寺尾, 道仁 ; TERAO, Michihito; 関根, 秀久 Author(s) SEKINE, Hidehisa. Citation Christchurch New Zealand, 98(1): 4

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1 \n Title ON CHARACTERISTIC TRANSMISSION AND COEFFICIENTS OF SPLITTER DUCT ATTE 寺尾, 道仁 ; TERAO, Michihit; 関根, 秀久 Authr(s) SEKINE, Hidehisa Prceedings : INTER-NOISE 98 : Nv Citatin Christchurch New Zealand, 98(1): 4 Date 1998 Type Cnference Paper Rights publisher KANAGAWA University Repsitry

2 ON CHARACTERISTIC TRANSMISSION AND REFLECTION COEFFICIENTS OF SPLITTER DUCT ATTENUATORS M Tera and H Sekine Kanagawa University, Rkkaku-bashi , Kanagawa-ku, Ykhama-shi, , Japan 1. INTRODUCTION The attenuatin f a splitter duct silencer has been estimated by applying such theries as P.M. Mrse (JASA, 1939), R.A.Sctt [1] and U.A.Kurze (JS&V, 1972), r by extraplating such an empirical chart f A.J.King (JASA, 1958). These are basically apprpriate nly fr the fundamental wave mde in the straight channel f an attenuatr f infinite length. Hwever higher rder mde waves are als imprtant because this type f silencers are emplyed in d'ucts f large width. T realize acustical estimatin f a finite length attenuatr including higher rder mdes, we investigated a determinatin methd f the characteristic transmissin and reflectin cefficients f an attenuatr by emplying the superpsitin principle f traveling waves, and carried ut tests by BE numerical simulatins and by an experiment n a tw-dim.ensinal attenuatr with tw air-channels fr the tw typical cnfiguratins; i.e., installatin in a straight duct and installatin between a cuple f right-angle elbws, at frequencies belw secnd rder mde cut-n in the far field. 2. DECOMPOSITION OF TRAVELING WAVE PRESSURES Taking an interface, crdinates and the rigin (,) as shwn in Figure 1 (a) in the straight duct regin f each duct cnnected t the attenuatr where evanescent mdes frm the discntinuities have died ut, the sund pressure p(x, y) is represented as N N p(x,y) = La(n) exp(-jk)x)c5(k)y) +Lb(n) exp(+jkn)x)cs(kn)y) (1) n=o n=o where n(= O,1,2,,N) represents the number f the pressure ndes in the y directin, here N is the highest number f the prpagating mdes fr the Christchurch New Zealand Nvember

3 frequency f interest in the straight duct. The quantities k) = nn / L y and kn) = {(c / c)2 - (kn»)2}1/2 are the wave numbers f the nth mde in the y and x directins, respectively. The quantities 8(n) and b(n) dente the utging and incming plane wave pressures, respectively, f the nth mde at the rigin. These plane wave pressures can be determined by slving simultaneusly a set f equatins (1) crrespnding t the sund pressures p(x, y) at 2(N + 1) psitins. These sund pressures were bserved by cnducting a BE numerical simulatin [2] fr bth cnfiguratins f Figure 1 (a) and (b), and by an experiment fr the cnfiguratin f Figure 1 (a). 3. DETERMINATION OF CHARACTERISTIC COEFFICIENTS The utging wave pressure, 8l(n), f the nth mde n a interface l is represented by the superpsitin f the cntributins f the incming waves, bl'(n') ' f all f the mdes n' f all f the interfaces t as L Nil 8l(n) = L L't'l(n)l'(n,)bl'(n')' fr l=i,ii,..,l,and n=o,1,2,,n l (2) i'=1 n'=o where Ni and Nil are the highest numbers f the prpagating mdes in the lth and l'th straight duct sectins, respectively, L dentes the number f the ducts cnnected t the attenuatr, and 't'i(n) t(n') represents the characteristic transmissin (r reflectin when.e = ') cefficient between an Q) :::J en Depth 5mm t t surce I Interface I Glass wl (32 kglm 3 ) (b) Installatin between a cuple f right-angle elbws -.,r--=i----1'---- I \5 25 b all(o) L.- 11() :::::::::::::: C"..., t"11(1}1(1) _.YaU(1)... -_ _ _ _.;;;.;b; ):......,.j ----a;:1 Interface I PVC - 3mm thick Q) :J Cl) Depth 5mm I surce I,I J- 3 '1 I' 9-.1 (a) Installatin in a straight duct sectin [Length in mm] Figure 1 Cnfiguratins f a splitter duct attenuatr and test setup Christchurch New Zealand Nvember

4 EO.41) 1 Cl.Q C\J Mea. & & &.6.UJ P ọ.. 1 \ TU(1)1(1) TI(1)1(1) \ u.em l '" Mea. t ". EO. -.. <; : --r-r;;1(1 7:'II CQ I COl! : 3 Mea. 2 4 e ) Frequency [Hz] Frequency [Hz] (a) Fundamental mde incidence (b) 1st rder mde incidence Figure 2 Transmissin and reflectin factrs fr installatin in a straight duct c 1 1 TII(O)I(1).UJ.(/) e. p.... Cl.2.Q C\J llll ll16 L\...== T U (l)i(q Thin lines rll(o)t( C\I 2 4 e Frequency [Hz] (a) Transmissin factrs fr fundamental mde incidence 9-' Cl 2! Frequency [Hz] (C) Transmissin factrs fr 1st rder mde incidence.fij - 1 Lt(O)I(O) 1:::II::)l:; I TI(O)I(l) =..:; I,--- 7:'1(1)1(1) \ \.. TI(O)I(1) I.....!'A r, /... " \... V LI(1)1(1),1 3 Thin lines Thin lines 2.a e '. 1e (b) Reflectin factrs fr fundamental mde incidence Frequency [Hz] Frequency [Hz] (d) Reflectin factrs fr 1st rder mde incidence Figure 3 Transmissin and reflectin factrs fr installatin between tw elbws Christchurch New Zealand Nvember

5 incming wave determine these cefficients, bi'(n') and its cntributin t an utging wave al(n)' T M deferent cases f the sund fields were generated and measured, where M = NI + Nil + +N L. Having and slving a set f M independent equatins (2) fr every (n), we can determine the unknwns 1"l(n) l'(n') f ttal M withut using anechic terminatins. 4. TEST RESULTS Installatin in a Straight Duct Figure 2 shws the transmissin and reflectin cefficients f the attenuatr installed in a straight duct sectin as shwn in Figure 1(a). The glass fiber blanket f 32 kg 1m 3 was used in its smaller directin f flw resistance, and its acustic prperties fr the numerical simulatin and the' Sctt thery were measured by an impedance tube methd. Disagreement between the cefficients by the measurement (Mea.) and the BE simulatin (BEM) may partly be caused by discrerpancy in the blanket acustic prperties between the measurement and the numerical simulatin. Fr fundamental mde incidence at this cnfiguratin, the reflectin cefficients, 1"1() I{O) and 1"1(1) 1()' are rather small and the transmissin cefficient crrespnding t the attenuatin by Scatt thery has gd agreement with that ( 1"1() 1(») by the numerical estimatin. It shuld be nted that the transmissin cefficient between the first rder mde, 1"11(1) 1(1)' is nt small enugh t be disregarded. Installatin between a Cuple f Right-Angle Elbws Figure 3 shws the transmissin and reflectin cefficients f the attenuatr between a cuple f right-angle elbws as shwn in figure 1(b). Cmpared t the installatin in a straight duct, every crrespnding reflectin cefficient becmes larger and the transmissin cefficients between the fundamental mdes, '11() 1()' becmes far smaller. Cmpared with this, thse f crss mde incidence, t"1i(1) 1(1) and II{O) 1(1) J becme imprtant. 5. CONCLUSIONS The transmissin and reflectin cefficients f a splitter duct attenuatr in tw typical cnfiguratins have been btained fr the fundamental and the first rder mdes. Fr the transmissin cefficient between the fundamental mdes, the attenuatin by Scatt thery f a lined duct channel f infinite length can be used as far as an attenuatr is installed in a straight duct sectin. But this des nt hld when it is installed between a cuple f right-angle elbws. The transmissin cefficients related t the first rder mdes cannt be disregarded cmpared with thse related t fundamental mdes. REFERENCES 1. R.A ScU, Prceedings f the Physical Sciety5 8, (1946). 2. M.Tera and H.Sekine, "A numerical analysis f sund field f a lng space by a sub-regin cupling apprach," Prceedings f Internise 96, Liverpl, England, 37 31, Christchurch New Zealand Nvember