Sound dissipation in porous media

SOUND I SStPAT ION POROUS IN MEDIA by -, K. ATTENOORO UGH Thes i s s ubmitted for the degree of Doctor of Philosophy March 1969 Dep ar tment of Civil Engineering The Univ ers ity of Leed s • •. ~ 6ound penetrates porous bodies more freely than would have been expected • • . On the other hand a hay- stack seems to form a very perfect obstacle. - Lord Rayleigh CONTENTS Acknowledgements Pag e 2 List of Symbols Chapter 0 Introduction and Abstract 6 Chapt er 1 Review of Literature 8 Chapter 2 Scattering Theory for fibrous media23 Chapter 3 Single scatterer theory 31 Chapter 4 Multiple scattering theory 41 Chapter 5 Exp~rimenta1 Chapter 6 Discussion and conclusions 61 Appendix A Waves in fluids 74 Appendix B Waves in solids 80 Appendix C Scattering coefficients Appendix D Expressions for computer program Appendix E Energy dissipation calculations 101 Appendix F ..Constants, basic expansions, List of samples 107 Viscoelastic absore~ scattering coefficients 110 Appendix H Appendix G results Spherical scatterer-solid stress components Photographs Resum~ of sample data Graphs References 56 95 I1b 1. Acknowl edgements I should l i ke to re cord my deep grati tude to Professor R. Ho Evans until re centlY9 Head of the Faculty of Applied Sci ence 9 for realising that there was room for such work as has formed the basis of t h i s thesi s 9 and f or maki ng avai l able t he ne c essary f i nan cial suppor t for three years . I am also indebt ed to my Uncle 9 Mr . Wi lliam Hough t on ~ Evans ? Seni or Le cturer i n Archit e cture 9 for his consi s t ently wise counsel and constant encouragement ; and both to Mr . Alan Yettram 9 at present Reader in Struc t ur es in t he Me chanic al Engi neeri ng Depar t ment a t BruneI UniversitY 9 and Mr. Paul Yaneske 9 my former "acous t i cal research" colleague 9 for many helpful discussions. The work involved in performing the relevant computations was made relatively simple by the assist ance of my col leagues Dr . Debendra Nath Hazarika 9 in the writing , and Mr. Dipeskanti Bhatcry ~ in the execution of the computer programs. I sincerely thank Mr. Roy Duxbury of the technical staff, for taking the photographs necessary and Miss Pam Bennion of the Earth Sciences Department for performing the arduous and complicated task of typing . Lastly ~ but by no means least ~ I ofe~ my sincere thanks to Dr. "John" L. A. Walker g Lecturer in Building Science and my supervisor throughout this work , for hiB careful encouragement of my often wild endeavours and for his painstaking checki ng of the substance of this work 9 despite the many and frequent other demands upon his time. 2. Li st of Symbols Superscripts : f refers to flui d proper t y B refers t o s olid property 1 defined in context below • c .0lII}?1 <ex; Subscripts : conjugate D si gni f i es dilat a t ional wave proper ty TH T b " " thermal " viscous shear II " bulk value " equi l i brium value " " normal inci dence absorbtion coefficient vi scous wave vector potent al A '" ABC D n n n n coefficients in expansion of seri es for wave potentials o Defined in 3.120unit vector in ~, 9 direction prODucts of f ibre r adius and propagati on constants, at normal incidence ditto , at oblique incidence radius of volume of integration (Appendix E) B B =(/ (JP/~ Cb Bulk modulus bul k wave velocity dilatational isothermal wave velocities specific heats ~ij product of fibre radius and axial phase constant d slab thickness dij rate of strain tensor Kronecka delta gradient operator ')!. ( ) di ver gence oper a tor \l j..( ) curl operator ("-J ~ i j strai n tens or E wave energy per unit normal area per unit time E incident wave energy per unit normal area per uni t time F surface g far=fi el d scatt ering amplitude ( as defined by Twersky) h1 gh2gh3 curvilinear coordina te paramet ers 0 i J n K gH Cylindrical Bessel functions of order n n axial phase constant K propagation vec t or ~ bulk propagation constant ,-oJ Kf KS thermal conducti vities • Kf Kf Kf D 91"H' ...,... S KS 9KS gK n 111 1"" } propagation constants of dilat ational , thermal and viscous (shear) waves . respectively l pro j ected di stance of any poi nt f r om origin of coordinate system (fi g 0 3 011 page 3/) L lengt h of cylindrical fibre N concentration of fibres N outward drawn normal from surface F (Appo E only) Nf . Ns coefficients (Ch o 3) . defined in Appendici es A and B p pressure t Pij viscous part of fluid stress tensor Pij total fluid stress tensor PAT atmospheric pressure r radial coordinate (fig A T 0 3.11 page j I) unit vector in r - direction 4. R radi us of fibre Re ( real . part of ( 1m ( imagi nary part of ( t time T temperature u speci fi c i nternal energy u di splacement vector ) velocity vector th of displacement 9 velocity u .v . JI J j v vol ume w t ime aver aged t otal energy loss i n volume V with surface F ~ o mponet vi acous and thermal energy l oss (time averaged) respectively in regi on of W. cart esi an coordinates (fig. 3.11 page 31) unit vector in z=direc t ion surface normal im pedance Z n coefficient of vol ume expansion ratio of specific heats (1 n :: cipher ( > (2 n 0 0 curvilinear coordinates polar angle measured in 'Xj plane angles between propagation vectors of scattered viscous (shear) waves and x- axis unit, vector in the a - direc tion I Lame's constants of the sol id +~ f ;f coeffici ent of vi s cosity )/ ki nema i c vi s cosi y 1 coefficient of compressi onal viscosity r ~ )f dens i ties ~ fb to al so i d stress (J-i-j r di ssipation cross- section s f } ([ ~ t hermal di ffusivities qJ angl e of incidence with res pect t o normal t o cylinder axis cP. 7 I.f1'H II) ~H 'f])5) + S 1 angles between propaga ion vectors of s ca tt ered flUid1 dilatat ional 9 t hermal waves and x- axi s solidJ s cal ar potenti als i nci dent potential A.f¢f 'f/ l> ) 1'H ) tJr; S) 11kS scalar y) X\f#- ) If(/" ~ I Y - '1.f'R) If-r <. s calar pot ent i als associated with viscous (shear) wave viscous and thermal di ssi pation functi ons 9 r es pectively f orward and backward t ravelling plane wave potentials reflecte d and t ransmit t ed p.l ane wave po tential amplitucies angular frequency tv ~ )i dilatat i onal...ana t hermal wave potenti als surf a ce and volume i nt egral respectively ).qv X1t time average compl ex conjugat e of x of t he order of approxi mately equal to 6. SOUND DISSIPATI ON IN POROUS MEDIA INTRODUCTION AND ABSTRACT 0. 1 . The particular f ield that has been the c oncern of this work is that of Bui l ding Sci ence . The porous media of i nterest are consequently those commonly used a s absorbents in Archi tectural Acousti~1) The ob ect of t he work has been t o formulate quantitati vely a theory of t he dissi pati on of sound in ~u c h materials, 60 that a basis can be laid for optimi s i ng and predicting t heir coef fi cient s of absorption. The theory has aimed at avoidi ng t he inclusion of empirical constants. 0.2 A review of literature is made involving a somewhat wider range of porous media , includi ng those of Engi neer i ng Geology . in t ers~ in the fields of Geophysics and Porous fluids, a term employed by A.B. Wood (in "A t extbook on Sound" Ch.3 ) , as they occur, for instance, in Underwater Acous ti cs are also considered . Further, . the literature concerned with sound propagati on in more general inhomogeneous and composite fluids and solids , is examined, where the theoretical techniques are relevant to our study. 0.3 It is found that the literature specifically related to sound absorbing materials and also to unconsolidated or consolidated granular media:(a) develops theories vhich are essentially macroscopic and do not allow adequately for the microstructure of non- isot ropic flexible framed media i.e. fibrous media . (b) provides l ittle realistic descri ption of the dissi pation in closed pore viscoelastic absorbers e . g. cellular rubber . 0.4 A theoretical technique, previously reserved for problems in under- water acoustics and sound propagation in suspensions is applied, as an alternative,to cases of fibrous and viscoelastic foam media. POROUS MEDIA FmOUS ROCKS FOf\QUS I t WOOD PLAST Ej FIBRE COMFDSITE MEDIA SOI LS , SA NDS PLATE I I I I CO NSOLI DATED o (j) J - I I fRAtvE w RIGID ?: I I I I I~ RIGID ELEMENT CL I ~O Ww O(f) 0::> i { 1 GONTINLOUS I E[AS CLOSE ) II ;.7 ~ / / I ~_ I II CLOSED OPEN IGm E ~nc ) SURFICE VISCQELASTfC FRAME ~ I'"" I ~I IGJ 2: LI NKS RE VIEWED Ll NKS tv1AR !NE SED:MEN TS I II AERDSOLS, II ' SUGGESTED I COt'-CE N TRA. TED /'- IMPURE OR FOLYCRYSTALUNE fROTHY V-iA-TE R SOLIDS I I MBEDDING FLUlD FRAM: SURFACE 9JRfi«E ElEMENTS fRAME fRAME ) I / / / I / / / / I / / / / (j) ~ ' I I UNCO NSOLI DATE D OR AGGREGATE U.- RIGIDl kLEXI BLE- CELLUL.AR Pa...YMER FOAM RUBBER FOAM STEEL WOOL, FELTS J I DIWTE . The predictions of absorb found t cor r eI a e on h s, obtained easonab y w' t h experi, en for f i brous me di a are data on g a_s f i bre bock srunp es o Fur her g an exp ana i on of t he physi s of s und absorbti on i n ce visc oe_as io mad ' a l it e a ture are s gges ed and con ular nsiona and observat i ons of pr evious orroborated . The literature is exami ned i n t he wider context previously ment i oned. As might be expect ed 9 t he field of interest has deter mi ned t he par ticular t ype of porous med ' um considered; t he model assumed ; and often t he theoret ical t echni que . The fo l d~ ut dia.gram shows t hes e links 9 toge t her with t hose discussed in t he t hes is and provi des a classifica i on for t he review. Models numbered i n t he chart are no w discussed. Chapter 1 Revi ew of Literat r e Thi s is the basic con ept ual model which under ies most of Model t he work on Bound absorbi ng materi als . Essent i a Y9 he porous medi um i s assumed to have a r i gid solid g continuous frame c ntaini ng a number of para!. el cylindrical pores open a t t he surface of t he material and normal 1.1 1 0 t his surface. Aft er i nitial work by Rayl eigh(3) and Crandall(4) dissi pation can be postul a ted to take place accordi ng t o such a model by : (a) vis cous l osses i n t he boundary l~er tube owing 0 of t he walls of each capillary r elative mo 'ion between t he cont ained viscous g conducting and compr essible f l u d and he solid walls; and (b) heat conduc ti on , i .e. exchanges of heat energy be t ween contained fluid and pore walls during cycles of f l ui d compres si on and rare - fac t ion . Thi s stems from t he He l mh o lt z ~ K i r c hof theory for s ound pr opagat ion in a rigid wal led tube con t aining a compre ssi ve, vi scous conducting fluid. 1 . 12 Zwikker and Kost en(5) ext end t hi s t heory t o a complete medi um cor- responding t o Model 1 . I n or der to allow for irregularity in t he pore cross sections ; viz. deviati ons f r om circul ar cross- section and changes in effective r adius causing a variation in t he fl uid particle velocity across each capillary pore (superimposed on t he vari at i on due to viscous drag at the pore walls) ; the flu i d particle velocity is replac ed by an average particle velocity over the cross- secti on of any pore . This can then be related to the volume flow through t he por ous materi al by Dupuit 's " Re I a t l.on (6) :~ volume flow = 1.13 [ porosity x average particle velocity] The viscous drag effects i n the separate capillary pores are combined(8) by the introduction of t he speci f ic resistance or f l ow res i stance c oefficient (0() for the porous material which relates volume flow with t he pressure var "ation through t he material . I ts st eady state val ue corresponds to the case wher e the equation of motion in the indivi dual pores reduces to that for Poiseuille flow . Then the equation of mo tion for vol ume flow i n the total medium i s Darcey ' s empirical law(6)(for non-turbulent f l ow) and the flow resistance is equival ent to an inverse permeability coefficient. The high frequency extreme where the Hel mholt z annular effect(7) can be considered to exist in the capill ary pores is given an approximate value by Crandal l. An expression for the flow r esi stance coefficient which applies at intermediate frequencies is computed by Zwikker and Kosten.(9) The characteristic impedance and propagation constant for a medium corresponding to Model 1. are derived considering viscous and thermal effects in 11 0 10 2 - a genera isat ' on of model 10 M o~ The porous medium is r egarded as having a continuous s o ' d fl exi ble frame contai ni ng pores g t he s r uc ures of whi ch ar e not s pecified , apart fro m the requi rement t ha t he me d 'urn be homogeneous and isotropic o 10 21 . Zwi kker and Kos en( 16 ) ex en d fr ame d model 1 the more general flexible frame model 0 t he concept of f actor =-.;~ vi s ous coup i ng be ween t heir hei r previ ous work f or t he r' gi d 0 2 9 by : ntr oducing This f ae or inc· udes the i ner ial and he so i d fr ame and he pore f ui d r es ulting from motion o The parame era of Egros i r ~ltive flow res istance coe ff ici ent are re ai ned i n modi i ed eq ' ati ons of mot ion and continuity f or bo h pore fl i d and solid fr ame o Thi s proc edure does not require t he more r i gorous calcula tions f or complex densi ty and s ti ffness f or the ri g ' d y framed model 1 0 The solution of t hese equations provi des f or ·he existence of wo t ypes of coupled wave s g which become decoupl ed i nto separat e compressi onal waves f or frame and pore f luid at cert ain f r equenci es dependi ng on t he compressi bi ity of t he frame 10 22 0 . . . 1 ar gener al f l eX1. bl e f r ame mo d e 1 a S1m1 Ot h er a u t h ors (, 17 ~ 18 9 1 9) uS1ng bas ed on a r igi d frame approach , define paramet ers l ess easil y i dentifi able with the s tructural properties of a ctual material s" In particular an effec ive dynamic mass factor (m) ' s i ntroduced 9 defined as t,he ratio of the effective mass of air i n he pores to t he mass of an equal volume of "fr ee " ai r o Th ' s is meant t o contai n t he effec t of the presence and motion of t he s o i d skelet on on t he mot i on of the fluid o Thi s paramet er reduces t o uni t y i" e 0 to he Bame value as Zwi kker and Kosten 0 s s t r ucture fac t or 9 for a medium correspondi ng to Model 1 wi h a rigi d frame and all the pores parallel 0 t he w~v e vec t or · of t he i ncident (pl ane) sound wave . Re t tinger(20) defines a slightly di f f er ent paramet er r e presenting t he amount of vibrating 12 . a ir mass pe r uni t vo ume of t he med "um o This differ s by a factor of bulk Bo l t ~1 8 ) density fr om the above defini ion due t o Morse and Beranek 1023 ( 21 ) appl i e s t he coup i ng facto a ppr oach t o model 2 t ype medi a o However 9 greater conc er n with f ibrous materials i s ShOWfi 9 "n that fric t i on between f ibres i s i ntr oduced a s a further considerat i on 9 i n the equa t ion of mot i on of t he soli d frame o Kosten and Janssen correctness of Beranek os hi s mi s int erpr etati on of ( 21 ) (22) expr ess doubt a s t o the he coupl i ng f a ctor and po nt ou deriva ti on of he str ucture f a ctor as a " dynami c Maas " coeffic i ent similar t o mo Further (22) t he co pIing f act or derivati on of Zwi kker and Kos t en(1 6 ) i s adapted to include the more compl ete express "ons for complex density and complex sti ffness of air .i n a It i s remarked t hat t o be f r i g i d ~ wale d po ~e5 ) l y r i goro s these expressi ons shoul d i n I fact be devel oped for a gas cont ai ned i n a flexibly walled cylindr i cal pore o Zwi kker and Kosten(23) compare t heir theoret i cal pr edic t ions for model 1024 2 with experimental observati on for (22) and J ans sen w o d~f ibre pl ate and hair- felt . Kosten furthe r compare the modi f ied theory wi th experimental results for more flexibly framed f i brous medi a o The correlation i n t hi s latter case is found good 9 if9 as is required for the earlier theory ~5) some s uitable estimation of the structure factor is made o Paterson(24) applies the Zwikker and Kost en(5)(16) theory further to fl y.~ ' 03 d ~satur ated granular 1IIaterials and obtains reasonable c orrelation wi th Model 3 - appli ed particul arly i n the cont ext of flui d- sat urated granular medi a , where concern i s more wi th sound propagati on t han sound absorbti on . The P!)2'o'US C 056 au eg y- pa 'ed sphe.eso tha a pp ~ ed as a. d ~ e ep er d ~' 3a - a t ed aggr ga'e of t; and .' n c::mt ac assWXl"d e as ar B 'i press o p ~c i 60 b t een ad j ac en and is ed ' IS cha."lge.s i he area of con'ac i s assumed incompressible spher So The .nvis ci do B and (25) "onsiders a randomly pa ked array of four di fferent s phere sizes such t ha larger s ' ze for s uch a 0 each sma e frac i on o A no a cons an ode1 i s size .::omple e y f 1 ~li s phere r ad ' us as a fun ion of s r e as - s near tained by calculating 0 t he voi ds of t he next B a n relat ionsh p he di 1at a ional deformation of he for e be ween he par 'c1eso A s'mi1ar model i s used by Duffy and Mind in(26 ), however t angential /' as weI as normal per cycl e due con~ a c ' pr essure is i nc1ude d g and t he frictional l oss s i p is considere d . 0 In both cases the va ocity of com ~ pressiona1 waves through such a frame is cal culated o 04 Model brous ma terials. - of The f i bre block i s consi dered as an which para11e fi br es of uni form d ameter suspended i n ai r 10 1 ~ fi b r e composite medi um i n and lengt h are Kawaaima(27 ) chooses ric ed per pendi c mode and a free y his model as an al ernative to models 1 or 2 for t he case of a f l exible and fi brous acoustic ma erialo is r ei ther bound ela stical ly t o fixe d positions i n space o 0 g air heoret ' cally t The: i nc i dent sound pl ane waves pr pagating i n t he dire ct ion ar to t he fibre axeso onded rigi d bar mode In are he elastically bound case 9 a s tring difer~n i ated o In t he former case t he fibres are assumed to move aBclamped s trings i oe o as t hough t he major frae i on ( g/1f" 'l- ) of hei r length were a rigi d bar 9 and the remainder fixed permanent ly at the equilibri um position o The latter case is a simp1ifi ca tion of t hi s 9 where he whole of t he fibre i s assumed rigid bar 9 cons r a i ned a ccording t o Hooke os Law o 0 oscillate as a 1. 2 The clamped string model i s devel oped as a general two cases are considered to be spe cial case s of this . tinui ty , introducing porosity, motion of fibres ~ G a s ~ and the Equations of 0 her on- and volume flow of air are deduced , together with equations of motion of fibres and air ? derived fro m HamiltonO s Princi ple , and incorporating a resistance coaf fi ient for each fibre . This resistance coefficient is given by Stoke Os law f or a "long e li psoi d of gyration" and gives a frequency depende nt expression for specific flow resistance when the equations of motion f or fluid and fibres are combi ned in terms of relative veloci t y . The general form . because of the model assumed ? predi cts to al absorbtion at resonant frequencies of the elasti ca! y bound fibres . 1 043 A similar model of an array of i den i al parallel rods uni f or mly spaced i n air has been adopted by Lang ( pro e r ~ t i es I O~) i n di s cussi ng the absorbtion of cellular plasters e . g. polyeurethane foam . is essent i ally based on Zwi kker and Kos en. The t heory used However. by choosing thi s particular model he avoids the use of their struc ture factor . Thermal dissi pation is not considered. 1.5. Model 5 - of particular importance in discussi on of sound pr opagati on through unconsolidated g granular ? fluid- saturated media where dissipati on is also considered . The medium is considered to consist of an elastic solid matrix . saturated with a compressible , viscous fluid . made about m i cro ~ struce. No specific assumptions are However, certain restrictions apply at some stages . 1 . 51 The literature based on this type of model is more concerned with the formal derivation of the equation of mo ion of the solid frame .via a stressstrai n constitutive relation than the previously discussed theo r ieB~5 , 7 , 8-23) A linear stress- strain relationship involving porosity is derived by Biot~28) Analytic expressions for the resulting six elastic coeffi cients are obtained from three equations of equilibrium and three equations , f D ' saw. 1 (6) B'10 t(29) ' representing a general1sed form oarcy : uses th 18 formulation together with a Lagrangian form of the equations of moti on to derive 1.52 Wave equations. , The coupling factor of ZWl.kker and Kos t en ( 16) an d .l.Lel'aneK,(21),1s spl't 1 "Q up into its inertial and viscous components for this mod~l .- The inertial . coupling is introduced into the equations of motion via t hree "mass" coefficients derived empirically . 'The viscous coupling is introduced allowing for the variation of the viscous dissipation with the across the pores i.e. v&PiatigR gf the weloi~y vis~o m i cro QissipatioR field I~th ~ velocity the mi~pg fiela aep9S6 the POF8& i.e. variation with frequency and with change of cross-sectional pore shape. These effects are combined into a viscosity function which depends on frequency, a characteristic dimens i on of the pore (extremes of shape being parallel walled sli ts and circular capillaries), tortuosity(6) ~d the kinematic viscosity of the pore fluid. The effect of tortuosity of the pores, in particular is contained by a structural factor . (30) Th e f requency d den t" ' a epen V1.BCOSl. t y f unc ti on 1.S more comprehensive, microstructurally sensitive representation of the , ' , dynamic flow res1stance coe f f1Cl.ent than that used by other authors.(16-21) 1 1.53 In a later paper, Biot(3 ) rederives the elastic stress- s t rain relation for a porous medium, considering both "closed" (fluid not free to circulate in and out of the medium) and "open" pore situations, (as do Gassman(32) and Paterson(24) for model 3). Anisotropy of the medium is also considered in this paper, together with the effects of heat conduction between grains and pore fluid and internal friction between the grains of the unconsolidated solid. 16 . The la er two cons ' dera i ons are i ntr odu ed under t he ga eral headi ng of effects" and previ ous t heories (33 ) of irreversible " heI'moe as i dynamics are evoked . resulting i n t he repl acement of by operators o The e q a io red c:'ng t he of he pore geome o f t h e coup ~ ' ng i ner i al par e ~ a sti c o t .i on ar . de :i ed in a di ffer en brae mass coeffi cients used previa f i cient dependent on he ry ~ . y 6 (29) t hermo ~ coeffi c ients f orm 9 o a s "ng e coef ~ i oeo mere s ' mply analogous t o t he f ac t or (o 6 . 21 ) Anot her paper (34) sees t h e introduct ' on of a vis codynami oper at·o in an equat i on of relative motion of flu ' d ' n he pores o generalisati on of a me thod previousl y us ed 0 obtain t he complex vis os ' y function The equa i on of rela t "ve mo i on 9 0 Thi s r epresents a oge her with an equat ' on of mo ion repre senting the ti me der ' vative of t he t o al momen t um of t he fluid so i d mi xture . are used rat her than separate equati ons of mot i on for f ui d and 60 ~ , d respe t ~ve ' be analysed as 1054 p re vioUS 1 Yo (28 . 29,31 ) lY9( 28 9 29 ~ 3 A wave equa t 10n " t h en d er1ve . d wh 1C ' h can ~s )into ,t wo compressional and one shear wave . Hardi n and Richart ( 35) poi nt out t ha t he elas ic constants i ntr oduced by Biot (28,29) are di ffi cult to measure i n practice o In t he ir t heory there- for t he Young Os modUlus of the elastic frame is derived f r omJDuf!y and Mind i n ' s analysis (26) a ccordi ng t o model 3 9 and this i s subst t t ed into the Biot theor y (24) 0 Brutsaer (36 ) extends analysis based on Model 5 t o the case of a three phas e me diu.a19 a porous granular medi um saturated with air 9 some grains bei ng covered with a wetting flui d o A Lagrangian approach v for the equati ons of mot i on ~ similar to that of Bi ot ' s i sotropic medi um theory(29) ~ i s used o The diff iculty with Bi ot es el astic c oef ic en t s(28 . 3 3 ~ 34) by emp oyi ng Br andt ' s approach(25 ) for elas ieity of Mode i s partially avoi ded 3 9 poi nting t 0 t hat the r emainder of Brandt Os theory for t he compressional wave velocity assumes t hat the fluid and sol id move toget her and exhi bit t he same di sp ace ~ 17. ments . Vi scous effects i n the air are i ncluded a ccordi ng t o Zwi kker and Kosten for Model 1~ func tio ~ ~ 9) 5) and t hose i n t he liqui d accor di ng t o Bi ot is v ~ s The e ffe c s of change of pore shape ~ osi ty ort uos i ty of por es . and heat conduction are not considered and i t i s assumed t hat t he t wo f' ui ds do not occupy the same pore at t he same time . 1. 6 Model 6 = represents a "finite element" approach t o the probl em. The medium is considered to consi s t of a number of i dentical e ements of vol ume or " cells" containing proportions of fluid and solid . 1061 Berank(~ A rigi d g porous 9 sound absorbing material i s compared by 7 ) to a model contai ni ng a series of rectangular cells divi ded i nto proporti ons of rigi d so i d and flui d a ccor di ng to he volume porosi y . is also i ntroduced int o a t heory i n which th ~ equations of Fl ow res istance ~ on ti n uit y and f lui d mot i on are deduced f r om f i rst pri nc i ples , maki ng no speci fi c assumpti ons about pore charac teristi cs but maki ng a number of rest r icting approxi mat i ons . Thi s model and approach were rejec ted by Ber anek (21 ) . n favour of Model 2 . 8 However 9 McGrath(3 ) poi nts out that an analyti c solution f or model 6 is possi ble without the number of approxi mat ions used and that r easonabl e results can be obtai ned with two acousti c materials with fai r l y rigid frames. 1. 62 Tyu eki n(39) devel ops a theory for a hypothetical rube ~ i ke material containing an array of parallel cylindrical ducts with their axes normal to the incident plane wave front o The medium i s treated as if i t consiated of an array of identical close-packed hexagonal "prisms" g each of which contai ns an infinitely long cyli ndri cal channel . Each "prism" or cell is approximated to a cylinder with a radi ally fas ened external surface for the purpose of applyi ng simple boundary conditions for continui t y of radial displacement and transverse stress of the boundary . The boundary conditions together with the requirement of continuity of direct and t ransverse stress 18. at t he channel boundaries fur ni sh a wave equation f or each cell cor r es ponding t o that f or t he propagation of axi ally symmetric elas t ic waves i n a so i d r od with a free surface . A cel l model approach is also used by Nesterov 1. 63 (40a ) o describe sound propagation in a concent rat ed suspensi on of heavY 9 rigid sol id part icles in a vi s cous flui d . Each cel l i s postulat ed t o consist of a double plug i. e . a cylindrical soli d plug surrounded by a coaxial cylindrical liqui d plug . For a r egul ar array of particles g t he assumed cylindrical s hape of the liquid plug i s an approxi mation to t he rectangul ar box shape obviously required for a represent ation of the act ual medium . Neglecting thermal effects 9 equations of motion for the liqui d and the soli d are obtained and used to defipe a comple·x dens ity for t he suspensi on which includes the viscous effects i n much t he same way as the complex density deri ved by Zwi kker and Bysova and Nesterov effects . ( 40b) ~osten(5) for the rigi d walled tube . extend t hi s cell model to in lude t hermal In this case the spherical shape is chosen f or t he l i qui d- soli d plug . A complex compressibili ty is thus defined expressi ng the thermal attenuation withi n t he concentrated suspension (c . f . Zwi kker and Kosten(5),s complex sti ffness for a medium with r i gi d walled por es) . theory(40a 9 b) i s suggested as being more applica ble to concentrated pens~o th an o 0 t her - t heor1es . (41) wh 1C " h d 0 no t ·1nc ] .u d e 1n " t erac t"~ onB · The SUB - be ' t-ween the particles . Model 7 appl icabl e to various t ypes of suspensions i n fluids ; and of inhomogeneous or composite soli ds . 1.71 The suspension or i nhomogeneous solid i s consi dered di stribution of di s conti nuities according t o the following (a) spherical fluid discontinui t i es 0 forms contai n a ~ - n a denser fluid medi um 19. (b) spherical f l ui d di s continuitie s in a less dense (c ) spherical ri gi d or el a s t i c solid di scontinuities i n a flui d f~Ui med ium d medi um . (d) spher ical or i8.rbi trariliY sha ped s oli d 9 flui d or cavi y discont i nui t i es i n a solid medium. When dissi pation is calculated for (a) 9 (b) or (c) i t i s based on t he solution of t he scattering problem for a single s cat t er er and extended by simple addit i on to the total medi um . i .e . t he s cat terer s are as sumed to have only a slight effect on the properti es of the imbeddi ng medium. 1 . 72 Rayleigh(42) develops the spherical harmon ic approach to the single scatterer , problem where the incident wavelength is large compared with the radius of the scattering obstacle . viscous effects and de ~iv. escatring Lamb(43) extends t hi s anal ysis to i nclude coeffi cients for a rigid sphere free to move, from rate of change of momentum considerations . The first appli - cation to a number of obstacles i s made by Sewel l(44) for a suspens i on of 'fixed 9 rigid 9 solid obstacles in a viscous fluid . A corr ect ion i s applied in this treatment to account for movement of the obstacles . A more rigorous approach 9 for suspensions of types (a) 9 (b) and (c) is given by Epste i n( ~ 5) This is revised and extended by Epstein and Carhart(46) for types (a) and (b) , to include the effects of heat exchange between particles and jmbedding medium . It is shown that the irreversible effect s due to viscosi t y and heat conduction are simply additive to wi t hin a close approxi mat i on 9 a t audio - frequencies. The attenuation coeffici ent for the medium is cor- respondingly calculated from dissipation func t ions represent i ng viscous loas and thermal l oss separately . Chow(47) considers s uspensions of t ype (a) and (b) and includes surface tension effects . The theor y (46) . 1S shown to be applicable even in the case of large displacements of the scatterers(47) 20. i oe. the boundary conditi ons us ed for ea h 6catter er retai n the same form when either t he o r~ g i n with the scatterer . of coordi nates i s fixe d in Wo d (2) cons d e s a ' t enua t'~on scattering procedure . spac~ or a lowed t o . b bbl y wa t er b y a 1n " (48) The same problem i s consi dered by Dev11n ~ on somewhat di ffer ent t echni que si de ri ng mo~e by a he pass age of t he sound wave aa a small perturbati on on t he volume of the bubble Cc. f . model 3) . The equations of motion are derived using generalised. coor di nates and a Lagrangian approach. Zi nk and De sasso ( 49a) a pp y t he Epstei n and Carhart suspensions t ype (c ) ; poi nting out Epstein Qs vis cous attenuati on i s consider 9~ ( 46 ) theor y to onclusi on(45) tha t when onl y dissi pation becomes almost independent of densitY9 (when the densi ty of t he solid obs a cles is much greater than t hat of he i mbeddi ng flui d) . I t i s furt her remarked t hat 9 in such media g l)sses due to (i) the mal effects ~ · t h i n he s cat erers ( ii ) relaxati on phenomena and ( ii ' ) s pherical s ca t ered wave format i on (removi ng energy f r om the i ncident p ane wave front = um.i rnpo t ant in reverberati on measurements ( 45)) are negl ' gible compared wi h t he eff ec ts of vis ous and thermal a ttenuat i on • • 73 Attenua ion i n i nhomogeneous so i ds of type (d) i s and Tr u~15 0) onsidered Yi ng The parti cular obst acles considered are (i ) isotropi c elastic sphere 9 (ii ) ri gi d sphere and ( ii ) s pherical cavity embedded i n an solido by ~ l aBtic The average energy removed as a fr a ction of t he incident energy per unit area per part icle by spherical compressi onal and shear wave f ormation in scattering, is calculated . The case of scattering of high f r equency s ound by arbitrarily shaped and ori ent a te d grai ns i n pol ycrystalline materi als i s developed by Bhatia(51) as a pr oblem of "slight " scattering Le. where t he properties of the scattering medium differ only slightly fr om t hose of the i mbedding medium . The effect 21. of mul ipl e s cattering . L e . t he e-ffe ctr of the material surroundi ng a singl e grain ~ being " granulat" r s and not bulk materi al ~ i s di s regarded be cause of the grai ns ' random ori entati on with respect t o each other . 1 074 Urick and Ament(41 ) consider the propagation o f sound i n a fini te slab region , (thi ckness d) 9 contai ni ng a concen ated s us pensi on f el a s t i c 9 solid particles , the :failbeddi ng fluid being v-iscous and non conducting o Model 7 condi t i ons are assttmed withi n the s l ab i oe. the wave i nci dent nor ma l yon the s l ab is assumed t o be a close approxi mation t o that i nci dent on each of the parti cles. The i nhomogeneous natur e of t he s ab medi um is brought i n 0 account by assumi ng plane reflected and t ransmi tted waves either s i de of the slab whi ch are respectively the sum of the backward scat tered and forward s cat t ered waves from the parti cles . The single s cat t erer oeffi ci ents for an elastic sphere f r ee to move i n a non- conduc t ing viscous f l ui d as calculated by Lamb (43 ) are used for each par . 1 le o The propagat ion constant and complex vel ocity of t he model are calcula t ed on the s uppos i i on that the scater i~ gfrom the i nhomogeneous slab region is i dent i cal wi th transmitted/ reflected components from the homogeneous s l ab o The vi s cous attenuat ion expression is shown to be the same as predicted by the relevant s ingle s catterer theory~43 , 45) Correlation is also shown with the derivati on of Uri ck(68) based on a theory of the vis cous drag process between fluid and partic les according to Stoke Os Law . Duykers (92) shows that t his i n t urn can be related to Biot ' s theory(28,3 1 ) for viscous attenuati on i n a relevant model . 108 Model 8 ~ a more concentrated version of model 7 . This model applies where the wave incident on each s cat t erer wi t hi n the suspension is not necessarily approxi mately the same as the source plane wave i . e. the obstacles do not scatt er i ndependentl y t o any reasonable approxi mation. 22 . Morse and Feshba ch (5 9 2) Wat erman and Tr ue .1 i nc ude t he i nteract i on be wee n s ca tteri ng 0 The firs (52 ) i s and bas es t he for echni que of s o he par i (53 ) and Twersky (54=56) es of model 8 caused by mul t i pl e on er ned wi h pr paga i on i n bubb y wa t er s i on he compl e e cas e o The ot her n ons t r uc i.on of a Gr een os f uncti on he ef erences (53=5 6 ) deal with a more general class of pr obl ems by cons tru t ng i n egral eq a i ons f or t he excit i ng fi el d on any s ea t erer i n t he medi um o Twer s / 5 6 derives b l k paramet e r s for a mu t "pl e s _at t ering s ab medi um c ont ain"ng a r andom array of s i milar y aligned 9 i dentical s cat ter ers of general sha peo these are comput ed i n t er ms of 4 In a f i r s t f ormalism(5 ) he propert i es of the i mbeddi ng medi m and the singl e s cat teri ng coeffi c "ent s f or an i sol a t e d s ca t t er er i n the imbeddi ng medi um 0 A s econd formalism(55 ) derives t he bulk parameters i n t erms of a generali sed i solated s ea t ering ampl itude correspondi ng t o each s catterer being exci ted by t he coherent mult ipl e s at eri ng f i el d but r adi a t ing i nto the imbeddi ng medi um. Thi s is consi der ed ,0 be more accurate t han the f i rst formali sm. Embleton (57) appli es t he fi rs forma i sm(5 4 ) t o the a t tenuat i on of sound by compr essional wave s ca t t ering i n f or est s by consideri ng t he case of s ound i nci dent on a s l ab region of paralle normal to t he i ncident plane wave vector o r i gi d cylinders 9 their axes Chap er 2 D i s~ a ~o n ' n F ' brous Med a Fi brous materials are now very commonly used as sound absorbi ng ' t ·arJ b r .i Pl d s are aval.'1 a bl e (. 1 ) ma t eri al s and a large var iety of proprie ' They a so provide t he best sound a bs or b tl.on audio- frequency range . " h arB. ,t erl.S l CS (5·9" , l.n . t he Thus, i t i s i mportant t hat there should be adequate theore tical work avai abl e t o explain t heir performance and lay t he basis for t he ir desi gn . 2 01 Cri ici sm of existing models ? theories and resulting parameters Although several authors speci fically concerned with Ar chitectural Acousti cs have suggested that their theories(5 y 17- 21 ) are applicable to fibrous media 9 certai n i nadequacies of their work can be poin t ed out . These stem basically from he differences between the mi cros tructure of the conceptual mode s (1 . 2) behi nd t he ir materials und er consideration . heore ti cal work and that of t he I t is proposed t ha the materi al parameters on which the dissipative capacity depends t end t o represent macros copic , ( 14) (17 18) propertl.es and are often frequen cy dependen t . ~ The most com- prehensive treatments are those due to Zwikker and Kosten(5) and Biot(~8-31) thus their parameters are the primary ones reviewed. 2.11 Struc ture Factors (10) , 1S i ntr oduced to allow for effects of orienta ' ion of pores and of side- holes . Also , at least for As remarked ' n 1.1 4 , the structur e factor that theory based on the rigi d framed Model 9 the factor is required to include the effect of the motion of t he fr ame( 'IO) io e . a slight frequency dependence is i ntroduced as in the effective air mass parmet~17 . 18.2) The result for all absorbing materials is a · factor which cannot be exactly measured(12) and is calculated s imply as a factor required to bring theoretical prediction into line with experimental obser vation . It can only be predic ad in fac t fo spe i al ases of "ar ifi ::: ' al medi a" e og o sacked glass stl"aws (59) wh 'ch are un ' ke y o mat erials used for sound absorb io n ~ 0 of c: r respond i n s truc+ure he hi gh fle xi bi i tY 9 low f ' ow , resl.S an e ca+agorY 9(5 ~ 21) s ch as g ass fibr e wool o The s tru ture fact or mater may be more viable for hi gh dens i y g i gh f low r esistance i a l~ as wood- fibre pIa e ; where i ' woul d ser e as a "persistence fae or " measuring the persis ence of a pore 'n di e i n a section paralle i on and cross- sectional area to t he " common" direc ion of t he fibres materials i t may be expect ed t hat he f i bres are contact for t he major portion of t heir au h (60 ) eng h wi h l ose ~ pa 0 c ked For such 0 ' oeo in her f i bres and thus , (10) such concepts as capillary pores 9 s de=holes g orient a tl.on and tortuosity(6 9 30) have meani ng o For 1 osely compacted fi bre wools 9 where the fibres cannot be expected t o be i n contact f or any appre ia.b l e port ion of their l ength s h concepts a ,9 these have I i t e place 0 Only t hat of tortuos ' y of streamline f1 w(6) retains any meani ngo 2 012 Flow Resistance This parameter is :ntroduced by authors (5 9 14- 21 ) to express t he viscous boundary lay~ a tion 9 a t he so ' d- flu ' d i nterfaces within porous materials 9 i n the equations of mot "on o It i s given an accurate r epresentation ( 1.13) as a frequency de pendent func t 'on onl y for Model 1 ( carrie d over t o Model 2) i oe o for t he case of periodic moti on of a vis cous 9 compressi ble f ui d in a fixe d g r i gi d walled 9 circular 9 cylindric al pore . For model 5 ( 1052) 9 a ~i sco it y func 1on( 29 ) is introduced to express t he more general periodic micro=velo ity f i eld s 'tuation a cross a pore of arbitrary shape 9 with a limi ted motion of he rigid por e wal1 9 i oe . uni di - rectional mot i on parallel to the induced pressure gradient o An analysis of incompressi ble 9 viscous flui d fl ow inside a cylindrical tube with el astic 25 0 wal whi h are massive and capab e of S9 b y Womsrs .ey ( 6 , 9 t h e ana y . Fi brous materi a t on b e 1ng " 1n so men ti one d ~ wou .d req ire an un ~ i s made h" a: ~ e d f~ an~ y s is C wo ...~ 1 " 0ns o o d~ vi s cous . 0 con d ct" ng flui d mo ion wi hin pores of arbitrary shape. 1& 9 possess ng massiv8 9 conducting e as ic f r t his C ear motion o erma 0 f i o n ~ mo in which cap ' llary pores can be "eii ned 9 such as the f ibre p1ates already compressib. hree d ' mensi na and for aL~ 13 9 capa Ie of three dimensional W' oos l y aompa ed woollen mat erials where capi llary pores cannot be defi ne 9 an analytic expressi on fo r t he dynamic flow r esistance even i n ....~ e rms 0 f t h e measurabl e i ntractable pr b em . as a paramet er 2 01 3 oeff 'oien 6 (5 14- 21 ) t he ories , . ne c essary in t he re . van t a '~ c t(. 62) presen t coe ff ~cen " 5 a 60mewh a t Thus t he pr ac 'icabili'Y of dynami c f ow resistance f or f exi ble g fi br o s absorbing me di a must be questionabl eo Conc e pt of a con inuously framed and i so ropi c medium The he ories of p opagation 9 based on models 2 9 3 and 5 9 predict pl ane " frame waves tl 9 coupled or deco p e d with the mo tion of t h e s a ura ting flui d o Thi s r equi es he exi sten e of a contin s solid fr ame t hrough t he medium or at l e ast a discontinuous frame which will transmi t the effec t of peri odic l oadi ng on t he "front" surface of t he mediumo r i gid s olid f r ame 9 of COurS8 The req irement of t he a c ousti A continuous 9 is an i n egral s pe ification of model 10 ontinui y of frame i s met i n ac t ual mat erials of pl aster 9 wood f "bre pIa e {fr m t he observa tions of 2 011 ) 9and granUlar 9 types 0 Where t he wood f bre an granular materia ls ma ~y be ~ expected t o show a non=linear el asticity due to t heir essenti al discontinuity poi nted out by some author s and by Jones~63) oncerned with models 3 and 5 (1 03 and 1 054) H owe v er ~ he corre a tion a h ' eved by Kawasima(27) be tween theory (64) and experi mental observation based on ';he unhonded versi on of Model on gl ass fib e materia.l 9 wou d s eem t o i ndicate t he smal hat. i n such materials l for disp ac ements and velocities i nvolved i n an acousti c disturbance t he f i bres reac frequenc:i quite i ndependently of each ot her , at least at high This is not t rue in t he pr essence of resin bonding . Further 9 he condition of iso ropy , required i n many of t he continuum me chani al t heories(5 ,1 9 921 , 35 936 ) based on models 2 9 3 and 5, is not ne cessarily satisfie d by f i brous materials if t he fibres have a "preferred" dire ction u Thus materials must come under the ca tagory of materi als not adequa ely deso i bad i n t heir sound abcor bi ng propert ies by t heories based on homogeneous, i sotropic med i a ~ 65 ) 202. !2EE cation of t he unhonded version of Model 4 2 021 K aw 6i~ a os By Theor~ - advan ages and di s advan ages hoosi lg a model for t he fibrous materi al as described in 104 Kawas i ma avo ' ds many of the difficu ti ea menti oned i n t he previous section. However 9 nE'W p oblems are introduced in a ttempti ng to apply me chani cal approa.ch to thi,g model. . t ( 66 ) 1.. q e coer' f'. Cl.en t he r e8 ' eta! . mo tion of a efie t can B. continuum An approxi mati on i s introd ced with 0 1m application of Stoke ' s Law for ody through a viscous f luid . Fur her v t he heat conducti on n y be introduced assumi ng a square array of i breB~67) Al.so v the compresGibi itY9 and t hree dimensional strain of the elastic fibres are not 2.22 aken into account by t he assumed rigi d fibre modelo A ~ .9 ~ t er~ t heor:l, Mode' 4 9 at least with unbound or l i ghtly bound conditions 9 may also be regarded as a version of model 7 or 8 consi dered to be a suspension of f i bres 0 Thus t he fibrous medium may be n air and the teChniques of analys r eviewed i n 10 7 and 1 08 may be applied . t o an el asti ~ c y lindr Each fibre may be compared conducting so id s catterer i mmerse d i n a i ca~ conducting , compressible flui d med ium. Vi S COUE The tractabi i ty of this approa ch is ens ured for most materials i n use as sound absor bi ng ma ' er ' als by the ver y small dimens i ons of the component material f i bre s (diameters be t ween 3 and mi qrons i n most i nstan ces) o Thi s me ans that t he harmoni c func ti ons 10 used i n the s ca t tering theory are rapi dly convergent and need only 'be expanded for the first few orders of the ir arguments ; L e . Rayleigh s ca t.tering conditions exist 9 he wavelength of t he i ncident sound being much greater than the radi us of t he s ca tteri ng obs t acles for t he frequency range (1006000 cis) of i nt erest . Dissi pat i on i n Fi brous. Medi a on a s cat t ering model 2 . 23 Epstein and Carhart c ondu cti ng ~ (46) ana yse sound propagati on in a compressible flui d medi um i nto three vis co us ~ ypes of waves; t wo compressional waves and one rot at i onal or shear wave . Within a small approximati on one of the compressional waves i s shown to be ascribable purely to thermal effect s g t he other corresponding to dilatati onal propagati on in an i nvi sid g no ~c ondu c t i ng f ui d . It is noticed t hat the shear (vi s cous) wave and the "thermal" wave are rapi dl y attenuat ed in ai r and water . The att enuati on of sound in i s passage through fl i d 6uspensions g as described i n 1.7 9 can then be attri but ed t o t he "mode conversion" of i nci dent pl ane dilatat ional waves int o t hermal and vi s ous waves by scattering at the various obst acles (suspended part icles) . 280 is (47 This mode conversion into vis cous waves / accepted by Wri Ck(68) , Chow ) and Cantensen and Schwan(69) as equivalent t o t he more convent i onal representation of viscous drag o Urick coefficient derived by Lamb (43 ) to the scattering l~s and ~ an viSCO~ (68 ) shows that t he attenuat i on be di vi ded i nto t wo part s relate d loss respe ctively 9 and proceeds to verify that the vis cous part can be deri ved independently by means of Stoke's equation o Chow (47) simi larly shows 9 that the first or der C l ow frequency) approximation of the viscous dnug (as a fun ction of re ative velocity) on water droplets in air 9 subje ct to an i ncident sound wave according to the Epstei n and Carhart (46) formulat i on 9is equival ent t o Stoke's law for spheres moving w i t~ the same vel ocity o Further . Chow (47) derives the first order (low frequency) approximation of the heat transfer rate for this formulat i o~ i ncludi ng heat conduction effe c ts fo r each . I droplet . as equivalent to the standard expressi on for heat t ransfer t o a sphere when Reynolds number tends to zero and heat exchange is by conduction only o It may therefore be expected that the expressions for dissipation given by mode conv'e rsion wi thin Model 4 (as a type of Model 7) are accurate representations of the mechanisms of diss i pation in an ideal fibrous medium previously inaccurately represented by theories using flow resistance etc o The main parameters in this approach are; radius of fibres Q their elasticity , number (average) per uni t volume and properties of the imbeddimg medium (air). Such quanti t i es are readi l y measurable compared with the less convenient ,parameters previously required , and i ndeed a scattering theory is more directly related to the mi crostructure of a fibrous material o 29 0 s c: a.t teri n&.. .heorx ~=sump 2 03 . Restri t ione and 2031 The un-bonded version of Model 4 ' s t he only on e whi . corresponds t o Model 7 0 Each fi bre i s a s sumed freely suspended i n air s o that boundary conditi ons of con inui y can be app ie f ibres j f pr essure j ve oei t o any poi nt on its surface o ne cessari y occurring ducing a systema tic error n a y ~ temperature and heat f ow Thus t he contac t be t ween ual ma. eri al s in so far as j i s neglected j i ntr o- hey i nduce "frame" waves o Fri cti on bet een fibres is al so negle ted , as is he e f fec t of resin bonding whi ch alters the apparent elastic ity of the f i bres , their orientati on and degree of contact wi t h ·each other o The influence of resin or cross bonding of t he f ' bres i s t o de crease the sound absorption at low frequenci es and in rease i t a t hi gh f r equ~ ei es(70) o It may be con- s i der ed t hat a bonded mat eri al is mor e amenable to analysis based on cont inuous frame mo dels (2 013) t han a l oose pi le o 2 032 When a s ingl e s ca t teri ng approach s pecif ' t o Model 7 i s used , t he t' me 8.verage of t he power di ssi pa.L ed per s atter er of t he i ncident ene gy .is calcu a.t ed t o give of t he medium. 9 as a fraction he a.ttenuat i on coeffici ent The power di e,eipated per s a t terer i s fo nd by in egrating t he dissi pat i on fun cti ons over a large v l ume surrounding each scatterero This volume mUt3t be at least large enough for t he viscous and t hermal waves t o have di ed 0 before reaching ' t s sur face o Thus a minimum r adius of t he volume of integrat ' oll must be t he wave de crement distance for t he viscous and thermal waves (roughly equal(46» 0 It follows that this me t hod of calculat ' on requires the s ca tterers t o be separated by a t least t wi ce hi s di stance so that the vi s cous and thermal waves do not i nterfere o The r equired separati on i n air has been calculated t o be 0.02 cm (46) •Y I n a f i brous block 9 t his r equir ement i s unlikel y t o be me t over t he whol e l engt h of every f i bre 9 and mus t pr ovi de another source of systemati c error o 2 033 In view of t he concen tra t i on of f i bres t o be expe cted in a c tual med i a and the var i at i on in thei r separati on 9 t he s i ngl e s ca t tering theory can only provide a crude approximat i on o The mul tiple s ca t ter i ng t heory appropri ate t o model 8 s hould i mprove on t h i s by i ntroducing t he possi bility of s ca tt er ed dilatat ional wave int erf e r ence . In t he multiple s catteri ng t heori es avai lable 9 however 9 for general s i tuationE, viscous and thermal wave mult i ple s cattering ar e not considered . The work of Waterman and TlU ell(53) would seem to indi cate that symmetr y arguments discount any effect of viscous (shear) wave mult i ple s ca t teri ng among spheri cal scatterers . requi re modi f i cati on of the More general situations 9 however 9 heoretical arguments used to allow for shear wave i nter ac ti on o Furt her 9 the firs t formalism of Twer sky c'54) 9 a1 t hough allowing for random spacing of the fibres , requi res them t o be parall el along t heir whole length . The extra refinement of s cattered wave i nteraction therefore limits the flexibility of t he s i ngle s ca t tering technique as regards the orientation of each fibr e with respect to the incident wave front . '/ for fre<}uenc,es > SSD Hz SCATTERING BY A SI NGLE FIBRE CHAPTER 3 0 The s cattering approach t o the problem of sound di ssi pation in a f i bre block 9 as indicated i n t he previous chapt er 9 r equires t he solution of t h e scattering probl em for a s ingl e fibre. Each f i bre may be approxi mated by an e a stic 9 conduct i ng soli d cylinder9 which i s suspended i n a viscous 9 conducting flui.d viz o air 9 f or t he cases of interesto Scattering by soli d cylindrical objects i s considered by Lamb Morse and Feshbach ( 82 CBO ) 9 Morse(81 ~ ) and LY&mshev (IIO) o Furt her t he cylindrical s cattering problem for pl ane acoustic waves is i nvestigat ed by White(77 ) and Tyutekin(78 9 79) both for normal and oblique i nciden e on flui d f illed or evacuated cavities • . . ( 77) The trea'tment of WhJ.te allows both shear and compr essi onal wave incidence o None of t he lit erature 9 howe'ver v consi ders di ssi pat ion 9 due to the scat- teri ng of viscous and thermal waves 9 around a cyli ndrical ob j ect ; this 9 t herefor e 9 must be i nvestigat ed o 3010 Scatt ering by an elastic. onduc t ~ . solid cylinder imbedded i n ~ viscous, conducting f 1 i d 3 0110 Oblique incidence ~2 I e / / / / / / f x .. 320 Fol lowing White (77 )9 consider a plane compressi nel waV'e i nci dent on the cyli nder 9 choosi ng a system of coordi nates as shown above 9 with thei r ori gi n i nsi de t he cylinder9 and t he z ~ ax i s coi ncident wi th t he axis of t he y i nder o For aimplicity( 77 979) the i nci dent wave vector i s chosen to lie i n the x~ - p l ane The i nc i dent wave potential may t herefor o be written us ing the convention tha t thi s r epres ents a progressive wave 9 travelling in the positive X directioD 9 and stipula ting tha t 1~ must sati sfy the s calar Helmholtz equat ion o 3 0 2 0 Ot her potent ials In general as both t he f l uid and the sol i d are all owed t o s upport shear . i t i s necessary t o consider more general sol t i ons of the vector Helmholtz equati on o For certain coordi nate systems [~\ )i~] 9 by (83) . it i s possible to . repr esent t he vector pot en tials of the velocity fiel ds i n t erms of two scalar potentials:- A where w = wlt) the coordinate and ~ 9 i s 8r unit vector parallel to t he axie representing f\ (with parameter unity in t he curvi inear system) 9 and both satisfy the scalar Hel mholtz equation o Thus it i s necessary to consider nine scalar potentials in t he scattering problem f or oblique inci dence 9 including t he dil ati onal . thermal and shear potent ials deri ved i n Appendices A and B for both f luid and sol id o 33 0 treamns According t o the conventional ( 80 ~8 1 ) is possible to expand i the incident plane wave together with scatter ed and induced wave p otenia1B ~ as soluti ons of the s calar He mholtz equati on 9 in cy ' ndrical harmonics vizo X = re..£)\'.9 writing I,( ¢. '" 11\. K~ I \. .6 = K; c.... S Ll E:-~I\. f4.'VlllI.2.) ~ where the K"; s'v-.. ,0 ~ ~G,wC) is E-j\. =- ,) LVr..C"g) 5 1) l :L, (\ (I.. ~ 0 >0 (3 0122) i ons of the first ki nd and order""-- ~ are cylindrical Bessel fun and the time dependence and '\ T",(K~ ~ ~ ,0 understood o The outgoing fluid scattered waves may simi ar y be written . fur t her suppressing t i me dependence : ¢yf ¢,: """ 0() 1<2.) L e.1<~( 'l 4\l (i.\("2) ~ Y! where the ~ \\, l l(~f-\) 'to- " L UlS ("Q) ~ O t , f~ ~ g. 'B 'l\.L f. . '" ~"l( t) tD\ ~\}) ~o (3 0123) (iYCz')t c! ~#. \·L, (..¢I-) ~s. (,.Q) (~\{"Z) ~"o D;·t H... (\(;r) ~l",Q) t, = ~ are Hankel functions of the first kind and or der ~ (repr esenting outgoing . waves) . The induced solid wave potentials may also be written i n expansi ns of cylindrical harmonics . again suppressing time dependence • .,t;. r~ r; f\! t J.,\(~} ~?l'z.) ~: +s t ~C\( , 4~(v3j ~o [, "0.<> ~ ,~ ~/ T.t c.osf f. ,-.. l'" J:(\~"') L" t -4?(;~Z) wh,e re ! 'g~ Ul~(.,G) ... ~o x.s, v'- w;(,~) " e. o , ~ K, \(~ "T tos. 0~ h ), ~" 1); ~"Jl<;\') T,J'(~\ -) ul2.(t-tf') ~GtQ) (3 0124) and the z dependence of all the waves relates the s cattered and refr acted angles to the angle of incidence, assuming a "generalised Snell ' s law,,(85) to hold viz. The expressions (3. 123) and (3 . 24) i ntroduce ei ght sets of unknown coefficients . However g t here are 8 boundary conditions which may be applied at the surface (r = R) of the cylindrical f ibre viz . continuity of pressure (three components), continuity of vel ocity ( three components); continuity of temperature and continuity of heat flow . 3.13 . Boundary conditions 3 . 131 Velocity (or displacement) I t is necessary t o deduce the components of vel oc ity \I f'/ or displacement u. of the solid of the flui d from t he form + c..ux\ A ( see Appendix A) "" in cylindrical polars . Now from (3. 121) the particular case of cylindrical polars gives(83) and G., "-;z.. w(~:) ~ I (unit vector in the Z direction) rv' Thus the expression required for ~ and ~ is Calling upon the l ') c..ur-\ e-w-\ &.i"(~) cw-\ l~) and the scalar Helmholtz equations satisfi ed by ( Vl- + The expression for ~\ c..w-l 1\ ,.., ~ ~[+ K~ ') (1) . = t and X viz :- 0 A: can be transfor med to /V ~th ~ ~\PX)]+ r \j~ e[l~"+ r ~h _ -~J'tzrfH:l h L~?- J (3.1:311) 35. Th us f r om ( 30 1 311 ) an d t h e ~() exprs ~o n ~ ,,() . cy l ~n ' dr~c . al po1ars (84 ) lor t Ln . f'.. ~l()+,z. "- ~r ') ~ \" V~ the components of f uid ve ocity or s lid di sp ac ement may be wri t t en r UlM<yClI.kt e to '2.. - ?J:. dl + d).4 I~ ""-'Yo (\(.-.\.(:: + t-~e l~ ~'& ", Suhstitut 'on of the relevant poten ' a ~) (3 01 31 2) + + \{~ ~"Z.) h. ~ + d~+ + ~ 1- KT 7" \" Q'b-~"Z -M ~po.'\t-, + ~r expansions (30 123) and ( 3. 124 )? where the solid f i bre velocity components are given by t he part ial derivative woroto time of the displacement comp nets fluid field must include both s cattered for example, for continuity of t"" _ + \10( e.~,« l~ ~< (\~) ~z.) oj, { ~o c.~ ~/l'\ ~! ~<":l) and in~ UJ~l,) de n t t" ~ + Z"';\~ t ~,;: t, A> ..:' T,\<~") [ Vr =- J ~r ,;V:;,) ..,(,\)) T. . 'l\(~r) r: D!"'~ , ""t.\)) + t~ "/l~\ l "'~ \\! t <~ r,.' «~ :-, t= R t~ R LoSl/\.GJ} ",C<..l + !\.\(! ~('.Kz) [-.."c,,,,,,) { potentials , t hen yi e ds . component of velocity ,,'l.&} ... H~ L\(~t) and r ememberi ng t hat t he t ot al ~ -)~fl'} t.\)'~ t) <0' > ' J\ ~ R ] ,<>R _,~ ['l ( .JU("l~\2) { ~_[: c~ " ~ -r; k:.st)~l,-8 + . . I\¥;~ /l~~}) ll'\t I."T" l\(~st-') Cc~l/\B) l J 's underst od for where t he not a tion he cylin drical Bess el Fun tions o Thi s boundary condi t ion through with the factor :- ~I\ R ay then · l. l- I\') .>L 'J(~l " e s i mp i fied by mu tip ica i on j c..o!':.l",,\i-} " -: \iC-z. ) 0 c-Ds l MQ ) such t hat t he orthlOlgonalH y of 1\ i s invoked viz . J -fo Je l,&') Co' (""'\)) '-", "h o II Thi s has t he e f f ec t of picking out t he ~ ~ t erms ~ may be used f or s i mplif ica t i on of t he other boundary cular t he continui y 8- f ompt of vel o . t y and 0 1.(1 ~ ~ R- "'" ~;SR ~' ' \1- ().. 'i \'C ,-~ :. C-. ',.~ onditi ons t-B ~ 'R. ~ R "'- to;.~ ::: IV [ - l\( [ 2. ~ I w::;'.,.S R __ c...I S I ms g- A~ \t~ l ~ ) J - b '~ ~ J "~ lb' /-) -+ ~ ~ c.: t c:.; \ L ~ I-~ ) -+ .... ~ t>:+1,J~ -iw ~ -c:. ~ ~, \ J~ ~ -b!.~A<;'J ..llb\s)+ ~1(Sc.: T: L C.? I\(.,s»;~S)}+ ~:.J')+ - Iv-) l B;"Jb\~ r\. [f\;'J. ~') -\- ~J eo" -:\" (AI) + Al ~,(c.') ~ = + ~ T~ lc:~+ = ~ ,--' '- R b~ The vel ocity boundary conditi ns f inal y yi e d the f - t.'£- [Eo" T~ in part i- 0 b(~ g ompt. of at es s s ~(\. lM.Q) r equire t he correspondi ng or t hogonal property of Introducing t he notat i n S mi ar fact rs 0 - ~w " l~S) .. -l\iCc.tI,~f) ] -,KL;J",(L\)] - ~ : H lt:~) ] + (I~ - I<~ [1\; T"Cc:s) -+ B;J,-C..bs) J- cWl~.-) -+ \'C~ ~ ~ : *~lc.'0 - 1C~ e' 5J):,'(i c1 "J2~ a TJLS) ~) 1 37. f rom Appendix Fu' t her po are it e nts may ~ t he t o 'al f l i d s ~ Als 9 B ~ whi ch where di s p a cement s are be seen ess may be wr:ll.t t <en ~. f r m t he expressions f o,r t he gi ven in Appendi x ~ c o m p on 9 A SI . ai n pp. y eq mponen -, i n y weI e placed by v ti i .c' he 0 ~ loci y in es for he f 1 i • ha t g= i den i cal forms f or C't'I ) csr.Q ) ot:"l- potent i als are r epl aced by correspondi ng ' eris 9 where he f ' id potentials and t he f 1 i d c nst ants are r epl aced by t he relevant ool i d on ants . • ~S:._ Use i s made in ea h case of / I Intr du i ng t he further n t a t ion K'K."" &: ? ,. the - on in i f; respective y o of s t· e 6 at h e f bre boundary gives 9 e sen oi al y = l~ A: 8~J,\()+ L I\.. +- y t he proced (~-J_t}rS) ~ ~Ls'() ] + ~ [b \ S1~ ' (\JS)-+ 45 [e::'"- [C:~Jl in - 1,(~-) ] -+ + --I- A.~[\ j)~< l(~ 'D~ (\ (r:~) -14,~)] [ Gr-I\:C2 f) -~P L g J~lL\) +- - ~ c:: [c~\1 - ij;\lc.'f-) J( j (f-t~)J"lb\ - J'"-L~ J \3; [~\" - l~) (21-) - 11" l~ J ~\.J() J - \-\" ~)J t) J 3 01322 : : I "-[ A~ G ~ "J,'l~) - Jr1~!) -+ Sr\~"T(I + "l\S Y,,\(,. LL. .l. ~: [~\SJ,(bi)-'o ;"Ln~ IS)..T"rls) ,J,,\c...) - <:': '-I",-c... ~lJ\f(t) >--rtll)] - ( \ -J,,\S- - Jlm 39. 3. 133. Temperature and Heat fl ow From Appendix A, the fo lowing expressi on for the temperature vari ati on within the fluid holds : - where and which from eqn (A.20) further, from Appendix B, for the solid T ~ where ~s ,...,., - (!.S~ '!. S ... _ lWf~ ex rt. s (Tn +- ~ ~ ) v'S _ on the assumption t ha o-~ Q.I\ cl. N S 0 -= J r_ CoS "l Thus temperature continuity at t he fibre surface gives ~ [~t-r",(f)+i\:l +F~:",Jf-) and continuity of heat flow v · z . '\(~ l tQ.~ [,-r:(~f) A.th~ + (~Q-)] ( ::: -Cx'E~T"U;!.:) ~ ~T ~ " .; ~r dT~ + rb\~B!lX)} ) t":R -= gi ves - ~ Ct n' 1 B;l",-'~ ~ ') ,0. 1 33 2 ) From the eight boundary conditions (3 . 1332) the coefficients ~ 1\", (3.1313) , (3. 322) , (3 . 1331 ) and required (see Appendix E) for attenuation calculations, for normal or oblique incidence, may be calculated (see Appendix C), showing their dependence on fluid and fibre properties (Appendix D). 3.2 . Attenuation due to a s i ngle cylindrical s catterer From Appendix E, the energy loss per s catterer per unit time is given by W= for a normally incident plane wave , for which the energy carried per unit time through unit area normal to its direction may be written (46 , 47) , ~pKf "i. 1° =E b 0 40 0 Thus a "dissipati on cr oss sect i on" ( u·) i for e a hun t l engt h of cylindri cal Bcatterer parallel to t he i ncident wave fron t may be def i ned by 410 Chapter ABSORPTION OF A NORMALLY INCIDENT PLANE SOUND WAVE BY A FIBROUS BLOCK , BACKED BY A RIGID PLANE . In t he dis US B ' on of for the f i brous bloc hapters " and 2 i t i s suggest ed that a mode s i milar 0 that used by Kawasima(27)may be chosen , In par i cu ar g if ' he bl ock is rega.. dE'd as a r egi on of space ( i mbeddi ng fluid) containing an ar a:y se para e and f r ee is s ui tabla. 0 f para! e mo e in he i nc Thi s appr B.ch may t e chni que assoc ate d ' t h mode f bres 9 whi ch are c ompletel y ent f i eld g a 6 at t ering a pproach e base d on either t he s i ngle scattering 7 or he mul iple scatteri ng techni que used with model 8. The speci al eir umstanee of a ri gi d plane backing is also considered i n order to corres pond wi h t he physi cal s itua i on of impedance tube measurement . 4. 1 . Si ngle Scat t eri ng Theo r~ 4. 01 Attenuati on cons tant Ea h f bre is considered to be a .ight e ir ular t he analysis of ylinder g so t hat hapter 3 app i e6 0 If t he wave ' nci ent on ea.h f'bre is assumed to be identical wi t h t hat incident on t he b 0 k sur ace and a 1 t he fibr es are assumed paralle1 9 hen t he t t al ener gy removed from he i ncident wave front by s cattering g per unit volume ·of t he fi re bl ock 's ( NW) g i . e . the pro duct of the average number of f i bres ross ng unit area normal t o t heir axes and the ti~e ~ average of the overall energy l oss thr oughout a volume t hat is large compared with E) . he decrement di s t an ce of Thus g if E repr sen s "he he t hermal and viscous waves (Appendix ' me averaged energy flux 9 then the energy loss during t raver s al of t hi ckne ss element dx of the block is given by )2. (from Appendix E and 3, 1. ) wh ere ,?$. i s t he direction of propa gati on of t he i n cident wave fr on t The s ol ution of 0 hi s i s and hence Ncr may Writing f ": b F' ~ egarded as the a 'tenua i on coe f ficie nt of t he medium . A ~ p ~ "if · - vJt- ) 0 repr esent t he internal fiel d pot enti al f or t he mode1 9 with bulk propa.gat i on constant, t he f or m of b may be dedu.ced i . e . b s i nc e t may b e not e d ( 49b ) ha t t his de initi on of b 9 i n cl u de s only t he effects of convers i on of t he incident plane wave i nt o viscous and her mal waves at t he f i bre boundari es . To e complete ot her eff ects , ment i oned i n 1 . 72 9 s hould be i ncluded . Of t hese , t h e dissipation due t o normal wave mo tion wi thi n t he imbedding flui d has a ready been neglected in t he deri vati on of W, by as suming t o be real. Further , t he t ime averaged power l OSs ident ifiabl e with the convers ion of t he i ncident pl ane wave i nto cyl i ndric al s catt ered dilatational waves a t t he fibre boundari.es , under Rayleigh s ca tt eri ng condi t i ons may also be negl ected , as f olw s~ ~ The time aver age d power per uni t lengt h of 6ca t terer 9 carr ied out ( t hrough s urface F 9 of a l arge volume V surrounding the s catterer) by t he scattered d i l a ' a~ i onal wave .an be shown to be given by The proc ess of i nt egr ati on us ed previously i n Appendix E ~ then gives t he followi ng expressi on for t h i s t erm g- This expression is of the same order as the s econd term of W (E 010) and thus may be neglect ed by t he argument of Appendix (E 03) 0 As remarked by Epstein(45) ~ t his wave conversion will not anywaY 9 represent energy "loss" i n reverberation measurements 9 s i nce the energy r e =enters the enclosure o It s hould be ment ioned that t he simi larity of the expression for the s ca ttered energy above 9 and t hat for t he t ot al ener gy loss in the r egion surrounding a s ca t terer , gi ven by Eo1 0 or Appendix E, i s fortuitous 0 The latter has consi dered vi scous and thermal dissipati on throughout a volume , and develops (Epst ein and Carhart) to t he form Eo1 o only after some mani pulation o The "Rayleigh Scattering" expression above is a first statement of the scattered energy carried across a surface o The subsequent evaluations of these expressions must di ffer since the dissipation integral includes both incident and scattered dilational potentials (~and whilst with t he s ca t tering i ntegral 4 012 ~f f; ), alone is involved o Surface normal impedance For a single scat terer therefore ~ (3 021) refresents the main part of the dissipat ion of energy from the incident beam o 4 I<' ~ 1<" Q ± From the abov e erg ment t he sing e /' CtJ ---+ / / s a flbtul.<S - 0 ering mode1 9 predi cts dissi pati on as i n (4 )u 0 As a f irs t approxi ma ti on bl()(..\( may be cons i dered as t he onl y ~ h i5 C'-- ('ill eff ec of t he pre senc e of t he fi bres on t he propert i es of x'" 0 he i mbeddi ng medi mo Thus v i n t he f oIl wing s i mp e analyti by a homogeneous a he f or ms . u t he r : br e block i s repl a c e enua ting medi um having a pr opagat ion cons tant given by ~ = .IUC? ( ~l,( + i.TNv \,(J = I(b Ass umi ng t rea m ~ 1> t~ )~ 4", 4? C:"k ~ ? l~ ~ l( t.)lU 1- ~ ~ '-<Io~) \.{ b ;X "l- l-i...:ll:') ~ 4 ') (4 01 03 ) for t he i nc "den wave ?( -' .:Jl) fo Y(-;...:ll:) q ~l- i.J t) f OT t he for ward wav e he re fl ec ted wav f or the ba kward wave } i ns i de t he s l a b/ the l a tt er pair assume he f i e d °nside t he fi r ous bock t o be plane and compr essi onal 0 Wr i ting ~ f or and vel ocit y along he to ' al f i eld pet en i al at any poi nt 9 t he pr essure he +ve d "r e ction are given a t any poi nt by lWp} - ± ~1 ;:) ~ and t he boundary c onditions of con i n ity of norma a :at.=' -& and z ero veloci ty a t s ituati on are given by )( -= 0 9 pressure and veloci t y ne es s ary f or t he "i mpedanc e tube" Thus t he surface normal i mpedance , Zt\. ~ - I 'Vxx v" )( L:::Pct f" -=- _ & 4'Y( '; ~) \(; \ ~?l-i'; ~ b' ~b ~ ~,(-i'\& { I + -4~(J;V<b) I - where Cu~ ~ ~ A ) - 4~'. ') + 'f-t ~'Y,( may be written + +11. ~ \'cv<~&.)} Cy li>{t~) +_ 4?l;>{b~) ~ .,; >J. ...&) - t- 4I(C,>{..,&')J ~ ' 1. ~( 1 ,"\(,i') ~ is t he complex velo i ty of sound waves i n the bulk medium . ~\ Z W iting n. t he normal i nci dence abs orb ion coeffi ci.ent i s rela ed to these expressions by( 91 ) where 4 02 t-f\. and ~r\. deri va from (4.1.5 ) Mult i p e s catteri ng t heory The s ingl e s ca teri ng approach previous y de tailed enables arrange- ment of the f i bres obli quely t o he surface of t he bl ock t o be considered as the s cattering coeffic ' ent s used in or may normal or oblique incidence condi t i onso be made appropriate to either Further , distributions of variously 46. inc i ed fibr es may als :2, N<L~ by an appropriate summation vi z 9 NiL i s 'he where 0(. f i bres i nc i.ned at angl e 0( block) and ~ e b 0 i s the s atte ction o f is um e o f t he h se f i bres empl oying e>fi'i c ' ent s . s cat t;ex'ing C t!- These s t rs.i ght fo ward toget he 'th . 1 .3 are only possible . 1.5 ) however i f t he i n e r a i nS 9 waves i nsi de t he b - c 9 'liari) t er ed compressi onal Be S are neg" eo ed o Restric i ng g for the momen he pr obl em 9 0 t he ontaining fibr es para! e s cat eriIlg theory may be improved Twer sky 5 be replaced no.of(paralle ) v o~ .:1 surra e ( p er uni ng c ass t he re evant ob i q e i n i de on a bl ock Nv be c ns : d r d 9 req i r i ng only tha t p In one of norma l incidence ock surface g t he s i ngl e by us i ng 'he fi r st for malism of (54) • By thi s appr ach f r' ~ r ar p ane waves: re presenting t he and hac war d bl ock 0 s i n ternal f i e d are derh'e a s i nt egral summations of t he for ward and backward s catt ered waves " i thi n t he r ' .le,vant s ca tter ng r egion . 4 021 S: ngle scatteri ng !llDp ',t ude Ini tally 9 i t i s n e cessar of interest io eo for a cy Considering only n or~ a i ndr i .~ der:, e the s ng e s ca tt er ing ampl itude f ~ bre i n t he form used by Twersky 9 incidence g t he s ea f i bre i s g iven by : ~ 2:, At l arge r, t he asympt ti used viz . 0 form A~ l~ H~(I: f - r) ered di a ~i ona l (54) 0 wave from any ~os9) he Ha:clI:e func t ion (Appendix E) may be I " ,.,-)..'1\.'" sinc e -'<..- (,, _Q)" '- - " l Thus t he far field form of ... an ;6pt is 1.. ("~ ("I<~ \'2. ~ V C'K;r) r) 06 L: A: ~ s, .. ~'" ((\li ') In genera1 9 Twersk/ 54 ) (eqn o (2 0'7) p o 702) g "ves j{ (1(/,) ~ where Et i-{(If~r) = hi s far fiel d form as (2- /~ ) G~r)i-u,?Ul\t r/QI~) and ~ ~( 9) ) being the angle be t ween t he d Orec t ion of i n i den e and observa ti on Comparing 402 0 and 4 0202 0 402 02 0 Internal field Nowg consideri ng t he general r ansmi ssion case fo b t he fibrous ck g as s hown 9 t he f orward and backward waves within t he slab may be t aken directly from equations (3 01) (~) of Twer sky(54) th suitable limit modific ations viz o (4 02 04) ~l\>(tX)[ 4~ I t'~) + S~ 4~ ;Jl. (time dependence underst ood ) Lt?~\(J' ~ l;\({)L-~'h/ " ')[~i+t&:r)14.l(J +~JT\ Jr} )J&( 48. wher ~ s at erers 9 from (4 02 . 3) f or n rmal i nc i dence 9 and cyl ' ndr ' ca c.. IN 1(" 1 ( 4) 'i> Z ( and t he axes chos en here differ from Twersky I S 5 i.e x.chosen to lie along he fi re axes 9 and ~ The t otal i i s as shown) . er nal field tL ~ -toT- -+ f- 9 now represents a mul t i ple scattering process i n t he i mbeddi ng f lui d 9 t he f ield at any fibre at point ~ i nside he i nci den he f i bre b ock 9 being assumed t o consist of con rib t i ons f r om pl ane wave ; t he backward eca and forward s cattered waves from fibres 4. 2 . 3 ered waves from f i bres beyond n f r on ' of £ • Bulk propaga i on constant If a bulk propagati on c onstan r egion 9 s o Kllis now a t t ri buted t o t he s cattering hat the in ernal waves may be written i n t he alternative form t he second differenti als (w ,\. t . x..) of (4 . 2. 4) and (4 . 2 . 5) give t ogether an expression for V(b viz . r. r, ~_I) . 't', L e. :r.. ( (l\(! + Kb = t \(!~ 1')~ - c:-~t). - .l~I(")FL ?J ''"r"!_I , + C:-(~'lc;)1"i = - -f-r v.~ I (4 . 2 . 6) Thi s may be separated out into r eal and imaginary components a and b viz . 6 where f\ :y and Q, rv ~ and ~ i ~J'- H-IM N~ - e.}. [Qe. A,~Qo -I- t+« - 'L (\l'~ - -rMAI~ e - [~Af Aul J lMA,t-+T"!Q~I\r] here are t aken t o have appr oxi mat e values 9 ~ \l\'" Af-+~ ~\ 0 I A}- - A~ t h t he approxi mation of E01 10 which are consis t ent -: 402 04 Low concent rati ons When Cg and Cg 1 are very sma )..- K'b io e o Kb 'V "V f-y ~ K~ ~ and can be negl e c ted abov e f irs t order, .1'~ C.(j~1l t .i : N~ - (4 0207) I(~ where use has been made of he B ~ ~ m ial expans i on . This s i tuati on corresponds t o very sparse concent rations where t he effect of mul t i ple scat tering i s l i kely t o be smal l. - IN ~'L attenuat i on constant given by agai n if the ~ \J I(, -~ 0 Thi s (4 02 07) indi ca t es an orresponds to (4 01 01 ) ~ are small and may b; neglec ted above f i rst order o However 9 it should be not ed that the real part of Kb in (4 0207) i oe o the phase cons tant 9 i s gi ven by _ 0.. =: V\; -+ ~ ~ ~ t(~ S\. \l ( 4 0208 ) This does not correspond to the assumpti on of the small perturbation theory i n 401 i oe o the singl e scat eri ng theory does no t completely correspond to t he l ow concentrat ion s i tuation as predic ted by the multiple scattering theory 0 Thus the mult i pl e s ca tt eri ng theory predicts a change i n phase as well as attenuati on of the incident flui d wave o 5 00 4 02 05 Surface norma_ i mpedanc e The coe ffi c ients in roduced in equati on (4 02 05) may be evaluated , for t he fibrous block s i tuation of i nterest , by us i ng boundary conditions 4 corresponding t o the t wo di mensi onal f orms of Twersky(5 ) modified for normal i nc i dence and s ca er ers symmetri cal t o re f lection in the surface(s) of t he slab viz o lj where The f irst t wo equati ons of (4 . 209) are gi ven by dire ct subs ti tution of (4 0205) int o t he f irs deri va ive of ( 4. 204) , namely , f:!1 The thi r d equa ' on ( It:: - &) ~ t:t ( ;1(; + ~) i ~ rf:r equires that the forward travelling wave at the surface of the s lab must be t he i nciden ' wave at the plane 0 Finally the fourth equation of (4 02 09) represents t he requirement that the backward travelli ng wave mus t cease to exist at the surface ~ At (4 2 09) may be solved for 0 -:: + &. . and the results substituted in the expression for the total internal field to give 1I where t~ + t- ~ ) G-Q )~ [t _ Q).-~ ~(I - l(t)J[ ml(~c9-]\ Simi larly, the r eflected wave amplitude is given by e.ll<';x+ Q ~K"tlJ-') J 510 (4 02 01 0 ) and he t ransmi tted wave amp i ude by [tT ~(>{}s. ) l ~ ~c\ f, If now t he i mpedance of t he s ab 0 ~ ~ ube s i t a · on is '" f+ (-&,f) ~ (I _QL) 4\, [t(\(b-({)JA]J> onsider ~ by repl a i ng he part &.. by a r i gi d medi um , prov i d ng a r i gi d plane a t ~ =0 t he method of i mages may be i nvoked t o compute t he new i n ernal and ref ec ed f i el ds o For a wave t he sur face ~ = - l' '" ~(cl of l l\~ ~ ( he s l ab g ¢ ': Q. ~ he i mage of i'l;:.f).1.) ) ,in i dent on he new i nternal waves may be wri t ten (4 02 01 1 ) It is easily seen t hat t he A-t) ~ are iden i cal for bo h ~ wave pot enti al ampli tudes , he boundary condit ions (4 02 09) are unchanged i oe o ~ fI and and ~T and 11 i n . dent o Thus the new i nternal he new reflected wave and are given by repl a cing ~ r ansmit ted wave by -~ i n the expressions of ( 4 02 01 0 ) 0 The me t hod of i mages requi res that the t o al i nternal and reflected field potentials (Ix and l.) f r eferred tO g should be gi ven by and r the i mpedance ube s i t uat i on previ ously 520 i . eo substitut ing for D (I--Q) <...;[I\!>-I(n el ~ .ed~ (( _ Q....- C\(b;:,1J Q.~\lb& (4 02 012) ) (4 . 2 013) Similarly 4 At this point 9 it is ne cessary t o note that in the notat i on of Twersky(5 ), which has been used her ~ the wave potentials 11 ~ etc . correspond to acoustic pressure and not t o acousti c velocity as required in Chapter 3 and 4 02 0 The potentials can be transformed by pressure and - rwf -' ~$ where ~ '" -~ d:R. fv ~ veloci t y - = refers to t he potentials so far used in 4 . 2 ~ and ~ to the potentials required o The surface normal impedance for the layer -&~ f rom 2.1\. "'- - 'P:o:.: at ::.t.- :. - ~ as i n 40.1 0 ~ ~ '0 now fo l lows The t otal potenti al required v~ for calculation of <P;w.lC and v;e. l~or- Thus { "'/ { fo~ .J. 'r may be taken to be ei ther ~ ~? (-l~) ('_ ' t&) _ ~?(-,lf&) lK"p (( _ + ~f(- , I(/~) f \ +:"!?(J , l(btl)} (t-fQ) l t - U?(.2')(Vl ) (I-G) -t rl!.. ;r: ~ or t. (Q - ~ 1,l\koct ) Z ,e...1.'1(bJ..) l; =t~J (402 01 4) J Thi s expressi on corresponds t o (4 0105) i f (l-~) may be taken to be 1+"- the relati ve characteristic °mpedance of the material on a multiple scattering : (I!~) theory vi z . i f sea tering theor y on an i deal ised model ? the Thus using f i brous block i s shown to behave l i ke a homogeneous medium g having a propagation constant Kb gi ven by (4 0206) and a bulk density given by = fb fQ~c.J c: t ll -Q) tl +0) substituti ng for Q and using the approximate form for f()~ V\b 'Kp'l- l~ K/ + l A,~c-+ C,"(l + 9-A!c.. + \.~>") :J and 9' ~I(» The expressi on (4 02015) i s derived by Twersk/ 54 ) for t he case of a slab region of s ca t erers 0 ~ ~& bounded by an infinite fluid o It can be s een t hat the conditions (4 0209) replace g effectively in the mu i pl e s cattering case ? t he boundary conditions (4 0104) for the single scattering case o 4 2060 0 ~ Oblique Incidence Jz ~7 _ _ x 1() • . 0 .0 v The mUltiple scattering theory has g t hus far g been restricted to normal incidence on t he fibrous block o Twersky 0 s,<54) theory 9 however 9 allows more generally for arbitrary incidence o (i) Oblique i nc idence i n the ~ plane For this situation g the incident plane wave front is still normally incident on fibres with their axes running parallel to the z axis o The forms of g and g 1 are therefore unaffected 9 and the only result of the oblique incidence is to introduce a phase dependence along the fibrous block surface normal to the oX: dire ction) 0 di rec tion (parallel to the y This alters t he expressions both for the propagation constant 54. in the scatteri ng regi on and for the effec t ive relat i ve chara c teristi c impedance . These are now given by I( ~ b and v.:~> == I) J+.;N(j + ( ~ - . ehCi. Lttol" l~p ( where s.~p! I(,f- t.;Js t> ( ii ) )l-(~ 1~_ ~ ,~) + L (%-=1Jt ) + ~Kb + (. l~ +~) + lKb .lJ ': for i ncidence at angl e , r:J. ~ lKt> ~IA\. '::: ~t- K)I- cc$.o(. ~ (4 Q2 017) (4.2 018) (4 . 2 019) tI- ~ i n the.J?..y plane 0 Oblique incidence i n t he £z.. pl ane For fibres with their axe s parallel t o the z axis obl i que i nci dence i n the :£.Z... plane will i ntroduce a phase dependenc e along th em according to the general theory of Chapter 30 Thus g and g 1 must be calculated from the obli que i ncidenc e scattering coefficients ~ Ao and f. A, given in (C 01.1) 0 The alteration i n these scattering ampli tudes must then be supe'r imposed on the expressions corresponding to (4 02 017 ) and (4 . 2 . 18) to give the relevant (iii) I(b and z.~ . Departure from continuum behaviour Both situations (i) and (ii) would i ndi cate that the behaviour of the idealised model used for the multiple scattering theory , departs considerably from that of a contJnuum slab . when the angle of incidence of the incident pl ane wave i s varied. This follows from the fact t hat bot h the propagation constant and the relative characteristic impedance attributable to the model vary with t he angle of incidence of i ncident sound . 550 A behaviour such a s t hi s is poss i bly t o be expected i n a model which allows t he i nc i den t wave t o penetrate 9 gradually al ter i ng i n phase and ampli tude 9 t he i nt ernal i ncoher ent f ield represente d by ~ bei ng a l i miting s t a te o This penetra ti on can easily be s een fr om t he simpli f i ed s i tuati on corresponding Twers ky (54 ) 0 s ma ll N (as i n 4 02 01 and using 40 2 019) gi ving f r om p o 708 io e o t he propagation constant i s t ha t of he i n i dent pl ane wave modified in phase and amp, i tu de by a s i ng e =s catter ing travers al of uni t thickness of t he material o It i s t o be expect.ed t hat t he behavi our of a c tual fibrous blocks wil l di ffer fr om t hat of conti nuum materi al s t o t he extent t ha t t he blocks correspond to t he i dealised model used o CHAPTER 5 ABSORPTION MEASUREMENTS AND THIDRETICAL RESULTS Experimental Proce dure 5. • 50 Absorption Coeffi i ent The absorpt i on c oeffic en i. e. t he frac tion of i ncident sound energy absorbed j is the parame ter of great est prac ti ~ a l s i gnificance in assessi ng t he performanc e of absorbing materi als i n vari ous s ituati ons . In order t o subst antiate t he 4. heory devel oped i n Chapt er i mpedanc e tube me t hod of measurement was used t o obt a i n norma t he i ncidence absorption coeffici ent s for several f i bre glass materi als . 5 0 02. St anding Wave Method Thi s method of measurement which requires re a ivel y sma 1 s amples of mater i al ~ probes the sound field 9 gener ated at dis crete frequen c ies withi n a closed tube o The sample . cut into t he shape of a disc j i s posit i oned at one end of t he tube o From the standard t heory(93 j 95) . the ratio of the magnitudes of the pressure maxima and m i nima ~ corresponding to the "pseudo,,(95) standing wave patt ern Cs nodes and an i nodes , may be used to ca culat e t he normal incidence absorption coeffici ent ~ o : - where n P P MA l< MIN 5 0 03 0 Materials Sampl es of glass fibre qui t. (as specified in Appendix F)j in layers of 2 054.cm and 5008 cm o.thickness and 3 cm . and 10 cm . diameter . ci rcular cross- section , wer e t ested o The vari ous bulk densities were weighing a known volume of each of the sample types o ~omputed by The weight of the 57. enclosed vol ume of ai r was no t t aken into a ccount , f or t he purpose of t he theoreti cal calcul ati on (Appendi x D) . 5.1.4. Apparatus Use was made of t he Br uel and Kjaer (B & K) Standing Wave Apparatus (Type 4002) , Basically t his apparat us i ncludes : ( i) a large t ube of i nt ernal di ame t er 1 0 cm which was found to be applic able in t he fr equency r ange 300 Hz - 1800 Hz ( ii ) a smal ler t ube of i nt ernal di ame t er 3 cm'9 applicable in the frequency range 1800 Hz ( iii ) ( i v) (v) ~ 6000 Hz . sample hol der s of appropriat.e and adjustable s i ze . a speaker wi th a cone approxi mately 10 em . di ameter . a condenser microphone with a wheeled carr iage and probe tube attachment s . The speaker was dri ven by a B & K Beat Frequency Osci llator (Type 1022) and the mi crophone was connected t o a B & K Frequency Analyser (Type 2107) Used as an ampli f i er . 5 . 1.5 . Error s Val ues for t he normal i n cidence absorption coefficients were read directly from t he calibr at ed s cales on t ~e B .~ K Frequency Analyser . This procedure requires corrections for air-absorption and non- rigidity of the tube walls and t erminat ions . Furt her i naccuraci es are i ntroduced due t o: (a) t he small di s turbance on t he s t andi ng wave patter n in the t ube p r obe caused by t he geome trical shape of theltube(93 . 94) . (b ) t he non- i nfinitely h gh i mpedan ce of the probe tube openi ng . (c ) t he contradictor y r equirements (i ) and for flexi ble . fi br ous mat erials . of (ii ) an airtight sample fi t avoi danc e of a situati on in which the i mpedanc e tube walls hampe ~ t he vi brati on of t he materials constit uent fibres by compress i on. (d) t he non- plane and (e) t he l eakages and resonances in the sys t em e . g. the leakage around no - ver tic al fir~nt surfaces of the samples used t he probe t ube channel passing t hrough t he speaker cone . (f) t he possi bi i ty of t he wavefronts . generated by the speaker . differi ng from p ane wave fronts at t he s urface of the samples: this is a large problem when t ransver se modes are excited 5 .1. 6 . (93 9 94) Alt er native Met hods Kosten and Janssen(22 ) revi ew a met hod in which the whole tube is filled with c' r cular di s cs of t he mat eria1 9 each di sc cont ai ning a triangular notch in its centre 9 to all ow t he passage of the probe tube . This allows the direct measurement of t he characteri sti c i mpedance of t he mat erial . which is another quanti t y of i nt eres t . Furt hermore . the probl em of sample fi t . in the standard method . is SUbst anti all y reduced . Anot her method suggeeted by TaYlor(96 ) . di spenses wi t h t he probe tube 9 and hence the error of 5.1.5(a). The method uses a microphone diaphragm 9 ae one end of the tube . and a pieton which can be used t o alt er the effective length of the tube . as the other end . 59. H o we the standard method used was consi dered adequat e f or ver~ observing t he vari a t ion with f i bre radius 9 s l ab density and s l ab t hi ckness of the frequency dependan measurement are B mat eri al absor pti ons . ffi ci ently a curate f or Further 9 t he results of ompari son with the s ca t tering theory predic tions . 5 . 2. Compari son of alcul a ted and measured absorpti on coefficients The measured absorpt i on=frequency characteristic s are shown i n graphs 1 and 2 . Some of t he ca c in graphs 3 ~ 7. 5 . 2. 1. a t ed charact eristics are pl ot ted for comparison Low freg ency di s crepancy The absorpti on- fre q en y characteristic s calcul ated for the free f i bre model have t he same genera s hape a s t hos e meas ur ed fo r t he r elevant materials. However 9 it can be seen fr om graphs 3=5 t hat consi derable differences in coefficient magni t ude s exist a t ow f r equencies . Thi s discrepancy is greater (gr aphs 3 and 5c ) for t he 2054 em. layers t han for t he 5008 cm . layers (graphs 4 9 5a and 5b) 0 Fur her f or t he l a tt er t hickness 9 better corr elation is obtai ned t he l ess dense 5.20~o t h ~ material considered . Dependence on assumed f i bre radius The r es pective (aver age) f i br e radii for t he Rocksil materials and the Rocksi l - K (resin bonded ) mat eri als are given to be 5 p (mic rons) and 3;U respectivel y . There for e t he comput er programs were designed (Appendix D) to out put a o values f or bo t h r adii (for t he same values of t he other variables) • 600 As may be seen f r om graphs 5a 9 5b and 5c the mul tiple s ca t teri ng calculation for the free f i bre model is sensi tive to t he assumed fibre radius; however , greates t cor relat i on i s obtained by t aki ng R = 5 microns for t he Rocksil - K materi als . 5 02 03 . The correlati on i s be t er 9 Dependence on s l ab dens ' ty and he t hicker t he l ayer consideredo t h ickn The calcul a t ed absor pt i on characteris es ~ c s show the expected improvement wi t h increased t hicknesa f r om 2 . 54 em t o 5 . 08 em . The impr ovemen i n absorption with s l ab density for 2 . 54 cm. layers (graph 1) is obtai ned with t he a c lated curves 9 i f R = for the Ro cksi - K mat eri als (gr aph 6) . 5 mi crons is used The measured eurves for the 5.08 em . layers show an inversion of t he rank orderi ng accordi ng to density in the range 1250 H ~ ~ 2500 H~ calculated curves bu a lower range viz . 600 (graph 2) . A s i mil ar i nversi on is observed in the ( usi ng R = 3 microns for t he Rocksil- K materials) in ~ 1600 Hz ( graph 7) . DI SCUSSION OF CHAPTER 6 Com~r 60 1. i s on ~ LTS AND CONCLUSIONS of s i ngle Bcat er i ng (SS ) and mul t i ple s ca tter i ng (MS) The 55 f o m of he a t enuation const ant for a part ie l a r dens i ty .o = frequency cha.ract eristic f i bres on t he f r ee f i b e model ( graph 9) is s i mi l ar to t hat obta i ned by Epstei n and Carhart( 46 ) and Chow (47 ) 9 f or aerosols. Thi s indi ca es that wi t h a scatt eri ng t heory applied both to t he fibre model and 0 a s uspension of liqui d dropl etsg neither the geometry nor the concentra t i on of s catt erers seriously a lters the f r equency dependence of t he at enua t i on o ' ca lcu Iat e d c harac ter1S ' t 1C ' ' an d Car har t (46, figo2) compare th e1r Eps t e1n with measured values o It may be observed that t he di fference bet ween t hese c u rv es ~ corresponds roughly to t he di fference between the SS and MS attenuati on characteristics on t he free fibre model ( graph 9) 0 Thus neglect of multi ple s ca t eri ng effect s by Epst ein and Carhart9 even at t he sparse concentrat i ons involved g must be a great er source of error t han t hey estimat e o MS a lso provides a consi der able improvement on SS for t he calculation of absorpt i on - frequency characteristics (graph 3) by the methods of Ch. 4 0 The assumpti on of a r eal density for sound propagati on g equal to the actual bulk densi y of t he fibrous block 9 i n the 88 t heory iS 9 t herefore g a 62. severely limiting one. I t would seem that the MS prediction of a complex density for sound propagation i n f i brous medi a i s more accurate o 6 020 1imitations of MS t heory f or flexible. fibrous medi a Motion of a single fibre 6 02010 For a s i ngle f i bre freely suspended and parallel to the incident wave front the mode of vibration resulting from the analysis of Chapter 3 may be taken to be t hat of simple oscillation without distort i on . io e. that derived . (45 . pp o 180=182) b y Eps t'eJ.n droplet 0 i n the analogous situati on for a spherical For oblique incidence where dependence along the f i bre axis is introduced . end effect s have been neglected by assumi ng the fibre to be infinite o Cleary ~ length end effects are also important where the fibres parallel to t he i nci dent wave front are bound . or must satis fy some boundary conditions of contact at their extremities o The s ituation i nsi de fibrous materials embraces both vari ously orientated fibres and fibres bound or in contact at randomly distributed nodeso Therefore the theory for a single sC.atterer should be extended to include end effects and thus to allow flexural end torsional modes of vibrati ono 6 0202 0 Macroscopic effects of bondi ng or contact From a more macroscopi c point of view the bondi ng or fibre contact will result in motion and di stortion of "groups" of fibres rather than i ndividual fibres o These groups of fibres wi ll represent individual "frameworks of motion" withi n the total slab mediumo The number of bonds defining a particular framework and the size of the framework will increase, wi ·h i ncr ease i n fr eq the ma u en c i es t e ri en gt h of t he i ncident sound , llnti he s l ab wi 1 t end t ~ move as a whole o This pic ture is consi stent wi th (K o and at l ow where t he wa velengt h i s much larger t han the t hickness of ~ al he wave J . )( 2 ~ whi ch a ccordi ng refi ned of t hose 0 he t heory of Kos t en and Janssen t he di s cussi on of 02 i s t he most oncerni ng fl exi ble sound absorbi ng mat eria ls and usi ng a con ".nuous framework modelo K & J pr edi c s t ha (a ) ~ a t hi gh fre quenc "es t he flexi ble skeleton is so i nert that it does not vibra e appr eciably L eo t he air and skele on are almost decoupled o (b) a t low f requn ci e s ~ the coupling between a ir and frame is so ti ght ·t hat t hey t end t o move t oget her o Thi s means j i n effect g t hat sound propagati on i nflexible fibrous medi a should be domi na· ed by "frame a ction" in t he low frequency range and by t he a ir ~ wave a t hi gher frequencies o On the ME t heory a compari s on of absorpti on characteri s ti cs based on f i bres (i) f r eely suspended and (ii) rigidly fixed in space ~ shows negli gi bl e differences above a clearly defined lower l i mit in the audible f r equency range o Si nce the ME t heory r epresents a purely "air- wave" for all frequencies in t he "rigi dl y fixed" case ~ the correspondence of (i) and ( ii ) above a lower lim.i t i ng frequency . confi rms predicti on (a) of Furt her t he di s crepancy between t he absorpti on charact eristics predi cted by the free f i bre MS theory and those measured (graphs 3- 5c) 64. is l a rges ' a This suggest s l ow fre quencies and i ncreases with decrease i n f r equ ency. hat K & J Os pre di i on ( b) is correct and that the neglect of fi br e conta ct and b ndi ng ( previ ously discussed i n 2.3) is the pr.i n ci pal error of indi ca es 6 . 2 03 . he he pr esent equi red e heory . Thus t he discussi.on of 6 02 .1 ens on of the scatteri ng t heory. Decoupling The MS t heo y for f ee fi bres makes some allowanc e f or frame a ction. Thus the frequ en i es at which t he MS absorption curves for fibres rigidly fixed in space begi n to fo low t hose f or free fibres and give an ind.i cation of the decoupli ng f equencies discussed by Zwi kker graphs 3 and 4 f or "Ac ous ic be approximately 1200 H~ 6. 2. 4. Zw~ker B l anket" ~ & Kosten(16) . From de coupli ng frequencies are seen to f or 2054 cm. , and 750 Hz for 5.08 em . layers . Impervious coveri nge . and Kosten ( 16) ( Z & K) consider the effect of closing the surface of an el ast ic porous layer with a t hin i mpervi ous covering, for t he s i mpl i f i ed case of uni constant for an elasti porosity . Havi ng deduced the propagation porous layer , Z & K derive a o for a closed layer by alteri ng t he boundary condit i ons a t the closed surface. These then express t he fact t hat t he enclosed f l uid and the solid frame are constrained to move together at t his surface . This ·procedure cannot be used with the free fibre model , as no such frame exi s ts . Moreover, t he presence of a solid skin at x =- d 9 introduces the complication of " r eflect ed" s cattered waves into the analysis of the s catteri ng theor y (4 0 2 . ) 0 6.3. Effec h e Radius The r es i n bond.i ng i n frequent.ly be lengt hs . 0 s erved he Ro k s i l~ K mat eri als (Appendi x H) can b Ond f i bres along the ma j or i ty of the ir 0 This " cl umpi ng" of f i bres along t heir lengt hs r epresents a form of " framework of motion" not di s cussed i n 6. 2 . 2 . As each f i bre " clump" wi ll move as an individual uni t the effect of " clumpi ng" must be to increase t he apparent physical size of the f i bres . Thus ~ assumi ng that small departu ,es from acyl ndrical cross section do not substantia 1.y a ffe type of bondi ng co the d be heory of Chapt er eluded i n t he MS t heo r y 3 ~ t he effect of this ~ by t aking t he effec ive r adi us of t he f i bres to be gr eater t han the a ctual mean r adi us . Evidence for t hi s argument i s gi ven by t he results di scussed i n 502 0 Usi ng MS t he o~y "effect i ve r adi g he r efor e ~ it must be possi ble t o choose an gi ving g eates t correlation bet ween calculati on aU g and measurement f or any gi ven mat eri al , whi ch radius will represent t he ext en 6. 4. of t hi s ype of f i bre contact or bonding . Predic i on of oblique i ncidence behaviour Zwikker and Kosten (97) argue that flex1ble , porpus layers should 0 be locall y reacting t o i ncident pressure variation i.eo the velocity component perpendicular to t he surface should depend only on the pressure and not on t he angle of incidence of the incident wave. This argument depends on the high predicted by t heir theory ~ · ~amping of the incident wave, and is affected only by the extent of int erc nne e i on of t he pores i n a "si deways" direction o Pyett( 98) 66. develops a "frame" heory for an anisotropic s i tua t i on g whic h predic s considerable departure from l ocal ly reacting behavi our . pre di ' on is made by Ford 9 Landa both di l a. a 'i ona A similar and West ( 99) by a t he oryg in whi h and s hear waves are allowed t o propagat e in t he soli d part of a f l uid=s olid mixtur e as a result of obliquely inciden t waves. Locally react i ng behavi our re quires that Z should be cons t an n and tha t the ob i q e i ncidence surfa ce i mpedance absorp i n coeff ' ' ent cos 0( (97 ) 0 from which t he a.(o() can be ca l cula t ed g should be gi ven by Z n A pl ot of ZO( i n t he complex pl ane t herefore g for a part ic l a r f equencY 9 should yi eld a straigh fo r a range of The MS ( Z ~ ) -g line of slope (X ~;In) 0(. heory g pr ed1 s 'angJ,.tIll.' dependent funct i ons for both gati on cons ant and cha ac er ' stic i mpedanc e (4 02 . 6 ( iii» . t herefore evi den t ha pro~ ~ It is he MS t he ory will pr edi c t consi derable depart ure from locally rea t i ng behavi ouro In fa ct g graph 11 shows that this departure r educes consi derably with i ncrease i n thickness of layer and reduces s lightly wi t h increase in f r equency . The tendency towards locally r eacting behaviour with increase i n thickness is consistent with the extremely high dampi ng of t he i n 'emal wave ca lculated on the MS theory (graph 10 ) 9 the effect bei ng parti cularly marked at low frequencies. The absorp ' on coeffi c.' ent for t he 10 27 cm. layer cas 'e i ncr eases rapidly with ~ i oeo t he random i n a somewhat higher value dence coefficient may be deduced to have hanQ,. o have calculated va ues of absorp constant wi t h 0( 0 for thi n l ayers . Thicker layers (5.08 cm) on coefficient which a r e roughly ax Dependence of Diss i pa ion ~n6H The f orm of t he expres s i ons (C o2 04 ~ (i i dica e tha A ~ f C o 3. ~) f or Ao {- and A, 9 represents t he thermal par t of t he diss i pation 0 ( ii ) A ~ , and single fibre 's associated p i ma r i ly wi t h viscous diss i pation. ns are deduced by Epstei n and Car hart( 46 ) for t he case These cone. us of a spher' cal dr plet o Thus t he fol owi ng int erpre a i n may be pl a ced on t he r.esul t s of t he ca l cula i ons (D. 2 ) for he Db i que i nci dence s ea t e ring coef- fi c;ients. ( by °obliq e O here i s meant t hat zer o) ypi a l forms (i ) 0 f i n fi g whi h a e shown i n gr aph ' he 3 .1 '1 i s ot her than 1 2 ~- hermal di ssi pa tion i ncreases s t eadily with obli q i Y of 'nc." dence ~ , he vi s cous d'ssi pat i on decr eases r api dl y wi th ob i qu 'ty of i ncidence and apparen y ends towards a limi ting condit ' on of zer o di ssi pation a t 900 i.e o gr azi ng i nci dence , (ii ) mi ght occur ', t w' h 'he forma tion of sur fa ce waves along t he cylinder , , d ence ( 100) were h by t h ere wou I d b e no re Iat 1ve ' at grazing 1nC1 moti on between t he cy i ndri a l f i b e and the imbedding f l i d . 6.6 . of MS 6.6 01 . heory Granular media. RoW. Morse ( 14) s gge ts he pos s i bili t y of using a "microscopic" s cat ering theory for r ' g ' d grai ned granular medi a . Indeed t he wave moti.on through a suspensi on of ri gi d. spherical sca tterers may be analysed by a MS heory. However 9 s uch a model departs considerably from granular m ed i a~ whe re grain con a t i s ' ne vit ab le ~ and problems of " i nterference" of the s ca ttered flui d vi scous and t hermal waves at and a r ound each i nt er =grai n boundary of contact must be considered . Any extension to elas ic grain s i tuations will introdu ce the problem (anal ogous to Further he fibrous one) of frame wave contributi on. he problem of fric i on bet ween the grains This is probabl y gr eat er oe considered. m us~ han i n f i brous media because of the r ougher surfaces i nvolved . (i ) Materi als such a.s acoustic plaster do not lend themselves very r eadily t o a wave analys ' s of t he type employed i n the MS theory. Consi der first a general pores is not normal wave fron. 0 c ase ~ where the persistent direction of the the surface of the model or the incident (plane) It i s ne cessary to choose wave functions inside fluid and solid whi ch must satisfy boundary c onditions both at the pore walls and a t the surface of the mat eri a L interface fo r The surface would be a fluid/solid he so i d waves and a pore entrance for the fluid waves. The latter s i t a t i on equires consi derati on of problems of diffraction effects at t he edges f t he por e entrances which wi ll interact with r eflected waves f r om t he solid surfaces and radiat ion f r om the pore .situa interior. i on ~ Further fo t he case which should represent a simplified where t he pore axes are normal to the surface (model . 1 with flexible frame) 9 one f i nds an ambi guity in the wave analysis . due to the eAistence of surface waves along the pore bo~ndares . This is (100) • Numerous au hors (1 0 • t'iO't ) ( 6 ) ~ flui d wave propaga i on in elasti have consi dered relevant cases of walled tubes. Ches er(1 02) has cons ' dere d propaga ion i n a rigid walled tube whose entrance is s urrounded by an infinite baffle ; a case 9 whi ch might be applicable t o Model 1 but t ends 0 i n rica e analys is . Apart f r om this~ a less refined appro a h could neglect nea r =surface diffracti on effec s by assuming hat wi h ' a few wavelengths of the surface t he wavelets would have recombi ned as an effec i ve plane wave. (ii) However 9 t he sea ering theory does give an explanation of the poor absorp i on eharac 'eri s tic observed with stiff framed 9 consolidated medi a when the i r fnont surfaces are sealed( 103) . In this s i tuati on 9 a reasonable model is one of a continuous "imbedding" solid frame contai ning a "sus pension" of cylindrica1 9 fluid- f i lled pores whi h do not eu he sur fac e. The pores will scatter waves propagati ng from t he so i d surface and the single scatterer situati on will correspond to t he " i nverse" of that analysed in Chapter 3. The energy cal c l a ion corresponding to Appendix E 9 t herefore? pr edicts a dissi pat i on cross·-section (rJ) dependent only on t he internal fr i tion of the sid. Even for very large concentrations of pores 9 the t ot al di ssi pat i on i s thus very small except when the solid is very elastic and has hi gh i nternal losses. 6. 603. Polymer oams Many materi als referred to as flexible "foams" have an opencelled structure which differs gr eatly from models 19 2 9 3 and 5 and are consequen ' ly UDBu ' t abl e for t he application of theories based on hes m o d e~s Taki' g po . o ca n d ' stingui s h i) bane foam a s an example of s u h medi a ne w hane f oam ( pla e Ri gid p I based on s eal d of por es 3) has i ts poros i t y c ompl et el y hus correspondi ng t o t he pr evi ous ly di s cussed cas e of cove ed conso i dat ed medi a (6 . 6 . 2) . ( ii ) Fl exi b e po yure ha ne foam nsist s essentially of a continuous three di mensi onal la t ice of polygons (usually hexagons) of polymer f i br e. Oc a s i ona " s i des " of t he l atti ce are f i lled i n wi th skins of t he polymer ( pIa e ha s t he i nt er conne cted., porous Wi th such a mi cr os structure of a f i brous mat er i a l i nt er s per sed wi t h clos ed or "seal edoff" por es 9 whe r e In vi ew of (a ) he po ymer s ki ns a re concentrated . he f ac ts ha t hese ma eria l s have absorption char a ct eri sti cs ver y s i mi lar t o t hose of gl ass f i bre and (b) o tha t he MS Lang ( 10 4) . obt a1ns r easonabl e result s with an analys i s s i mi lar of Kawasi ma ( 27 ) {see heary shou d a so be applicable . However 9 t he p ob em of a cont i nuous ne t wor k of " f i bres " is there f r om the outset and some knowledge of t he pol ymer elast icit y i s requi red. I t i s possi ble t ha t an equi val ent geome t rical f orm ( q . v . 6 . 3) could be devi s ed whi ch woul d make t he s ca ttering probl em tract able . Further 9 s ca t ter i ng by geome t i cal f orms ot her t han spher e or cir cular cylinder must be solved. 71 . 6. 7 • • Zw i ~ Thi s type of absorber is menti oned by r 1 and Kosten ( 05) and I t is described as consisti ng of a rub Furrer (1 06 . e r ~ , l i k e s olid matrix con 'a i ning a random dist ribut ' on of closed por es. porous medium model i s no o f ~ . t s per ormance . ~ s suggested . A qua lita ive . t erms , ( 05) 1n g1ven 0 A particular asemn t ~ f 1n . t er na 1 f r1' Ct ~o ' n only ~ ' ascr1' b 1ng a complex s iffness ( r bulk modul us) to t he material. 6. 7 02 0 MS des n A viscoel asti c mat eri al may be regarded as havi ng complex propagation cons t ants for both di a a i onal and shear waves signi fyi ng bot h compressional ~ and shear viscosi es. A rubber - l i ke ma t eri al in very little effec due 0 to shear (1 0 7 par ti cular~ exhi bi ts compressi onal vi s cosi y compar ed with t hat due • Thus a pore- di s on i nui ty i nside such a materia l wil19 from the s ca ttering viewpoint 9 alter part of any shear wave b mode con ersi on a t i t s ( t he shear wave bei ng damped ) . in b ounda ~ ident ry wave into ~ ~ owpresinal thus caus i ng dissi pation Thi s mechanism is simi lar to t hat previ ously cited for a fi bre i mbedded in a vis cous fluid , but now wi th the solid as i mbeddi ng medium • . ( 108) VOVk 9 Past ernak and Tyut ek1n suggest the controlled manufacture of such absorbents with hi gh absorption. Their main advantage over the tradi tional fibrous or " foam" absorbents would be ? of course ? that their surfaces are impervi ous i. e . they do not rely on the penetration of the i ncident fluid wave for heir absorptive propert y . This means that their su,rface a c s as a vapour and/or dust barri er 9 these be i ng important c ons ~ . d er a ~ ' ons i n fact ~ say .n ~ , .~ ng sw . t a 1 s o Vov k e t sec ( 108) poo I s and h osp~ ~ m ons i der t he specia l ca s e of a mat eri al c ontai ning cylindri cal channels normal rad he au f a ce and 0 i 5 ~ l ' f a s t ened g followi ng t he cell model t heory of Tyu ek i n(39) (see 1 06) 0 The ma erials should still absorb g however g a.ccording t o the MS des cr i ption g. whe. her or ne t he pores cut observa i on he s urf a ce o Thi s s t a t ement i s consis ent wi th t he y K 6 en( 93 ) hat ve y f lexi ble materi als i oeo materials l i ke sponge rubber ( wi h a visc oelasti c frame) do not have t heir vering t he i r s urf a c e wi t h a l i gh abs orp 'ion impaired by Indeed g is s a e d ' coat i ng 0 hat coa ' ng i mproves the absorption a t l ow fr e quenci es g and i s not pa r ticularly det r i mental a t hi gh frequencieso Thus choosing a " s us pens i.on" model for such a viscoelastic abso rb enables a deduction of i ts absorb i ve ! behavi our i n terms e ~ of:-(i) 'he numbe of di scontinui t i es per unit volume ( i i ) their di mensions and ( i ii ) elasti c ma~ rix o he elastic properties of the imbeddi ng I n Appendix for a s i ngle scat t e r e Gg v is c o ~ an. outline is gi ven of the theory i n this absor ber fol lowi ng t he work of Chapt er 30 For simpl i c ' tY g the.rmoelasticity i s neglected g and the pore dis continuities are assumed evacua t ed o More over a phen o menol g g i~ aldescr i pt i on is used for t he v ' s coelas ic behaviour o The t heory t herefore requires modif ca i on for a more exac h~ o ry of vis oela~ti ~ and t he pre.enc e of air c ity in the pores o The pr opagat i on constant and characteristic impedanc e for a s l ab r egion of such di s c on i nuities may then be derived g following 40 2 0 The impedan e of a layer of viscoelastic ab so rbe ~ agai nst a r i gid back i ng g may subseq ent y be cal cula t ed by assumi ng t he layer to be 73 0 hom a gen e o u s ~ con ' a i °ng h e propagati n constan derived r efl e e war d and backward ( plane ) waves , wi h and cha r a c' eri s t.ic i mpedanc e as pr evi ously 0 This p a c ed he ma 0 e '~ as c pic pic of s ea men i oned i n 6 02 03 h weve 9 i ntr oduces s ome i n consi s t ency i nto re as he s u rfa ce of t he l ayer will a ct a s a er ed waves 9 hence a probl em a:;'m'o a r 0 t ha 74. APPENDIX A. SMALL AMPLIT DE WAVE PROPAGATION I N A VI SCOUS , CONDUCTING , COMPRESS BLE FLUI D F or a compress h e equa t 1' n 0 f con t 1" n U1 t 'y (71 ) , may b e WT1. tt en:- 1 e I., , ' d 9 o &;" l f r~:) t hus wher e &p~ and f. ~ lX,,, ~ . '£ : ~ ~l -+- dt ~l t hen + a-t C~to! 4- f i- ). + (~ft) de: Y..- . ~ lJ:,v'!.~ 0 (A. 1 ) If the usual assumpti on for a normal acous t ic di sturbance is made i . e. tha t al t he velo 'i ti es t he produc t s of he pert l densi t y alt ers wi h and the product (a~!-) di s pl acements , et c . are small, such that 9 bations intr oduced may be neglec t ed e.g . the s f o~ + ( ~ . ,~ ) c\ p~ , ( fot- being ' t he equ ' 1ibrium value ). m.a y be neglec t ed; then the equation of conti nui ty be omes, d.f~ fct Ai\j '! + clt 0 (A. 2) Negl ec ting body for ce g t he equa i on of moti on(72 ) i s, l JtJ"" where .. ~ 'i)' \ d~ ~' ' J - ~ h ., + ~ ~l ~ $. ~ \. It + ir~ ) I ~ ' y ct,\{ V,..,. + f~ V'l... Y, 75 . he s ummati n conven 'ion is impli ed . and the viscosity Here 9 coeff 'cien s 1 9 ~ have een assumed 'sotropic o , and the s ma] disturban e assump ' on , t he above equation of motion may be trans f o ed t o: - The energy equati n(73) may be written wher The s ma 1 and :: B con iat che r .h .s. of t rm i n At he e ergy equation, may be ignored , by the an e ass mption , leaving, (A. 4) Further, f or Thus with (7 ) u C~ )ef ( ~V) c... v (A o6) ':: '<,,, \" l~}Ft <Jp - c...v T~B - '\> - = 10 ~ ~ ~1. C!:, I f" C- / 76. ( A. 5 ) becomes and the mod" fied en f} &.;" eq a. on ( A. 4) gives U C~-0 '!... tv + lc." ~T ~ Similar y :f p "" p II I ) J - ,, / \1':>-1 ~ 0 (A . 8 ) Jt t hen <9.,\, c\.\:; ~ (01) ) d:.e. + + (u \ (j( ~l \ cit ~T ~f T (A . 9) ctt and with (A.2) and (A06 ); (00 9) becomes (A.10) a."1d from (A.6) Now and not" ng that the s al der i~ ative "th resp c the time der" va tive f d":'sturbance assumption makes t he total and partial to time approximately equal e.go ( A.3) becomes (A.11) 77. Pos t 'l a 'ing a e dependenc e exp (- i wt) v where all t he variables take on t he s· gn . f" c ane; where T' de var." ati on from eq 11'· br· urn v-alues e. g . f a'llpli val e T pre epr es& s t he t t a ~ i ous ly used g then ( Au11 ) may be written , + p} -+ ~ p .., lvJ 'II - N;w'I J~ , ~ + ;vJV""'! 'I. C'[ J( 'f..) '( ( A. 12 ) -0 and ( Ao ) may be writt n :vJ -\ -+ (J" , \I~T - (y;-;') d~" '!- ;. 0 ~ Furt he g if t he parti , e ve oci y i s writt en i n terms of 'Iv "=' - 'l rf -t- ~t ent i al funct i ons e.w\ A ( A. 1 4 ) 0 J.;-.rl\ where ~ and ~sa i "e . t · sf Y he scalar and vector Hel mholtz equations respectively l \] l- + 'ij) ~ ( q ~ + '\)..) hen f r om ( Ao1 ) -= 0 1- 0 } (A. 15 ) 78 . V ,. . , The RoH oSo of thi s eq a t ! n may be transfor med to This is ze y ( Ao 0 i. r w ~f\ - iv.wV1A,. . / ] l',......, 5 ) if (Ao 17) wher e epr es ents t he vi~ The LoHoS o of ( Ao16 ) may l gi 'V e t he f 11 'ring oY~l-w ~ onst ant o hen be us ed w"th (Ao 1 ' ) 1~= i harm ni c f or m for p- \:/"(;>-{~wN)v'tf+ wave propaga _ n ~8 + ~\w l{- I} elimi nate T and 0 1 --c P -- ',wN ') \/l-1 +l,w ~ ( whi ch can be seen t o correspond to t wo di latati onal waves satisfying \ \jl- + K \ h)f,~o l ~ + ~t) and 1~ = 0 ~\ where ~ 4 ~ f :0 and e and f repre e nting t he s ma11 quanti ties which are both < W -= NvJ and r-./ cr>- t- = ; W c:.J ~ at 1000 Hz (as an upper limit) 10 tJ ~ ~b- , [I - ~l"(~y Hence - ~ where IL _5 - (~)>- n~) + :t ] ~t se haa been made of t he bi nomial expansion and the small quantities e and f have been negl e Inspection shows t hat ~: ed above f i rst order " may be i dentified compl etely with thermal wave 790 propagation? and ~, with the usual compressi onal wave 9 i .e. one may write with f urther appr ox o I(t = (A.19) 'j> and (A.20) These two expr essi ons correspond t o t hose derived by Epstein and Carhart(46) . The expressi on for f . KT 1 8 ~so . ( 75) der1ved by Mason 0 80. APPENDI.!J1. SMALL AMPLITUDE WAVE PROPAGATION IN A LINEAHj_,ELAS!!,S,. CONDUCTING ~ The equa t i ons of moti on and ener gy balan ce for s uch a soli d can be wri tten a ~ 76 ) g~ ( Bo1 ) and (B . 2 ) respecth'e y ; where T refers t o t he steady state t emperature such that T 0 repr esent s t he di ffer ence ( T I ~ I T ) . T bei ng 0 he actwel t emperature at time t . As with t he flui d ( Appendix A) 9 postulati ng time dependence exp( - i wt) and consider i ng and s i milar mani pula t ions yi e. d f r om ( B. 1 ) :- From t he RoB oS o of this equation ; t he sol uti on sat i sfying t he vector Helmholtz equation for A gives 'V (B.4 ) representing t he s t andard shear wave pr opagati on constant i n the s olid , whi ch is r eal i. e 0 non- dissi pa ive for a linear . elasti.c s oli d with no rel axation . (B .2 ) may be trans formed to · s S _ tw f c.. " -r (l.,SQg , + l 0 10 \_ ,,-:"J lW V Y which together with the L.H.S. of (B .3) and t he thir d relati on of (A.6) yields a biharmonic equation for ¢ o where ~ ~ r) (A + J- ':: " f\ . , the isothermal d il in the soli d , may be obta'ned from setting T = at t iona~ (B.5) velocity of sound ° in the L. H.S. = ° of (B.3) and use has been made of ( A.6) . Analysing in a similar aanner to that of Appendix A, the resultant propagati on constant s luti ons may be : _(~\ where + B~ c~ c. o~) .l ~ WC1 wr it ten ~ - ~~ 1~ _l~{) l cj). ({'- I) and the small quantity ( os '-s >has been negle c ed above first order i n binomial expansions (~" Under the further approximati on that (~s 5 x 10'5 , a-' 'W , ::.. cm/sec). _ \ ) is small (this is necessary as cW,) J \ c: L (, + ~ \ u ) l~ .l(Js'6~ ~ ~ c.~ ~-i) ( .lf ~S ~ J )~ S \ + ~l'() Using these in L. H. S. bracket of (B . 3) 1.. = 0, with f S~)' - I) ( ¥ = 1\+cf~ j l c1>- SS"" c;).), the propagation constants may be separated out as ~s 'a' (B.6) 82. + 't 4\ -+- ~ 4 :l- 9 say It can be seen from (Bo6) and ( Bo7) that 9 i t is convenient t o make t he approximation ""s __ \ u ( a reasonable one for s oli ds ) 9 for then the propagation cons t ants may be i denti f i ed with di latational and thermal waves respecti'Vely 9 as in the flui d cas e (Appendix: A) viz . (B.B) and the temperature expressi on re duces to where o as for Here the resul ts for corresponds to that for ~ Ki I<: ,-S are standard and t he f or m for p C0 · G\ b ave, . I\! (Appendix A). and l''' .,. '1\ / APPENDIX C CALCULATION OF SI NGLE FI BRE SCATTERING COEFFICIENTS E1.astic Fi bre Co" o The boundary condit' ons (3 01 3 1 3) ~ (3 01 322) 9 <3 0.1 331) and (301332) fo r oblique i nci dence 9 may be collected together as ,below t [ (O-..J:.(CA'I) / { - ~ ~ ~c.lf-[€, A i · \.J~-) -1- + ~&\Y('<l) T~().\- [C,. T",\~ ~l "~(r-) -+ tb 'f ~ r ) + f\i~)J (e-.. . T..l~C.) A! K,(~ 4- + [c:., : r ~ lc.' ~ ) /' l~"'[c.fJ() '= (~ -+ t~1 ' "Xb) + ~: I[ 1\~ '§-l\(~ -KW{ t\-':T,(2~) t ~lT:Cc") ~ - ,{-) l\~ct: - ~"J;.'1 -+ I " r ~(l':"cL-Jdn l ! +8~(" 5 (f\;~ I\. il · :(~) s 'Jc:~ I'! ~ - \.\( (~ ...~'J"(o.l) - \(C.:[~ST,'2t) eft. -'I<~ ~) - "JC:') +A:[C~I().- - 1\~u:)J - 1.J~ -+ -+ - 1(;t.,sJ>~' c.~·t-v;l') A:l 6~ (~"I 14:( e.'~ -+ g1 ~ ; SJI\'(b) !:b'~ I- )+I\K, 1\(~;{2"'C. 1 I-) :f~'lIHi,\-) 1] - cl.'~- · t\~)J"L 2 1) - J,~) b'S) - 'J.J ~t) -+ - &~ - "'-) \)c.'f~j" ~b '~ J ..I('cJ't) +(' ~ ..'t ~I-) J -+ ~rL; I(:~c')-C." -+ .).. J) - 'f"Ce'l)) A!"'~ 5 ... C.2Q ) ~ H~lr.')+ - t"~V)J' - "#.(c~)] - l,(C:[~t.rf) Q:.:9 c ~ ".J2') ] + ; K ~i-l Ctl1~'c. + £~ -+ 4- ~s-rto.:) · )1 ",S~ + ' K.shc~T'l(S,) 1('... - .:,(K!)-"'lC~J T .. ~r -+ ( ~}.- + ,~j);J.(L} J - 1-~P.'Dl"(2I) r,~I)J +e,;J"l~) £-~) k~) ~l() lh'9 ~) ~M(:blH,'t-) ~ .<:c'l0T + + A-.hl"CcJ) +£ill,Jb'r)1 + ~ (C .101 ) t,1Jb IS (;_h e~'3 "'~t2 g!«.Jb'(J -It.<c!~ =-\'"{,, (A;r,~94-i\( \(c:r.,t - b S &;J,:t~9 \'- ) ~ = - -+ S_ ~ 3f\- ;T,\l~'") t - &-~':JI\CbS) i ~: (~\01 -~/hl('o\) -= - \\.J fL .~ -+ ":-H,\c' - -<; 1>1 ( c.:'T(~) 1. \1'- K')I~W + ~c. •. (~\. - .c-) 'JX2~) " - i\~"(Ll) )~ ..\ ~ ~)} ~l'T."t) / + :~,t"J)! -+ ~ J. ~!-r .. ~l)} j 84. In parti cular t he so ut ' ons f or normal i ncidence for the elastic fibre case are required (Chapter 4). ()? At normal i ncidence the axi al phase constant K i s zero , equation 3.125) and thus the Bessel funct10n arguments a If Furt her it can be seen t hat with k lose their dashes . Ve of (C.1 .1 ) represent' ng t he cont i nuity of if C",+ and Gros, are potentials yf ~ ~ C ' ) ~ g If 0 in etc . the equat ions and PiO can only be satisfied en t = C arbitrary ( r trivial viz. and ~ II J if ) ~ ) L e . the ....ave S-:. 0 I1 representi ng t he par t of t he , vector fi eld 9 , normal t o the r coord1na t e surface (77) are redundant and t herefore may be rejec t ed. Thus the equations of (C. 1 . 1) have t he i r z dependence removed and reduce to the six below :- , [to" T" lcf) 4- ~ l ~ ~ ~ lL,~Ic.-)+ A~lcf)j . T... lCAI-) l (C. 1 .2) t"lc:r~)-T +~ '" r ~ [tI. { :0 / ~ -+ ~:tl.JGn4- [b~:lf) S) rl €:.- A~ [().gJ,' s!\l~(J) - lw{-cfA!r~(S) -~). ~.J\i) ~ ) . \- l~{ A~(a.':J)+ 4 LtI. ~ A; (e:rJ~) ; ~l)] - ".J-)] 4- Di l~cP {sJ:. [fr,,'lo.'-)-rJof)J + -A:~h\(c.) - II,JIcf)] :~(J-) g:[blrJi) 'k) lc. ~ -~)Jcfl r ~ Jb~l+ 4- t>:T. (c')1 1 -) lI .. (~ t )J { -rt)-r..(Io!)] S J -J .Jt-") ( ~ ~ ~/ <' ~D T:lt~ J -+ f!, ~ Lb ~ J ..'Ch~-T( t\~r'b,{[ + !\t[bF-"~(If) \ ;\(~ -r; (~) -+ t\.~ + ~; ••lb')] -t. ~ )J.c/ + bSD:-r~l'n - \(~C.I' + + l~ ,~f-})Jt - "-"'+(A;J~) (~) ' ~ - \(~ G-b ~ -+ ~iJ\) ~tJ.-!)i,l - G- ~T" t\~ lb~)] -i = I~ ~Jb) ~lJ)1-+h ~ :L!'o.J,\~)-+ I\.l~ f~: \\"lo.()J + oK./ J:>~ ~ [~ 4"(l: -!~ - ".Jc) II .(~S{t!:\);J I\~(l)O a ti For t hi s s it on ~ 1>; the onl y possi ble s ol t ion f or Otherwi se y ~ ~) y: t h and s ixth equa t i ons of (C o1. 2) f rom t he f o ':D: and 1>; i s t he t r i vi al one are arbi t rarY 9 t heir contr ' bu tion for ~.o "\),,1 -=- 0 e vani shi ng anyway fr om t he rema ' n ' ng equat i ons . (' (46 ) This means that as in t he spher i cal case ' 9 t he zero order c oeff i ci ents are independent of the shear wave potenti al s of both f l ui d and soli d . The boundary conditions may now be written :- f [ -ro(o.') -+- f\:f\otct)] + f ()Q~\ o \n~) &~J "'" - o \b t ) (C . 2. 1 ) I<~t to. ~ [J~len-+ ~Itl)J l - e! Tu'(o.~) r~ + A-of~(c. t[~T,'lc-) \ ~ b e used for fun ~ }J ':: - r;: s c.bS~ ~tJl;(b) - =- J +~! ~ l. ~lcn + [ o.~ - "~\(/)+ r roCa.S)] + B; [b!J ~ ~ : [ctJo'(e.9)+ == / Furthe -\ \=b~&h'ol)} -' IW~-c! _ ;T.'(cl) S J~l\' S) - \n;Jo'()] l~.)c'J+ £! [~J ~ \~I) + (~)Vol} ~ l'o ~ ) + ( ~'- )"r.( b')] the fo l owing sma 1 argument Bessel Function expansi ons can ot ():~ .. c ~ l. On5 Tol;x:') 1 ,-v c...s and S <!... ; \ H Q (~) - - -:f,(lIQ) ~ -)LJul~ = also all of whl., c h are ver y small ( ""' 10- S) ; "J~l) , ~'Tl)1.+"J == - - =i hen e J o1'()Q.) ~ - ~ l1,b) where use has been made of the small argument relation (86) J.JlQ.) [ (C . 2 02) or 1 -7 ~o \ l:t) "'i . . -, M . and the Recurrence relat i ons 9 which appl y where R~ H... (87) :r.. :l R~'t1l.) ~l*-) "'- (\ Q"l~) -+ e~ ') " ~J1I'.) - ~ ~oI.+J!lQ) - :(.l"~) i s either ~ 86. and thus the se t (C .2. )becomes : 1+ \= '\St \\Jl:f) ~ [ \ + A~ ~olcI) ~t\-e!L u~ -~ /1l(C.~)-f; l~ + f\;~ - ~!"\lc.)J ;. - ex~ - ~\)l1o} lJ')] - ~SC.bl'i3;TJ) '" b~\l) 4- ; -rvlhS) rgo~ -\ A!~\.l()-u - lw h \tJ)] '" / t-i (c. S)._ (!..1 + tf\.; + \,Ss:TJ~!)} 6;(fl\"t'o)-~bJ ~ 1\; + '¥,; [ ~l Jot ~i) - \Fr, (\,l) Jj say and thus in the first equation say where X, Yt W and Z have straightforward connotations . In the third equati on , substitut i on appreciably to the coefficient of for~: f\: ;!. Y ~ and~! and -! ~) wil l not add Z. both being L..:<.. c.~" ,(c.f-) . Sl.nce -t an d -\ ~ F nf I-l,C,r) '" v are b 0 t h very sma11 ,an d ..-v_ - .l_; ; fUrt her , " By i nspe ction, as particular with terms i n the f irst bracket on t he RoH. So; and neglect i ng again t .£ compared wi t h C. and with : 1. (thi s time)- ~ - ~ ~ • ~C!} t: S where p- c.; -+ ~,lo.) LH .lb~) ~ .S - \(~b",l) bt-~ ~} .(b~) ",(o.n v-q b~ "I, (\:f) Jo(b~) ] / r ( C. 2.4) For this case 81.1 six equations of the set ( C.1.2) must be retained, 8 however. the fol lowing smal l argument forms(86 9 7) f or the Bessel functi ons . involving q.. Co.. • s e-. and s c... may be used : - 88 . ~ Q, ' ( 'I1.) - [ - ~ QI' (" » +l\ Furt her )2.l- e,l. In particular 9 when R ~ I b ec this J ome s ~ ~ ~[ t-o.l-[l+ ' -Co.~ [ 1-+ ~) J -+ ~Itl)] ~ ,I- 1+ ,tb~) f I\f ~I(- ) J - b~f '"- - &~, = L~ D/I\:lc-f ~ .=- - rt 1().: e.~ -+ A~[lt,'(o!-) t ~ r' A~ ~ r~Cc!'3 l A~[ ~ -+ tt, a. ~,Il< +(~l- H , (t~ ) _ bg vV f A t -+ ~ ,3 - \)",l~fJ f -{) + ~.' Ilf) _ 11)11 r[I'(I) l(I)04~ \l ,It. J -\ ~I ~", ~ - ", ,S T,'(~ \ )41(, ~ T, \~ " } - + O,~[b·'(l)+ - ~ t J)~[t.I\ \ - .1t'"~li) t. ~)J1 j = S~A( C\.~O 9 s i nce If I i s small (approx . 10- 7 at 1000 cis) , ~ r l~ J- .~o"1l:} ~\S(,s-'l" + "'~!t. From thefi rst t wo equat i ons of (C . 3 . 2 )9 i t is pos sible t o vri te, As with H,('J)j ~ J.\- ,k')] [ "J,'lt')t(t' - I}r, (b ~ \j 1(';; ~ ? 1 (~1 _ 1) [(,~Ilc.) J \ ~~ ) + "'"lI-D,!- ( = 1 ) :- 1C ! ~\, s eh~(b1) - H,'CIJ) -+ "}'l\~ -+ e,h\lb'') - - ~.f 4- 3 J, ( ~f) = - lW{ - o. S i~ ct-'t A!",\~) R: (>C) o.~ H- r = 2 f or ~ H~(S,'91)} -+ ~) J - R..,()Q} l ~ lC...- ;>' 0 The set (C.' .2 ) may then be wr' t t en (where -+ l.e,c~") '~ I f- [e5--\ I\~ \I}~ ~") - ~)Q.,l - t.\/,) ' ("\) - ~,Il1 7r 1>,~ ~ SUbsti tut i on of (C o3.3) i n ei t her of the fir st t wo eq ~f (C o 3 o ) does not appr eciably affe ct the coeffici ents of c..i- H, (cf) ~ -~ t ) or the constant t erms ( "'i\ ) Ct t L f M. _1If- may of ( for i ns t an ce be writ ten then and the smal l size of I I (~IO to indicate that S~ also has a negligi b e effect on 9 ~ ons 0 Fur t her from the f i f t h equat i on cf (C o30 2 ) if ~If. ~ u at at 1000 cIs ) may again be cal led upon he l as'l:; wo equations of (C o302) o Thus f r om these, It can fur t her be seen , that i n subtract ion of the l ast two s~ equations cf (C o3 02) and neglecting ~ at 1000 c/s) , ~ ~ compared with 1 (<,-s I' 1. 10- 1 :1- l- e:., terms vanish anyway , and .-....' terms do not contribute anything appreciable compared with other terms (by the previous arguments), thus, AI' c.~i4-r ~fc. ~ S "1rS l\s e r!l~ ~ I-Y )- \lJ'>k~ + i.e. in the second equati on of (C o30 1 ) , r1 T ~ \ \ let) - K '(~c.f-p" ~~ \ 'Cc.t-) \ .c - r-~c!\If) ~ L l J~.v S~). f'C- '«-!}~ 1- ~; may be e i minated ) Llo.' c.. 9 /- , \ r-:~\tc) - __.l.. \ ~ - "t c. ~ ~\lC);tI-D • / -_ L ~C.3 giving, finally , with further use of relevant recurrence relations, . 6) . '\ 90. I t may be not ed t hat he PI" ce d r e us ed f or Act more r efi ned argument g i n ea ch i nvolves a some wha ase g t han t hat ~~ employed by Epst ein and Carhart (46 ) f or t he spheri al Where g f or the c orresponding through by I<~ f responding ~\ 9 Aor 9 and t he t er m i n A: a.bove and (flui d/fl uid ) . 'he hea t fl ow equat i on ' s di vi ded .t r t hen neglected ; a..'1d for t he cor - , t he temperat ure and heat flow equa tions are made i denti cally zero , negl ec ting B; , and ~: , terms i n t he remai ni ng equat i ons. c. 4. Fibre rigidly f i xed i n spac e For investi gating the effect of vari a ti on i n t he angle of i ncidence of the incident pl ane wave on the s catteri ng coeffi cients At , it is c onvenient to c onsi der the simpl ifi ed case of a f i bre rigidl y fixed in spac e. This case i s also , of i nterest , when consideri ng t he effect on the absorption characteri stics of a f i bre bl ock of resin bondi ng (Chapter 5 ) 0 For a rigi d fibr e , the potential s the coefficients A,.,':. 9 ~t\ , L; f~ 9 ¢~ , ~f ' and"J); are all zero . and S X vanish 9 Le o Thus if the fibre is also fixed in space , the boundary conditions whi ch may be applied, are merely , continui ty of t emperat ure (i . e . zero vari at i on) and the necessity for zero veloc i ty (di splacement) at the f i bre boundary. The temperature gradient is not ne cessari ly zero a t thi s boundarY 9 however , the condi tions cited are suffici ent t o evaluate the Af From the set (C . 1 .1 ) , the relations (C . 2. 2) and t he argument of Co2 which concerns the ~, , t he rel evant condi t ions may be written: ·· 91. f[ ~ [~ - \~[ ~ o ~H J~)] I -t -4- A:H\lc.'~) -+ \ + AJ Hotc:~) t= ;HJ~I) ~ = ] -t\}~SI(bl) - +- ~"I-l\o) ] + (I:~- ~l-) \(c. 0 \ ~cl1.r)", 0 1 J (c . 4.1.1 ) LJHol21-) ~ 0 From t he f ' rs t eq a ti. on ~o ~ - fHolb\~) ~ S I + Ah.l~f) 1 ~ which can be subst it t ed i n the remai ni ng t w equat i ons , negl ec t ing ~ terms in of Co2 , givn wherever possi ble i oe. i n a s imilar manner t t he procedure : ~ rL""A_I~ r'6\hJ,tb~)] _ ~ l. - + Afoc.t\-lG~/'·J( 11 H.l'o'r-) I' 'r 1\ r\)= D l~ [ Hence ~P - _ H'~,l1()]-\ .l. r - ~tJc.'f) L " lV(,h_\~)Hoi!- ,,:>-c...\f-H,lc.'I) ( 'I:/"-W()H.l~f ~ "olb't) - ~" "c.t~) ".(~') J Co 4 2 o 0 Similarly from the set (C o . 1) and t he rel at ions ( Co301 ) - ~[+ ~ - l A\~flt)J + A,!-I1\~)J -+ ~t"J) - Io\fe,;tl)~ -=+ 0 ~Kc:.'H\(ef)C} + V(T~f>/ \-l(L~) '=- 0 ~ C.401o2) 92. The first equat ' on of thi s se t ! again shows that the effect of (),fin the other equati ons is ent i re y negli gi bl e . Thus from the l ast equation of (C . 4. 2. 1 ) . L \' =- I (~_'«). ~ [i!.I~ \ IW'-) + Atl-\o~) 1 J SUbstitu ti on of this into the sum of t he second an.d thi rd equati ons of (C . 4. 2. 1) then gives , after rearrangement fand finally? substi t ution for both L, , and~, ~ i nto the thi rd equation gives: - Aft (C . 4.2 . 2) C.5. Normal Incidence For ~ =0 the boundary conditions exi s ting and required reduce to three and four in number for expressions for ~: putting k = 0 A,~ and (l = 0 and 1,\ " \ , respectively , and the for normal incidence are simply given by in (C . 4. 1. 2) and (C . 4. 2. 2) ; and removing the dashes , viz : - At- A-- 0 .1 "'~ and ~ _ c..b \ f + b~\() c..I'-\\ ,leY) \= l1J.b~) ll..V) (l \ Cc "'--' Ct\ lr!-) + J-~ \ Lt.I-) \\,'kf) ] 1. H-Jt.~ ) 6 o ' he a ppr oxima t i ons je t nvo v Ylg ma e p evi A~ C06. The e ffe t of t he s at erer proper ties on the It i s easy 0 s ee tha s yo f\ " he expressions ( C05 01) and C02 . 4) f o nor mal in .idence differ only as F"oll,t)di ffers fr m S f Thus allo'w' ng !S.s -;> 0 i mmedi ate_y makes t he two expressi ons i den ical 9 V\ i 06 0 al, owing the f ' bra A~ dependence of It s ~ on a become i nf ' n ' te y cond c he fibre p so of in erest Now per i es o exami ne t he effect = and using (\ b!. -> Thus as he asymptoti ng 9 removes t he +~ ) ( ~: f l - ';> 06 I fS) ~ «. 06 f orms of t he Bessel funct'ons i ntroduced i n Appendi x E9 and i oe o Hence 9 t he expect ed r esu t t hat t he va ue for flt t ends to that for a f i br e rigidly f i xed in space as t he density of t he sca tt er i ng fi bre i ncreas es . C. 6 2 0 Comparison of t he expressions ( C. 5 . 2) and (C . 3. 6) for , 'd A/' a t normal ' 1nC1 ence 9 shows t hat they d1ffer f r om each ot her onl y as t he t erm l~/ roSe).. di ffers from zero . Now lWrc..r '). r ~ ;w s .... c; t! l( ~r r . ) (~ (7t ) ~ ~ pt- l R1- from t he expressi ons (Append1x A and B) for (", ~ ) v.: T ~ of L f , (.. S and t he definit i ons (Chapter 3) . ~ Hence 9 again t he expec t ed result t hat allowing e. -') 0 p~ implies the r educ ti on of A,f t o the value for a fi bre rigidly fix ed i n space . 95. APPENDIX D CALCULATION OF ABSORPTION COEFFICIENTS AND SCATTERING COEFFICIENTS; COMPUTER PROGRAMMES D. 0 Normal Incidence scattering coefficients Using the expressions (C2 . 4), (C3.6) , (C5 01 ) and (C5 02) , from Appendix C, together with the approximate expansions of Besse Functions given in Appendix F , t he real and imaginary parts of the relevant scattering coefficients for an elastic fibre and a rigi dl y fixed f i bre may be calculated as follows :Elastic fibre - " N ( (~ ~ ' K~ Ao ::.. - W8 ) J- ( Al- -+ -1M. At-0 _ f>- -= - " N l wA + (Z€:,) + ,,0: .l (A~+ If ~ where IN ~ ~ A and ~ where ~y ~ L R~ :0 CbP-~,(\! )J 'ttJlb \ - 1\ == II and J ef (I - ~ k) -w e..(- -=- ~>- =- ~ I M t4. o (.kF) [tP H,('o~)J 1:M [ T.(") l S b T\ (b ) ~ ) -iM. At\ = po- W ' r 1:~ = -+-2 ) J.. (flll-4- BI>-) where - M lZZ) + ' S ) ~ kotb~) : Re.- (\\IY R<- [ '.Cb') -= 5~) (WZ.-RY) -M- ~ ""- Q.. L () ~) ) ~ I \t- J.y (1 -~ t.') + -SI R.~A i~ c- C Al -=- J...'X (l-~k) X = R<-l - \ - gl' ~ ,(e') J e!'HJJ) 6' y = -:: :rM[ H,lt') J c..f uJt) J R~ R· s i d f i br (! ~} - TtN = "LV\ ~ o ~ - - ( ~L.+vJM) "- (~+ fv\l.-') "J)- - liN ((ZM J-(t:+M~) J.t- Itl~ 1<..e.. hy - 1\ 6. (:1)(- \ )~ D.2 WL) + 4- yl.- Obl i que Incidence s cattering coefficients Simi l arl y fro m t he expressi ons (C4 . 1. 2) and (c4 . 2. 2) t he oblique i nci dence scat t eri ng coeff i cien ts for a f i br e rigidly f i xed i n space may be calcula te d as f ollows : - Re. A: : A where tA + \3.]) A).- -+- ()l.- C. -:: -lM A0f ~ ~+H \-).. - c..: - E... +G. ' .1- ~ -=- - R~ \' i ~ I- Il.~r) L~ ] ~Jbl) ) t= '" - Ifv\ J) [£~ bI t· ",t~)J 1\-,,( b't) and the appr oxi mations H~ l~ + ~ c.:; ~) 0 ~ ~ (+ b~ - 1-\ - ~ z.. ) H Cx-L - ~ 1\ 97. have been made ~ whic h rely on the definitions of tw~ cr~ Chapter 3 and the fact that ~;. c.:;>- and b'r-~ given i n <:..' f- are small quanti ties. This is an equivalent approxi mation to the assumpt i on that v(J real. Similarly f<t A\~ ~ - ,~ ~[ ~(b+J.Y) - blo-.+J?< -J)] lz.' where c! H:(J-) H.cd-) -z.. (0--+ .JX-()~ I ' t10n ' 'b~ and th e approx1ma concern1ng Do3 + (b+~Y)" ? is agai n assumed C Absorption Coefficient The Twersky theory in general (see Chapter 4) gives : ~b .... ~). '=' Ks> - .iN where K/ c.o~ 4-~Q and eX. -+ C-l{+~)(J" ~ ~ ~l A~ 1 ~I = ~\I lY'~: The form of g is t o the approximation suggested in Appendix Eo = ~: and I A~ may be evaluated from D.1 and thus v:~ - ~ - ~ : N ~ 0... + ~b - k-c.l- fit A~ is 98. where generally ~:>- + If---N - tpi RQ.. ~ I~ - - 4-~} ?{-c-}- [R [~ ~ A!Q,-~ ~ ~} - I-1A~l& ~J fI; Ql1. AI~ J Il.dr ~ + T~ To be consistent with t he previ ousl y used conventi on for a forward travelling wave (Chapter 3 ) g(a) must be real and positive o Further from Chapter 49 t he r elative characteri stic i mpedance is given by ~~ ) +~ J('Qe.A~ cl - b and the surface normal impedance Z ~ fI.. where f/ C} (I{ + ~ W) ( (( + ~ ~) {' I + f I - .e:-.l\:'J (lo$. e. - .2~J (~ ~c.! \S~ -+ Jo-.& -+ J \~ J~cl) ~cl)] Thus finall y 4- (vI-~w) CVR - S, W + \)~ The following comput er ~ogram. English E1~ctri -+ (R. vJ + V ~ )~ written in Al gol 60 for use with -the - Leo- Marconi KDF- 9 machine at Leeds University, was 99. Used for calculation of normal inci dence absorption coeffi c ients a ccording to expressi on (D . 3) . The various materials for whi ch experi mental value s existed , were typed in terms of mean fibre radi us 9 slab density and slab thi ckness and t his data toget her with the constan t s tabled in Appendix F were input . ~o It should be noted that the program , as wri t t en , outputs va l ues of ' for a given slab densi ty and th i ckness, for each of the values of mean fibre radius fed in . 100 0 This program was modifi ed to calculate absorpti on coeffici ents for oblique incidence in the XY plane 9 by including an extra loop for 0 0 and by replacing t he va ues of i ncident angle from 0 to 90 expressi ons fo r Kb 9 ~o as i n ' D03 by devi at ions c f them ' V and W 9 9 based on express i ons 4 02 01 7 to 402 01 90 Similarl y a progr am was wr itten to comput e oblique i nci dence s cat t ering coe ffi c: en t s ac cording to the expressi ons (4 02 01 7 to 4 20 19)0 0 . begin comment , Calculation of normal incidence absorption coefficient against frequency for fibre glass by a scattering theory developed by Attenborough. Variation with fibre radius -: slab density and slab thickness is also considered.; libra r y Ar', A6; integer i,j,k,n,m,s,kt,1,f O,f1,f2,f3; real w,gamma,rof,ros,kf,ks,cpf,cps,muf,mus,pi,rog,b,af,Cof, Cos, N,nuf ,kdf ,C,es, c, Z, Y, W,R2, L,M, A,B,ReAOf, ImA( 'f, P ,Q, S, T ,LL, YY, X,x,Re A1f, ImA1f , ReRigA('f, ImRigAnf, ReRigA If, ImRigA 1f ,yxz ,V, gg ,WW,RR,SS,kfg,a(' ,NN,Reg,Img,d,e,ff,g,mm, pp,AA,BB,AAA, BBB ,sigma f,sigmas,delta,a,bb,ll,yy,xx; open(2(' ); open(7 n ); f O:= forma t (12s+d .ddd1o+ndl); fl:= format (12s+nddd.dddddl); f2:=format (12s+d.dddw+ndcl); f3:= format(12s+ndddl); Cof:=read(2 n); garnrna :=re ad(2 (\); rof:=re ad(2 (1 ); rog : = ros: =re ad (2 (\ ); kf :=re a d (2 (1 ); k s : =re ad (2 (1) ; I cpf: =r e ad (2 " ) ; cp s: =re ad (2 (1 ); Cos: =read (2 0 ) ; muf:=read(2 (1); mus:=re ad(2 () ); n:=read(2 (1); m:=re ad(2 0); s:=read(2 (1 ); kt:=read(2 0 ); b egin in tege r a rray f[l:n); arra y R[l:m], t[l:s], for 1 : =1 s t ep unti l n do f [ i] : =re a d ( 20 ) ; for i:= 1 step until m do R[i) :=re a d(2 0 ); f or i:=l ste p until s t[l] :=read(2 (1); for 1:=1 step until kt do ro [ 1 :kt]; .. do .. ro[i):=re ad(2 ('); close(20); p i:=3.142; nuf:=muf/rof; sigmaf : =kf/(rofxcpf); sigma s:=ks/(rosxcps); wri te text (7c.',112cli12s1J12s1112s1k12s1V16s1WWl6s1a16s1 bb[6s ]a(' [ 4s] ImRigA 1f' [2c]]); - - -------..- -- -----= ... fo r begin i:= 1 step 1 until w:=2xplXf [i]; n . do N:=( gamma-l)xwxs i gmaf/Cof i 2; kdf:=w/Cof; for j:=l step unt il m do begin b:=sqrt(w/(2xs i gmas))XR[j]; af:=wxR[j]/Cof; cs:=sqrt(ros/mus)XR[j ]XW; c:=( w/ nuf)XR[j] i 2; Z:=(-12-bi 4/36)/(48+b i 4); Y:=(48+2Xb i 4)/(48xb i 2+bi 6)x(-1); x:=wXR[j] i 2/(4xsigmaf); W:=2/pi+x; xx :=(ln(sqrt(w/sigmaf)/2XR[j])); R2 :=2x/pi-4X( +( ~ . 5772) ; L:=O. 5+(xx+(). 5772) x2x x/pi-2xx/pi; M:= (~ . 5xx- (xx+(' . 5772)X2/pi; A:=L-kf/ksX(R2XZ+WXY); B:=-M-kf/ksx(R2XY-WXZ); delta:=w/nufxR[j]i2; , '- . , Re P..( l f: ::::-piXNX (R2XP..- VJXB) / (2X(A i 2+Bi2) ) ; ImA(\f: ::::-p iXafi2/4+piXNX(VJXA+R2XB) /(2x( Ai2+Bi2) ); (l• • 5772+'1 yy o::::( • r • . • "J~t;vln(('»_ln(2) \ I -- I \. J- j P:=c/(2xpi)-C/pixyy+ct2/32;Q:=2!pl-Ci2/8xl /p ix (-5/4+yy)+c /4; S:=c/pi-c t 3/48xl/piXYY-C i 2/16; T:=4/pi+Ci2/(16xpi)X(4xyy-3); X:=(PXS+QXT)/(S i 2+Ti 2); LL:::::mufxw/musxCS i 2; AA:=2XXx(1-deltaxLL)-1-deltaxLL; YY:=(TXP-QXS)/(Si2+Ti2); BB: =2XYYx(1-deltaxLL); Re Alf:=-plXBBX(1-deltaxLL)xafi2/(2x(AAi2+BBi2)); ImAlf:=ReAlfXAft ! BB; ReRigAOf:=-o .5xpiXNX(R2XL+WXM )/(Li2+Mi2); ImRigAOf: =-pixafi 2/4-piXN/2X(R2XM-VJXL)/(Li2+Mi 2); ReigAlf:=-pxa2XY/()~4t; ImRigAlf:=O.5xReRigAlfX(2XX-l)/YY; fo r 1 :=1 s tep 1 un t il kt do beg in NN :=ro [1 ]/( p iXR [j ]i 2xrog ); C:= 2xNN/kdf;Reg: =ReAOf+ReA1f; Img : = ImAOf +ImP. 1f; Aft fl :=kdfi 2+4xNNXlmg-4xC i 2X (Re A1fxRe AOf - ImA 1fXIm A()f) ; BBB:=-4xNNxReg-4xCi2X (Re A (~ fxIm A 1f+ImAOfxRe A1f) ; a:=sqrt( O.5X(AAA+sqrt(AAAi2+BBBi 2))); bb:=BBB/(2xa ) ; d:=kdf+2xCxImAlf+a ;e:=2xCxReAlf-bb; ff:=kdf+2xCXIm AOf+a;g :=2xCXReAOf-bb; for k:=l step 1 unti l s ~ beg in yxz :=exp (-2xbbxt [k]); 11:=1+yxzxsin(2xaxt[k)); mm:=yxzxcos(2xaxt[k]); pp:=1-yxzxsin(2xaxt[k)); gg:=gi2+ffi2;kfg:=ppi2+mmi2; V:=(exg+dxff )/gg;vnv :=(dxg-exff)/gg ; RR:=(11xpp-mmi2)/kfg;SS:=mmx(pp+ll)/kfgj a l' : =4x( VXRR-SSXVlVl) /( (VXRR-SSX\oJltl+ 1) i2 +(RRXWW+VXSS)i2); wrlte( 7l) ,f3,1) ;wrlte(7n ,f3,j) ;wrlte(7() ,f3,1) ;"'1!'lte (7 0, f3,k) jwrl te (70 ,f 1, V) ;vlrlte( 70 ,f1, WW) ;wrl te (70 ,f1, a ); wrlte(7() , fl , bb) ;"'1!lte(7 ~" fl,a O ) ;wrlte(70 ,f2,ImRlgA1f); end; Reg :=ReRlgAof+ReRlgAl f; Img:=ImRlgAOf+lmRlgA1f; AAft :=kdfi2+4xNNxlmg-4xCi2X (ReRlgAlfxReRlgAOf-ImRlgA1fxlmRlgAOf); BBB:=-4xNNxReg-4xCi 2X(ReRlgAofxlmRlgA1f +ImRlgAr'fXReRlgA 1f) ; a: =sqrt ( () . 5X( AA A+sqrt (A/I./li2+BBBi2) ) ) ; bb:=BBB/(2xa ); d:=kdf+2xCXlmRlgA l f+aje:=2xCxReRlgAlr-bbj fr:=kdf+2xCxlmRlgAOf+aj g:=2XCxReRlgAOf-bbj for k :=l ste p 1 until s do beg in yxz:=exp (-2xbbxt[k]); 11:=1+yxzxs in(2xax t[k]); 1nIl1 : = _ y~"'{zxcos (2xaxt [k] ) ; pp :=1-yxzxsin(2xaxt[k] ); gg :'=gi2+ffi 2; kfg :=ppi2+mrni 2; V:=( exg+dXff )/gg ;\VW :=(dxg-exff)/gg; RR :=( 11xpp - rnmi2)/kfg; SS:=rrmx (pp +ll)/kfg; a 0 :=4x(VXRR- SSX\fW) /((VXRR-SSXWW+l)i2 +(RRXWW+VXSS) i 2); wri te (7(1,f3, i) ;\,lri te (7{) , f3, j) ;wri te (7(1 ,f3, 1) ; write ( 7 (1 , f 3 , k) ; wr it e ( 7 ( I , f 1, V) ; wr i t e ( 70 , f 1, WW) ; wr it e ( 7 (), f 1, a) ; wri te(7( ' ,fl,bb) ; vvrite(7",f2, a n); end end~ end end end end ;close(70 ) 101 . APPENDIX E. Eo1 . ATTENUATION DUE TO A SINGLE CYLINDRICAL SCAT~ The Di ssipation Integral Briefly outlining Epst ei n and Carhart Os approach 46 ) fo r fluid spheres , the time average of t he overall ener gy l oss consists of viscous and thermal parts ·· viz : = VJ where and W r + w" W = f Jf ~, Wo- L( ~ . I CV'T) ~v -=: \\J 'l J~ ~ i n which N s i gnifies t he comp nent of ~iJ J" \,? \f . ~'2 -\- I\v A\I A~ ~-J i n t he dire tion of t he out ward normal drawn from the sur face F of a large volume V surrounding t he s c a t t~re concerned. and 'V N' conta ined i n the time average are complex (-iwl:) ~ and have time dependence ~ i and i ntroducing the complex number nota~ Rememberi ng that y~ •• The energy l osses ar'e gi ven by t he integrati on and time average OJf the viscous and t hermal di s sipati on functions ( \~ and 1 i s t he di ssi pa t i ve part of the t otal stress tensor) respectively . Tbe viscous dissipati on function T~has R~l&igh ("Theory of Soun~'vl used by Mason (71) . o the standard form introduced by I . Ch. 4) . the tensor eXl,reasion above being The t hermal di ss pation function and Fine (Rev oMod. Phys . gQ. 51 , 1948 )0 ~r i s derived by Tolman 102 . the time mean can be cons i dered from as t he t er ms i n e - .:Ii.we and e ,l ' w t C ±R~(x \ ) vani sh in the : me a.we-r aging . Thus t h e dissipati on expr essions above can be writt en i n Wr ~ ~ t IF "t v ,~r Wo- iT b R ~ ~ K~jl'\1-(3}f -+ f v ~-fr - KLI~V ex {'oJ. ) ~ e~ = he forms ·~ ~v ~\ 1 c\v~ The analysis ( 46 ) shows t hat t he sum of t he second terms ( i. e . t he vol ume i nt egrals) of each expressi on i s zero whils t t he fi r s t t erm of V~ i s negl igi bl e l eavi ng t he total W ~ ~ RIL J V .' ~\>N· d..f (E. 1 ) .1 f J ~ i n which t he time dependence is now suppre s s ed . This argument i s unaf fec t ed by change from a s pher ' cal t o a cyli ndri cal coor dinate s yst em , t hus for a s i ngle cylindrical fi br e choosi ng V to be a l arge c mcentric cyl indrical vol ume 9 r adi us B and surface F, the i ntegral (E . 1) can be evaluat ed agai n f llowing Epstein and Carhart (46) wher e the contributions of t he surface i n egrals over the i nt erior and ext eri or surfaces F~ due to the continuity o f .both t he ( t he ~ K~ and F~ V· ~ of t he s ca t ter er c an cel ~ and))N ' acros s t he s catterer boundary ~ bei ng equal and opposi te for F2 and F ) . 3 If the radius B i s chosen sufficient l y l arge t he hi ghly damped ther mal and vis cous potential s will not contribute at F1 and i n the expressi ons f or '/4 an d ~ rN~ ~ vi z . (3 . 1312 ) and (3 .1 321 ) terms i n t- may be n eglec t e d c ompared 'Wi th 't erms i n l"" a ga i nst t- -\ terms 0 9 and r _l- terms neg ected 0 Thus . t i s poss ' te e to ' and wher e F r , arge r, obvi usl y ompare d with to obtai n on ~ vtYrr ) he product v :~s a s it decr eas es wi t h I n general ( 89 ) the e , ement of are a on i n curvi inear coordinates ~ , '~ tt f i. e . for cy i ndrical polar's Furt h e r t h en and 9 de fini ng ~, }.) ~3 may be negls ~ ). (e .g. 9 t hus it i s r e quir e d he s urfa c e ~ , w' t h parameters \-... \ ) \.- ~ :: ).~ k 3 ~ ~ ~ ~ L -= 15&t'JA.'Z. and , ~d [, QF , :: f, -;:0 ) h.~ cons t . is J"l « J.Q J\.. cl~ 0 • 104 where 0 1\.. '" 0 1\.>0 Now assuming K~ .. d " a t a i onal wave ~\ is r eal i oeo neg ec i ng t he order of sma >.1 \ ..- ~ "At< LI ~ 04 Then usi ng t he r ecurrenc e x'KX)+I\.~l and t he + ~: ~ (~IS)J re l a t'o~ -.:. ',(~ 8 7) ... _,(x) . wher e i t ' 10n8 (90) 9 arge argumen t · approx_ma H;'}(l<) l ;x)~ )( ~ .~ Co3 (;x)I- ~ :0 f he f id q ant t' es g see Appendi x A) , [Go" TI'(~Y,) -r"G) ex) Sill'dlarly he dampi ng (~I\ - ~ ~'yr," [~"J,lYiB) + ~'1 R,, ::: J" G- t1\(~ C)( - ~ or H" J ~) \TI (I\+ ok)] - lj~ - \ - ~ -r")~, \I:(C~ri)J The secon d t e m vanishes anyway for (\ ... 0 ot herwise i f of T. . . and H" and may be n agle ; ed he condition ')( - > 00 ~ which allowed t he appr oximate f orms l5e be 01 d ~ a i ns 0 e (as I(;~ It shou d be n ote d hat i s small ) vi z . \OC.~ B "" equ i r e s B hi s argument 0 'S 1 0 \cc..(S and B rv 10 c..M . a Further f rom t he "generalised Snel be very large a · 50 "f(c. ~ . • s l aw" of (}o 125) and also f rom Appendix Ao thu s us i ng where t he ~,~ : . - ""I'I:) C ( E o3 ) ( - =0 ( he fit: hav e Z· ~ - r- , p..... (E o 8 ) n) U( K~ E 08 ) i n (E o2 ) W ~ - .1wfof- (I + H-S~@ t h eir ob For the part i e K ",~ .,~ ~) LQ~ .6 [~o ~ (1\1 + t,~1\ q e ' ncidence values ( Appen di x C4) ar e as e of n or mal i nc iden c e r~ C1Dmpon ent s of t he v~ and '\>r{J nor mal i n i dence va u es (Appendi x C) and q uired vanish) the ~ A/) J 0 in Chapt er 4 9 A~ ake on t hei r he t ime averaged ener gy l oss per s c a tterer of leng th L i s given by IN = J~foLR- ~ [g~(Af+ ±,~;A*)] (E o 10) As can be seen f r om Appendix C. t he '\I'a! e s of ~ or ders of higher powers of 0.. a s I\. i ncreases e . g . ~ Thus as Q .vIO range of i nter est g and Carhart (46 » to he lit I~ -5 ~: will i nvolv e ~ will be at least c..: Af- I< ", over the audio f r equency it seems a sati sfactor y approxi mat ion (followi ng Epstein o n s~ ' d er nly t h e va es 0 f "~ no an d AI~ ~o , e . 1' t i S poss i b' e to writ e (E o11 ) 106 where further sh01ftl:i.~g th at ~; A!t has bf!'sn n<"'glected compare d wit;1, ~ 0 {ca1culation 106. where further showing that 1\ ~Ah tty\. ,.. has been neglected compared Wi th n~ r. ~ (calculation even at higher frequencies). \ 107. APPENDIX F Ao Material Constants used in AIR -~= 0.1825 x 10=3 g cm~se ,- - Lv - 3044 X se,_~1 104 "'m v 9 1) 111 GLASS 41 x 1010 dyne.~rlcm2 ~1 1017 x '10=3 g em =3 f~ c:~a.tiSl(4b) (fibre) '0- 749 x 105 em sec ,"1 ~ 00 0000-155 cal sec ~ ·-1 .,c ~1 em ,,1 6 0 21 x 1010 dynes em-2 0 PAT 400 x 1011 dynes cm->2 (fibre) 1/T (absolute) 205 x 10-5 B. 112 ) 203 g <:m'-3 '~ B =~- J °K~·1 Approximate expansions of Bessel Functions Using the well-known expansions(8?) for the cylindrical Bessel function of the first kind and the Weber Bessel function of the second kind respectivelY9 ioe. 108. where '( = 0.5772 (Euleros constant) 00. the following approximate expressions can be derlwsd ;,-, (F 0 -:) h were c f2.(' = lO and c f is small such that f or higher powers than 4 it may be neglected compared with 10 ~ Similarly '-".(J) , c ~ g _ r _r l '6' -+~ i6 Il-f-n \. . ~ -\",1') -+ ~ ~).-, ~st. J - ~ [*' -+£1i~(r\"l)3J ~ (F.2) Further (F.4) ~'" where (r.6) (\+)l:$R-.~6 \ ,say, and b is small such that powers of b higher than the fourth may be negleoted compared with 1. !he expression (r.1) for c f , also applies for b order L\;,~ '" (1 + i) (~])iR J f which is of the same Further \-\J\f) ~ 0·5 4- .l~ (t.le y') - ~ 1\ 4- ~ ('0+,) 1090 where the zero order Weber Bessel function being gill·en by .... l ') K similar expressions to (Fa?) also being valid for c: f v Co Fibrous materials examined Rocksil and Roeksil -K resin bonded materials were used, supplied by Cape Insulation Limited in 1" and 2" thick circular disc samples, as specified below:~ Name and spec~fid (lb/ Density ) Roeksil-K Measured Density go 011- 3 Average fi bra radius R 1 000184 .3 " " 105 0 002462 3 2 0 0 0288 3 " (MoDoS.) 5 000825 3 " (HoDoS.) 6 0 0 0863 3 Rocksil Building Slab 0 0 0874 5 Rocksil Acoustic Blanket 000636 5 110 0 Appendix H YiBcoelasti,e Absor~ Following Nowacki (Dynamics of Elastic Systems :963 Chapman and Hall Ltd.) neglecting thermolasicy~ t.he const.it.l1tiy«'l relation, for the standard viscoelastic mode1 9 may be written" ~ S "'-f (r~ + t )..)t l ) ('-H, ~ ) (60'0 situaon~ for a periodic ~ ~ -lw and on comparison with the tensor ~I:- constitutive relation for a normally elastic 5Q,lid vbv Ot~ +). .Q~K Jr-SeiJ = ~ it can be seen that the effect of viscoelasticity is to replace the elastic real constants b, frequency dependent c.omplex variables Leu if -? where rI and \ I are given by (601) the form of the constitutive relation, propagation constants and strese expressions for the normally elastic problem may be retainedo This corresponds to the Voigt case as given by Kolsky (Stress Waves in In detail and \+ {-+ r >-1 C'-lwt~) \ (! - lwt-,) :: ,\ [,i. + E:3r lr .1 So .'l S (t-lwt,.')], (\ - iW~1 ') 111. :::- :=. [A 3rs - I( S(prl)] }; G- ~r\] I -l- ~ \ ~ -:: ~ F1 X - "1 i. e. . I in this case therefo.re it is only necessary to retain one frequency dependent elastic coefficient. N•B 0 found to be roughly true for rubber-like materials as the effect of 1 • B 1S iJ 1 1 this relationship is equivalent to putting B "" 0 i [ B -;> B( 1 + B very much smaller than for I fA- . The wave propagation constants may then be written = ';(1)1/£5 W ~ [f ~A(\ +>.') + (lA+,.f 11/ r'rf w [ + l ~ when » r \{,. :: ~I(+r 15 ') J \ !. .,(~ and ]:t (r,r (~A. r:/ :;~ '" [ < + r"(t+fJ w \(,.~\ y. Cr(Jrl')' ( I \ 4 r, ) r-- 112. ,~ontiu.6 Consider now the model of the hypothetical absorber to be a matrix~ viscoelastic solid containing a random distribution of spherical or cylindrical cavities sealed off from ea.ch other and fr.:>m t.he surface by intervening layers of viscoelastic materialo susceptible to the s~e This model is obviously kind of scattering approach as that employed for the fibrous materialo Firstly ~ it is necessary to consider the problem of .scatt.ering by a spherical or cylindrical. cavity imbedded in a semi.,"inf:ini h viscoelastic Solido (a) S.Eherical For simplicity the amplitude of the incident ·;,;'s.vre may be taken as unity (it is i_aterial to the expression for the attenuat.ion du.e to a single scatterer)0 Then the coefficients representing the scattering inside a viscoelastic Solid are given simply by the boundary c:onditi.ons for a ca'd ty inside a normally elastic solid with the real elastic coefficients and constanta ( by the relevant complex ones vizo r~placed ~vr.?- + (~S + Jr"Ht\(~C'!.kcV)<lLY [c.'~) N O(kS, ~: N'\~. (o...Vl:'!.) (e-~) -+ j + o...l/U ~c.M >. I/~ AVi! k~ ~ lo.v~j) + 1\.( I\. ~ \)~ (o...'lU) -\- "{~I' (c....IH) ] I\lM I) ,\"'-! h,,,, lc.'1~ =- 0 J 113. ==- e which express the fact that 'the radial udl 0 ' campt. of stress at the boundary of the cavity mast be zero. From these equations) where use has been. made of the Te.lation ~e for n =0 the C\.V'iS ')- N'(.~ ~(8. 9 J~\ ?~\) t\ := (~+ )?" (~9) dependence vanishes (o."f,~) .f. l t'J.V~) J c...'lkS AVI<S~'u\ -+ A:f:S [ ~ + thus [ .''' 't"S:(~) [ Cl-.V~S v~,P·NVf;. Jc..VfS ).. vla~! >-'Nlt\.~ ~"\'; lo..'ln)-+ ~1\.,(HkSlo9) ~I (c."f~) lcxVH ) ] C,-H) =- 0 j J 114. Now the functions JI\. and k . . satisfy BessePs differential equati.on II /J "'", 4- and as J", (\C<.v<s'I from Further the small argument forms for Epstein and Carhart may be used. vizo . ( J>l 0<. V~) t"v . (l;~ similarly e,...'f=-":. ,.. + A,(tS [e.t~ + H- ~I N«~ ~ \I • JQ 11\ ""-' . l VH) Vf:S. c( iE~ (c,..VH) + l o.'1t~ [\-.I i~ ((Am) 3 ......... ~l- \5 -t ) ~ (c...VU ) 'J,' (o..vtg) I(C +- J,\Yl. JJc.. VU) >. vt:\: lc..I'f..l) + ~ {f\ .lo.'fH) J '"N 'Il:.'l ,,-,hec..Vf.9) -t ~'f r l~ "" C>.. - c..... t!h,' (~vt) ] 0 1150 r'" { U' (.''') - ""'j,' l~ ,.,)] + ~S [ ll-. I + Cc-vr:s) - .. A.' [h.l.''') - ",'" h: C~,) -.l C_·,~S ~'" (e. J ," vf-~ vkS) 1 -= 0 3/ - c... /15 For gentlrali ty oblique 5ln,::;iden.ce is cOXlsidere:L cylindrical J Howe1rer ~ this meane that even when the surface of the material is ,dosed with an impervious la.;yer 9 the existence of end effects at, the cylindrical channel intersections with the materialboWldaries is a complication which requires consideration over and above the folwing~ where they are lIILeglectedo From Appendix ( c ); again neglecting thermoelasticity 9 where the symbols have obvious meanings, dropping superscripts9 where redundant:- for ~ r \J~ ~ c,.o) + ~:- r {- '.e [~'r.() and for ,~ K» J. C.') + A, [ .. ~ (.') +(\ -,(~ + Aoe.' "~(.) J + <'t. (K~ -K~ I\,(~) ]. t« '('.";'(,') } 0 H:C,') J < 0 116. a[~J,'ld) + (~-KI)J;] A, [~I() -+ -+ (C-~_I) .+.'\(C,..'l-t,H,"(c...') l[c.'J,I(a..I) -1JC:')] IC,'i\ -t H,(CA')] [c..I«,I(c.I ) + A\[c.IH,(~)_J ~ H,(c.' )] "- 0 -\- lKt,[c.-I",I(c. I ) - H,LC')] J>, [c....1 Hi (1:- 1) -I- HI'\(C. I) 1 c.... ). - - H, (c') ] -- 0 Appendix Go Solid stress in spherical polar coordinates for a normally elastic solido and for the axially symmetric problem (ind.of~) .et~ ~ ~9.f '"' 0 thus orealso and thus V-, ~ - ~ ~B u'S- - 1 ~4 0--;:, .2 .. T' r Te -1 h). - ':. - :!:J¥ - ~ r dr~& ~$':" 1\,1') (t: \ _I . ~ (£~\). ~ ~e of ~.;B 1- i (s.~ ~N -+ A&~I and thus after some manipulation o-,Q ~ r~, >- ti l .~!) ~8 C~ t ~G -\- l~ r ~ (_\. ~ ~.A!)-\ !.-S II\) . A~) ~ .s IA. ~ ~ ~s._ ~ rl-~\ Plate 1 Plan view of sample of Rockeil Acoustie Blanket Plate 2 Side elevation of Acoustic Blanket showing the tendency of the fibres to lie in layers parallel to the surtace (netting) of the sample. Plate 3 Plate 4 Surface of rigid polyurethene foam sample (approx 5 x magnifc.to~) Sample of coarse ~ flexible polyurethene foam in which all "skins" have been dissolvedo The basic "fibrous" lattice framework i.e clearly shown o - Plat.,2 Finer sample of flexible polyurethene foam in which the "skins" are retained o Plate 6 S~face of Rookail-K Resin Bonded sample 9 showing the tendency towards "clumping"" 1170 / RESUME OF DATA RELATING TO FIBROUS MATERIALS Name and specified Densi.ty (lb! f~3) A1I'erage fibre radius R (microns) 1 000184 3 " 105 0.02.462 3 Ii 2 000288 3 " (M.DoS o) 5 000825 3 " (HoD.So) 6 000863 3 Rocksil Building Slab 000874 5 Rocksil Acoustic Blanket 000636 5 Rocksil-K " Measured Density go cm~3 . __ .. -=_ -.-..10-- ___..==-"":'" _ _,: _-::-._ -----. _ -- ___ __ ~_ 2 ·54 em. Roeksi l Samples - - -. I . I ~M .': i! ..... . 70 z !:!:! ~ It LLI 0 u 60 2: 0 ~ a.. 0:: 0 til - en 50 4: w u z w 0 u ~ ~ 40 -' + c::: 0 ~ z * /; -L 30 A -u~ --0--0_ 20 - -><"'~ 10 AcoU'Sl ic S(anJ.,r;! B~ldi ng "'·Slb/cu-ft." 1\bJc u. ft 7 ! ,000 r 2000 3000 FREOUENCY ' H! Solan .. 2Ib/cu.ft. " ! 4C~O 50:J0 -~. -- -r-------. - - - - - - : : - - - - - - - 2 .. 5.08 em. Rocksil Samples 100 90 to .:. -~ - .... - -,-_ ,. •.... II II *4 D W II. u I, 60 LL LL I II w 0 II z ~ o 50 ,~ 40 UJ c U z ~ -' • - - ) ( - - .. t lb /eu.ft." II « z '~ 0 1 co u '.1 Ip - III w -. II u et: -.,.. ..... 7 Z "0 ----- ...... ., _t... __ ....... ..... D- ..... --'-- '3 rr: z 0 ~f l --0-- "'·5loa/e u.ft ," -0-- .. 2 \bJeu .ft. " - --4---- . Slb/eu .f\. . -- .....--~ou5tlc • / Slan ke t .. 6Ib/eu·ft." L-b0~IO2V to L-- ~-c=OAs FREQEtlC Y Hz _-- ~: -r- [ ~ - - - ~-,.r I -~ I ;-r - !.J: --:;-t-·..- . - -.r. t . L.:..l..., . . -I r ,; .. '/ . ~-,O I I .... C> t-~ / ..--~. o o J MULTIPLE SCATT. R1GID APPROX SINGLE I SCATT. I I I 0- ' __ 2 ~ ~ f ~ ~ t- l f ," b- ~ ~- EXPTL. CURVE /;/ I [:- _I_I 8 -. ~ ,{ -?" ~ .~ • - 1 -- -' - - t --.- *-4--.-- - [ _____ . _-.. __ . .: ---'I< _~= ~ [:..;. - ~- -.,~ 3'.COMPARISON OF SINGLE AND MULTIPLE S.CATTERING . -iHEORIES FOR 2·5[, C~I ACOUSTIC BLANKE T ~ -!- 8 t • .~- - ~ - . ~ VI m < r: t ~ w f: ~ [ Z ~. ~ · '. lli ~:r t::-:: r~ -t -~ =t r--(---= :--T---.---------___.____ 4 ____ - - - 3000 FREOUEN:V HZ ~ - - - - - - - -. - - - - - - - - - - - - - - - - - - - - - 400D 5000 600'0 7000 -~ , Oom.p ari son of theory and expt . 5- 08 em. Aeoustf e Blanket 90 ~ ~ ,-. ...,.::= ~-= -~ - - . ... - - - ...;. '-_-I'" -- ..- -" , "-'---., I 1, .... t- Z .- L!.I U u:: I:b 0 u 60 z 52 o t- o.. L o Q cxptl. § ~ 50 - - - - - - - - - - - - fflze t itre mOde l L!J U r- Z UJ Cl - - - - - - - rig id approx. U ?; ~ "0 -' Q: .0 z 30 20 10 FREOUENC Y HZ --- Sa , Comparison of theory and expt . It 2lb/cu. ft" ~ ~- o -,....., ___ ~. ~"-r,. r'----.- ~ -.-- . ....--_. .. ~- --t -,.~ __ t- _ _ . . -0 -- -' _ ---- ---- ---'" - .. -----<0 - - - - < 0 - - - ---------- - Exp:l. 5 · OSem. R: 5XIO·'cm, 20 10 o lOCO 2000 30Q<l FREOUENCY Hz '000 5000 roOD ---_..l , r 5b, I I " 1lbl cu , ft " 5.08 em. ---,- ---- "-j- ~. _ ' ___ --;;.... ,..:::: .; , ·1· , . , ! ,! I I , 0 , . ! I .f i i .'..1,I I : , ,!. . , I ! I r :,t. -T l ' ~ .1 I.. 1 I I. .1 , I I I I I ·1 , ! "I; ./ lit I - i . I •• , " , 1 I I i '"I EXPTl, I 'I I 1I +- '~·I-; I : I~' ' I " . I"" ~o 1- I f I j 1 . I, '. ,000 2000 frequency 3000 I Hz ' 5c. 2lb/cu.ft," II 100 / / Y 90 '" 2·5L.cm, ---- - ,.-,---------..._./ / - - ' ""- ,-- ,...,-- ...... .:: // t • so I 70 / / / , '" 0 / I I J / I I I liD // I I r I / I ,I I I I I /' I I I r , i o~ I / I 30 o ,I ,I I .4 R; 5Xl0 em. Calcula ted variat i on of 00 wi th Qen si ty 6,. 2·54 em layers ;:' 90 i/ 1 ,--\. ~\ ;;1 . .. - ' . C /1 - ~ / I. - // 70 / j 60 /' I ' / / /1 5 / / c 40 d 30 20 10 / / / // o / / / / / / / / / / . I , I / / / / A-A •. Ilb/cu.tt." B-8 " 2Ib/cu,f\ ," C-C Acoustic 8ICllke I / A I o ,00 200 400 WI 271 SOD ,iCO 3200 Calculated variation d ~-; with density · 5" 08 em. layers ". -. Qo _ 100 -1- - A ~. B C F tt _ 11 ~o 70 -+~Ov.2u40Ae163G 30 • 2Ci W/21f 8. RELATIVE CHARACTERISTIC r , ,I IMPEDANCE. • I I I . .J ACOUSTIC I I 1 . • " ! BLANKE T I. I I .. I· ..I I I , I. - . I I I i , I I !- 2{) I I I' L . I I I ·1 • - 1... .~ I , I .,_ I· j .I 1- .', ri _•. I·S I I I + I' , ,. 1 : - I V"6 . .. ~ ..,u I -I * I a::: 1'4 " 1·2 1,0 I 0·3 0'4 0,5 0·6 I 0'7 o·s r ..:. . I • '.: 1 I I . .l . , I "l''',j "I' J I I. : l-.. I . I' • 1•• ~ - • :. l .' I " • 0 _.L. • I ' --. t .. • I :, 1 9. A'f.TEf'WA TION ---- b - = N6"' "2 b= B single scattering multiple scattering (appendix Q) 2a b _0 _ _ 1·" • o·~ 0'7 -- o T 0·2 .- o 2000 4000 6000 FREQUEtlCY Hi --~ 8000 N <=> OUl rt _ I 6 C> C> .. 1 I I - -I N C> o C> w C> 0 C> -t. (D ..0 C (j) ~ J:-. 0 C> C> n '< I N Ul C> C> C> en C> C> C> ....:I C> C> C> co C> 8 r _. I .- - 11, J..... VARIATION OF SURFACE IMPEDANCE WITH ANGLE OF INCIDENCE IN")(Y" PLANE (Acoust ic Blanket) ~ , , , , I' , .~ I,. • , " --><'--.- ..... 1"4 'i N 1'2 1· • Re Zot 500 1000 i I , I I '"T 1 I • I ' I . .. I E , i" , 1 t ~ .. 0 en I, I, I I I , I 0 <SJ I 1 / I / I . ... w I z I tZ W .-. <.) ...- I I I -l I eL I : Z <..') 0 Z U 0 ~ ill W rt- ,- Z H X g W ~ ~ « a u (j) 1.0 I IJ\ U ~ I 0 0 LL Z lL LLI W 0 <..) - to- I / N X 0 I I <{ (j) I I I I , I I <:( I I I I -l en I 0 I " I I f f I I I N N ....o I I I / / ,- ,- / / / ,- / ,- ,- ,- ,- / / QJ ,I' U C 0 ./ of OJ -0 U C / 0 0 ~ '" 0) c: <:( I I I N rI :r: 01 01 01 -, j 1 / .j I I -i 1 -.1 0 N --1 I . 1 J 1 References 1. EVANS, E.J. and BAZLEY , E.N. "Sound Absorbing Materials" . HMSO London, 1960. 2. WOOD, A.B. 3. LORD RAYLEIGH . "Theory of Sound" . 4. CRANDALL, LB. ''Vibrating Systems and Sound" . 5. ZWIKKER and KOSTEN . 6. CARMAN, P.C. 7. RICHARDSON, E.G. 8. ZWIKKER and KOSTEN . 9. ibid. pp . 26-28. 10. ibid. pp. 20-21. 11. ibid . pp. 12. ibid. pp. 22 and 76. 13. ibid. pp. 48-51. 14. MORSE, R.W. 15. NAKAMURA, A. pp. itA Textbook of Sound". Vol. II. "Sound Absorbing Materials " . ''Flow of gases through porous media". "Technical aspects of sound". Vol. I. "Sound Absorbing Materials" . p. 20. 42~. J.A.S.A. 24 696 (1952). Mem. Inst. Sci. and Ind. Res. Osaka Univ.18 (1961) 1~3. 16. ZWIKKER and KOSTEN. "Sound Absorbing Materials". Ch. III. 17. MORSE, P.M., OOLT, R.H. and BROWN. 18. MORSE, P.M. and BOLT, R.H. 19. SCOTT, R.A. 20. RETTINGER, M. J.A.S.A. ~ 21. BERANEK, L.L. J.A.SoAo 22. KOSTEN , C.W. and JANSSEN, J.Ho 23. ZWIKKER and KOSTEN. J.A.S.A. 12 217 (1940) Reviews of Mod. Physics. JO 09-150 (19 41..) Proceedings of the Physical SOCiety, Lond. SS 53-59 (1936). 12. SSt> (1947) Acustica 372-378 "Sound Absorbing Materials". Z (1957). Ch. VI. 358 (1940) 240 PATERSON, N.R. 250 BRANDT, Ao 260 DUFFY , J. and MINDLIN, RoD. ~ Geophysics 69-714 (1956). ~. Journal of Applied" Mechanics Jo 479 seqo (1955)0 Mech o Amo Soco Mech. Eng Apl~ 0 24 585 (1957)0 0 0 v. KAWASlMA, Ac stiea 1Q. 208 (1960) 280 BlOT, M.A. J o Applo Phys. 26 290 BlOT, M.A. (i) and (ii). 30. ibid. 31. BroT, M.A. 32. GASSMAN, F. 33. BlOT, M.A. J. Appl. Phys. 34. BlOT, M.A. J.A.S.A. 35 . HARDIN, BoO. and RICHART, F.E. 0 182 (1955)0 JoA.S.A. 28 179 (1956). ppo 185. Eng.; .2Z 1482 "(1962). J. Appl. Phys • Geophysics 2i ~ 673~85 4 (1951). 1385 (1954) and S1. 462 (1956). ~ 1254 (1962). Proceedings of the Am. Soc. eiv. Mechanics Division, So i~ 33 (1965)0 ~(i) ~ 36. BRUTSAERT, W. J. Geophyso Research )70 38. BERANEK, L.L. J.A.S.A. 14 McGRATH, J.W. J.A.S.A. 24 305 39. TYUTEKlN, V.V. 400 (a) NESTEROV, V.S. (b) BYSOVA,N , L•• and NESTEROV, V.S. ibid. 419 (1959) 41. \ 42. 43. LAMB, H. 44. SEWELL, 45. EPSTEIN, P.S. (1942). (1952). Soviet Physics - Acoustics Soviet Physics URICK, R.J. and AMENT, W.S. LORD RAYLEIGH. 2 243 (1964). ~ J.A.S.A. ~ 3 309 (1956) . Acoustics £1 2 344 (1959). 115 (1949). "Theory of Sound" Vol. 1I Art 395 . ''HydNldynamics'' pp. 647. C.J.T. Phil. Trans. (A) ~. 239 (1910). Applied Me ... hanica Theo. Von.Karman Anniv. Vol. p. 558 46. EPSTEIN, P.S. and CARHART, R. R. 47. CHOW , J.F.C. 48. DEVLIN, C. 49. (a) ZINK, J.W. and DELSASSO. L.P. (b) ZINK, J.W. ~(i) J.A.S.A. J.A.S.A. J.A.S.A. §. 553 (1953) • 2395 ( 1964 ) • (1959). 1654 2 2.1 J.A.S.A. 765 (1958). lQ U niv. Cal if. Ph.D. Dissertation, October 9 1957. 50. YING, C.F. and TRUELL, R. 51. BHATIA , A.B. 52. MORSE , P.M. and FESHBACH, H. ?:11086 (1955). J.Appl.Phys. J.A.S.A. 2.1(i) 16 (1959). "Methods of theoretical Physics" Vol. II p. 1495. 53. WATERMAN, and TRUELL 9 R. 54. TWERSKY, V.F. J. Math. Phys. 2. 700-715 (1962) • 55. TWERSKY , V.F. J. Math. Phys. 2. ,56. TWERSKY 9 V.F. J.A.S.A. 2§.(ii) 1314 57. EMBLETON, J.A.S.A. 40(1) 58. DOMBROVSKII,R.VSov. Phys ics ~ 59. ZWIKKER 60. J. Math. Phys. ~ 724~3 (1961 ) . 512 (1962) • 667 (1964) • (1966) • Acoustics 1£ 419 and KOSTEN "Sound Absorbing Materials" (1967) . p. 109. BRISTOL SYMPOSIUM ON POROUS MATERIALS (Colston Papers) p. 151. 61. WOMERSLEY, J. 62. ZWIKKER and KOSTEN 63. JONES, R., LISTER, N., THROWER, E.N. W.A.D.C. Technical Report "Sound Absorbing Materials" and Foundations" . 12 KAWASIHA g V. 65. HARRIS, C.M. and MULLOY , C.T. 66. KAWASIMA , V. 67 . ibid. 68. URICK . R.J. Acustjca "The Dynamic behaviour of Soils & Tech. April, 1965q 64. Acustica p. 79. Symposium on Vibration in Civil Engineering held in the I.C. of So. p. 215. TR 56-614. p. 209. (1960) . J.A.S.A. 10 212. (1960). 24 1-7 (1952). 69. CARSTENSEN , E.L. and SCHWAN, H.P. 70. B. R.S. RepDrt No. 36 21 J.A.S.A. 185 (1959). "Sound Absorbant Treatments" . F~b. "Building Materials" 1965). 71. MASON , W.P. "Physical Acoustics" 72. ibid. po 50. 73. ibid. p. 17. 74. ibid. p. 45. 75. ibid. p. 64. 76. ibid. p. 75. 77. WHITE , R.M. 78. TYUTEKIN , V. V• Soviet Physics 79. TYUTEKIN , V. V • Soviet Physics - Acoustics 80. LAMB 9 H. 81 . MORSE , P.M. 82 . MORSE, P.M. and FESHBACH 9 H. 83. ibid. 84. see for example RUTHERFORD , D.E. "Vect or Methods" 85. REDWOOD , M. 86. MORSE, P.M. and FESHBACH , H. J.A.S.A. (As given in Volume I Part A. p. 9. (1964) . 2Q 771-785 (1958). ''HydrodYnamics'' ~ Acoustics 2 105 pp 645 97 ~ (1959). ( 1960) . (1916) . ''Vi brati on and Sound" . "Methods of Theoretical Physics" pp . 1376. Vol. I I p. 1766. p. 69. "Mechanical Waveguides". "Methods of Theoretical Physics" Vol. I I p. 1563. 87. See for <example SNEDOON , I.N. "Special Functions of Mathematical Phys. and Chem". pp. 107-1 08 . 88. i bid . p. 139-140. 89. See for example 90. for example SNEDOON , J.N. as above p. 139-140. 91. KINSLER , and FREY. 92. DUYKERS, L.R., J .A.S.A. 41 RUTHERFORD , D.E. "Vector Methods" p. 70. "Fundamentals of Acoustics". 1330 ( 1967) p . 133. 930 See f or ex . BeranfCkO Acoustic Measurement s O p. p. 317 et seq . 94 . Zwi kker and Kosten "Sound Absor bi ng Mat erials" o 1949 0 Ch . 5 . 95 . Mor se 9 P. M. and Bolt9 R. H. Rev . Mod . Phys . 96 . Taylor9 HoD. J. A. SoA. 249 701 9 (1952). 97. "Sound Absorbing Materials". pp o 166 et seco 98 . Pyett, E. T. Acustica 29 375=382 16 9b (194-4) (1 953) 0 99 0 For d g P.g Landau g B.Vo and West g H. g Jo A. S.A. 44 g 2 g 531=536 (1968) . 100. Goodier g J oN. and Bishop v R.EoDo J o Appl o Phys o 101. Junger g M. Co v J. Appl o Mech o 102. Chester 103. "Sound Absorbing Materials" o p. 70. g ~g 2 27~ 23 1 12g 124- 126 9 (1952) . (1 955 ) . W., Proc o Roy o Soc o ! 2039 33=429 (1 950) 0 104 0 Lang v W., Pr oc . of Int; • Symp o on Acous tics . Ed. L. Cremer (1959) . 105 . "Sound Absor bing Mat er:\.a1s" o pp o 13=1 8 . 106 . Furrer g N. "Room and 107. Kol skY g Ho "Stress Waves i n Solids " o p o 1000 1080 Vovk g Past ernak g and Tyut eki n . Sovo Phys. Ac o 109. Bi ot9 MoAo) J o Appl. Physo g ~9 997 (1952) . 110 . Lyamshev , L. N., Sov . Phys. Ac. 5,56, (1959) . B~ il d i ng Acoustics and Noise Abatement" p. 64. i 22 ( 1958) . Ill. MEREDITH, J . 112. KAYE, G.W . C. and LA:&Y, T.H . "Physical and Chemical Constants " p.63. Text. Inst . 45 T 489 (1954)