a l l s e i t i g e n Einfall mittels einer Nachhallmethode mit konstanter Tonh6he. Die A b b . 5, 7 u n d 8 zeigen in wie w e i t f 2 1 b e r e i n s t i m m u n g b e s t e h t zwis c h e n T h e o r i e u n d E x p e r i m e n t . O f f e n b a r t r i f f t die T h e o r i e in u n g e n f i g e n d e r W e i s e zu.
§ 1. More t h a n once already one has derived mathematically a formula for the absorption coefficient of porous walls, expressing its dependence upon the porosity, the angle of incidence, the specific resistance of the air and the thickness of the material 1). Most complete of all in this respect is M o n n a's work, in which, moreover, the other publications are criticized. The formula, derived by M o n n a , is rather complicated. The quantities introduced by him to describe the absorption are, apart from the thickness and the angle of incidence, which are defined in the usual way, the cavityfactor h, being the percentage of the volume occupied by air, and the specific resistance ~ of the porous material for a steady flow of air (therefore, - - grad p = a v). In order to make a comparison possible between the theoretical results obtained and the values for the absorptioncofifficient following from measurements on the reverberation, it was necessary to compute from the values of the absorptionco~fficient a~ belonging to the variable angle of incidence e, the average value d by means of - -
Physica \: I I I
1 4 9
- -
I 0"*
150
j . VAN D'i~N E I J K AND C. Z W I K K E R
the formula ,a-/2
f a~ 2r~ sin ~. cos c~d d = o ~r/2
f 2re sin ~ cos ~ d ~
7r/2 = 2f%
sin ~ cos ~ d c,..
0
0
Owing to the fact t h a t a~ is a complicated function of cos ~., it was not possible to calculate d a c c u r a t e l y b y an integration. Taking into account, however, t h a t the weight-factor sin ~. cos ~ has a m a x i m u m value for ~ = 45 °, one can assume as a first a p p r o x i m a t i o n d = a4s. We have tried to t e s t . M o n n a's results experimentally. To that end, we have d e t e r m i n e d directly the cavity-factor h and the specific resistance for air a of a few materials and substituted the values, found in this way, in M o n n a's formulae for a o and a45, we have then e x a m i n e d w h e t h e r the values, c o m p u t e d in this m a n n e r agree with the values of the absorptionco~fficient, d e t e r m i n e d acoustically for normal incidence and for incidence from all directions respectively. §2. Availing ourselves of B o y l e ' s law: p l V l = P2V2, we d e t e r m i n e d the c a v i t y - f a c t o r h by means of the a p p a r a t u s shown in fig. I. The specimen of known e x t e r n a l volume was placed in the vessel of volume Vo. In order to get rid of a n y w a t e r kept back b y capillary forces, this vessel was a few times e v a c u a t e d and refilled with d r y air. The increase of volume was effected b y opening tap F and b y introducing a few of the spheres of volume V ' etc., which was done b y lowering vessel B. The change of the pressure was measured with the paraffin m a n o m e t e r E. Meanwhile the level at C was k e p t at the same heigth, with a view to eliminate a n y changes of volume in the m a n o m e t e r . F o r t h a t reason tube D was applied. The accuracy of the m e a s u r e m e n t s was a m p l y sufficient ; we were able to measure h as far as two decimal places. In the above way only those pores, c o m m u n i c a t i n g with t h e air outside are measured, but those are precisely the ones, with which we are concerned. § 3. We measured the specific resistance of the air with the aid of an a r r a n g e m e n t , given in outline in fig. 2. On the specimen A is placed a box B in which, b y partially pumping away the air, an u n d e r - a t m o s p h e r i c pressure is created. This pressure is m e a s u r e d
ABSORPTION OF SOUND BY POROUS MATERIAL
15 1
with a F u e s s-micromanometer (D in fig. 2). The velocity of the current is measured by a current-meter C, 'which is calibrated beforehand. f
(J (, "~
E
('
ii jl t
I
Fig. 1. A p p a r a t u s for m e a s u r i n g t h e c a v i t y f a c t o r .
In order to p r e v e n t any radial currents in the specimen box B is surrounded b y a wider box F, in which the pressure is kept at exactly the same value as in ]3. This equality was checked b y means of the drop of alcohol H, which was to remain at rest inside the capillary J and the iinmobility of which was observed with a microscope. Box P Physica VIII
10"
152
j. VAN DEN E I J K AND C. ZWIKKER
was fastened along the edge on the specimen A by means of plaster of Paris. The under-pressure applied was of the order of 2 to 0,02 mm H20; the pressures in B and F could be adjusted to the same value with sufficient accuracy, as appeared from the smallness of the spreading in the measuring points. The measured values for the air-resistance per cm a were of the order of 10000 dynes sec/cm 4 for a certain kind of woodfibre plates and for plates of acoustic plaster they were of the order of 100 dynes sec/cm 4.
"
3
G
Fig. 2. A p p a r a t u s for m e a s u r i n g t h e p e r m e a b i l i t y .
The air-resistance turned out to be indeed independent of the value used for the under-pressure. Some caution was necessary in connection with the watercontent of the specimen. Immediately after the plaster of Paris was applied, the results obtained for woodfibre plates appeared to be irregular; this must in all probability be ascribed to the swelling of the material under the influence of its increased watercontent. § 4. The measurements of the absorptionco~fficient for normal incidence were carried out according to the method of stationary waves 2), the sound pressure in the wave-system being measured in the places of the maxima and minima. The area of the specimina used was 400 cm ~. The details of our arrangement will be published elsewhere. For the measurements of the average absorptionco~fficient
ABSORPTION OF SOUND BY POROUS MATERIAL
153
specimina were used of 5,5 m 2, placed in a reverberation-chamber of 164,5 m a. We performed series of experiments ~t constant frequencies and during the emission of sound the loudspeakers were revolving, while the microphone was moved in the reverberation-chamber In order to average over the places of nodes and anti-nodes the reverberation time was determined at each frequency for 100 different positions of loudspeaker and microphone. The arrangement started and stopped working automatically, a counting mechanism gave us the number of reverberation-periods and the added-up time of reverberation was read from an electrical clockwork. The measuring arrangement is shown in its general outline in fig. 3. i I I
Q
Fig. 3. S e t u p for t h e m e a s u r i n g of t h e r e v e r b e r a t i o n t i m e .
The output of a soundgenerator reaches after due amplification via a relay the revolving loudspeakers, mounted in the reverberationchamber. After the loudspeakercircuit is opened, the intensity of the sound in the chamber begins to decrease, thereby causing a decrease in the amplified and rectified voltage of the microphone. At a certain pre-determined value of the voltage the clockwork is set going, and at a lower voltage, also pre-determined, it stops, while at the same time the catch of the counting-mechanism moves and the loudspeakers are switched in again. Fig. 4 shows the relay-arrangement in more detail. 1 and 2 a r e thyratrons. The three discs in the figure are in reality only one disc, set into a sliding motion by a small motor (not shown). With the lowering of the sound level in the reverberation chamber the rectified tension is also lowered, whereby thyratron I comes into action. The plate-current operates relay A, the clockwork is set going and the grid-circuit of 1 is opened, so that no grid-current can occur. A few moments later 2 comes into action, by which relay B is operated, the grid-circuit of 2 is opened, the clockwork stops, the counting mechanism gives a jerk and the catch is pulled out of the
154
j . VAN DEN E I J K AND C. ZWIKKER
n o t c h i n the disc, so t h a t the l a t t e r no longer slides b u t starts revolving in the indicated direction. F r o m this it follows t h a t the plate circuits of the t h y r a t r o n s are opened and t h a t the relays A and B resume their rest positions. The catch t h e n falls back on the disc, so t h a t this is s t o p p e d after one revolution. R e l a y C is t h e n operated, b y which t h e loudspeaker circuit is closed. A f t e r one revolution of the disc the rectified tension is once m o r e a m a x i m u m ; shortly a f t e r the disc has been caught, the loudspeaker circuit, therefore, has been o p e n e d and tension once again exists on the plates of the t h y r a t r o n s , t h y r a t r o n 1 comes into action etc.
a
b
A
°
l 0,25 M
I00 V
T
Fig. 4. R e l a y a r r a n g e m e n t as u s e d in m e a s u r i n g t h e r e v e r b e r a t i o n t i m e .
§ 5. Measurements on wood/ibreplates. Measurements were performed on woodifbre plates of a thickness of 12 ram. F o r the average result of a great m a n y m e a s u r e m e n t s on this material we found : c a v i t y f a c t o r h = 0,80 air-resistance, per cm a, ~ = 6000 d y n e sec/cm 4. On substituting these values in M o n n a's formulae our compu-
155
ABSORPTION OF SOUND BY POROUS MATERIAL
tation of the absorptionco~fficient yields for 500 Hz, in the case of n o r m a l incidence . . . . . . . . . . . in the case of 45 ° incidence . . . . . . . . . . . .
a 0 = 5 °/o a,s = 70/0 •
These values are considerably lower t h a n those generally found from direct m e a s u r e m e n t s of the absorption. As is well-known it is not the same w h e t h e r the plates can v i b r a t e freely over their full e x t e n t or not, b u t also for plates stuck with their entire area on a h a r d backwall, values for the absorptionco~fficients have been measured, lying between 15% and 200/0 . Our own m e a s u r e m e n t s yield results for this m a t e r i a l of which the comparison with M o n n a's theoretical values is shown in fig. 5.
30~ a
e THEORY a o
I 20
A ,,
o 400 °
a~s
250
I
SO0
t
r
t000
2000 HZ FREQ.
Fig. 5. C o m p a r i s o n of t h e m e a s u r e d a b s o r p t i o n c o e f f i c i e n t s (full line) o f a w o o d f i b r e p l a t e w i t h t h e t l l e o r e t i c a l p r e d i c t i o n of M o n n a.
We have e x a m i n e d w h e t h e r a n y possible inhomogeneities of the material might be the cause of the discrepancy between the measured and the c o m p u t e d absorptionco~fficients. These inhomogeneities are of a twofold nature. In the first place the pores are not, all of them, of the same size and in the second place it m a y be t h a t the o u t e r layers of the plates are more tightly compressed t h a n their inner parts, owing to the way in which the plates are made. If the spreading in the d i a m e t e r of the pores be such, t h a t a few large openings occur, its influence on the result will be especially noticeable, for t h e n the measured value for the resistance of the air will Come out too small. This means t h a t we should have to substitute in M o n n a's formulae a higher value for a than the measured one. This, however, would lead to a still smaller value for the comp u t e d a (cf. fig. 6), so that the discrepancy would only become still more pronounced.
156
J. VANDEN'EIJKANDC. ZWIKKER
T h e assumption of a h a r d surface-layer leads also in the w r o n g direction. F o r the outer layer will have a g r e a t e r influence on the absorption of sound t h a n the i n n e r parts. In our d e t e r m i n a t i o n , is averaged o v e r the entire thickness of the plate and this value is lower t h a n the one for a more rigid top layer, if t h a t should be present. A n y possible correction would, therefore, consist in introducing an increased value for , in M o n n a's formulae and this, as is clear from fig. 6, leads in the wrong direction.
,oo'~
T.Eo~Y ~-o,.¢,.
a*e5BO
I
'
I
1 f
~
l .°~o r 40
50
E 400
~ 500
4000
5000
40 000
Fig. 6. The absorption coefficient for a 45 degree attgle of incidence as a function of the specific air-resistance or, frequency v and plate thickness l for a cavity factor 0,80 according to M o n n a. F r o m the foregoing we draw, therefore, the conclusion t h a t the damping due to porosity, as c o m p u t e d b y M o n n a and others, is in the case of woodfibre plates not the most i m p o r t a n t one, but t h a t there must exist, besides, o t h e r effects, to which at least an equal a m o u n t of damping must be ascribed.
§ 6. ~/Ieasurements on porous plaster. We made some porous plates of the kind, commercially known as ,,Sorbolite", thickness 3 cm. The average value of the Cavity-factor h t u r n e d out to be 0,63, while the average result of a n u m b e r of determinations of the specific airresistance was 5] d y n e sec/cm 4. The value of ao, c o m p u t e d with these values according to M o n n a's formula is c o m p a r e d below (see fig. 7) with the value measured b y us according to the interference method. The average a was measured on specimina of a thickness of 1,5 cm and for a great m a n y frequencies according to the r e v e r b e r a t i o n method. The little circles in fig. 8 indicate the theoretical e x p e c t a t i o n
ABSORPTION OF SOUND BY POROUS MATERIAL
157
according to M o n n a's f o r m u l a e w i t h h = 0,63 a n d ~ = 51 cgs for e = 45 °. T h e a g r e e m e n t b e t w e e n the c o m p u t e d and m e a s u r e d t00~
SORBOLIET 4cm h=0,63
o THEORY ao 80
a
2. x
m
~
' o=Stcg~
I~ . ~ "
7"-,.
. m
a
6O
40
20
SO0
250
400
4000 ~-
Hz
FREQ.
Fig. 7. C o m p a r i s o n of t h e m e a s u r e d a b s o r p t i o n c o e f f i c i e n t s (full d r a w n c u r v e s ) of a n a c o u s t i c a l p l a s t e r w i t h t h e t h e o r e t i c a l p r e d i c t i o n of M o n n a's. N o r m a l i n c i d e n c e .
Ioo% o = THEORY d = 5 0 cgs h = 0,6
8O
= 03'5"
( = '1,5 c m
6O
40
/
20
J
0
I
n
100
250
500
I000
2000
~-OOO Hz
FREQ.
Fig. 8. Comparison of theory and experiment for an acoustical plaster. The absorption coefficient resulting from reverberation-measurements is compared with the theoretical a-value for a 45 degree angle of incidence. values is s a t i s f a c t o r y for the higher frequencies. F o r lower frequencies this is no longer true and this t i m e again the c o m p u t e d values are considerably lower. As is well-known, m e a s u r i n g of the a b s o r p -
158
A B S O R P T I O N OF S O U N D B Y P O R O U S M A T E R I A L
tionco~fficient according to the reverberation-method is liable to show a wide spreading in the results especially at lower frequencies, but even after having duly reckoned with this fact, there remains, none the less, a pronounced difference between the results. In a second article we hope to enter into the cause of the discrepancy between theory and measurements. R e c e i v e d D e c e m b e r 3rd 1940.
I~I£FERENCES 1) R a y l e i g h , T h e o r y of Sound 1I, p. 3 2 9 ; C . Z w i k k e r , l)e I n g e n i e u r 4 7 , A'177, 1932;V. l ( f i h'l und E. M e y e r , U n t e r s u e h u n g e n i i b e r d i e \ . V i n k e l - u n d F r e q u e n z a b h ~ n g i g k e i t der S c h a l l s c h l u c k u n g you por6sen M a t e r i a l i e n . M i t t e i l u n g aus d e m H e i n r i c h H e r t z I n s t i t u t for S e h w i n g u n g s f o r s c h u n g , S-B. A k a d . Wiss. Berlin, phys. m a t h . NI. 1932; A. F. M o n n a , P h y s i c a ~i, 129, 1938. 2) For v a r i o u s i n s t r u m e n t a l e m b o d i t u e n t s see D a x ' i s : Modern Acoustics, p. 211, London, 1934; For t h e o r y see W e n t e a n d 13 e d e 11: The Bell S y s t e m T e c h n i c a l Journal 7, 1 - - 1 0 , 1928.
ABSORPTION
F e b r u a r i 1941
OF SOUND BY POROUS MATERIAL
I
by J. VAN DEN E I J K and C. Z W I K K E R Laboratorium voor Technisehe Physiea. Technische Hoogeschool, Delft
Zusammenfassung Von einer Holzfaserplatte und yon einer P l a t t e aus a k u s t i s c h e m P u t z s i n d die P o r o s i t t i t u n d L u f t w i d e r s t a n d g e m e s s e n . Mit H i l f e d i e s e r Messw e r t e s i n d u n t e r B e n u t z u n g t t l t e r e r T h e o r i e n die zu e r w a r t e n d e a k u s t i s c h e Schluckzahlen berechnet. A n d e r e r s e i t s s i n d diese S c h l u c k z a h l e n d i r e k t g e m e s s e n ; ftir s e n k r e c h t e n E i n f a l l u n t e r B e n u t z u n g e i n e r I n t e r f e r e n z m e t h o d e u n d ffir a l l s e i t i g e n Einfall mittels einer Nachhallmethode mit konstanter Tonh6he. Die A b b . 5, 7 u n d 8 zeigen in wie w e i t f 2 1 b e r e i n s t i m m u n g b e s t e h t zwis c h e n T h e o r i e u n d E x p e r i m e n t . O f f e n b a r t r i f f t die T h e o r i e in u n g e n f i g e n d e r W e i s e zu.
§ 1. More t h a n once already one has derived mathematically a formula for the absorption coefficient of porous walls, expressing its dependence upon the porosity, the angle of incidence, the specific resistance of the air and the thickness of the material 1). Most complete of all in this respect is M o n n a's work, in which, moreover, the other publications are criticized. The formula, derived by M o n n a , is rather complicated. The quantities introduced by him to describe the absorption are, apart from the thickness and the angle of incidence, which are defined in the usual way, the cavityfactor h, being the percentage of the volume occupied by air, and the specific resistance ~ of the porous material for a steady flow of air (therefore, - - grad p = a v). In order to make a comparison possible between the theoretical results obtained and the values for the absorptioncofifficient following from measurements on the reverberation, it was necessary to compute from the values of the absorptionco~fficient a~ belonging to the variable angle of incidence e, the average value d by means of - -
Physica \: I I I
1 4 9
- -
I 0"*
150
j . VAN D'i~N E I J K AND C. Z W I K K E R
the formula ,a-/2
f a~ 2r~ sin ~. cos c~d d = o ~r/2
f 2re sin ~ cos ~ d ~
7r/2 = 2f%
sin ~ cos ~ d c,..
0
0
Owing to the fact t h a t a~ is a complicated function of cos ~., it was not possible to calculate d a c c u r a t e l y b y an integration. Taking into account, however, t h a t the weight-factor sin ~. cos ~ has a m a x i m u m value for ~ = 45 °, one can assume as a first a p p r o x i m a t i o n d = a4s. We have tried to t e s t . M o n n a's results experimentally. To that end, we have d e t e r m i n e d directly the cavity-factor h and the specific resistance for air a of a few materials and substituted the values, found in this way, in M o n n a's formulae for a o and a45, we have then e x a m i n e d w h e t h e r the values, c o m p u t e d in this m a n n e r agree with the values of the absorptionco~fficient, d e t e r m i n e d acoustically for normal incidence and for incidence from all directions respectively. §2. Availing ourselves of B o y l e ' s law: p l V l = P2V2, we d e t e r m i n e d the c a v i t y - f a c t o r h by means of the a p p a r a t u s shown in fig. I. The specimen of known e x t e r n a l volume was placed in the vessel of volume Vo. In order to get rid of a n y w a t e r kept back b y capillary forces, this vessel was a few times e v a c u a t e d and refilled with d r y air. The increase of volume was effected b y opening tap F and b y introducing a few of the spheres of volume V ' etc., which was done b y lowering vessel B. The change of the pressure was measured with the paraffin m a n o m e t e r E. Meanwhile the level at C was k e p t at the same heigth, with a view to eliminate a n y changes of volume in the m a n o m e t e r . F o r t h a t reason tube D was applied. The accuracy of the m e a s u r e m e n t s was a m p l y sufficient ; we were able to measure h as far as two decimal places. In the above way only those pores, c o m m u n i c a t i n g with t h e air outside are measured, but those are precisely the ones, with which we are concerned. § 3. We measured the specific resistance of the air with the aid of an a r r a n g e m e n t , given in outline in fig. 2. On the specimen A is placed a box B in which, b y partially pumping away the air, an u n d e r - a t m o s p h e r i c pressure is created. This pressure is m e a s u r e d
ABSORPTION OF SOUND BY POROUS MATERIAL
15 1
with a F u e s s-micromanometer (D in fig. 2). The velocity of the current is measured by a current-meter C, 'which is calibrated beforehand. f
(J (, "~
E
('
ii jl t
I
Fig. 1. A p p a r a t u s for m e a s u r i n g t h e c a v i t y f a c t o r .
In order to p r e v e n t any radial currents in the specimen box B is surrounded b y a wider box F, in which the pressure is kept at exactly the same value as in ]3. This equality was checked b y means of the drop of alcohol H, which was to remain at rest inside the capillary J and the iinmobility of which was observed with a microscope. Box P Physica VIII
10"
152
j. VAN DEN E I J K AND C. ZWIKKER
was fastened along the edge on the specimen A by means of plaster of Paris. The under-pressure applied was of the order of 2 to 0,02 mm H20; the pressures in B and F could be adjusted to the same value with sufficient accuracy, as appeared from the smallness of the spreading in the measuring points. The measured values for the air-resistance per cm a were of the order of 10000 dynes sec/cm 4 for a certain kind of woodfibre plates and for plates of acoustic plaster they were of the order of 100 dynes sec/cm 4.
"
3
G
Fig. 2. A p p a r a t u s for m e a s u r i n g t h e p e r m e a b i l i t y .
The air-resistance turned out to be indeed independent of the value used for the under-pressure. Some caution was necessary in connection with the watercontent of the specimen. Immediately after the plaster of Paris was applied, the results obtained for woodfibre plates appeared to be irregular; this must in all probability be ascribed to the swelling of the material under the influence of its increased watercontent. § 4. The measurements of the absorptionco~fficient for normal incidence were carried out according to the method of stationary waves 2), the sound pressure in the wave-system being measured in the places of the maxima and minima. The area of the specimina used was 400 cm ~. The details of our arrangement will be published elsewhere. For the measurements of the average absorptionco~fficient
ABSORPTION OF SOUND BY POROUS MATERIAL
153
specimina were used of 5,5 m 2, placed in a reverberation-chamber of 164,5 m a. We performed series of experiments ~t constant frequencies and during the emission of sound the loudspeakers were revolving, while the microphone was moved in the reverberation-chamber In order to average over the places of nodes and anti-nodes the reverberation time was determined at each frequency for 100 different positions of loudspeaker and microphone. The arrangement started and stopped working automatically, a counting mechanism gave us the number of reverberation-periods and the added-up time of reverberation was read from an electrical clockwork. The measuring arrangement is shown in its general outline in fig. 3. i I I
Q
Fig. 3. S e t u p for t h e m e a s u r i n g of t h e r e v e r b e r a t i o n t i m e .
The output of a soundgenerator reaches after due amplification via a relay the revolving loudspeakers, mounted in the reverberationchamber. After the loudspeakercircuit is opened, the intensity of the sound in the chamber begins to decrease, thereby causing a decrease in the amplified and rectified voltage of the microphone. At a certain pre-determined value of the voltage the clockwork is set going, and at a lower voltage, also pre-determined, it stops, while at the same time the catch of the counting-mechanism moves and the loudspeakers are switched in again. Fig. 4 shows the relay-arrangement in more detail. 1 and 2 a r e thyratrons. The three discs in the figure are in reality only one disc, set into a sliding motion by a small motor (not shown). With the lowering of the sound level in the reverberation chamber the rectified tension is also lowered, whereby thyratron I comes into action. The plate-current operates relay A, the clockwork is set going and the grid-circuit of 1 is opened, so that no grid-current can occur. A few moments later 2 comes into action, by which relay B is operated, the grid-circuit of 2 is opened, the clockwork stops, the counting mechanism gives a jerk and the catch is pulled out of the
154
j . VAN DEN E I J K AND C. ZWIKKER
n o t c h i n the disc, so t h a t the l a t t e r no longer slides b u t starts revolving in the indicated direction. F r o m this it follows t h a t the plate circuits of the t h y r a t r o n s are opened and t h a t the relays A and B resume their rest positions. The catch t h e n falls back on the disc, so t h a t this is s t o p p e d after one revolution. R e l a y C is t h e n operated, b y which t h e loudspeaker circuit is closed. A f t e r one revolution of the disc the rectified tension is once m o r e a m a x i m u m ; shortly a f t e r the disc has been caught, the loudspeaker circuit, therefore, has been o p e n e d and tension once again exists on the plates of the t h y r a t r o n s , t h y r a t r o n 1 comes into action etc.
a
b
A
°
l 0,25 M
I00 V
T
Fig. 4. R e l a y a r r a n g e m e n t as u s e d in m e a s u r i n g t h e r e v e r b e r a t i o n t i m e .
§ 5. Measurements on wood/ibreplates. Measurements were performed on woodifbre plates of a thickness of 12 ram. F o r the average result of a great m a n y m e a s u r e m e n t s on this material we found : c a v i t y f a c t o r h = 0,80 air-resistance, per cm a, ~ = 6000 d y n e sec/cm 4. On substituting these values in M o n n a's formulae our compu-
155
ABSORPTION OF SOUND BY POROUS MATERIAL
tation of the absorptionco~fficient yields for 500 Hz, in the case of n o r m a l incidence . . . . . . . . . . . in the case of 45 ° incidence . . . . . . . . . . . .
a 0 = 5 °/o a,s = 70/0 •
These values are considerably lower t h a n those generally found from direct m e a s u r e m e n t s of the absorption. As is well-known it is not the same w h e t h e r the plates can v i b r a t e freely over their full e x t e n t or not, b u t also for plates stuck with their entire area on a h a r d backwall, values for the absorptionco~fficients have been measured, lying between 15% and 200/0 . Our own m e a s u r e m e n t s yield results for this m a t e r i a l of which the comparison with M o n n a's theoretical values is shown in fig. 5.
30~ a
e THEORY a o
I 20
A ,,
o 400 °
a~s
250
I
SO0
t
r
t000
2000 HZ FREQ.
Fig. 5. C o m p a r i s o n of t h e m e a s u r e d a b s o r p t i o n c o e f f i c i e n t s (full line) o f a w o o d f i b r e p l a t e w i t h t h e t l l e o r e t i c a l p r e d i c t i o n of M o n n a.
We have e x a m i n e d w h e t h e r a n y possible inhomogeneities of the material might be the cause of the discrepancy between the measured and the c o m p u t e d absorptionco~fficients. These inhomogeneities are of a twofold nature. In the first place the pores are not, all of them, of the same size and in the second place it m a y be t h a t the o u t e r layers of the plates are more tightly compressed t h a n their inner parts, owing to the way in which the plates are made. If the spreading in the d i a m e t e r of the pores be such, t h a t a few large openings occur, its influence on the result will be especially noticeable, for t h e n the measured value for the resistance of the air will Come out too small. This means t h a t we should have to substitute in M o n n a's formulae a higher value for a than the measured one. This, however, would lead to a still smaller value for the comp u t e d a (cf. fig. 6), so that the discrepancy would only become still more pronounced.
156
J. VANDEN'EIJKANDC. ZWIKKER
T h e assumption of a h a r d surface-layer leads also in the w r o n g direction. F o r the outer layer will have a g r e a t e r influence on the absorption of sound t h a n the i n n e r parts. In our d e t e r m i n a t i o n , is averaged o v e r the entire thickness of the plate and this value is lower t h a n the one for a more rigid top layer, if t h a t should be present. A n y possible correction would, therefore, consist in introducing an increased value for , in M o n n a's formulae and this, as is clear from fig. 6, leads in the wrong direction.
,oo'~
T.Eo~Y ~-o,.¢,.
a*e5BO
I
'
I
1 f
~
l .°~o r 40
50
E 400
~ 500
4000
5000
40 000
Fig. 6. The absorption coefficient for a 45 degree attgle of incidence as a function of the specific air-resistance or, frequency v and plate thickness l for a cavity factor 0,80 according to M o n n a. F r o m the foregoing we draw, therefore, the conclusion t h a t the damping due to porosity, as c o m p u t e d b y M o n n a and others, is in the case of woodfibre plates not the most i m p o r t a n t one, but t h a t there must exist, besides, o t h e r effects, to which at least an equal a m o u n t of damping must be ascribed.
§ 6. ~/Ieasurements on porous plaster. We made some porous plates of the kind, commercially known as ,,Sorbolite", thickness 3 cm. The average value of the Cavity-factor h t u r n e d out to be 0,63, while the average result of a n u m b e r of determinations of the specific airresistance was 5] d y n e sec/cm 4. The value of ao, c o m p u t e d with these values according to M o n n a's formula is c o m p a r e d below (see fig. 7) with the value measured b y us according to the interference method. The average a was measured on specimina of a thickness of 1,5 cm and for a great m a n y frequencies according to the r e v e r b e r a t i o n method. The little circles in fig. 8 indicate the theoretical e x p e c t a t i o n
ABSORPTION OF SOUND BY POROUS MATERIAL
157
according to M o n n a's f o r m u l a e w i t h h = 0,63 a n d ~ = 51 cgs for e = 45 °. T h e a g r e e m e n t b e t w e e n the c o m p u t e d and m e a s u r e d t00~
SORBOLIET 4cm h=0,63
o THEORY ao 80
a
2. x
m
~
' o=Stcg~
I~ . ~ "
7"-,.
. m
a
6O
40
20
SO0
250
400
4000 ~-
Hz
FREQ.
Fig. 7. C o m p a r i s o n of t h e m e a s u r e d a b s o r p t i o n c o e f f i c i e n t s (full d r a w n c u r v e s ) of a n a c o u s t i c a l p l a s t e r w i t h t h e t h e o r e t i c a l p r e d i c t i o n of M o n n a's. N o r m a l i n c i d e n c e .
Ioo% o = THEORY d = 5 0 cgs h = 0,6
8O
= 03'5"
( = '1,5 c m
6O
40
/
20
J
0
I
n
100
250
500
I000
2000
~-OOO Hz
FREQ.
Fig. 8. Comparison of theory and experiment for an acoustical plaster. The absorption coefficient resulting from reverberation-measurements is compared with the theoretical a-value for a 45 degree angle of incidence. values is s a t i s f a c t o r y for the higher frequencies. F o r lower frequencies this is no longer true and this t i m e again the c o m p u t e d values are considerably lower. As is well-known, m e a s u r i n g of the a b s o r p -
158
A B S O R P T I O N OF S O U N D B Y P O R O U S M A T E R I A L
tionco~fficient according to the reverberation-method is liable to show a wide spreading in the results especially at lower frequencies, but even after having duly reckoned with this fact, there remains, none the less, a pronounced difference between the results. In a second article we hope to enter into the cause of the discrepancy between theory and measurements. R e c e i v e d D e c e m b e r 3rd 1940.
I~I£FERENCES 1) R a y l e i g h , T h e o r y of Sound 1I, p. 3 2 9 ; C . Z w i k k e r , l)e I n g e n i e u r 4 7 , A'177, 1932;V. l ( f i h'l und E. M e y e r , U n t e r s u e h u n g e n i i b e r d i e \ . V i n k e l - u n d F r e q u e n z a b h ~ n g i g k e i t der S c h a l l s c h l u c k u n g you por6sen M a t e r i a l i e n . M i t t e i l u n g aus d e m H e i n r i c h H e r t z I n s t i t u t for S e h w i n g u n g s f o r s c h u n g , S-B. A k a d . Wiss. Berlin, phys. m a t h . NI. 1932; A. F. M o n n a , P h y s i c a ~i, 129, 1938. 2) For v a r i o u s i n s t r u m e n t a l e m b o d i t u e n t s see D a x ' i s : Modern Acoustics, p. 211, London, 1934; For t h e o r y see W e n t e a n d 13 e d e 11: The Bell S y s t e m T e c h n i c a l Journal 7, 1 - - 1 0 , 1928.