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Absorption of sound by porous material v
Physica X, no 4
April 1943.
A B S O R P T I O N OF SOUND BY POROUS MATERIAL V b y C. Z W I K K E R ,
J. V A N D E N E I J K a n d C. W. K O S T E N
I . a b o r a t o r i u m v o o r T e c h n i s c h e P h y s i c a , T e c h n i s c h e H o o g e s c h o o l , Delft
Zusammenfassung lm zweiten Teile wurde eine theoretische Formel fiJr den akustischen Scheinwiderstand' yon por6sen Schallschluckern gegeben, in welche Forreel als Materialkonstanten die PorBsitAt h, der Strukturfaktor k und der Luftwiderstand a eingingen. Messungen zur Pr[ifung dieser Formel werden beschrieben. Die Schallschlucker waren Proben aus Glasr6hrchen yon Durchmesser 1,6, 4,0 und 9,4 m m und eine Probe aus durchl6cherten, in einiger Entfernung yon einander Iixierten Brettern. Eine vorztigliche Obereinstimmung zwischen die theoretisch und experimentell gefundenen k-Werte wurde festgestellt. Der gemessene ( d y n a m i s c h e ) a - W e r t war ffir die Glasproben etwa 1,53 real der H e 1 m h o 1 t z'sche, also etwa gleich dem K i r c h h o f f schen \,Vert. Der gemessene ¢r-Wert fiir die Holzprobe war 2,8 mal der H e I m h o 1 t z'sche Weft, eingerrnassen in f3bereins t i m m u n g mit in der Literatur fiir Resonatoren aufgegebenen "vVerte. § i7. Introduction. I n n u m b e r [ I t) of this series of p u b l i c a t i o n s a n e w t h e o r y was p r o p o s e d for t h e 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 . A c c o r d i n g to this t h e o r y t h e a c o u s t i c b e h a v i o u r of p o r o u s m a t e r i a l s w i t h a rigid s k e l e t o n c a n be c o m p l e t e l y d e s c r i b e d w i t h the aid of three c o n s t a n t s , viz. : the specific resistance to the air , , t h e c a v i t y - f a c t o r h a n d t h e s t r u c t u r e - f a c t o r k. I n n u m b e r I I I ~) the results were c o m m u n i c a t e d of a p r e l i m i n a r y t e s t i n g of this t h e o r y on s p e c i m i n a of s o u n d - a b s o r b i n g m a t e r i a l , m a d e f r o m l e m o n a d e - s t r a w s a n d f r o m s m a l l glass tubes, p l a c e d e i t h e r at r i g h t angles or at an angle of 60 ° to t h e w a v e - f r o n t . W e shall n o w give t h e results of m o r e d e t a i l e d e x p e r i m e n t s , likewise c a r r i e d o u t on samples, c o m p o s e d of glass tubes. This t i m e we did e m p l o y t h r e e different d i a m e t e r s , n a m e l y 1.6, 4.0 a n d 9.4 m m . F i n a l l y , we describe in § 24 m e a s u r e m e n t s c a r r i e d o u t on a test-piece, w h e r e we h a v e t o deal e x c l u s i v e l y --
239
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C. ZWIKKER, J. VAN DEN E I J K AND C. W. KOSTEN
with ,,pores" provided with lateral cavities. Of this testpiece too, the structure-factor could be computed as well as measured. § 18. The disturbing in/luence o/ the interstices between the glass tubes. In these experiments a complication arose from the presence of the more or less triangular interstices between the tubes, as these act as an (specific) acoustic impedance z2 in parallel with the impe•
-~ ~
,
,,
.
.
.
+S
+40
.
.
::d
:
~1
I
1
I
I
I
-20
-I0
-S
Fig. I. Impedance-curves
for a sample of glass tubes, tested at normal incidence : a) for simple piling with tubes closed at the back with paraffin (curve z) b) for another piling with tubes closed at the front (measurement of
i m p e d a n c e of i n t e r s t i c e s z2). a = a b s o r p t i o n c o e f f i c i e n t , ~ ~ p h a s e j u m p of t h e s o u n d a t t h e f r o n t of t h e s a m p l e , pc = w a v e r e s i s t e n c e of air. F r e q u e n c y of t e s t i n g is n o t e d near the measuring points.
dance zt of the tubes themselves. In the case of the widest of the three types of tubes we were able to fill these interstices with melted paraffin; for the other diameters, however, this was no longer feasible owing to the very large number of very narrow cavities. All tubes re, this time, closed at the back with paraffin, as this proved to be
ABSORPTION OF SOUND BY POROUS ~VIATERIAL
241
necessary in order to avoid secondary effects. This provided us at the same time with the opportunity of measuring z2 separately, namely by piling up the same tubes with their cloSed ends to the flout. This measurement yielded the surprising result that the disturbing parallel impedance was much smaller than we can expect it to be if we imagine the triangular interstices to be replaced by their inscribed circular cylinders or by circular cylinders oi the same Grosssection as the triangular cylinders and if we then apply to these the above-mentioned theory 1). Our supposition that the influence of to
+5
+20
+40
+90 A + 180 - 90
-¢0
-20
Fig. 2. D u p l i c a t i o n m e a s u r e m e n t s , see s u b s c r i p t i o n of fig. I.
this parallel impedance could be neglected ") § 13, has, therefore, turned out to be wrong. At the same time, however, it appeared that, whereas a repeated measurement on one and the same specimen furnished practically the same impedance-curve, a new piling-up led to a value of z2, which differed appreciably from the former one. This is very likely caused by the (although small) deviations of the radii of the tubes from their mean value, so that we do n o t obtain an ideally compact pile of the circular cylinders, but one in which the Physica X
16
242
C. ZWIKKER, J. VAN DEN E I J K AND C. %~r. KOSTEN
intervening spaces are connected with each other here and there. I n a fresh piling up the mutual order of the tubes is changed; this m a y lead to ~ different shape of the volume mentioned and in this way to a different value of z2. This can be clearly seen from fig. 1 and fig. 2, .in which, on the right, the impedance-curves z2 are drawn for two different pilings. On the left in these figures the impedance-curve for the total z is drawn, likewise for two different (other) pilings. We'observe .that for the two pilings equality of this total z holds in the case of resonance, whereas in the case of anti-resonance it does nol. This is caused by the fact that for 400 Hz (resonance) z2 is so large that it does not influence z, appreciably, so that z m zl, whereas for 775 Hz (antiresonance) zt and z2 are of the same order of magnitude. The reader is also referred to fig. 3, in which for these two different pilings the coefficient of absorption, due to the total z only, is plotted against the frequency.
/\ so
/
_
J
,( ,,,]
0
,J
0
f
f
200
400
600
800
Fig. 3. A b s o r p t i o n c o e f f i c i e n t a as a f u n c t i o n of f r e q u e n c y , c o r r e s p o n d i n g t o i m p e d a n c e - c u r v e s z of fig. 1 (crosses) a n d fig. 2 (circles).
§19. Simpli/icaHon o[ the /ormula to be checked. In connection with the above we shall restrict ourselves, as regards the various conclusions to be drawn from our measurements, to the resonancepoint, that is to say, the point in which the impedance-curve intersects the real axis for the first time. The theoretical formula to be checked furnishes, namely, in the special case of first resonance a value, which can be approximated very closely by a very simple expression.
243
A B S O R P T I O N OF SOUND BY P O R O U ~ MATERIAL
The original formula for the impedance was (article II mula 8) z=~
1 ~/k~-
I/
" ~
1--]~p-co cothjcol
Vk - ~ J/;;1
1), for-
°
i kpco
I11 our measurements the values of the quantity a/koo~ turn out to be so small that we can in all cases replace x/l--]~/kpo~ by 1 - - j,/2kpo~. As a/K/p = c, where c is the velocity of propagation of sound in free air, we can now write
z/pc-
x/~ (l - -
i./2kOo~) coth io~ cl Vk (1 - - j./2kpo~)
which for resonance transforms in close approximation into
(z/pc) ....
-
h
" 4
"kp~,.
Here, p is known, k and h are obtained from the dimensions of the component parts of the specimen, while the angular frequency o~,c, and likewise (z/pc),,s follow from the impedance-measurements. In all cases to be discussed the value for o/k found from this formula appears to deviate much less than 1% from the value for this quantity, which can be computed from the complete formula for z. § 20..Results/rom the measurements on glass tubes, pertaining to the dynamic resistance to air. The measurements showed that the dynamic resistance to air was not equal to the one, following from H e 1 m h o 1 t z' formula, but larger. If, now, we take also into account the influence of the interchange of heat with the wall and if, as did K i r c h h o f f, we adopt the point of view that, practically speaking, the current of heat does not experience any resistance in its transition from air to wall (see, however, article IV 3), we must complete H e l m h o l t z ' formula with a factor
For diatomic gasses (air) × = cp/c~ = 1-4 and B = 1.95 lo). Substituting these values, we obtain for the factor mentioned the .value 1-46.
244
c . ZW ! K K E R ,
J. VAN DEN EIJKAND
C. V¢. K O S T E N
F r o m o u r m e a s u r e m e n t s , however, still higher values result ; these are collected in table I. The spreading in the m e a s u r e d v a l u e s - o f z/pc relatively to their m e a n Value and, th.erefore, also in those of the ratio ~ m e a s u r e d / a H e 1 m h o 1 t z, a m o u n t s to a b o u t 4 { % , so t h a t it c a n n o t explain w h y we find for the" tubes of 9.4 m m d i a m e t e r such a strongly deviating value. L e a v i n g out the latter, we obtain for ~ m e a s u r e d / a Helmholtz a m e a n value of 1 - 5 3 i 0 . 0 7 . TABLE I F r e q u e n c y I i m p e d a n c e a n d r a t i o of m e a s u r e d to c o m p u t e d ( H e 1 m h o I t z') res i s t a n c e to t h e a i r in g l a s s t u b e s of 1-6, 4 . 0 a n d 9 4 lure d i a m e t e r ill r e s o n a n c e . T u b e s a t r i g h t a n g l e s a n d a t 60 ° to t h e f r o n t of t h e s p e c i m e n O 1-6 n l n l , l e n g t h 20 c m
~
.L 4"0 r a m , l~2ngth 2 0 c m
~ 9'4 ram, l e n g t h 17"5 eln
at reson, freq ....
(~/pc)res
.
.
.
measured o" H c l m h o l t z
.
.
90 ° 4124 0"63 1-55
at
60 °
410 1'27 1"54
at
90 °
437..,
"t a t
I
60 °
440
0"132
0"28
1 "48
l "54
at
90 ° 500 0 047 1'76
§ 2i. Comparison with the measuremenls carried out by others. On c o m p a r i n g our values for a m e a s u r e d / a H e 1 m h o 1 t z with the m e a s u r e m e n t s carried out b y others, it appears t h a t the f o r m e r agree with the results o b t a i n e d b y E. W a e t z m a n n and L. K 1 e i b s 4) for a d i a m e t e r of 0-6 cm. F o r , t h e y find, namely, a v e r a g e d over a frequency-region from 400 to 1400 Hz, a value, 1.08 times t h a t of Kirchhoff, t h a t is to say 1.58 times the H e l m h o l t z one. E x c e p t for the widest tubes our m e a s u r e m e n t s refer to 400 H z a n d for this f r e q u e n c y W a e t z m a n n and K l e i b s obtain a b o u t 1.13 times K i r c h h o f f ' s value or 1.66 times t h a t of H e l m holtz. Other investigators found higher values. Those, found b y T r 5g e r s) for a t u b e of 6.6 cm d i a m e t e r are 3 0 % higher t h a n the ones c o m p u t e d according to K i r c h h o f f. F r o m their m e a s u r e m e n t s with tubes of 0 . 0 4 c m d i a m e t e r H . L. P e n m a n a n d E . G. R i c h a r d s o n 6) draw the conclusion t h a t the viscosity is, perhaps, as m a n y as 8 times as large as for direct current. In t h a t case the d y n a m i c , is, therefore, v/8 or 2.8 times too large. As t h e y have al-
ABSORPTION OF SOUND B Y POROUSMATEI~IAL
245
r e a d y applied the correction according to K i r c.1~ h o f f, we obtain a f a c t o r 1.46 x 2.8 = 4.1 relatively to the H e 1 m h o 1 t z value. Winfried WisotzkyT) finds in the case of spirally w o u n d c o r r u g a t e d c a r d b o a r d t h a t , in order to a d a p t the t h e o r y to his m e a s u r e m e n t s , the measured s t a t i o n a r y resistance of the air must be nmltiplied b y ~/v/10. Now the expression for the d y n a m i c a is V~'~po~/r, t h a t is a b o u t V'v/530 times the s t a t i o n a r y value 8~/r 2, so t h a t the m e a s u r e d values are ~/53 - - 7.3 times too large. F r o m m e a s u r e m e n t s on resonators one deduces likewise a value for t h a t is m a n y times too large. A n t o n i o G i g 1 i s) for example finds, although not v e r y accurately, a factor 4 to 6. V i 1 h. L. J o rd a n 0) obtains a factor 3 to 4 for resonators with one hole. F o r those with more holes this n u m b e r diminished; however: ,,Es kann als best~tigt angesehen werden, dass die wirkliche Reibung wesentlich grtisser als die theoretische ist". These large ratio-values for resonators are possibly due to the high velocities and the large displacements of the air at the exitopening, which m a y cause, for example, t h a t the q u a d r a t i c t e r m s in the losses become of appreciable importance. As our e x p e r i m e n t s were also carried-out for resonance, it is not improbable t h a t our results have t u r n e d out too high for that same reason. § 22. Change in the stru,ct,z~re-/actor arising/rom placing the tubes at a certain angle to the normal to.the wave-/ront. On p u t t i n g the tubes slantwise, z is multiplied, as follows from the impedance-formula, b y a factor x/k. For an angle of 60 ° we have k =- 1/cos 2 60 ° ---=4. It appears from table I t h a t for resonanceofrequency of tubes with an inner d i a m e t e r of 0.16 cm the factor is 2-02, for tubes with an innend i a m e t e r of 0.4 cm slightly more, n a m e l y 2.08, so t h a t within the e s t i m a t e d limits of a c c u r a c y of a b o u t 4{-0/o, the t h e o r y is confirmed, which could indeed h a r d l y be e x p e c t e d to be otherwise. § 23. The velocity o/propagation in the direction o/the tubes. Theoretically, it is quite thinkable t h a t there is some connection between the q u a n t i t y , on the one h a n d and the decrease in the velocity of propagation in the direction o/the tubes (which has nothing to do with k) relatively to t h a t in free air on the other. According to our theory, however, (see part II 1)) this effect is of a higher order o/ magnitude, t h a t is to say that, owing to the values of ,/koo~ occurring in our
246
C. ZWIKKER, J. V&N DEN E I J K AND C. W. KOSTEN
experiments being so low this decrease cannot be measured with a satisfactory reliability b y means of our apparatus. In this context it is, perhaps, advisable to observe that the correction to the velocity of propagation turning out to be of a higher order leads to the conclusion that in some respects an improvement of our theory it still necessary. For it is well-known ~1) that even in a first approximatiol, the velocity of propagation is changed on the introduction of ~. This refinement of the theory, however, will be justified only when it has been ascertained what mental picture one has to form of the origion of ~, and how one has to imagine the flow in the pores to take place. For the time being we are of opinion that we had better refrain from refinement as well as from verification of the theory in this respect. § 24. Measurements on a specimen, containing exclusively ,,pores with lateral cavities". The structure-factor k can become larger than I not only because the tubes are slanting but also because they are provided with lateral cavities. These diminish, namely, the apparent rigidity of the air, so that the velocity of propagation is decreased 1). (In this way, for example, the absorption-peak of perforated Celotex, occurring at comparatively low frequencies, can be explained). ~..,...~
..liL
....................
.If'==
~.~.
Fig. 4. S~mpl¢ of perforated planks; check of our conception concerning
the structure factor k. We have made a test-piece of which for this reason k > 1 by placing 5 small planks of 20 × 20 × 0.4 cm 3 each, one above the other at mutual distances of 0.4 cm, and b y then making in the specimen, obtained in this way, 232 holes of 0.38 cm diameter, each perforating all 5 planks (see fig. 4). Part of the interstices between the planks is occupied by the supports of the construction. Taking this into account the geometrical value of k (that is ,,the total volume of air divided by the volume of the small canals passing through the planks") turns out to be 7-84 and that of the cavity-factor 0.521. The resonance-frequency was 750 Hz. On applying the correction
¢
ABSORPTION
OF SOUND BY POROUS MATERIAL
247
for the o p e n end, this points to a velocity of propagation c' in the test-piece of 4 × 4. 1 × 760 or 12300 cm/sec. Th'e velocity of propagation m e a s u r e d b y means of the i n t e r f e r o m e t e r (that is, therefore, the one in ,,free" air) is equal to 34850 cm/sec, so t h a t for the specimen in question we obtain a s t r u c t u r e - f a c t o r equal to (34850/12300) 2 = 2.842 = 8.0, agreeing v e r y satisfactorily with the geometrical k
(7.84). F o r the c o m p u t a t i o n of G we fall back once more on the formula in § 3: h
4
kp~
The measured value for z/9c at 750 Hz is 0-500, which on substitution yields a/k = 0.675, whereas according to H e 1 m h o 1 t z we h a v e a/k = ~ / ~ ) / r = 0-2375, so t h a t our measured ~ is too large b y a factor 2.8. The deviation as regards a is, therefore, much larger in this case t h a n for the glass tubes; there is now a closer agreement with the d a t a for resonators, to be found in the relevant literature. It we take into account t h a t , p r o p e r l y speaking, the wooden walls of the small canals cover only half their length, the discrepancy becomes still larger, although not twice as large. l
RElVERENCES 1) 2) ,31 41 5) 6) 71 8) 9) 101 1 I)
P h y s i c a B, 4 6 9 - - 4 7 6 , 1941. P h y s i c a B, 1094--1101, 1941. P h y s i c a B , 1102--1106, 194l. Ann. IJhysik (5) °:I, 247--2.56, 1935. Phys. Z. :11,26, 19,30. J. At'. Soc. Am. 4 , 3 2 2 , 1933. Hoehfr. Techn. und E l e k t r o a k u s t i k ~8, 93, 19,39. Alta F r e q u e n z a 1940, p. 717. A k u s t . Z. ~, 7 7 - - 8 7 , 1940. Chapman, PhiI. T r a n s . A 211, 433,1911. I. 13. C r a n d a I 1, V i b r a t i n g s y s t e m s and somld. L o n d o n 1927, pag. 239.
April 1943.
A B S O R P T I O N OF SOUND BY POROUS MATERIAL V b y C. Z W I K K E R ,
J. V A N D E N E I J K a n d C. W. K O S T E N
I . a b o r a t o r i u m v o o r T e c h n i s c h e P h y s i c a , T e c h n i s c h e H o o g e s c h o o l , Delft
Zusammenfassung lm zweiten Teile wurde eine theoretische Formel fiJr den akustischen Scheinwiderstand' yon por6sen Schallschluckern gegeben, in welche Forreel als Materialkonstanten die PorBsitAt h, der Strukturfaktor k und der Luftwiderstand a eingingen. Messungen zur Pr[ifung dieser Formel werden beschrieben. Die Schallschlucker waren Proben aus Glasr6hrchen yon Durchmesser 1,6, 4,0 und 9,4 m m und eine Probe aus durchl6cherten, in einiger Entfernung yon einander Iixierten Brettern. Eine vorztigliche Obereinstimmung zwischen die theoretisch und experimentell gefundenen k-Werte wurde festgestellt. Der gemessene ( d y n a m i s c h e ) a - W e r t war ffir die Glasproben etwa 1,53 real der H e 1 m h o 1 t z'sche, also etwa gleich dem K i r c h h o f f schen \,Vert. Der gemessene ¢r-Wert fiir die Holzprobe war 2,8 mal der H e I m h o 1 t z'sche Weft, eingerrnassen in f3bereins t i m m u n g mit in der Literatur fiir Resonatoren aufgegebenen "vVerte. § i7. Introduction. I n n u m b e r [ I t) of this series of p u b l i c a t i o n s a n e w t h e o r y was p r o p o s e d for t h e 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 . A c c o r d i n g to this t h e o r y t h e a c o u s t i c b e h a v i o u r of p o r o u s m a t e r i a l s w i t h a rigid s k e l e t o n c a n be c o m p l e t e l y d e s c r i b e d w i t h the aid of three c o n s t a n t s , viz. : the specific resistance to the air , , t h e c a v i t y - f a c t o r h a n d t h e s t r u c t u r e - f a c t o r k. I n n u m b e r I I I ~) the results were c o m m u n i c a t e d of a p r e l i m i n a r y t e s t i n g of this t h e o r y on s p e c i m i n a of s o u n d - a b s o r b i n g m a t e r i a l , m a d e f r o m l e m o n a d e - s t r a w s a n d f r o m s m a l l glass tubes, p l a c e d e i t h e r at r i g h t angles or at an angle of 60 ° to t h e w a v e - f r o n t . W e shall n o w give t h e results of m o r e d e t a i l e d e x p e r i m e n t s , likewise c a r r i e d o u t on samples, c o m p o s e d of glass tubes. This t i m e we did e m p l o y t h r e e different d i a m e t e r s , n a m e l y 1.6, 4.0 a n d 9.4 m m . F i n a l l y , we describe in § 24 m e a s u r e m e n t s c a r r i e d o u t on a test-piece, w h e r e we h a v e t o deal e x c l u s i v e l y --
239
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~40
C. ZWIKKER, J. VAN DEN E I J K AND C. W. KOSTEN
with ,,pores" provided with lateral cavities. Of this testpiece too, the structure-factor could be computed as well as measured. § 18. The disturbing in/luence o/ the interstices between the glass tubes. In these experiments a complication arose from the presence of the more or less triangular interstices between the tubes, as these act as an (specific) acoustic impedance z2 in parallel with the impe•
-~ ~
,
,,
.
.
.
+S
+40
.
.
::d
:
~1
I
1
I
I
I
-20
-I0
-S
Fig. I. Impedance-curves
for a sample of glass tubes, tested at normal incidence : a) for simple piling with tubes closed at the back with paraffin (curve z) b) for another piling with tubes closed at the front (measurement of
i m p e d a n c e of i n t e r s t i c e s z2). a = a b s o r p t i o n c o e f f i c i e n t , ~ ~ p h a s e j u m p of t h e s o u n d a t t h e f r o n t of t h e s a m p l e , pc = w a v e r e s i s t e n c e of air. F r e q u e n c y of t e s t i n g is n o t e d near the measuring points.
dance zt of the tubes themselves. In the case of the widest of the three types of tubes we were able to fill these interstices with melted paraffin; for the other diameters, however, this was no longer feasible owing to the very large number of very narrow cavities. All tubes re, this time, closed at the back with paraffin, as this proved to be
ABSORPTION OF SOUND BY POROUS ~VIATERIAL
241
necessary in order to avoid secondary effects. This provided us at the same time with the opportunity of measuring z2 separately, namely by piling up the same tubes with their cloSed ends to the flout. This measurement yielded the surprising result that the disturbing parallel impedance was much smaller than we can expect it to be if we imagine the triangular interstices to be replaced by their inscribed circular cylinders or by circular cylinders oi the same Grosssection as the triangular cylinders and if we then apply to these the above-mentioned theory 1). Our supposition that the influence of to
+5
+20
+40
+90 A + 180 - 90
-¢0
-20
Fig. 2. D u p l i c a t i o n m e a s u r e m e n t s , see s u b s c r i p t i o n of fig. I.
this parallel impedance could be neglected ") § 13, has, therefore, turned out to be wrong. At the same time, however, it appeared that, whereas a repeated measurement on one and the same specimen furnished practically the same impedance-curve, a new piling-up led to a value of z2, which differed appreciably from the former one. This is very likely caused by the (although small) deviations of the radii of the tubes from their mean value, so that we do n o t obtain an ideally compact pile of the circular cylinders, but one in which the Physica X
16
242
C. ZWIKKER, J. VAN DEN E I J K AND C. %~r. KOSTEN
intervening spaces are connected with each other here and there. I n a fresh piling up the mutual order of the tubes is changed; this m a y lead to ~ different shape of the volume mentioned and in this way to a different value of z2. This can be clearly seen from fig. 1 and fig. 2, .in which, on the right, the impedance-curves z2 are drawn for two different pilings. On the left in these figures the impedance-curve for the total z is drawn, likewise for two different (other) pilings. We'observe .that for the two pilings equality of this total z holds in the case of resonance, whereas in the case of anti-resonance it does nol. This is caused by the fact that for 400 Hz (resonance) z2 is so large that it does not influence z, appreciably, so that z m zl, whereas for 775 Hz (antiresonance) zt and z2 are of the same order of magnitude. The reader is also referred to fig. 3, in which for these two different pilings the coefficient of absorption, due to the total z only, is plotted against the frequency.
/\ so
/
_
J
,( ,,,]
0
,J
0
f
f
200
400
600
800
Fig. 3. A b s o r p t i o n c o e f f i c i e n t a as a f u n c t i o n of f r e q u e n c y , c o r r e s p o n d i n g t o i m p e d a n c e - c u r v e s z of fig. 1 (crosses) a n d fig. 2 (circles).
§19. Simpli/icaHon o[ the /ormula to be checked. In connection with the above we shall restrict ourselves, as regards the various conclusions to be drawn from our measurements, to the resonancepoint, that is to say, the point in which the impedance-curve intersects the real axis for the first time. The theoretical formula to be checked furnishes, namely, in the special case of first resonance a value, which can be approximated very closely by a very simple expression.
243
A B S O R P T I O N OF SOUND BY P O R O U ~ MATERIAL
The original formula for the impedance was (article II mula 8) z=~
1 ~/k~-
I/
" ~
1--]~p-co cothjcol
Vk - ~ J/;;1
1), for-
°
i kpco
I11 our measurements the values of the quantity a/koo~ turn out to be so small that we can in all cases replace x/l--]~/kpo~ by 1 - - j,/2kpo~. As a/K/p = c, where c is the velocity of propagation of sound in free air, we can now write
z/pc-
x/~ (l - -
i./2kOo~) coth io~ cl Vk (1 - - j./2kpo~)
which for resonance transforms in close approximation into
(z/pc) ....
-
h
" 4
"kp~,.
Here, p is known, k and h are obtained from the dimensions of the component parts of the specimen, while the angular frequency o~,c, and likewise (z/pc),,s follow from the impedance-measurements. In all cases to be discussed the value for o/k found from this formula appears to deviate much less than 1% from the value for this quantity, which can be computed from the complete formula for z. § 20..Results/rom the measurements on glass tubes, pertaining to the dynamic resistance to air. The measurements showed that the dynamic resistance to air was not equal to the one, following from H e 1 m h o 1 t z' formula, but larger. If, now, we take also into account the influence of the interchange of heat with the wall and if, as did K i r c h h o f f, we adopt the point of view that, practically speaking, the current of heat does not experience any resistance in its transition from air to wall (see, however, article IV 3), we must complete H e l m h o l t z ' formula with a factor
For diatomic gasses (air) × = cp/c~ = 1-4 and B = 1.95 lo). Substituting these values, we obtain for the factor mentioned the .value 1-46.
244
c . ZW ! K K E R ,
J. VAN DEN EIJKAND
C. V¢. K O S T E N
F r o m o u r m e a s u r e m e n t s , however, still higher values result ; these are collected in table I. The spreading in the m e a s u r e d v a l u e s - o f z/pc relatively to their m e a n Value and, th.erefore, also in those of the ratio ~ m e a s u r e d / a H e 1 m h o 1 t z, a m o u n t s to a b o u t 4 { % , so t h a t it c a n n o t explain w h y we find for the" tubes of 9.4 m m d i a m e t e r such a strongly deviating value. L e a v i n g out the latter, we obtain for ~ m e a s u r e d / a Helmholtz a m e a n value of 1 - 5 3 i 0 . 0 7 . TABLE I F r e q u e n c y I i m p e d a n c e a n d r a t i o of m e a s u r e d to c o m p u t e d ( H e 1 m h o I t z') res i s t a n c e to t h e a i r in g l a s s t u b e s of 1-6, 4 . 0 a n d 9 4 lure d i a m e t e r ill r e s o n a n c e . T u b e s a t r i g h t a n g l e s a n d a t 60 ° to t h e f r o n t of t h e s p e c i m e n O 1-6 n l n l , l e n g t h 20 c m
~
.L 4"0 r a m , l~2ngth 2 0 c m
~ 9'4 ram, l e n g t h 17"5 eln
at reson, freq ....
(~/pc)res
.
.
.
measured o" H c l m h o l t z
.
.
90 ° 4124 0"63 1-55
at
60 °
410 1'27 1"54
at
90 °
437..,
"t a t
I
60 °
440
0"132
0"28
1 "48
l "54
at
90 ° 500 0 047 1'76
§ 2i. Comparison with the measuremenls carried out by others. On c o m p a r i n g our values for a m e a s u r e d / a H e 1 m h o 1 t z with the m e a s u r e m e n t s carried out b y others, it appears t h a t the f o r m e r agree with the results o b t a i n e d b y E. W a e t z m a n n and L. K 1 e i b s 4) for a d i a m e t e r of 0-6 cm. F o r , t h e y find, namely, a v e r a g e d over a frequency-region from 400 to 1400 Hz, a value, 1.08 times t h a t of Kirchhoff, t h a t is to say 1.58 times the H e l m h o l t z one. E x c e p t for the widest tubes our m e a s u r e m e n t s refer to 400 H z a n d for this f r e q u e n c y W a e t z m a n n and K l e i b s obtain a b o u t 1.13 times K i r c h h o f f ' s value or 1.66 times t h a t of H e l m holtz. Other investigators found higher values. Those, found b y T r 5g e r s) for a t u b e of 6.6 cm d i a m e t e r are 3 0 % higher t h a n the ones c o m p u t e d according to K i r c h h o f f. F r o m their m e a s u r e m e n t s with tubes of 0 . 0 4 c m d i a m e t e r H . L. P e n m a n a n d E . G. R i c h a r d s o n 6) draw the conclusion t h a t the viscosity is, perhaps, as m a n y as 8 times as large as for direct current. In t h a t case the d y n a m i c , is, therefore, v/8 or 2.8 times too large. As t h e y have al-
ABSORPTION OF SOUND B Y POROUSMATEI~IAL
245
r e a d y applied the correction according to K i r c.1~ h o f f, we obtain a f a c t o r 1.46 x 2.8 = 4.1 relatively to the H e 1 m h o 1 t z value. Winfried WisotzkyT) finds in the case of spirally w o u n d c o r r u g a t e d c a r d b o a r d t h a t , in order to a d a p t the t h e o r y to his m e a s u r e m e n t s , the measured s t a t i o n a r y resistance of the air must be nmltiplied b y ~/v/10. Now the expression for the d y n a m i c a is V~'~po~/r, t h a t is a b o u t V'v/530 times the s t a t i o n a r y value 8~/r 2, so t h a t the m e a s u r e d values are ~/53 - - 7.3 times too large. F r o m m e a s u r e m e n t s on resonators one deduces likewise a value for t h a t is m a n y times too large. A n t o n i o G i g 1 i s) for example finds, although not v e r y accurately, a factor 4 to 6. V i 1 h. L. J o rd a n 0) obtains a factor 3 to 4 for resonators with one hole. F o r those with more holes this n u m b e r diminished; however: ,,Es kann als best~tigt angesehen werden, dass die wirkliche Reibung wesentlich grtisser als die theoretische ist". These large ratio-values for resonators are possibly due to the high velocities and the large displacements of the air at the exitopening, which m a y cause, for example, t h a t the q u a d r a t i c t e r m s in the losses become of appreciable importance. As our e x p e r i m e n t s were also carried-out for resonance, it is not improbable t h a t our results have t u r n e d out too high for that same reason. § 22. Change in the stru,ct,z~re-/actor arising/rom placing the tubes at a certain angle to the normal to.the wave-/ront. On p u t t i n g the tubes slantwise, z is multiplied, as follows from the impedance-formula, b y a factor x/k. For an angle of 60 ° we have k =- 1/cos 2 60 ° ---=4. It appears from table I t h a t for resonanceofrequency of tubes with an inner d i a m e t e r of 0.16 cm the factor is 2-02, for tubes with an innend i a m e t e r of 0.4 cm slightly more, n a m e l y 2.08, so t h a t within the e s t i m a t e d limits of a c c u r a c y of a b o u t 4{-0/o, the t h e o r y is confirmed, which could indeed h a r d l y be e x p e c t e d to be otherwise. § 23. The velocity o/propagation in the direction o/the tubes. Theoretically, it is quite thinkable t h a t there is some connection between the q u a n t i t y , on the one h a n d and the decrease in the velocity of propagation in the direction o/the tubes (which has nothing to do with k) relatively to t h a t in free air on the other. According to our theory, however, (see part II 1)) this effect is of a higher order o/ magnitude, t h a t is to say that, owing to the values of ,/koo~ occurring in our
246
C. ZWIKKER, J. V&N DEN E I J K AND C. W. KOSTEN
experiments being so low this decrease cannot be measured with a satisfactory reliability b y means of our apparatus. In this context it is, perhaps, advisable to observe that the correction to the velocity of propagation turning out to be of a higher order leads to the conclusion that in some respects an improvement of our theory it still necessary. For it is well-known ~1) that even in a first approximatiol, the velocity of propagation is changed on the introduction of ~. This refinement of the theory, however, will be justified only when it has been ascertained what mental picture one has to form of the origion of ~, and how one has to imagine the flow in the pores to take place. For the time being we are of opinion that we had better refrain from refinement as well as from verification of the theory in this respect. § 24. Measurements on a specimen, containing exclusively ,,pores with lateral cavities". The structure-factor k can become larger than I not only because the tubes are slanting but also because they are provided with lateral cavities. These diminish, namely, the apparent rigidity of the air, so that the velocity of propagation is decreased 1). (In this way, for example, the absorption-peak of perforated Celotex, occurring at comparatively low frequencies, can be explained). ~..,...~
..liL
....................
.If'==
~.~.
Fig. 4. S~mpl¢ of perforated planks; check of our conception concerning
the structure factor k. We have made a test-piece of which for this reason k > 1 by placing 5 small planks of 20 × 20 × 0.4 cm 3 each, one above the other at mutual distances of 0.4 cm, and b y then making in the specimen, obtained in this way, 232 holes of 0.38 cm diameter, each perforating all 5 planks (see fig. 4). Part of the interstices between the planks is occupied by the supports of the construction. Taking this into account the geometrical value of k (that is ,,the total volume of air divided by the volume of the small canals passing through the planks") turns out to be 7-84 and that of the cavity-factor 0.521. The resonance-frequency was 750 Hz. On applying the correction
¢
ABSORPTION
OF SOUND BY POROUS MATERIAL
247
for the o p e n end, this points to a velocity of propagation c' in the test-piece of 4 × 4. 1 × 760 or 12300 cm/sec. Th'e velocity of propagation m e a s u r e d b y means of the i n t e r f e r o m e t e r (that is, therefore, the one in ,,free" air) is equal to 34850 cm/sec, so t h a t for the specimen in question we obtain a s t r u c t u r e - f a c t o r equal to (34850/12300) 2 = 2.842 = 8.0, agreeing v e r y satisfactorily with the geometrical k
(7.84). F o r the c o m p u t a t i o n of G we fall back once more on the formula in § 3: h
4
kp~
The measured value for z/9c at 750 Hz is 0-500, which on substitution yields a/k = 0.675, whereas according to H e 1 m h o 1 t z we h a v e a/k = ~ / ~ ) / r = 0-2375, so t h a t our measured ~ is too large b y a factor 2.8. The deviation as regards a is, therefore, much larger in this case t h a n for the glass tubes; there is now a closer agreement with the d a t a for resonators, to be found in the relevant literature. It we take into account t h a t , p r o p e r l y speaking, the wooden walls of the small canals cover only half their length, the discrepancy becomes still larger, although not twice as large. l
RElVERENCES 1) 2) ,31 41 5) 6) 71 8) 9) 101 1 I)
P h y s i c a B, 4 6 9 - - 4 7 6 , 1941. P h y s i c a B, 1094--1101, 1941. P h y s i c a B , 1102--1106, 194l. Ann. IJhysik (5) °:I, 247--2.56, 1935. Phys. Z. :11,26, 19,30. J. At'. Soc. Am. 4 , 3 2 2 , 1933. Hoehfr. Techn. und E l e k t r o a k u s t i k ~8, 93, 19,39. Alta F r e q u e n z a 1940, p. 717. A k u s t . Z. ~, 7 7 - - 8 7 , 1940. Chapman, PhiI. T r a n s . A 211, 433,1911. I. 13. C r a n d a I 1, V i b r a t i n g s y s t e m s and somld. L o n d o n 1927, pag. 239.