An improved tube method for acoustic impedance measurement

Publisher’s version / Version de l'éditeur:

Vous avez des questions? Nous pouvons vous aider. Pour communiquer directement avec un auteur, consultez la première page de la revue dans laquelle son article a été publié afin de trouver ses coordonnées. Si vous n’arrivez pas à les repérer, communiquez avec nous à [email protected].

Questions? Contact the NRC Publications Archive team at

[email protected]. If you wish to email the authors directly, please see the first page of the publication for their contact information.

https://publications-cnrc.canada.ca/fra/droits

L’accès à ce site Web et l’utilisation de son contenu sont assujettis aux conditions présentées dans le site LISEZ CES CONDITIONS ATTENTIVEMENT AVANT D’UTILISER CE SITE WEB.

Building Research Note, 1977-03

READ THESE TERMS AND CONDITIONS CAREFULLY BEFORE USING THIS WEBSITE. https://nrc-publications.canada.ca/eng/copyright

NRC Publications Archive Record / Notice des Archives des publications du CNRC :

https://nrc-publications.canada.ca/eng/view/object/?id=6278aa96-6f69-47f3-aa3a-001fe4798732 https://publications-cnrc.canada.ca/fra/voir/objet/?id=6278aa96-6f69-47f3-aa3a-001fe4798732

NRC Publications Archive

Archives des publications du CNRC

This publication could be one of several versions: author’s original, accepted manuscript or the publisher’s version. / La version de cette publication peut être l’une des suivantes : la version prépublication de l’auteur, la version acceptée du manuscrit ou la version de l’éditeur.

For the publisher’s version, please access the DOI link below./ Pour consulter la version de l’éditeur, utilisez le lien DOI ci-dessous.

https://doi.org/10.4224/40000591

Access and use of this website and the material on it are subject to the Terms and Conditions set forth at

An improved tube method for acoustic impedance measurement

AN IMPROVED TUBE METIIOD FOR ACOUSTIC

IMPEDANCE MEASW REMENT

by

W.T. Chu

INTRODUCTION

The t u b e method f o r s p e c i f i c acoustic impedance measurements

i s we1 1 known. l - b A b r i e f review of the current procedure involving extrapolation o f t h e measured d a t a to t h e r e f l e c t i n g surface was

g i v e n recently b y K a t h u r i y a and ~ u n j a l . The authors t h e n proposed a

more a c c u r a t e method that made u s e of t h e positions o f t h e pressure

minima and t h e values a f t h e standing-wave ratio, SWR, at t h o s e points.

Both methods, however, involve t e d i o u s measurements of a number o f minima

,and a n a l y s i s either by graphical e x t r a p o l a t i o n as theoretical manipulation. An accurate measurement of at l e a s t one pressure maximum is a l s o

s e q u i r c d ; moreover, the methods fail a t low frequencies where no maximum pressure can b e o b t a i n e d in a particular tube. These methods

can g i v e both t h e impedance o f t h e material and the attenuation

coefficient of the t u b e simultaneously; however, the l a t t e r parameter

is not a quantity t h a t needs to be determined o f t e n .

To overcome t h e low frequency problem, Fluntley6 recently suggested measuring sound pressure at a poinr XJ8 from the minimum pressure p o i n t instead of the maximum pressure p o i n t . This simplifies

t h e calculation somewhat. The method r e p o r t e d is similar t o Huntleyls

proposal b u t i s generalized to utilize any distance from t h e minimum,

t h u s t a k i n g advantage of t h e improved precision when t h e second p o i n t i s c l o s e t o t h e maximum.

In this r e p o r t an i t e r a t i v e scheme is presented t h a t i s based

on the exact plane-wave a n a l y s i s of the standing-wave pattern in the zubc. To use t h e method in its proposed format, t h e t u b e attenuation

coefficient i s assurncd known eirher from theoretical prediction o r

p r i o r experimental determination. The speed of sound can be calculated

i f the temperature o f the a i r in the tube i s measured. Alternatively,

t h e wavelength can be measured directly b y measuring the distance

between t w o minima in the tube. Two sound pressure measurements,

at t h e minimum and one o t h e r point, are then all that is required.

TI-IEORY

--

Consider a plane wave of pressure p i , i n c i d e n t on the

a b s o r p t i v e material as i n d i c a t e d in F i g u r e I . The combination of t h e

jncident wave pi and t h e reflected wave p, produces a standing-wave

p a t t e r n i n s i d e t h e t u b e ,

me

magnitude of the total acoustic pressure a t a distance x from t h e sample surface can be expressed by the f o l l o w i n g

w h s r c

Ip+l

is t h e amplitude of the incident pressure wave, _- a is t h e a t t e n u a t i b n coefficient whose value depends on the side wall of - t h e tube

and -the gas i n s i d e it, k = 2 r J A is the wave-number and R = I R ] ~ - " Y i s t h c con~plex r e f l e c t i o n c o e f f i c i e n t o f t h e sample at x = 0. The normal

a b s o r p t i o n c o e f f i c i e n t ( u ) and t h e seal and imaginary p a r t s [r and $) of

t h e s p e c i f i c acoustic impendance [ Z J p c ) o f t h e sample a r e given by the

following equations:

and

D e n s i t y and s p e e d - o f sound in the tube are represented by 0 and c respectively. A t high f r e q u e n c i e s , when a number of minimum pressure p o i n t s and t h e

SWR

can b e determined accurately i n s i d e t h e tube, b o t h the impedance Z and t h e a t t e n u a t i o n c o e f f i c i e n t a can be o b t a i n e d simultaneously by t h e previous methods.

ow ever,

a t Isw frequencies

when only one pressure minimum i s attainable i n the t u b e , a must b e

e s t i m a t e d from Kirchhoff's formula7 or b y extrapolation fr& data a t

h i g h e r frequencies. Experimental evidence 3 indicates t h a t i n c r e a s i n g

t h e coefficient of Kirckhoffts formula by S per cent results i n a more

accurate p r e d i c t i o n of a .

_-

'TI;(!

b a s i s o f the method proposed in t h i s paper i s an iterative pro-

cess performed en Eq. 1 where t h e a t t e n u a t i o n c o e f f i c i e n t a is assumed knoxn.

Tile wave-number or wavelcngxh can be obtained from t h e f r e G e n c y and t h c

specd of sound, the l a t t e r to b e d e t e r m i n e d from the temperature

measurement i n s i d e t h e tube o r by the separation of two pressure minima at one p a r t i c u l a r frequency. In principle, Eq. 1 can be. s o l v e d f o r t h e

three unknowns lp+

1

,

I

R

I

,

and

r

if pressures a t t h r e e locations a r e measured. This has been tried by Gatley and

ohe en"

w i t h l i t t l e success because o f s h e interdependence of y and [ R ] i n t h e i r iteration procedtrrc.

It i s recognized, however, t h a t y can h e estimated from The positinns of

any o n e of the pressure minima or maxima. The proposed method is h a - e d

on this fact. T h e r e l a t i o n s h i p for t h e position of t h e p r e s s u r e

minimum or maximum can be o b t a i n e d by differentiating E q . 1 and s c t ~ ing

7hus f o r the first pressure minimum, we have

y = -TI + 2kd + s i n

-

1 [(a/2k(~I) = (cZad - [ R \2e-2adll

where d i s the d i s t a n c e from t h e surface of t h e sample to t h e f i r s t pressure minimum. To complete t h e iteration scheme, we need o n l y one

more pressure p o i n t i n a d d i r i o n to the f i r s t minimum pressure p o i n t . The following r a t i o i s s h t a i n e d from Eq. 1 :

The n e x t s r c p i s t o f i n d v a l u e s f o r

J R I

and

r

t h a t s a t j s f ~ - E q . 7 and E q . 8 simultaneously. I n p r i n c i p l e , one can s o l v e for

I R ~

f i r s t and t h e n u s e E q . 7 t o o b t a i n y . S o l v i n g for

I R I

i n c h i s manner

requires an i t e r a c i o n scheme such as t h e Newton-Raphson method.

Another simple i t e r a t i o n procedure has been found t o bc equally effective. Since aJk i s u s u a l l y v e r y small the arcsine term in E q . 7 does not c o n t r i b u t e significantly to y except when y approaches zero,

and d r o p p i n g t h i s term p r o v i d e s a good first a proximation f o r y . Then

E q . 8 is _{s o l v e d a s a quadratic equation f o r}

I R

7

.

_{Only t h e positive} root o f

I R

1,

which is <1. i s u s e d . A new y is t h e n computed u s i n g the f u 1 l Eq. 7 and another v a l u e of

I R

I

is o b t a i n e d by s o l v i n g E q . 8 a g a i n .

Thc iteration process is continued u n t i l

~ R I

and y reach their 1 imi t i n g

v a l u e s .

Heuristic argument based on E q . 7 and E q . S shows t h a t t h e present iteration procedure w i l l be convergent p r o v i d e d t h a t t h e o t h e r prcssure p o i n t is chosen between h / S and X/4 from tFlc minimum p r e s s u r e p o i n t . Sarnplc computer s i ~ n u l a t i o n s a r e p r e s e n t e d i n t h ~ n c s r s e c t ion.

COMPIJTER SIMULATIONS

-

To test thc proposed method, a computer experirncnt u n s performed t o simulate impedance t u b e d a t a corresponding to a range 01- a c o u s t i c impedances, For the experiment, s p e c i f i c acolistic iml~ednnct.5 o v e r a very wide range ( 0 . 1 4 < r I 10 and 0.25 1. I 1 5 3 wcrc t e s t e d . U s i n g t h e relationship

and Fqs. j,4 and 5, b o t h [R

(

and y werc determined and used t o

generate t h e theoretical standing-wave p a t t e r n s w i t h an appropriate va!uc far t h e a t t e n u a t i o n coefficient a . Two f r e q u e n c i e s , 100 H z and 1000 Hz,

were used. From these standing-wave patterns, both t h e f i r s t m i n i m u m

pressure p o i n t s and other p r e s s u r e p o i n t s were chosen

and

fed into the iteration program to learn t h e degree of accuracy of t h e recovered s p e c i f i c a c o u s t i c impedances. S i n c e t h e iteration procedure calculates

(nl

and y a n d t h e error of the _{recovered [ R ] and y will have d i f f e r e n t}

c f f ~ c s s on r and Q, it is more appropriate to study t h e magnitude o f

t h e s p e c i f i c acoustic impedance r a t h e r t h a n its components. The o t h e r

pertinent parameter, the absorption c o e f f i c i e n t a, is discussed i n t h e following paragraph.

About 4200 coinbinations of r and q , o v e r t h e r a n g e 0 . 1 4 I r

-< I 0 and 0 - 2 5 2 ib 1 1 5 , were t e s t e d w i t h the second pressure p o i n t s e t at 0.2X _{from the minimum, In nearly a l l t h e cases} c o n s i d e r e d ,

I R

1

and Q settled to t h e i r limiting values up to 3 decimal places accuyacy

in three iterations. The accuracy of t h e recovered parameters depended

on t h e precision assigned to the p r e s s u r e magnitudes and t h e positions

u s e d . Since t h e r e wcre no significant differences between the 100 Hz

and t h e 1000 Hz cases, r e s u l t s f o r t h e 100

Hz

c a s e are presented.

F i r s t , both the pressure magnitudes and d i s t a n c e s were p i c k e d from t h e theoretical standing-wave pattern up t o f i v e d e c i m a l places accuracy. The recovered a b s o r p t i o n coefficients were compared w i t h

t h o s e compwred from t h e o r i g i n a l parameters used. F o r each combination

o f r and $, an e r r o r percentage was computed to the nearest 0 . 1 per c e n t

The e r r o r s are plotted in t h e "error map" of F i g u r e 2 . The superi~posed

s o l i d curves represent the magnitudes o f a. It can be observed t h a t f o r

n 3 0 . 0 1 , the e r r o r i s l e s s t h a n 1 p e r cent. E r ~ o r s in t h e magnitude

of t h e s p e c i f i c acoustic impedance werc a l s o computed and found to be

evcn less.

Next, t o simulate the accuracy a t t a i n a b l e i n a t y p i c a l

measurement; t h e d i s t a n c e s were rounded to the nearest 0 - 0 1 crn and the p r e s s u r e r a t i o s to the n e a r e s t 0.005. In t h i s case the e r r o r s became s i g n j f i c a n t l y larger. In o r d e r to make t h e " e r r o r mapw easier t o visualize, e r r o r s below 2 per cent were represented by -, and t h o s e abnve 10 p e r cent by *. F i g u r e 3 shows the error p e r c e n t a g e s to t h e

n e a r e s t 1 p e r cent f o r a and F i g u r e 4 shows t h e corresponding results f a r t h e magnitude of t h e s p e c i f i c acoustic impedance. h e can observe t h a t , in general, t h e errors of the magnitude of t h e s p e c i f i c a c o u s t i c irnpedancc are less than 5 p e r cent and those of u are less than 10 p e r cent for a > 0.15. Two additional sets of data were a l s o obtained when I n a114 20 p e r c e n t e r r o r in t h e a t t e n u a t i o n coefffcient a was

d e l iberatelp included in the iteration program- Results i n d i r t e t h a t ,

i n general, the e r r o r s i n both a and the magnitude of t h e impedance i n c r e a s e d by 1 and 2 p e r c e n t , respectively.

An interesting observation can a l s o b e made c o n c e r n i n g

w11crc t h e crror in a exceeded 10 per cent. T h i s is attributable to t h e normal precision o f measurement. It i s d o u b t f u l t h a t t h e accuracy of + 0 . 0 1 for a specified by Ref. 7 c a n be obtained i f d i s t a n c e s can o n l y l>c measured t o t h e n e a r e s t 0.01 cm and t h e pressure r a t i o s t o t h e n e a r e s t 0.005 o r less in a c t u a l measurements.

EXPERIMENTAL STUDIES

As another examplc, experimental measurements o f t h e standing-rcave

p a t t c r n s were carried out i n an impedance tube at d i f f e r e n t f r e q u e n c i e s . TIlc t u b e was t e r m i n a t e d by a resonance absorber formed from a p e r f o r a t e d

p l a t c ( 8 . 6 per cent opening), backed by a 4 I J 2 in. layer of g l a s s f i b r e .

']'he p l a t e and the glass f i b r e were separated by an a i r space of 1 1 / 4 i n . The first pressure minimum was estimated from the experimental results and t h e o t h e r pressure p o i n t was chosen at 10 p e r cent below t h e maximum.

The speed of sound was estimated from room temperature measurement and

t h e tube attenuation coefficient was obtained by the following formula :

~ ~ h c r c F and D rcpycsent frequency and t u b e diameter rcspcctively. L s j l l g

this information, b o t h

I R ]

and y were calculated by t h e iteration process +

Thc calculared

I

R

1

and

v

neTe used in E q . 1 to generate the standing- wave pattern. Two twical cases arc presented in F i g u r e s 5 and 6 w h c ~ e

the computed p a t t e r n s are compared with t h e experimental data to sl~oi\r

a close correspondence that g i v e s s t r o n g s u p p o ~ t to t h e proposed method.

TFlc conventional method could not have been used in this t u b e for t h e

100 Hz case shown i n F i g u r e 5 because no pressure maximum was available within t h e t u b e length.

CONCSUS ION

An improved tube method f o r a c o u s t i c impedance measurement h a s

been presented. It is simple and accurate. I t s accuracy depends o n l y

an how precisely t h e pressures and t h e i r p o s i t i o n s can be measured. The method r c q u i s c s t h e f i r s t minimum pressure p a i n t and one ather p r e s s u r e p a i n t , provided t h a t the tube attenuation coefficient i s known. The iteration process involvcs s o l v i n g a quadratic equation several times. If the other psessure point i s c l ~ o s e n to be aho11.t: 0 . 2 h from t h e minimum pressure p o i n t , o n l y three iterations are requircd t o reach final v a l u e s , Such computation can easily he done with a IlancF-

h e l d s c i e n t i f i c calculator.

Over t h e r a n g e of a c o u s t i c impedances s t u d i e d , e r r o r s of 10

and 20 per c e n t in t h e estimate of the t u b e attenuation coefficient a o n l y resulted in an increase o f 1 and 2 pcr cent e r r o r s , rcspcctivcly. -

determined accurately o n l y once for any p a r t i c u l a r tube.

f i e present computer experiment a l s o indicates t h a t it i s d o u b t f u l that an incremental precision of +O.OL in n , as specified hy t h e ASP4 standard ,7 can be achieved f o r a < 0.2 in o r d i n a r y

laboratory measurements. REFERENCES

S a b i n e , H . 3 . Notes on acoustic impedance measurement. 3 . Acoust.

S O C . Am. 1 3 , 143-150

_-

(1942).

Scott, R.A. An apparatus f o r accurate measurement of the acoustic

impedance af sound a b s o r b i n g m a t e r i a l s , Proc. Phys. Soc.

-

58, 253-264 ( 1 9 4 6 ) .

Beranek, L . L . Acoustic measuremenfs. New Yerk, Wilcy, 1949. 914 p.

L i p p e r t , W,K.R. The practical representation of s t a n d i n g waves in

acoustic impedance t u b e . A c u s t i c a 3 , 153-160

_-

(19531.

Kathuriya, M.L. and Munjal, M.L. A c c u r a t e method f o r t h e experimentaI

evaluation of the acoustical impedance of a b l a c k box. J. Acoust.

Sac. Am.

_-

58, 451 -454 ( 1 9 7 5 ) .

IIuntley, R . Private communication.

ASTM C 384-58 (Reapproved 1 9 7 2 ) . Standard method of t e s t f o r impedance and absorption o f acoustical materials by t h e tube

method.

Gatley, W-S. and Cohen, R - Methods f o r evaluating the _performance of s m a l l a c o ~ ~ s t i c filters. J . Acoust. Soc. Am

_-

4 6 , 6-16 ( 1 9 6 9 ) .

A B S O R P T I V E P -411- FIGURE 1 T H E I M P E D A N C E TUBE F I G U R E 2 E R R O R M A P O F T H E RECOVERED A B 5 0 R P T I O N C O E F F I C I E N T , Ot, O V E R A W I D E R A N G E O F V A L U E S O F r A N D $ . N U M E R A L S I N D I C A T E E R R O R S TO T H E N E A R E S T 0 . 1 % W H E R E A S B L A N K I N D I C A T E S C A S E S O V E R 1 % . S O L I D C U R V E S S H O W T H E M A G N t T W D E O F a . F R E Q U E N C Y U S E D I S 100 H Z . N O T R U N C A T l Q N I N I N P U T D A T A .

F I G U R E 3

ERROR M A P OF T H E R E C O V E R E D A B S O R P T I O N C O E F F I C I E N T , a , O V E R A W l D E R A N G E O F V A L U E S OF r A N D #'

.

N U M E R A L S I N D I C A T E E R R O R S TO THE

N E A R E S T I?=. - I N D I C A T E S C A S E S UNDER 2 % W H E R E A S

*

I N D I C A T E S C A S E S

O V E R 1 0 9 b . S O L I D CURVES SHOW THE M A G N I T U D E OF Q . F R E Q U E N C Y USED IS 1 0 0 H z . I N P U T D A T A T R U N C A T E D TO S I M U L A T E E X P E R I M E N T A L A C C U R A C Y .

F I G U4E d

FRROR M A P O F THE R E C O V E R E D M A G N I T U D E O F T H E SPECIFIC A C O J S i l C I M P E D A N C E O V E R A W l D E R A N G E OF V A L U E S O F A N D $ FOR THE S A M E

0 E X P F R l M E N l A L D R i A E X P L R I M F N I 4 L D A T A U 5 E D

\

I N C O M P V T A ' I O N

-

C O M P U T E D 5 r A N D I N G i 4 J A V E P A T T E S N a

-

0.32 r = 1 . 2 9 Ilr --3 .27 O l S T A K C E FROEA F A C E OF S A M P L t , c m F I G U R E 5 C O M P A R I S O N OF C O M P U T E D S T A M D I N G W A V E P 4 T T E R m W I T H M E A S U R f D O A T * I I I S T A N C E FROM F A C E OF S A M P L E , r m F I G U R E b C O M P A R I S O N O F C O M P U P E D S T A N D I N G ?:AVE P A I T E R F A W l T l k M E A S U R E D D A T A