Silence is the only homogeneous sound field in unbounded space

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Marcus Donald Powers
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1 Cha.5 Soues of Sound Slene s he onl homogeneous sound feld n unbounded sae Sound feld wh no boundaes and no nomng feld 3- d wave equaon whh sasfes he adaon ondon s f / Wh he lose nseon a he on of = he wave equaon s edue o he Lalae equaon beause saal devaves ae moe sngula / f / f Sne em s moe sngula Nose Engneeng / Aouss - -

2 Soues of Sound Slene s he onl homogeneous sound feld n unbounded sae Consde he small volume suoundng he ogn s fne V sufae S of aea 4 V f / f / s n f 4 4f ds / as f / 3-D dela funon δ() V ( ) dv If V nludes = ohewse Nose Engneeng / Aouss - -

3 Soues of Sound Slene s he onl homogeneous sound feld n unbounded sae Boh sdes of equaon s se o be zeo on he homogeneous wave equaon f / 4f - onl homogeneous ondon of - f= Geneall eq. an be wen as ' ' q( ) In sound feld q = In soue egon q Nose Engneeng / Aouss - 3 -

4 The defnon of a sound soue The sound wh ve weak dsubanes Thee dmensonal wave equaon s q Soues of Sound In he sound feld q mus be zeo. In he soue egon q s nonzeo f q f q f q q f q Sound feld : Soue feld : The wave feld does no onan suffen nfomaon fo s soues o be denfed. Bu he sound feld geneaed b a soue feld q s unque. ' f ( q) q ' f ( f ( q) q ) Nose Engneeng / Aouss - 4 -

5 Nose Engneeng / Aouss Soues of Sound The defnon of a sound soue Readed me The sound avng a a me mus have been launhed fom he soue a he evous me τ = - - / whh s usuall alled he eaded me. If he soue egon(v) s oma he soue egon s neglgble fo a dsan obseve( ㅣ ㅣ ㅣ ㅣ ). f f dv 4 q 3 4 d q V V d q q 3 q q 4 / I-I V

Nose Engneeng / Aouss - 6 - Soues of Sound The defnon of a sound soue Monoole and dole soue dsbuon The monoole soue dsbuon q() s sad o be degeneae no a dole dsbuon Afe dffeenaon

6 Nose Engneeng / Aouss Soues of Sound The defnon of a sound soue Monoole and dole soue dsbuon The monoole soue dsbuon q() s sad o be degeneae no a dole dsbuon Afe dffeenaon wh ese o he 3 4 d f V V f d f ) ( 4 / 3 Sufae S enloses volume V n whh les all he egon whee f Soue ) ( f q ) ( f q : monoole dsbuon : dole dsbuon 3 4 d f V So Equvalen monoole fomula

7 Soues of Sound The defnon of a sound soue Pon monoole and doles ) on monoole R X Obseve q( ) Q( ) f ( ) O Soue : Soue sengh q d V 4 3 Fo oma soue Q / 4 omn-deonal ) Pon Dole q( ) ( F( ) ( )) ( F ( ) ( )) Dole sengh wh deon Nose Engneeng / Aouss - 7 -

8 Nose Engneeng / Aouss Soues of Sound The defnon of a sound soue Combned soue sengh Soue : F()=Q()δl Q dole as Q as dv as ( 3 3 l l Q l l l l Q Q δl F d 4 / 3 4 d f V F F d 4 os

9 The defnon of a sound soue Noe Soues of Sound ) he sound feld of dole soue has an angula deendene ) no dsubane a 9 o he dole as(he sound eal anel) ) Pessue dsubane have a mamum magnude on he dole as v) P`dole = Nea-feld em (~/ )+ Fa-feld em (~/) P`monoole = Fa-feld em onl (~/) v) Dole has some anellaon of he aous feld so doles ae less effen adaons han monooles. (b M ) Nose Engneeng / Aouss - 9 -

10 Soues of Sound The defnon of a sound soue (Eamle) Two on soues ; () monoole () dole a = adaes sound of fequen ω. Deemne he ao of mamum essue dsubane a =ε( /ω) o ha a =l /ω) fo he wo soues. ) essue eubaon of monoole A ' ( ) e 4 ( ' A ma 4 ' A ma l 4l ) Rao l Nose Engneeng / Aouss - -

11 The defnon of a sound soue ) essue eubaon of dole Soues of Sound ' ( ) os B e 4 ( ) ' B ma ' ma l 4 B l 4 Rao l εω/ so l l Nea-feld dsubane deas ve fas so does no does no adae fa awa as sound. Nose Engneeng / Aouss - -

12 Aous soue oess Soues of Sound Wha mehansms do odue hese soue dsbuon n naue? Soue oess one n whh hee s a fong em n he wave equaon Volae he homogeneous wave equaon!! Common soues Vbaon of he sufae Beakdown of he sae equaon =f(ρ) (unsead hea addon o flud) Nose Engneeng / Aouss - -

13 Aous soue oess Combuson nose Soues of Sound When hea s added o a flud he dens of he flud a funon of wo vaables of essue and hea ρ=ρ(h) Hene d d dh h h d h dh Fo a efe gas RT h RT T h RT Fnall d dh d The lneazed equaon desbng moon of a efe gas wh unsead hea s h Nose Engneeng / Aouss - 3 -

14 Aous soue oess The soluon of wave equaon s / h d 4 Soues of Sound The sound feld s foed b he ae of hange n he hea addon ae. Sead heang s slen 3 Auall he sound of he monoole soue aused b unsead heang s uel due o he unsead eanson of he flud a essenall onsan essue. h Monoole sengh = ) ( V Rae of hange n he faonal ae of volume gowh Nose Engneeng / Aouss - 4 -

15 Aous soue oess Soues of Sound Sound geneaon b he lnea eaon of mae and eenall aled foes The aous soues b he mass eaon dv v m The lneazed momenum equaon wh aled foes v gad f The dens would be funon of a essue and he newl eaed mass. Suose ha oal dens volume s onssed of eaed mass and es of he flud m ( m) m f Nose Engneeng / Aouss - 5 -

16 Nose Engneeng / Aouss Soues of Sound Aous soue oess The dens fluuaon wh ese o me s So The wave equaon above hese elaon s I s onl he volume oued b ha newl eaed mass ha dves sound feld m f m m dv f Aeleaon em Eenal Fong em

17 Nose Engneeng / Aouss Soues of Sound Aous soue oess Noe ) sead eaon of mass s slen (onl aeleang gowh makes sound) ) Monoole sengh 3) dole sengh The soluon of wave equaon s If he flow feld has non-lnea effes sound waves ae also geneaed.(quaduole) d f d V V volume un f

18 Lghhll s aous analog Sound geneaon b flow Soues of Sound Sound Weak moon abou a homogenous sae of es Soue of sound b Lghhll (95) The dffeene beween he ea saemen of naual laws and he aousal aomaons The ea saemen mass onsevaon v The ea saemen of momenum onsevaon The manulaon of above wo equaons v v v v v Nose Engneeng / Aouss - 8 -

19 Nose Engneeng / Aouss Soues of Sound Lghhll s aous analog Suba dens nd ode devaves and Lghhll s equaon s deved T s he soue and s alled Lghhll sess enso The sound feld b he ea equaons s T s of ouse zeo n he sound feld f we have lnea moon onl Howeve T anno be vanshed n he ubulen flows T v v T dv T V 4

20 Soues of Sound Lghhll s aous analog Noe ) No aomaon made so fa ) Real maeal moon aous moon Quaduole Monoole Doole Non- lnea Lnea ) The sengh of he quaduole s T v) Double dvegene - double enden fo soue elemens o anel Dole feld - he dvegene of a monoole feld Quaduole feld - he dvegene of a dole feld - he double dvegene of a monoole feld Nose Engneeng / Aouss - -

21 Soues of Sound Lghhll s aous analog Je Nose Consde a e flow of velo U and damee D Je seed U dens ρ Sound waves oagae o () a seed T a (=l-l/) Tubulen eddes of haaes seed U and sale D geneae sound of sale λ=d/m Chaaes fequen and wavelengh s defned b dmensonal oeffen of e of flow and damee D U DM Nose Engneeng / Aouss - -

22 Lghhll s aous analog Nose Engneeng / Aouss - - Soues of Sound Fo low Mah numbe (λ D) he soue egon s oma. he eaded me an be neglgble The sound feld s dmensonalzed b belows : o Togehe hese gve he esmae fo low Mah numbe M 4 D M D 3 dv D T U Low Mah numbe e ae aousall ve neffen as ae all oma soues M 8 D when M<< Eghh owe low

23 Soues of Sound Lghhll s aous analog Fo hgh Mah numbe sueson es (λ D) he sound feld s domnaed b eaded me Fa-feld aomaon ds = sufae =onsan as obseve a () O Nose Engneeng / Aouss - 3 -

24 Nose Engneeng / Aouss Soues of Sound Lghhll s aous analog The sound feld wh fa-feld aomaon s In ode o bng ou he eaded me vaaons The haaes magnude of he dens eubaon n he sound ndued b sueson es s esmaed b seng Dens and he mean squae nose level s dv T 4 T T d ds T 4 s s D ds D D M D M T U U D d

25 Lghhll s aous analog Soues of Sound Auall hee s one ohe effe o do wh he fa ha he numbe of eddes ha an be head a an me neases wh Mah numbe whh hanges hs law o he velo ubed deendene ha s n ageemen wh he eemenal obsevaon Nose Engneeng / Aouss - 5 -

26 Soues of Sound The sound fom he saona foegn bodes The foegn bodes n he flow T Evaluae dv V 4 Though some mahemaal oedues ( e) Nose Engneeng / Aouss - 6 -

27 Soues of Sound The sound fom he saona foegn bodes When hee ae foegn bodes n he flow s somemes onvenen o lum ogehe he quaduoles dsbued ove he neo of foegn bodes. Fs he devaves of wh ese o he obseve oson ae ewen n ems of he devaves of wh ese o he soue oson T T T T an be e-aanged as([] mles ha funon of eaded me) T Dffeenae wh ese o he gves T vv v vv v Nose Engneeng / Aouss - 7 -

28 Nose Engneeng / Aouss Soues of Sound The sound fom he saona foegn bodes The oedue desbed above shows ha Afe usng he equaon of mass onsevaon. Hene follows ha Then negae ove a saona volume v v v v v v v T ds ds v v dv T S S V n v

29 Soues of Sound The sound fom he saona foegn bodes The esul allows Lghhll s aous analog Noe 4 dv..quaduole 4 4 S S V T vv n v n ds ) The quaduole feld neo o he bod = monoole + dole ds..dole..monoole V T dv S vv n ds S v n ds Nose Engneeng / Aouss - 9 -

30 Soues of Sound The sound fom he saona foegn bodes ) If he foegn bodes ae moonless and gd (n v=) 4 s n ds n oma bod 4 F F n ds S F s Foes aled o he bod => Dole soues F U M D D M 6 M Bod foes ae moe effen o geneae sound b a fao of M - han low Mah numbe fee ubulene 3 D D Nose Engneeng / Aouss - 3 -

31 Soues of Sound The sound fom he saona foegn bodes 3) monoole soue 4 Q Q S v n ds M D and M 4 D Moe effeve han dole Nose Engneeng / Aouss - 3 -

32 Nose Engneeng / Aouss Soues of Sound

calculating electromagnetic

calculating electromagnetic Theoeal mehods fo alulang eleomagne felds fom lghnng dshage ajeev Thoapplll oyal Insue of Tehnology KTH Sweden ajeev.thoapplll@ee.kh.se Oulne Despon of he poblem Thee dffeen mehods fo feld alulaons - Dpole