3: Theory of Pressure Gradient Microphones

Transcription

1 On h Tho of h Scond-Od Soundfild Micohon Phili Coll : Tho of Pu Gadin Micohon In an iooic mdium uch a ai, h ound u a a oin du o h oagaion of a ound wav i indndn of h dicion in which h wav avl hough ha oin [] [89]; hnc, a anduc which ond o h ound u a a oin i omnidicional, onding quall o wav incidn fom an dicion. Ral u micohon onl aoima hi idal, inc h a no ul oin dvic. Such a micohon ond o h u ov i diahagm, and an omnidicional chaaciic i obaind onl bcau ound wav diffac aound h caul o ha h u on an fac of h caul do no dnd on h dicion of incidnc. fqunci wh h dimnion of h caul a comaabl o h wavlngh, acouic hadowing occu and uch micohon nd o bcom dicional, onding mo o fonall incidn wav and l o ound aiving fom oh dicion. Nvhl, i i oibl o conuc mall u micohon ha a ubaniall omnidicional ov h full audio bandwidh; ha i, fom H o H. Th ali micohon w nominall omnidicional, u-onding dvic. Fo man coding alicaion, i i diabl ha micohon hould no b omnidicional, bu hould hav oh dicional chaaciic. Th fi dicional micohon aa o hav bn h ibbon micohon dvlod a RC [69]; h fi of h wa mad availabl a a commcial oduc in 9 [7]. Such micohon a uful bcau h allow h ound fom a cific ouc o b mo adil diinguihd fom ambin o bacgound ound, which ofn coniu unwand infnc [7] [74] [76] [8]. Th a alo quid o facilia h coincidn micohon chniqu ofn ud whn maing o and uound ound coding [] [7] [4] [5] [75]. Dicional micohon ma b claifid a bing of ih h infnc / wav o h u gadin [7] [74] [76] [78]. Pu gadin dvic a b fa h mo commonl ud, and will b dicud h in dail. Th ond o h diffnc in ound u bwn wo o mo oin aad b dianc which a mall comad o h wavlngh. B cona, infnc micohon mu hav dimnion a la comaabl o, and uuall lag han, h wavlngh. Thi cla includ uch dvic a aabolic flco, 9

2 On h Tho of h Scond-Od Soundfild Micohon Phili Coll acouic ln, lag-ufac micohon, and lin alo nown a hogun and infnc ub micohon; of h, onl h lin micohon i in common u [76]. Bcau of h long wavlngh aociad wih h low ach of h audio fqunc ang aoimal.7 m a H, i i no uuall acical o mlo uch micohon whn icu ov h full audio bandwidh i quid [7] [74] [76]. Fuhmo, h ola an hibi idlob which va wih fqunc, o ha off-ai comb filing ffc occu; h a uuall ufficin o nd micohon of hi unuiabl fo high quali coding [7] [7]..: Fi-Od Pu Gadin Micohon Th ho of fi-od u gadin micohon will now b dvlod. Poof ha cific of micohon.g., h ibbon micohon ond o h u gadin a abundan in h liau [] [6] [69] [7], and uch andad maial will no b dulicad h. Rah, h following dicuion i concnd wih ul which follow fom h conc of mauing h u gadin, gadl of h anducion mchanim mlod. B u gadin i man h a of chang of ound u wih c o a mall movmn in ac; ha i, h divaiv of ound u wih c o dilacmn. Tha dilacmn ma b in an abia dicion, which will dnd on h oinaion of h micohon. In h oinion of h auho, h dicional divaiv i hfo h obviou mahmaical ool o uili fo h anali of uch micohon. Howv, hi mhod ha no gnall bn adod in h liau alhough on occaionall find ha ubaniall hi aoach ha bn mlod, bu wihou lici u of h noaion of vco calculu;.g., [5]. lhough h ul nd in hi cion a no hmlv nw, i ma b claimd ha h divaion a mo aifaco and mo gnal han ho availabl in h iing liau. In addiion, h amn h of fi-od micohon will ovid h bai fo a imila amn of cond-od u gadin micohon which, c fo cain cial ca, hav civd lil anion o da. L b h ound u a a oin in ac dfind in m of om wold

3 On h Tho of h Scond-Od Soundfild Micohon Phili Coll coodina m. Conid a lan wav oagaing in a dicion cifid b hical ola coodina angl,, a hown in figu.. Figu.: Dicion of Wav Poagaion in Wold Coodina Sm L b h wav vco co co in co in. wh h wav numb π λ c. Uing h coml hao noaion co co in co in.

4 On h Tho of h Scond-Od Soundfild Micohon Phili Coll and co co in co in co co in co in co co in co in.4 Now l û b a uni vco oining in a dicion dfind b hical ola coodina angl, ; uˆ co co in co in.5 Th dicional divaiv of in h dicion û i hn uˆ co co co co in co in co in in [co co co co in co in co in in ] [co co co in in ].6 Th faco [co co co in in ] i h cala oduc of h wo uni vco û and /, and i hfo qual o h coin of h angl bwn hm. Thu, i i qual o h coin of h angl bwn h dicion in which û oin and h

5 On h Tho of h Scond-Od Soundfild Micohon Phili Coll dicion of oagaion of h wav. Thi ul ha bn d in m of a wold coodina m, in which a micohon ma b lacd a an abia oiion, bu i i convnional and ofn mo uful o wo in m of a coodina m cnd on h micohon. I i convnin in hi ca o dcib h lan wav in m of h dicion fom which i i incidn, ah han h dicion in which i oaga. L h wav b incidn fom a dicion,, a hown in figu.. Figu.: Dicion of Incidnc in Micohon Coodina Sm W dfin h wav incidnc vco co co in co in.7 and h ound u a h oin m i hn givn b m.8

6 On h Tho of h Scond-Od Soundfild Micohon Phili Coll no h chang of ign in h onn, o ha co co in co in.9 L û oin in h dicion givn in h micohon coodina m b angl,. Thn h dicional divaiv uˆ [co co co in in ]. I i convnin o oin h micohon coodina m uch ha û i coincidn wih h ai, i.., o ha u ˆ. Thi i conin wih h convnion ha h ai oin ˆ fowad, inc û dfin h dicion in which h micohon oin. In hi ca, and uˆ co co. Th amliud of hi u gadin dnd on h dicion fom which h wav i incidn, bu i alo ooional o fqunc; addiionall, h u gadin lad h ound u b 9. Th unuiing conqunc of diffniaion can b liminad b aoia filing; cificall, b ingaion wih c o im and mulilicaion b c; c u ˆ d c co co c co co co co. No ha h dndnc of h amliud on fqunc, and h ha hif, a liminad b h ingaion; mulilicaion b c iml mov a conan cal faco ha would ohwi aa. I i includd h fo convninc, bu in a al micohon caul i ma 4

7 On h Tho of h Scond-Od Soundfild Micohon Phili Coll b conidd o b on faco conibuing o h oal onivi; hnc, i i no of fundamnal imoanc, and h fac ha h vloci of ound ma va lighl wih amohic condiion i no ignifican o fa a h n oiion of ho i concnd. Thi will b fd o hncfoh a h qualid u gadin on. In mo ca, h ingaion i inhn in h oaion of h anduc. Th ola an of hi micohon i M, co co. Thi i h wll-nown diol, figu-of-igh, o bidicional ola on aociad wih a u gadin micohon. I ha on oiiv and on ngaiv lob; oiiv in hi con imli ha h ouu i in ha wih of h am olai a h ound u, whil ngaiv indica ha h ouu i in aniha wih of ooi olai o h u. Th wo-dimnional lana ola an i obaind b ing o ha co and M co.4 To obain oh dicional on, a micohon which ond o a wighd um of u and qualid u gadin ma b ud; oof ha cific caul dign hav uch a on ma again b found in h liau [] [57] [7] [77] [9]. Th ouu of uch a micohon i givn b [ a bc ˆ ] v G u d.5 a b o, b ubiuing fom quaion., v G a b GM, [ a bco co ].6 5

8 On h Tho of h Scond-Od Soundfild Micohon Phili Coll wh a and b a conan ha dfin h aio of h u and qualid u gadin comonn, G i an ovall onivi conan, and M, i h ola an M, a b [ a bco co ].7 M i nomalid o uni in h dicion of maimum on; hi dicion, dfind b uˆ, i ofn md h dicivi ai of h micohon. Th ola an of all fi-od micohon hibi aial mm ; h h-dimnional an ma b obaind b oaing h wo-dimnional on abou h dicivi ai. Equaion.6 ma quivalnl b win a v G [ a buˆ d ˆ ] a b.8 wh dˆ i a uni vco oining in h dicion fom which h wav i incidn, o ha dˆ.9 and [ a uˆ d ˆ ] M, a b b. Thi fom i convnin whn woing wih val micohon caul wihin a common coodina m, and fo ha aon will b uful la in hi hi. Hncfoh, a u fi-od gadin micohon, a dcibd b quaion., will b dcibd a a fi-od gadin micohon, whil a micohon of h dcibd b quaion.5 will b md a fi-od micohon. Hnc, a fi-od gadin micohon i a fi-od micohon wih a. Whn a b, a o-calld cadioid imling ha-had on i obaind 6

9 On h Tho of h Scond-Od Soundfild Micohon Phili Coll G v cadioid [ co co ]. whil uing a and b ul in a hcadioid an v G [ co co 4 hcadi oid ]. ucadioid i obaind b ing a and b 5, whil an an fo which a > b i md ubcadioid. Plo of cadioid, hcadioid, ucadioid and naiv ubcadioid ola on a givn in figu.. U of hi nomnclau i no nil andadid; om auho dcib a hcadioid o ucadioid an micohon wh a < b, diinguihing bwn diffn an b h angl a which h null occu o b h aio of h fon and bac on; hnc, a micohon wih a and b ma b dcibd a a -null hcadioid o a a 6 db fon-o-bac hcadioid. Cadioid micohon a omim dcibd a unidicional, alhough hi i ah difficul o uif, inc a cadioid on i onl 6 db down a 9 off ai. In h ca of a al micohon caul, h valu of G, a, and b ma b fquncdndn, o ha h onivi and / o h ola an va a funcion of fqunc; uch bhaviou i uuall conidd o b undiabl [7] [74] [9]. W obv ha h ola an of an omnidicional micohon i a oh-od hical hamonic, whil fom quaion. i can b n ha h ola on of a fi-od u gadin micohon i a fi-od hical hamonic. Fuhmo, fom quaion.7 i can b n ha h ola on of a fi-od micohon onding o a wighd um of u and u gadin i h quivalnl wighd um of a oh-od and a fi-od hical hamonic. If a fi-od figu-of-igh micohon i oad o ha u ˆ, hn h uling ola an i ˆ M Y, in co. 7

10 On h Tho of h Scond-Od Soundfild Micohon Phili Coll Figu-of-igh a, b Cadioid a, b Hcadioid a, b Sucadioid a, b 5 Subcadioid a, b Subcadioid a 5, b 4 Blu oiiv in ha Rd ngaiv aniha Figu.: Rnaiv Fi-Od Pola Pan 8

11 On h Tho of h Scond-Od Soundfild Micohon Phili Coll whil if uˆ ˆ hn h ola on i M Z, in.4 Now, an oh valu of û ma b d a a lina combinaion of ˆ, ŷ and ẑ. I follow ha h ola on of an fi-od gadin micohon ma b d a a lina combinaion of co co, in co and in ; ha i, a a lina combinaion of h h indndn fi-od hical hamonic. Fuhmo, if h ouu of h co-locad muuall ohogonal fi-od gadin micohon a availabl, hn h ouu of a noional gadin micohon in an oinaion ma b nhid b foming a uiabl lina combinaion of h h caul ouu ignal. Sinc h on of a gnal fi-od micohon, a in quaion.5, coni of h wighd um of an omnidicional comonn and a fi-od gadin comonn, i fuh follow ha h ola on of an fi-od micohon ma b dcibd uing h oh-od and h fi-od hical hamonic, and ha, givn h ouu of an omnidicional micohon and h muuall ohogonal u fi-od gadin micohon, all co-locad, h ouu of a noional fi-od micohon of abia ola an and oinaion ma b nhid. I will b aciad ha i i no in ali oibl o aang fo uch a of micohon o b acl co-locad; hi will b dicud in a la cha..: Scond-Od Pu Gadin Micohon ˆ Th cond-od dicional divaiv of ound u i givn b u ˆ u ; i i h dicional divaiv in h dicion û of h cala fild which i h dicional divaiv in h dicion û of h ound u. In gnal, uˆ ˆ u; howv, w will conid fi h ca wh h wo vco a qual. L m, hn uˆ uˆ ˆ in h micohon coodina 9

12 On h Tho of h Scond-Od Soundfild Micohon Phili Coll uˆ ˆ u ˆ ˆ ˆ co co co co ˆ co co co co.5 on would c, qualiaion of a cond-od gadin on qui a fil which i a doubl ingao. Th micohon ouu i hfo givn b uˆ ˆ u v c dd.6 ling uch qualiaion, w obain, in h cun ca, an ouu ignal v co co.7 Micohon having uch aial quaduol, o cond-od figu-of-igh, ola on w fi dcibd b Olon [7] [76]. No ha h an ha wo lob, bu a wll a bing naow han in h fi-od ca, h a boh oiiv. If wo cond-od figu-of-igh micohon a oiiond wih hi dicivi a oind civl in h and dicion and hi ouu a addd, h uling ignal i ˆ ˆ ˆ ˆ v c [ co co in co ] co dd.8 Micohon having hi cond-od ooidal ola an hav bn invigad in h con of hi onial alicaion o confnc lhon [5] [78] [8] [8]. I i oibl o conuc fi-od ooidal micohon, bu h qui boadband 9 hahif cicui and hav a ha on ha vai wih h dicion of incidnc [95]. W now conid a ca wh u ˆ. L uˆ ˆ and u ˆ ; hn h micohon ouu ˆ u ˆ 4

13 On h Tho of h Scond-Od Soundfild Micohon Phili Coll v c c c c c ˆ ˆ dd ˆ co co dd co co ˆ dd co co in co co in co dd co in co in co dd.9 Thi hibi h al quaduol, o clov-laf, ola an conidd b Gon [7] [8] [9]. Th lana al quaduol an i hown in figu.4; no ha aial mm i no n, and ha h h-dimnional on can no hfo b obaind b oaion of h lana ola an. I ma no b immdial aan ha hi dicional on i v uful and, indd, in iolaion i obabl i no. No, howv, ha i coond o on of h cond-od hical hamonic; i will b dmonad la ha icu an of hi fom a uful in conuncion wih oh micohon having diffn ola an, and a alo cificall quid fo cond-od ambionic coding. Blu oiiv in ha Rd ngaiv aniha Figu.4: Plana Tal Quaduol Clov-Laf Pola Pan 4

14 On h Tho of h Scond-Od Soundfild Micohon Phili Coll v lag numb of ola an ma b obaind b combining oh-, fi-, and cond-od comonn. Ending quaion.5 o includ a cond-od gadin m a in quaion.6, w obain fo h ouu of uch a micohon [ a b c ˆ d b c ˆ u, u, ˆ u, ] v G dd κ. wh h conan a, b and b, which cif h laiv wighing of h u, fiod gadin and cond-od gadin comonn, a wll a h h dicion vco, ma all b chon indndnl. Th nomaliing conan /κ ha h am uo a h a b faco in h fi-od ca, bu i i no ncail qual o a b b. If all h dicion vco a qual, a, and b, hn h ola an obaind i b h um of a co-locad fi-od and cond-od figu-of-igh wih a common ai and qual aial onivi; v G G [ co co co co ] [ co co ] co co. Thi ola an i omim fd o a cond-od unidicional, alhough hi m ha alo bn alid o ola on achivd b combining h fi-od and cond-od gadin lmn in diffn aio, oibl including a u comonn a wll [7] [74] [79] [8]. Th m cond-od cadioid i alo ud, alhough h ha ha i no long aan. I will b nod ha hi on i v much mo dicional han h fi-od cadioid, having null a 9 and 8, and v mall ngaiv a lob figu.6 a h nd of hi cion. I ha vioul bn hown ha h ouu of an fi-od micohon can b nhid b aing a lina combinaion of h ouu of fou coincidn micohon having aoia ola an. W now conid h nion of hi conc o condod micohon. L 4

15 On h Tho of h Scond-Od Soundfild Micohon Phili Coll 4 [ ] T ˆ u. and [ ] T ˆ u. W ma hn wi ˆ ˆ ˆ u u u.4 n cond-od u gadin ma hu b d a a lina combinaion of i cond-od aial divaiv. In h lan wav ca co co.5a co in co in co.5b in co in co co.5c co in.5d in in in co in.5

16 On h Tho of h Scond-Od Soundfild Micohon Phili Coll in.5f and o uˆ ˆ u [ M, co co co co in in in co in in co in co co ].6 wh M, C co co C C co co in C C in co in C 5 co in co 6 4 in co in.7 wih C C C C 4 C 5 C 6.8a.8b.8c.8d.8.8f Th Lalac i anion of M, ha cofficin C C4 C6 C6 C C4,, C.9a.9b.9c.9d.9 44

17 On h Tho of h Scond-Od Soundfild Micohon Phili Coll, 6 C 6 C4 B, B, C5 B, 6 C.9f.9g.9h.9i Thi ma b ablihd ih b uing h fomulæ givn in Cha quaion., o b maniulaion uing igonomic idnii. No ha h h fi-od cofficin a qual o o, gadl of h valu of h lmn of û and u ; hnc, a cond-od gadin micohon nv ha a fi-od hical hamonic comonn in i ola on. No alo ha ˆ uˆ uˆ.4 Thi how ha h ola an of a u cond-od gadin micohon can includ a oh-od hical hamonic and, indd, will alwa hibi uch a comonn unl u uˆ ; i.., unl h wo vco a ndicula. I i no oibl, howv, fo a ˆ ingl cond-od gadin micohon o hav a ul oh-od ola an hi i ovd in ndi. Th a onl fiv cond-od hical hamonic in h Lalac i. Hnc, h i aial divaiv which aa in quaion.4 ul in an ion wih onl fiv cond-od m, along wih a oh-od m. Thi can b accound fo b h igonomic idni co co in co in.4 o b noing ha a lan wav aifi h Hlmhol quaion.4 wh 45

18 On h Tho of h Scond-Od Soundfild Micohon Phili Coll.4 I hould b nod ha if h ound fild do no aif quaion.4 hn quaion.4 canno b d in m of oh-od and cond-od hical hamonic in h mann dcibd abov, inc h ound u and cond-od divaiv can no hn b aumd o b linal dndn; hi i an imoan obvaion, which ha ignifican imlicaion fo cond-od ambionic m, and will b dicud in mo dail in ubqun cha. I hould alo b nod ha no fuh lina dndnci can i bwn h ound u and h cond-od aial divaiv, bcau h hical hamonic a linal indndn. Th ola an aociad wih h aial divaiv in quaion.5b,.5c and.5 a cond-od hical hamonic. I ma b n ha h maining wo condod hical hamonic a aociad wih combinaion of u and cond divaiv uch ha co co in.44a.44b In ndi, i i ovd ha no oh cond-od aial divaiv of ound u ul in ola on which a cond-od hical hamonic. Givn an omnidicional u micohon and fiv cond-od u gadin micohon wih ola on coonding o h fiv cond-od hical hamonic, all co-locad, an oibl cond-od gadin micohon in an oinaion ma b nhid b foming a uiabl lina combinaion of ouu ignal. If h fiod u gadin micohon coonding o h fi-od hical hamonic a alo includd, hn an oibl cond-od micohon of h fom nd b quaion. ma b nhid. No, howv, ha whil h hical hamonic dcomoiion of a ola on i uniqu, h i ofn mo han on wa in which ha ola an ma b obaind in m of h cond-od aial divaiv; ha i, h 46

19 On h Tho of h Scond-Od Soundfild Micohon Phili Coll ma i a mulilici of cond-od u gadin micohon aangmn which giv h am on o a lan wav, bu canno b aumd o giv qual ouu in on o abia ound fild. Fo aml, h ooidal on of quaion.8 ma alo b obaind uing an omnidicional micohon and an aial quaduol wih u uˆ ˆ. ˆ In aing, i ma b obvd ha wo of h ola on chaaciic dcibd abov ma ail b gnalid o nh-od [7]. n nh-od figu-of-igh on i obaind b aing h nh-od dicional divaiv wih all n dicion vco qual; h ola an i givn b n n M, co co.45 Th um of figu-of-igh on of od n and n having a common ai and qual maimum onivi i an nh-od cadioid fo which M, n n n n [ co co co co ] n n [ co co ] co co.46 Plo of nh-od figu-of-igh and cadioid ola on fo valu of n u o and including 6 a givn in figu.5 and.6. Th ola an all o aial mm. 47

20 On h Tho of h Scond-Od Soundfild Micohon Phili Coll n n n n 4 n 5 n 6 Blu oiiv in ha Rd ngaiv aniha Figu.5: Fi-Od o Sih-Od Figu-of-Eigh Pola Pan 48

21 On h Tho of h Scond-Od Soundfild Micohon Phili Coll n n n n 4 n 5 n 6 Blu oiiv in ha Rd ngaiv aniha Figu.6: Fi-Od o Sih-Od Cadioid Pola Pan 49

22 On h Tho of h Scond-Od Soundfild Micohon Phili Coll.: Th Dicivi Faco On wa in which h dicional oi of a micohon ma b quanifid i b h dicivi faco, dnod b γ o omim b Q. Thi i dfind a h aio of h man qua ouu volag gnad b h micohon in on o a lan ound wav incidn fom h dicion of maimum on o ha gnad in on o a fcl unifom diffu ound fild having h am oal acouic ow [] [6]. Hnc, h dicivi faco mau h abili of h micohon o mhai h did ound ouc ov ambin ound whn ha ambin ound i dicionall homognou; hi i ofn aoimal h ca, fo aml, fo vban ound. Wh h i a mall numb of dic, aiall localid ouc of unwand ound, h ac ha of h ola on, and aiculal h locaion of h on null, ovid a mo aoia indicao of diciminao abili. Th dicivi faco i givn b γ π π / π / 4π M, co dd.47 Whn h ola an hibi aial mm, h dicivi faco ma found fom h lana ola on uing h iml fomula γ π M in d.48 Thi ali o all fi-od micohon; howv, man cond-od ola an a no aiall mmical. Evaluaing γ fo an abia fi-od micohon wih ola an givn b quaion.7 giv h ul γ a b a b.49 5

23 On h Tho of h Scond-Od Soundfild Micohon Phili Coll I ma b hown ha h maimum oibl dicivi faco aainabl uing an nh-od u gadin micohon i givn b [4] γ ma n n.5 Hnc, h maimum dicivi faco which ma b achivd uing a fi-od micohon i 4; hi i in fac obaind wih a -null hcadioid. Fi-od figu-ofigh and cadioid micohon boh hav dicivi faco of and an omnidicional micohon, b dfiniion, ha a dicivi faco of. I i cla fom quaion.5 ha cond-od micohon ma hav dicivi faco a high a 9; hi i on aon wh hi u ma b conidd diabl, inc i indica a ubaniall ga abili o dicimina again ambin and vban ound [76]..4: Rlaionhi Bwn Paicl Vloci & Pu Gadin Fi-od u gadin micohon a ofn fd o a vloci micohon ; in h oinion of h auho hi i unfouna and oniall confuing, inc i i h u gadin and no h aicl vloci ha acua h micohon. Th minolog can nvhl b uifid b fnc o h laionhi ha i bwn h u gadin and aicl vloci fild in a ound wav. on of h fundamnal quaion govning h oagaion of ound wav, w hav [] [68] ρ v.5 wh v i h inananou aicl vloci and ρ i h aic undiubd dni of h mdium; no ha hi quaion do no dnd on h ound fild bing a lan wav, bu i of gnal alicabili. W hfo hav fo h dicional divaiv of ound u 5

24 On h Tho of h Scond-Od Soundfild Micohon Phili Coll uˆ ρuˆ v.5 Ingaing wih c o im and muliling b c, w obain c v ˆ u d c ρ uˆ d v cρ uˆ d v cρ uˆ d cρ uˆ v.5 Th qualid u gadin i hu ooional o h comonn of aicl vloci in h am dicion, o ha a u gadin micohon ma b aid o mau h aicl vloci. Th conan c ρ i nown a h chaaciic cific acouic imdanc of h mdium. oibl miconcion aiing fom u of h m vloci micohon i ha a cond-od u-gadin micohon mau h aicl acclaion. Thi i no h ca; inc h aicl acclaion i b dfiniion h divaiv wih c o im of h aicl vloci, w ma aang quaion.5 o giv uˆ v ρ uˆ.54 fom which i i cla ha h aicl acclaion in a givn dicion i ooional o h fi-od u gadin wihou qualiaion in ha dicion. Manwhil, aing h qualid dicional divaiv in h dicion û of ach id of quaion.5 giv c uˆ ˆ u dd c uˆ c c c ρ cρ uˆ ρ uˆ ρ uˆ uˆ ˆ u v d v d ˆ u v d uˆ vd.55 5

25 On h Tho of h Scond-Od Soundfild Micohon Phili Coll howing ha an qualid cond-od dicional divaiv of u i ooional o a fi-od dicional divaiv of h comonn of aicl dilacmn in a givn dicion..5: Th Poimi Effc I i wll nown ha h low fqunc on of a micohon having a u gadin comonn i accnuad whn h micohon i oiiond clo o a ound ouc. Thi i nown a h oimi ffc, o inc i i h low fqunci ha a mhaid h ba boo ffc ; i i gnall conidd o b undiabl, alhough in om muical alicaion h ffc ma b ubcivl laing []. On moivaion fo ing an undanding of hi ffc i ha i facilia h dign of comnaing fil. Howv, iciv of whh uch fil dign i inndd, h oimi ffc i a ignifican faco in h fomanc of gadin micohon, and an undanding of i i quid o ol chaaci h cd bhaviou of h cond-od oundfild micohon. n anali of hi ffc will now b nd. lhough om of h ul gading h fi-od ca a andad, h amn nl availabl in h liau wih which h auho i familia a omwha unaifaco; vn h mo accabl naion uch a Ban [] lac gnali and fail o adqual dcib h divaion. Fuhmo, man dicuion a nd wih fnc onl o a ingl of anduc, o ha h incoc imion ma omim b cad ha h oimi ffc i a dfc of a aicula micohon dign, wha i i in fac an inviabl conqunc of mauing h u gadin. v fw ul laing o cific cond-od micohon ma b found in h liau [9] [76] [7]. Th hav again bn givn almo nil wihou divaion; addiionall, inconinci in h ul nd in [9] ugg ha h ma no b nil coc..5.: Poimi Effc fo Fi-Od Micohon Unil now, h anali nd ha bn concnd cluivl wih lan wav. Clo o a mall ouc of ound, i i nca inad o conid hical wav [] [89]. 5

26 On h Tho of h Scond-Od Soundfild Micohon Phili Coll 54 uming a oin ouc locad a h oigin, h ound u fild i dcibd b.56 wh i h amliud a uni dianc fom h ouc, and i h dianc fom h ouc o h maumn oin. Th divaiv wih c o.57 In hical ola coodina, h gadin of a funcion f i givn b [68] f f f f co.58 o ha u u ˆ ˆ.59

27 On h Tho of h Scond-Od Soundfild Micohon Phili Coll wh û i a uni vco in h adial dicion; ha i, oining dicl awa fom h oigin. I i convnin now o conv hi ion ino caian coodina. h oin,,, uˆ co co ˆ in co ˆ in ˆ.6 figu.7. Figu.7: Caian Comonn of Radial Vco û Subiuing fo û in quaion.59 giv co co in co in.6 I i cla fom hi ha dnd on h oin a which h maumn i an. Thi mu of cou main u whav coodina m h gadin i d in, bu a 55

28 On h Tho of h Scond-Od Soundfild Micohon Phili Coll can b n fom quaion.59, h dndnc do no aa licil whn h ul i win in ola coodina; ah, i i imlici in h u of u ˆ, which ilf dnd on oiion, a i vidn fom quaion.6. Now, a bfo, l û b a uni vco oining in a dicion, ; uˆ co co in co in.6 Th dicional divaiv in h dicion û i hn givn b [ co co co in in ] u [ co co co in in ] ˆ.6 which i imila o quaion.6, alhough h coml gain i a mo comlicad funcion in hi ca. I hould howv b mmbd ha h oagaion dicion, dnd on h oin a which h gadin i an, wha in h ca of a lan wav h dicion of oagaion i vwh h am. I i again uful o hi ul in m of a coodina m cnd on h micohon. Conid a oin ouc locad a dianc fom h oigin, wih coodina,, ; hn.64 L b h dianc fom h ouc o an abia oin,, :.65 Th u a uch a oin i hn givn b 56

29 On h Tho of h Scond-Od Soundfild Micohon Phili Coll and, b h chain ul,.67 Sinc h coodina m i cnd on h micohon, w valua hi a h oigin, o ha and ; ˆ O.68 wh ˆ i a uni vco oining in h dicion of h ouc; hnc, if ha dicion i dfind b angl,, hn in co in co co ˆ.69 doing a bfo h convnion ha û i alignd wih h ai of h micohon coodina m, h u gadin i givn b

30 On h Tho of h Scond-Od Soundfild Micohon Phili Coll ˆ O co co.7 and h qualid u gadin b c d co co ˆ O co co.7 o ha h ouu of a fi-od u gadin micohon in on o a hical wav i v gad G co co G co co.7 Th oimi ffc ba boo faco i dfind a h aio of h amliud of h u gadin micohon ouu o h amliud of h ouu of an omnidicional u micohon, whn boh a oiiond a h am oin in h ound fild, and h u micohon i uch ha i on o a lan wav i idnical o h on of h gadin micohon o a lan wav aiving on ai. Th ouu of h u micohon in hi ca i hfo v omni G.7 I hould b nod ha h i lighl diffn dfiniion of h ba boo faco, which dnd on uing a u micohon wih diffn onivi a h bai fo comaion; in mo ca, h ul obaind uing alnaiv dfiniion uch a ho quod in [76] diff onl b a muliling conan. L h ba boo faco b dnod B, hn 58

31 On h Tho of h Scond-Od Soundfild Micohon Phili Coll B v v gad omni G co co G co co co co.74 Conid h ca of on-ai incidnc, which i h mo acicall imoan ca and h on mo ofn mniond in h liau; co co and B.75 Whn i lag comad o h wavlngh of h incidn ound wav, i lag comad o uni and B.76 Thfo, a dianc fom h ouc which a lag comad o h wavlngh, h i no ba boo. Thi i o b cd, inc a a dianc fom a mall ouc which i lag comad o wavlngh h hical wavfon ma b aoimad ov a mall gion b a lan wavfon. Whn i mall comad o wavlngh, i mall comad o uni and B.77 Hnc, clo o h ouc h ba boo vai invl wih ; a a fid dianc, h micohon ouu i b 6 db fo ach ocav do in fqunc. Thi i a wll-nown 59

32 On h Tho of h Scond-Od Soundfild Micohon Phili Coll ul [] [9] [76] [7]. Fo oh dicion of incidnc, h on vai wih in h am wa, alhough h ba boo i cald in accodanc wih h ola an. Thi caling i, of cou, a conqunc of h dciion o coma wih a u micohon having a lan wav on qual o h maimum on of h gadin micohon. No ha fo ±9, B, bcau h gadin micohon do no ond o wav fom an dicion in ha lan. W now conid h gnal ca of a micohon having a mid u and u gadin on. Subiuing quaion.66 wih and.7 ino quaion.5 giv a micohon ouu v G a a b bco co G a bco co a b G a bco co a b.78 o ha B a bco co a b a b bco co a bco co a b a b a b bco co b co co co co a bco co.79 Thi allow h ba boo o b calculad fo an fi-od micohon and an dicion of incidnc. Conid a cadioid, wih a b : 6

33 On h Tho of h Scond-Od Soundfild Micohon Phili Coll co co co co B.8 Fo fonal incidnc, hi giv 4 B.8 which i no diimila o quaion.75. Fo lag hi again nd o uni, a on would c. Fo mall B.8 Thi again giv a 6 db i fo v halving of, bu 6 db blow h on of h u u gadin micohon. Thi i anoh andad ul; h low boo occu bcau onl h u gadin lmn of h on i accnuad b h oimi ffc, and o h ooional inca in h oal ouu i l. Fo incidnc fom h a 8 and, o ha co co, B.8 Th cadioid an ha a null in hi oiion, and fo lag, B bcom ngligibl mall, aoaching h o on ha would b oducd b a lan wav. Howv, fo clo ouc, h imlicaion i ha h on no long how a null in hi oiion. Thi i a aonabl concluion; h null i du o cancllaion bwn h u and u gadin comonn, and dnd on hm bing qual in magniud and in ac aniha. Clo o a ouc of hical wav, h u gadin i incad in amliud and ha-hifd, o ac cancllaion do no occu. Thi aoning i alo alicabl o oh comoi micohon; hnc, i ma b concludd ha h ola an of a micohon which ond o a combinaion of u and u gadin will chang whn h micohon i lacd clo o a mall ound ouc. 6

34 On h Tho of h Scond-Od Soundfild Micohon Phili Coll 6.5.: Poimi Effc fo Scond-Od Micohon W now ocd o conid h on of cond-od gadin micohon o hical wav. Fom quaion.67, w now ha.84a.84b.84c Diffniaing a cond im giv 4.85a 4.85b 4.85c 4.85d f vioul, w now aum ha h maumn oin i h oigin, o ha,, and 4.86a 4.86b

35 On h Tho of h Scond-Od Soundfild Micohon Phili Coll c 4.86d f Eing h ouc locaion in hical ola coodina fom, w obain co co.87a co in.87b in.87c co in co.87d in co co.87 in co in.87f Finall, alicaion of doubl-ingaion filing giv dd c co co co co.88a dd c co in co in.88b

36 On h Tho of h Scond-Od Soundfild Micohon Phili Coll 64 dd c in in.88c dd c co in co co in co.88d dd c in co co in co co.88 dd c in co in in co in.88f To ablih h on o a hical wav of an abia cond-od gadin micohon, w conid h qualid vion of quaion.4: dd c dd c dd c dd c dd c dd c dd c dd c ˆ ˆ u u.89 Subiuing in h qualid aial divaiv fom quaion.88, w obain

37 On h Tho of h Scond-Od Soundfild Micohon Phili Coll 65 M dd c, ˆ ˆ u u.9 wh in in co in co in in co co co in co co co, C C C C C C M.9 wih, C, C c., dfind a bfo b quaion.8. Th Lalac i fo, M in hi ca ha cofficin obaind a bfo b uing h fomulæ givn in Cha o b igonomic and algbaic maniulaion 6 4 C C C.9a.9b [ ] 4 6 C C C.9c,.9d, C.9 [ ] 4 6 6, C C.9f, B.9g 5, C B.9h

38 On h Tho of h Scond-Od Soundfild Micohon Phili Coll B, 6 C.9i Comaing wih quaion.9, w ha h cofficin hav h am valu a in h lan wav ca, c ha v cond-od cofficin i mulilid b h faco. Th on of a cond-od gadin micohon o a hical wav i hfo clol lad o i lan wav on, and indd can b found uing h Lalac i dcomoiion of i ola an; fom quaion.9, including an ovall onivi conan G, w ma wi fo h ouu of a cond-od u gadin micohon in on o a hical wav v G M,.9 wh M, i h cond-od hical hamonic comonn of h ola an; i.., M, [in B, ], in co in B co co in,, in co co co.94 Fom hi, w obain fo h ba boo faco B M,.95 conqunc of hi ul i ha h al quaduol i h onl cond-od u gadin micohon fo which h ola an i indndn of in h hical wav ca, inc v oh cond-od ola an includ an omnidicional comonn. B wa of aml, conid h cond-od figu-of-igh on dcibd vioul. Th Lalac i anion of h ola an in hi ca ha h non-o cofficin:.96a 66

39 On h Tho of h Scond-Od Soundfild Micohon Phili Coll 67.96b 6,.96c Thfo, h ba boo faco co co co co co co co co co co co 6 co co in co co 6 in co co ] [in 6 co co 6 ] [in B.97 Thi can b n o b h ul ha would b obaind uing quaion.88a dicl o find h on of h micohon. Gah of B fo fonal and 9 off-ai incidnc, fo boh fi-od and cond-od u gadin micohon, a givn in figu.8.

40 On h Tho of h Scond-Od Soundfild Micohon Phili Coll Fonal incidnc -od nd-od 9 off-ai incidnc -od nd-od Figu.8: Ba Boo fo Fi-Od and Scond-Od Pu Gadin Micohon 68

41 On h Tho of h Scond-Od Soundfild Micohon Phili Coll If fi-od comonn a alo n, o ha w hav a micohon of h dcibd b quaion., hn h on o a hical wav i v G M M,,.98 and h ba boo faco i B M, M,.99 wh M, i h fi-od hamonic comonn of h ola on and M, i, a vioul, h cond-od hamonic comonn. No ha dic maumn of u, a wll a h cond-od divaiv, ma conibu o h oh-od comonn. n aml of a micohon having hical hamonic comonn of all h od i h cond-od cadioid dcibd b quaion.. Th Lalac i in hi ca ha cofficin 6,, 6.a.b.c.d o ha h ba boo faco i 69

42 On h Tho of h Scond-Od Soundfild Micohon Phili Coll 7 [ ] co co co co co co co co co co 6 co co co co co co co 6 in co co co in co co 6 in co co co co 6 in co co co co in 6 co co co co ] [in 6 co co 6 B. Gah of h ba boo fo fonal, 9 off-ai, and a incidnc a givn fo fi-od and cond-od cadioid micohon in figu.9.

43 On h Tho of h Scond-Od Soundfild Micohon Phili Coll Fonal incidnc -od nd-od 9 off-ai incidnc -od nd-od Figu.9: Ba Boo fo Fi-Od and Scond-Od Cadioid Micohon 7

44 On h Tho of h Scond-Od Soundfild Micohon Phili Coll Ra incidnc -od nd-od Figu.9: Ba Boo fo Fi-Od and Scond-Od Cadioid Micohon coninud.6: Th Blumlin Diffnc Tchniqu Th Blumlin diffnc chniqu, o namd b Gon [7] [5], i a mhod of uing micohon of a givn od o obain dicional on of high od. Th condod oundfild micohon dnd on hi chniqu o gna cond-od on uing fi-od caul. W conid fi h u of wo u micohon o div a fi-od u gadin on. Th dicional divaiv of h ound u fild ma b aoimad b h diffnc in u bwn wo oin aad b a mall dianc δ ; δ uˆ δ uˆ δ ˆ u. 7

45 On h Tho of h Scond-Od Soundfild Micohon Phili Coll Phicall, hi ma b alid b lacing wo omnidicional u micohon a ho dianc aa, and aing h diffnc of hi ouu. Thi mhod wa ud b Blumlin o obain a diol on bfo h availabili of individual u gadin caul [6]. Combind u / u gadin on ma b obaind b foming a uiabl wighd um of h qualid diffnc ignal and h dic ouu of on of h individual caul o, fo b ul, of a hid caul locad midwa bwn hm [7] [8] [7]. Sinc uˆ uˆ δ uˆ δ uˆ uˆ ˆ u δ. an aial quaduol ma b obaind b aing h diffnc bwn h ouu of wo fi-od u gadin micohon oiiond uch ha ach ha h am dicivi ai û and h laiv oiion of on wih c o h oh i δ uˆ ; figu.. Figu.: ial Quaduol a Diffnc of Two Fi-Od Pu Gadin Micohon Thi wa dcibd b Olon who alo nod ha h chniqu ma b ndd o high od, inc an nh-od u gadin ma b obaind b aing h diffnc of wo gadin of od n [7] [74]. I will b aciad ha h am ul ma b obaind b aing h um of wo coaiall oiiond diol oining in ooi dicion. If gnal fi-od micohon a ud inad of diol, w obain omiing fo clai h G a b cal faco 7

46 On h Tho of h Scond-Od Soundfild Micohon a bc ˆ u d a bc ˆ u d a δ auˆ bc uˆ ˆ u δ uˆ bc δ uˆ δ δ uˆ δ uˆ ˆ u d δ uˆ δ uˆ δ uˆ d Phili Coll.4 H h diffnc of u comonn ild a fi-od gadin m, whil h diffnc of fi-od u gadin conibu a cond-od gadin lmn. In h aicula ca a b, a cond-od cadioid i obaind; hi wa alo dcibd b Olon [7] [74] [8]. No ha ca i ndd wih uch aangmn o nu ha aoia qualiaion i alid o ignal ning diffn od of gadin. W ma ma a mo gnal amn han ha d b quaion. a follow: uˆ uˆ δ uˆ δ uˆ uˆ ˆ u δ.5 wh h wo vco ma b chon indndnl. al quaduol ma hfo b obaind b oiioning wo diol uch ha hi dicivi a a aalll, o ha û and û a ndicula; Gon dcibd hi aoach [7]. W ma gnali fuh b coniding h diffnc bwn h ouu of wo fiod micohon of abia ola on. v lag numb of cond-od ola on ma hn b obaind; h oibilii ma b fuh ndd b combining h qualid diffnc ignal wih h ouu fom a hid micohon locad bwn h Blumlin diffnc ai. Som aml a dicud in [7] [8]. I will b aciad ha, in incil, h ola on and oinaion of ach of h micohon could b chon indndnl; hi will no b uud fuh h, inc h incil a adqual dmonad b h iml ca alad conidd. aifaco Blumlin diffnc ignal can b obaind fom a ai of micohon wih an givn aaion onl ov a icd fqunc ang. fqunci wh h dianc bwn h micohon i no long ufficinl mall comad o h wavlngh, h did ola on i no obaind. Gon ad ha h chniqu fom aonabl wll u o fqunci wh h micohon aaion i qual o 74

47 On h Tho of h Scond-Od Soundfild Micohon Phili Coll half h wavlngh, alhough uch a udgmn i ncail omwha inac, inc i dnd on h amoun of vaiaion fom h nominal on which i conidd accabl. low fqunci, h diffnc in ound u bwn h micohon bcom mall, and h Blumlin diffnc ignal hfo bcom mall comad o uiou ignal comonn aiing fom diffnc bwn h micohon. Th limiaion a dicud in dail in [8] and [9]. Each of h fi-od u gadin in quaion. could b divd b uing a ai of u micohon ah han a diol; hi would n a m mloing a lina aa of fou u micohon o obain h cond-od u gadin. Scond-od micohon bad on hi incil hav bn invigad [78] [79] [8], bu w will no conid hm h inc hi aoach i no dicl lvan o h cond-od oundfild micohon. Oh chniqu which hav bn ud o obain cond-od micohon includ h u of micohon caul mbddd in h id of a hollow clind [8], and h lacing of a ingl diol clo o a lan ufac o loi h bounda ffc, ffcivl aing h diffnc bwn h ouu of h al micohon and i acouic imag [5]. W will now conid h u of a cangula aa of fou fi-od caul o div a al quaduol in a mann which ma b conidd o b an nion of h Blumlin diffnc chniqu. I will b n in Cha 6 ha aangmn of hi o occu in h cond-od oundfild micohon. L h fou caul b dnod FL, BL, FR and BR. Th oiion of h caul a FL BL FR BR [ d d ] T [ d d ] T [ d d ] T [ d d ] T.6a.6b.6c.6d and hi dicivi a a uˆ uˆ [ co α in α ] T [ co α in α ] T FL BL.7a.7b 75

48 On h Tho of h Scond-Od Soundfild Micohon Phili Coll uˆ uˆ [ co α in α ] T [ co α in α ] T FR BR.7c.7d Thi aangmn i hown in figu.. Figu.: Caul Configuaion fo Divaion of Tal Quaduol Ron L h ound u a h cn of h aa b O.8 Th ound u a ach of h caul i hn givn b FL BL FR BR d ˆ FL d ˆ BL d ˆ FR d ˆ BR O O O O.9a.9b.9c.9d and h caul ouu ignal a v FL G a b dˆ FL [ a buˆ FL dˆ ] O.a 76

49 On h Tho of h Scond-Od Soundfild Micohon Phili Coll v v v BL FR BR G a b G a b G a b dˆ BL [ a buˆ BL dˆ ] O dˆ FR [ a buˆ FR dˆ ] O dˆ BR [ a buˆ BR dˆ ] O.b.c.d L V B b h diffnc ignal V B v v v v v FL FL v BL BL v FR FR v BR BR. Subiuing in h caul ouu V B G a b G a b dˆ FL BL {[ a bufl dˆ dˆ ] [ a bubl dˆ ˆ ˆ ] dˆ FR BR [ a bufr dˆ dˆ ] [ a bubr dˆ ˆ ˆ ] } O dˆ FL dˆ BL dˆ FR dˆ BR { a[ ] dˆ FL dˆ BL dˆ FR dˆ BR b[ uˆ FL uˆ BL uˆ FR uˆ BR ] dˆ } O. Now inc,, BR FL FR BL ubr ufl ˆ ˆ and uˆ uˆ, o FR BL V B G a b G a b G a b dˆ FL dˆ BL dˆ BL dˆ FL { a[ ] dˆ FL dˆ BL dˆ BL dˆ FL b[ uˆ FL uˆ BL uˆ BL uˆ FL ] dˆ } dˆ FL dˆ FL dˆ BL dˆ BL { a[ ] dˆ FL dˆ FL dˆ BL dˆ BL b[ uˆ FL uˆ FL uˆ BL uˆ BL ] dˆ } { a[ co dˆ FL co dˆ BL ] b[ uˆ FL in dˆ FL uˆ BL in dˆ BL ] dˆ } O. If h caul aaion a ufficinl mall comad o h wavlngh, hn h cala oduc d ˆ and FL d FL ˆ a mall comad o uni, and w ma wi 77

50 On h Tho of h Scond-Od Soundfild Micohon Phili Coll 78 [ ] [ ] { } O O O O O O BL BL FL FL B d d b a Gb d d b a Gb d d b a Gb d d d d d d d d b a Gb d d d d b a Gb b a b a G V co in in co co co in in co in co in co co co co in co in co ˆ co in in co co in co in co co co co co in in co co in co in co co co co ˆ co in co co in co co in co co in co ˆ ˆ ˆ ˆ ˆ α α α α α α α α α α α α α α α α α α d d d d u d u.4 which i a al quaduol on.