Some Analy Resuls fo he Suy of Boaban Nose Raaon fom Wngs Poelles an Jes n Unfom oon *. aassa an J. Case NASA Langley Reseah Cene Hamon gna Absa Alan Powell has mae sgnfan onbuons o he unesanng of many aeoaous oblems n aula he oblems of boaban nose fom es an bounay layes. In hs ae some analy esuls ae esene fo he alulaon of he oelaon funon of he boaban nose aae fom a wng a oelle an a e n unfom fowa moon. I s shown ha when he obseve o mohone moon s suably hosen he geome ems of he aaon fomula beome me neenen. he me neenene of hese ems leas o a sgnfan smlfaon of he sasal analyss of he aae nose even when he nea fel ems ae nlue. o a wng n fowa moon f he obseve s n he movng efeene fame hen he oelaon funon of he nea an fa fel nose an be elae o a sae-me oss-oelaon funon of he essue on he wng sufae. A smla esul hols fo a oelle n fowa flgh f he obseve s n a efeene fame ha s aahe o he oelle an oaes a he shaf see. o a e n moon s shown ha he oelaon funon of he aae nose an be elae o he sae-me ossoelaon of he Lghhll sess enso n he e. Ea analyal esuls ae eve fo all hee ases. o he ases une esen onseaon he nluson of he nea fel ems oes no noue aonal omley as omae o esng fomulaons ha ae lme o he fa fel.. Inouon Alan Powell s aee n aouss sans a eo of moe han ffy yeas. He has mae moan onbuons o he weneh enuy s mao aeas of aous eseah nlung anom vbaon ege ones e nose an bounay laye nose. Alan Powell was one of he fs eseahes o eognze he moane of alng ege nose []. oeove he has ke u amably wh evelomens n all of hese aeas. Hs ublaons onvey a ee unesanng of he hyss of aous henomena. He wes lealy an elans eas luly wh a maseful alaon of analyal ehnques. I s wh gea leasue ha he auhos eae hs ae o Alan Powell fo hs lfeme ahevemens. He s he Gan eahe o us all. hee s a long hsoy of eseah n boaban nose aaon fom es an sufaes see he efeenes n many haes of []. oay s aeonaual oblems ha eque * Publshe n Inenaonal Jounal of Aeoaouss ol. No. &. 5 5.
boaban nose eon ae foun o esen some of he mos hallengng aeoaous oblems o ae. I s fa o say ha he aous analogy has laye a mao ole n he analyss of suh oblems. uh nsgh an unesanng have esule fom suh analyss [ 5 6]. Howeve he lak of suffen measue aa as well as he omley of he oblems hemselves has ause eseahes o make assumons ha smlfy he analyses an ene aomae fomulaons. wo of he mos ommon smlfyng assumons ae ha of an obseve well no he fa fel an a soue oung a low ah numbe. he analyss esene heen wll emonsae a meho by whh boh of hese assumons an be elae. uhemoe he auhos feel ha hee s a nee o emove some o all of he aomaons ha ae ommonly mae n boaban nose eon. In oe o e a anom sgnal ha esuls fom a boaban soue s logal o evelo an aous fomulaon ha s base on sasal analyss. Pevous auhos have eve moan esuls wh he use of sasal analyss e.g. [ 5 6]. uhemoe beause of he uen nees n afame nose een eos on sasally base eseah ae aeang n he leaue e.g. [7]. o he esen auhos knowlege hee ae no ublshe esuls ha ae smla o hose eve heen. In he esen ae some analy esuls ae esene fo he alulaon of boaban nose aaon fom wngs oelles an es n unfom elnea moon. he obeve s o oban ea fomulaons fo suh alulaons wh he assumon ha some sasal oees of he flow ae known on he sufae of he wng o oelle blae o whn he volume of he e. Beause ea esuls ae ese any nea fel ems ha ou n he evaon ae nlue. No assumon s mae on he soue see ohe han a eson o subson moon. I s shown ha he nluson of he nea fel ems oes no a moe omley o he evaon of he fnal fomulaons. I s also moan o ealze ha he nluson of he nea fel ems oes no fuhe omlae numeal algohms ha ae use fo nose alulaons. he followng seon esbes he analyal aoah o he hee fomulaons eve n hs wok. Eah evaon begns wh an aoaely hosen fom of Lghhll s aous analogy [8] wh s usual soluon.e. wh ese o an obseve o mohone ha s fe wh ese o he meum a es. A suably hosen obseve moon hen esuls n he me neenene of all he geome ems n he aaon fomula. Suh a esul gealy smlfes a sasal analyss of he aae nose even when he nea fel ems ae eane. he auooelaon of he aous essue an hen be ely elae o he oss-oelaon of he sufae essue n a smle fom ha an be neee hysally. In Seon he analyal meho esbe above s fs ale o a wng n unfom elnea subson moon wh fluuang sufae essue. o an obseve fe o he unsube meum he aous essue s gven by he soluon of he loang em of he fows Wllams-Hawkngs W-H equaon [9]. he lose fom soluon o hs equaon s avalable an s he founaon fo he analyss. If he obseve s eloae o a fame of efeene ha moves wh he wng hen many of he ems n he soluon beome me neenen yelng a smle esul ha avals self o a sasal fomulaon. A oelaon funon of he aous essue s hen eve
ha s val n he nea an fa fels. he esulng ea fomulaon an be smlfe by makng vaous geome o hysal aomaons fo he evaon of ohe useful esuls. In Seon he meho by whh he oelaon funon s eve n Seon s eene o a oelle n unfom fowa moon by loang he obseve n a efeene fame ha oaes a he shaf see of a oelle. he me neenen funons aeang n he aous essue oelaon funon fo a oelle an be omue numeally an no analyally as n he ase of a wng n moon. he analyss begns smlaly fo a e n fowa flgh n Seon 5 wh an ea esul fo he aous essue a an obseve oson fe o he unsube meum. he ansfe of he obseve o a fame movng wh he e esuls n many me neenen ems. he aous essue oelaon funon s foun by he same oeue ha s use fo a wng o a oelle n moon. Some of he onsequenes of he eve esuls ae susse.. Analyal Aoah he me oman aoah use n he esen wok s base on Lghhll s aous analogy [8]. he govenng equaons ae always solve n a efeene fame ha s fe o he unsube meum. he sae-me vaables ae assoae wh hs fe efeene fame. In aon he analyss eques an obseve n a movng fame of efeene wh assoae sae-me vaables. Saal loaons whn hese wo efeene fames ae elae by he oonae ansfomaon. he aous essue n he fame fe o he unsube meum s enoe by. In he movng fame he aous essue s. Noe ha an ae eesene by ffeen analyal eessons an heefoe he esen auhos hose o use ffeen symbols fo hese wo funons. All of he esen evaons begn wh an equaon of he fom Q whee eesens he lnea wave oeao an Q s he aoae soue sbuon fo a gven oblem. Eq. eesens he oagaon of soun o an obseve n he efeene fame ha s fe o he unsube meum. An analy eesson s now ese fo he aous essue eeve by an obseve n a movng fame of efeene. g. shemaally shows wo ahs o he ese soluon n he movng efeene fame. akng he fs ah along he sol lne leas mmeaely o an equaon of he fom L Q
g.. Soluon Dagam. he oonae ansfomaon an be nooae befoe solvng he equaon sol lne o afe he soluon s obane ashe lne. whee L s a omlae wave-lke oeao an Q s he assoae soue em n he movng efeene fame. In mos ases he soluon of Eq. s moe easly obane han he soluon of Eq. beause of he smly of he fee-sae Geen s funon fo he wave oeao. heefoe he seon soluon ah s hosen as shown by he ashe lne n g.. o he oblems une esen onseaon he soluon n he fe efeene fame s ealy avalable. he soluon n he movng efeene fame s hen obane by alyng a suable oonae ansfomaon o he fe-fame soluon.e. o an aoaely hosen oonae ansfomaon all of he soluon s geome ems n he movng efeene fame beome me neenen hus falang he subsequen sasal analyss. he eque ansfomaon s easly sefe fo he ases une onseaon n he esen ae.. A Wng n Unfom Relnea Subson oon As ealy as 95 Powell sue bounay laye nose fom a g bounay as a sbuon of fluuang oles []. In 96 he sue hs oblem fuhe [] eenng o some een he wok of Cule []. In hs seon he nose fom anom sufae essue fluuaons on a wng n moon s onsee. Sufae essue fluuaons ae ause ehe by he neaon of he wng wh amoshe ubulene
o by ubulen fluuaons n he bounay laye. he aous essue an be alulae fom he soluon of he W-H equaon eanng only he loang em: [ n δ f ] Hee enoes he lnea wave oeao wh onsan soun see s he fluuang essue on he wng sufae an δ s he Da ela funon. he wng sufae s efne by he funon f suh ha f on he sufae of he wng an f > n he egon eeo o he sufae. he noaon n eesens he -h omonen of he ouwa un sufae nomal n f. In hs equaon an henefoh unless ohewse noe he summaon onvenon s assume fo eeae nes. he soluon of Eq. fo a wng n unfom moon s [ ]: n [ n π S e f f ] he flgh veloy veo [ ] s ee along he -as. he obseve sae-me vaables ae enoe n he efeene fame fe o he meum a es. Howeve s moan o emembe ha he sufae negaon s efome wh ese o he saal vaables n he movng efeene fame. he vaable s he me ae of hange of he fluuang sufae essue as measue by a ansue on he wng sufae an s a omonen of he un aaon veo. he quany s he ah numbe n he aaon eon efne by whee s he ah numbe veo base on he fowa veloy of he wng. he subs e enoes evaluaon of he bakee ems a eae me. Noe ha he fa fel aaon s govene by he me ae of fluuaons of he sufae essue. A fame of efeene ha moves wh he wng s now onsee See g.. Le hs movng efeene fame one wh he fame fe o he unsube meum a he me. o fn he aous essue a he obseve oson n he movng fame mus be elae wh n Eq. 5. If an abay soue on on he wng n he movng fame of efeene s esbe by he oson veo hen he followng wo funons ae me neenen an an be wen n ems of he vaables an : e S 5 n n π e π e 6 See [5 Se. 5] fo eals of he analy eessons fo he wo funons n Eq. 6 n ems of he vaables an. he me neenene of hese wo funons gealy smlfes he sasal analyss of he aous essue. Noe ha he emsson sane 5
η g.. he efeene fames use n he analyss of a wng o a e n unfom moon. he -fame s fe o he meum. he -fame moves wh he wng o he e a see along he -as. he leang ege of he wng faes n he osve eon. he lne segmen has lengh s he soue oson an s he obseve oson n he movng efeene fame. o a oelle he -fame s aahe o he oelle blae oaes aoun he -as an moves fowa wh see along he same as. [ ] e as well as ohe geome vaables suh as ae also me neenen an an as susse n [5 Se. 5]. g. shows an be eesse as a funon of he geomey assoae wh aaon fom a soue a oson n he movng fame. he ue onsuon shows ha hs geomey s neenen of me. he angle ABC somemes alle he Gak angle n he leaue of lnea unseay aeoynams s seen by an obseve osone n he saonay fame of efeene. he shae an he mensons of hs angle o no hange wh me. heefoe Eq. 5 an be ewen wh ese o he movng efeene fame n he followng smlfe fom: Usng he noaon f [ ] o enoe an ensemble aveage n me he auooelaon of he aous essue n Eq. 7 an be wen: S 7 6
7 [ f f η η S S ] 8 whee he me noaon on he quanes an enoes he elaemen of he vaable n hese quanes wh. Now efne a sufae essue oss-oelaon funon by R 9 Usng he ego hyohess he vaable eenene of he oss-oelaon funon n Eq. 9 an be eesse: R R Le A enoe he oelaon aea as a funon of. hen he aous-essue oelaon funon n he movng efeene fame an be wen: ψ η g.. he geomey of aaon as seen by an obseve fe o he meum a es. A an B ae he soue an obseve osons a a gven obseve me. C s he emsson oson of he soue an CB whee s he sgnal oagaon me fom C o B. Alaon of he law of osnes o angle ABC yels as a funon of neenen of me as ae an. Noe ha he symbol enoes soue me n hs fgue an no elay me.
[ R R f A R] S S whee an enoe esevely he fs an seon aal evaves wh ese o elay me. Eq. s he man esul of hs seon an s ea fo he alulaon of boh nea an fa fel loang nose fo a sufae n unfom elnea moon. A few emaks ae n oe a hs on. Noe ha he oss-oelaon funon R n Eq. 9 an be obane fom measuemens by ansues on he wng sufae. uhemoe alhough he assumon of egoy s easonable one anno fuhe assume ha he eenene of R on he oson vaables an an be geneally eesse n ems of he seaaon veo. nally s neesng ha n he fa fel Eq. akes he followng smle fom: R S S f A Howeve beause he funons an an easly be omue numeally he eson o he fa fel s no neessay bu meely a onvenene fo aoae oblems. If qualave esuls ae ese e.g. a ah numbe eenene ule fo an obseve n he fa fel hen fuhe smlfyng assumons an be mae n he above equaon. I s lea fom Eq. ha fo a eals oblem when he obseve s n he nea fel a ah numbe eenene ule ha s eve fo a smle ole n moon anno be eee o hol. hs s beause of he omlae eenene of he funons n Eq. on an. Howeve numeal evaluaon of he negal on he gh-han of Eq. s que feasble an an gve a goo ue of he nea fel behavo of he boaban nose fom a wng n moon. By analyzng eemenal aa analyal moels an be eue fo he sufae essue oss-oelaon funon on a wng. As n e nose analyss suh moels ae vey moan. If avalable he eemenally measue sufae essue ossoelaon funon an be ely use n a nose alulaon. I s neesng o noe ha he numeal evaluaon of he oelaon funon of he aous essue n he nea fel s no moe fful han n he fa fel omuaon.. A Poelle n Unfom owa lgh In oe o nlue he effes of oaonal moon he soluon of Eq. beomes slghly moe omlae han he soluon fo he wng n Eq. 5. o a oelle wh fowa flgh veloy he aous essue a an obseve loaon a me s gven by he followng eesson [ ]: 8
whee π f f n [ n e S ] s he -h omonen of he loal sufae aeleaon veo e S v ve by he soun see. he eae s agan emne ha he sufae negaon s efome wh ese o he vaable n he movng efeene fame. he obeve n hs seon s o eve a esul smla o Eq. fo he moe omlae ase of a oang oelle n fowa moon. Noe ha he smly of Eq. eens on he me neenene of an n Eq. 7. I s naual o ask whehe hee ess a saghfowa eenson of Eq. 7 o oaonal moon. If he obseve s lae n a oaonal efeene fame ha s aahee o he oelle blae an me neenen funons an agan be eemne? he answe o hs queson s affmave by he followng easonng. Le enoe he angula veloy veo of he oelle an s unfom flgh veloy. In a smla fashon o he wng oblem above assume ha a efeene fame ha oaes wh he oelle an a efeene fame ha s fe o he meum a es one a he me. Le enoe he obseve oson n he oang efeene fame an be he oson veo of a on on he oelle blae sufae n he same efeene fame. Noe ha he elave oson of hese wo veos s he same fo all obseve me. Whn hs famewok efne he followng wo funons: n π π e e n π I emans o show ha he wo funons n Eq. as well as ae neenen of he obseve me. An analogous esul o Eq. 7 wll hen mmeaely follow. he me neenene of an an be emonsae n he followng way. Refeng o g. a a gven obseve me feeze he oson of he obseve n he efeene fame fe o he meum a es. he onbuon o he aous essue a he obseve oson fom a soue oson on he oelle blae sufae eques he evaluaon of he loaon of he soue oson a he eae me. he eque loaon of an be vsualze by leng he on ae s hsoy bakwa e Eq. as shown hee has been oee fom ha ne on age n Inenaonal Jounal of Aeoaouss ol. No. &. An eaum has been subme o he ounal. 9
% 5 *& - &'!"#$. 5678569 η 6 :;56<5 g.. he geomey of aaon fom a soue on a oelle blae as seen by an obseve fe o he meum. A an B ae he soue an obseve osons a a gven obseve me. C s he emsson oson of he soue an CB whee s he sgnal oagaon me fom C o B. C s foun fom he elaon CA helal see helal ah numbe whee CA s he sane along he hel fom C o A. hs shows ha an ae funons of. Noe ha he symbol enoes soue me n hs fgue an no elay me. ω n me on a hel See g.. he emsson on of hs soue les on hs helal ah an s always n he same hysal loaon elave o he obseve oson neenen of he obseve me. In hs susson he obseve s osone n he fame fe o he meum an obsevng he oagaon henomenon. Usng Eqs. an whn he afoemenone famewok of a oang efeene fame he aous essue a an be eesse n he followng fom: [ ] f whee an ae efne n Eq.. Noe ha fo a oang oelle he funons an as well as ohe geome quanes anno be eemne n lose fom even f hee s no fowa flgh. Howeve hese funons an easly be omue. S 5
hus has been esablshe ha he gh-han ses of Eqs. 5 an 7 ae enal n fom wh me neenen funons an. heefoe Eq. s also val fo he aous-essue oelaon funon a an obseve n a oang efeene fame ha s aahe o he oelle blae wh he unesanng ha an ae gven by Eq.. oeove all of he ommens followng Eq. fo a wng n unfom moon also aly o a oelle n fowa flgh. he measuemen of soun by a oang mohone esens some eemenal ffules suh as he esene of wn nose. Howeve s ossble o use avane sgnal analyss ehnques o oveome hese ffules. he esuls n hs seon ae esene o bng aenon o he smly of Eq. fo a oang mohone. 5. A Je In Unfom Relnea Subson oon A e n unfom elnea subson moon s now onsee. Lghhll s [8] e nose equaon s: 6 whee s he Lghhll sess enso. he e s assume fee of shoks he mlaons of whh wll be susse below. Conse a fame of efeene ha moves wh fowa flgh of he e an moves wh a unfom elnea subson veloy. In [6] an ea soluon of Eq. 6 s eve ha nvolves only he obseve me evaves an nlues he nea fel ems: π Hee an e e e 7 s he ah numbe n he eon of nose aaon base on he fowa veloy. he vaable eesens he ubulen volume n he e whle enoes a volume elemen. I s moan o noe ha he obseve me evaves hee ae efome whle keeng he obseve loaon fe. In fa s eemhasze hee ha he obseve n Eq. 7 s esbe n a efeene fame ha s fe o he unsube meum. uhemoe all he veloes n he Lghhll sess enso ems ae evaluae elave o he fame fe o he unsube meum. Howeve he negals n Eq. ae efome n he movng efeene fame. hese fas an all be nfee fom he evaon of he above equaon. Benne [7] has eve an equvalen eesson o Eq. 7 when he me evaves ae aken nse he negals:
e e π 8 e 5 6 In Eq. 8 a o ove a symbol enoes a soue me evave keeng he vaable fe.e. he me ae of hange of he aamee as measue n he movng fame. he le noaon eesens a ah numbe base on he fowa veloy. he above equaon sealze o he oblem of oang blae nose has been use suessfully by Benne [7] o omue hgh-see heloe oo blae nose. In a manne smla o he evous wo seons f he obseve s n he movng fame a he oson hen many of he ems n Eq. 8 ae me neenen. he aous essue n he movng fame s heefoe: 9 H G [ ] whee hee me neenen enso funons ae efne as follows: e e e H G 5 π π π δ π π π δ π Hee δ s he Koneke ela. As n he evous wo seons e ] [ * s also me neenen. he veloy ems n he Reynols sess em of he Lghhll quauole enso ae now onsee. Le u an U enoe he veloy omonens of he e n he movng an he saonay fames esevely. hen he ems nvolvng he veloy ou n he Reynols sess enso n all he above equaons mus be U U ρ as elane above. In ems of he veloy omonens n he movng fame usng he elaon U u δ he veloy ems of he Reynols sess ae:
u u U U u u u U U u u u U U hus he aous essue n he movng fame an now be eesse n ems of quanes ha an be measue n he movng fame of efeene. uhemoe he only me eenen quanes n Eq. 9 ae he Lghhll sess enso em an s me evaves. he analyss fo he wng an he oelle n fowa moon s now eeae. he auooelaon funon of he aous essue an be eesse as follows: [ H H G H H H G G G G H G ] whee he vaable s elae by he vaable n he me quanes. Now efne he oss-oelaon funon of he Lghhll sess enso by: R lk hen f C s he oelaon volume Eq. an be wen as:
C [ H H R R G G R G G H H R G H H G R ] whee e. enoe aal evaves wh ese o elay me. o fa fel aaon wh he obseve n he movng fame he auooelaon of he aous essue assumes he followng smle fom: [ R 5 ] C η heefoe o alulae he auooelaon of he aous essue n he movng fame he oss-oelaon funon of he Lghhll sess enso R mus be moele base on eemenal obsevaons. In geneal suh moelng s a fful ask. Howeve he mahemaal famewok gven hee an lea o he evelomen of analyal moels ha an be use o make eons o omae wh aous measuemens [8]. Noe ha alhough Eq. 7 s val fo a e wh shoks Eqs. 8 an 9 ae only val fo shok-fee es. he eason fo hs eson s ha he obseve me evaves n Eq. 7 mus be eae as genealze evaves n oe o nehange hose evaves wh he negaon [9 ]. When he genealze evaves ae bough nse he negal any shok sonnuy n a e wll oue aonal ems ha ae no nlue n he esen e nose analyss []. he analyss of es wh mbee shoks s moe omlae han wha s esene hee. nally noe ha he esuls esene n hs seon ae only a elmnay sasal analyss fo e nose. he woks of Rbne [] fows Wllams [] an ohes an be use n moelng he oss-oelaon ems n Eq.. Conlung Remaks Some ea analyal esuls have been eve fo he eon of boaban nose aae fom a wng a oelle an a e n unfom subson moon. o an obseve n an aoae fame of efeene he oelaon funon of he aous essue an be ely elae o he sae-me oss-oelaon of ehe he sufae essue on a wng o oelle blae o he oss-oelaon of he Lghhll sess enso n he ase of a e. I has been shown ha he nluson of he nea fel ems oes no a o he omley of he analyss. he evelomen of hese esuls eques ha he obseve be n he movng efeene fame n he ase of a wng o a e n fowa flgh o n a
oang efeene fame n he oelle ase. hese equemens noue many me neenen funons n he negans of he aous fomulaons esulng n onseable smlfaon of he sasal analyss. hese analyal esuls wee eveloe wh aal alaons n mn. In aula he goal of he esen analyal evelomen s o oban sasal fomulaons fo whh he eque nu aa ae no unusual o fful o measue. I s shown hee ha hs an be aheve by he seleon of an aoae fame of efeene n whh he obseve oson s esbe. Alhough some of he esen esuls may aea omlae hey ae oneually smle fo numeal wok. One of he avenues oen o uen eseahes nvolves eemenng wh analyal moelng of he oss-oelaon funon n aous fomulaons usng measue aa fo guane. In aon eemenal aa an always be use ely n he esen analyal esuls when hey ae avalable. As a fnal emak noe ha an analyss smla o ha esene heen an be use o oban a oss-oelaon funon ha oelaes he sufae essue o he aous essue. he evaon of suh a oss-oelaon funon was esene by Son []. As eoe by Son suh an analyss an be vey useful fo he enfaon of nose geneang egons. he aeoaouss ommuny owes muh o oneeng eseahes lke Alan Powell who le he es of us o he moane of enfyng an solvng sgnfan aous oblems. We wsh hm all he bes fo many yeas o ome. Refeenes. A. Powell: On he Aeoynam Nose of a Rg la Plae ovng a Zeo Inene J. Aous. So. Am. 959. 69-65.. H. H. Hubba Eo: Aeoaouss of lgh ehles heoy an Pae olume Nose Soues Aousal Soey of Amea 995.. W. C. eeham: Sufae an olume Soun om Bounay Layes J. Aousal Soey of Amea 7 965. 56-5... E. Son: Sufae Dole Sengh by Coss-Coelaon eho J. Aousal Soey of Amea 5 97. 69-6. 5. W. C. eeham: heoy of A ame Nose: J. Aousal Soey of Amea 576 Pa II 975. 6-. 6. W. R. lle an Wllam C. eeham: Lage Sale oel easuemens of Afame Nose Usng Coss-Coelaon ehnque J. Aousal Soey of Amea 7 98. 59-599. 7... Books an W.. Humheys: la-ege Aeoaous easuemens an Peons J. of Soun an baon 6. -7. 8.. J. Lghhll: On Soun Geneae Aeoynamally. I. Geneal heoy Poeengs of he Royal Soey of Lonon A 95. 56 587. 5
9. J. E. fows Wllams an D. L. Hawkngs: Soun Geneaon by ubulene an Sufaes n Abay oon Phlosohal ansaons of he Royal Soey A 6 969... A. Powell: houghs on Bounay Laye Nose Aeonaual Reseah Counl Gea Ban Reo A.R.C. 677.. 6 Al 95.. A. Powell: Aeoynam Nose an he Plane Bounay J. Aous. So. Am. 8 96. 98-99.. N. Cule: he Influene of Sol Bounaes on Aeoynam Soun Po. Roy. So. Lonon A 955. 55-5... aassa an G. P. Su: he Peon of Heloe Roo Dsee equeny Nose ea 7 98. 9-.. K. S. Benne: Peon of Heloe Roo Dsee equeny Nose- A omue ogam nooang eals blae moons an avane aous fomulaon NASA ehnal emoanum 877 Oobe 986. 5. J. Case an. aassa: A New me Doman omulaon fo Boaban Nose Peon In. J. of Aeoaouss. 7-. 6.. aassa an K. S. Benne: he Uses an Abuses of he Aous Analogy n Heloe Roo Nose Peon J. Amean Heloe Soey ol. 988. 9-6. 7. K. S. Benne: An Effen an Robus eho fo Peng Heloe Hghsee Imulsve Nose J. Soun & baon 997. 87. 8. P. J. os an. aassa: Aous Analogy an Alenave heoes fo Je Nose Peon AIAA J. Al. 67-68. 9. I.. Gelfan an G. E. Shlov: Genealze unons: Poees an Oeaons olume Aaem Pess New Yok 96... aassa: Inouon o Genealze unons Wh Alaons n Aeoynams an Aeoaouss NASA ehnal Pae 8 99 Coee Al 996 Avalable a: h:eheos.la.nasa.govlspd8.f.. aassa an. K yes: An Analyss of he Quauole Nose Soue of Hgh See Roang Blaes Comuaonal Aouss Saeng Gaussan Beams an Aeoaouss D. Lee A. Cakmak R hnevesky Eos ol.. 7-99.. H. S. Rbne: heoy of wo-pon Coelaons of Je Nose NASA ehnal Noe ND-8 976.. J. E. fows Wllams: he Nose fom ubulene Convee a Hgh See Phlosohal ansaons of he Royal Soey of Lonon ol. 55 96. 69-5. 6