Chap.4 Ray Theory. The Ray theory equations. Plane wave of homogeneous medium

Transcription

1 The Ra theor equatio Plae wave of homogeeou medium Chap.4 Ra Theor A plae wave ha the dititive propert that it tregth ad diretio of propagatio do ot var a it propagate through a homogeeou medium p vae i thi diretio over the ale of oe wavelegth Diretio of propagatio p e ( t (, )) iω θ wavefrot : the magitude of p i the ame at all poit alog the wave frot (, ) oθ iθ Noie Egieerig / Aouti - -

2 The Ra theor equatio Ra Theor O the other had, a wave propagatig through a regio with a lowl varig oud peed will have a lightl urved wave frot Ra are defied to be urve whih are alwa ormal to the wavefrot Ra are ueful oept if L λ, i.e. high freque wave. The the ol effet of variatio i the oud peed i that peed of propagatio of the wavefrot alog the ra. ra wavefrot at time tδt(,) wavefrot at time t(,) Noie Egieerig / Aouti - -

3 Ra Theor The Ra theor equatio p (,, t) (, ) e ( t (, )) iω (Note) (i) varie beaue of gradual hage i the oud peed. p (ii) The variatio of our over the legth ale over whih the oud peed varie (geerall log ompared to wavelegth) (iii) The phae i a futio of poitio ad aout for the variatio p over the ale of wavelegth (iv) f, L, the ale o whih the oud peed varie, i MUCH greater tha the wave legth, the i lowl varig ompared to ad the movig urfae, t- (,) otat to be wavefrot Noie Egieerig / Aouti - 3 -

4 The Ra theor equatio Tramiio through the tratified medium Ra Theor A otiuou variatio a be oidered a a approimatio of N jump δ Wave frot θ Noie Egieerig / Aouti - 4 -

5 The Ra theor equatio Ra Theor Whe a plae wave propagate from the oe medium to aother with a differet oud peed, Sell law i valid i tramiio. iθ otat iθ Sell law a be ued to determie the ra path ad hee to diover where oud i heard The ra bet to the diretio of dereaig of ireaig dereaig θ Night Da Noie Egieerig / Aouti - 5 -

6 Ra Theor The Ra theor equatio (Eample) the oea Sell' Law : iθ iθ C C where d d d C iθ i i ( ) θ θ α d d d { ( ) } C d Noie Egieerig / Aouti - 6 -

7 The Ra theor equatio (Eample) the oea d d ( ) (( ) )( α) i θ) d d d ( α)iθ ± d { ( α) i θ }, ( i θ) otθ α ( α)iθd ± { ( α) i θ} oθ { ( α) i θ} ± α iθ α iθ oθ α iθ α iθ ( ( α) i θ) α i θ Ra Theor o otθ ( ) α α e θ α Noie Egieerig / Aouti - 7 -

8 Ra Theor The Ra theor equatio Cotiuoul varig medium i otat withi eah lab : Noie Egieerig / Aouti - 8 -

9 The Ra theor equatio Ra Theor Refletio Coeffiiet of Plae wave at the iterfae with oud peed ad deitie, The preure & veloit (ormal diretio) hould be iterfae R oφ oφ oθ oθ R θ θ φ T oφ i θ R ~ N Noie Egieerig / Aouti - 9 -

10 Ra Theor The Ra theor equatio ompat laer (L λ) No phae differee Tramiio & Refletio deped ol o the et hage i aouti propertie 'aro' the laer. (Neglet the hage of aouti propertie withi the laer) o - ompat laer (L λ) Large phae differee betwee iidet ad refleted wave. Eerg refleted from eah iterfae R Noie Egieerig / Aouti - - N N iterfae E R ~ N

11 The Ra theor equatio f N Ra Theor " the atual otiuou variatio i oud peed ad o eerg i refleted. " (ER ) No refletio eerg for a high freque oud ra propagatio through a medium i whih the oud peed varie otiuoul. Eerg flu i otat alog the ra tube of oud propagatig through a medium i whih ad var lowl ( L λ) Covergig ra High oud preure Ra theor a be ued to determie the level of oud heard Noie Egieerig / Aouti - -

12 Noie Egieerig / Aouti - - Ra Theor A more rigorou derivatio of Ra theor Derivatio of Ra theor Coervatio of ma tate that Mometum equatio of eah diretio i Thee two equatio a be ued for ubtitutio Ra erie i derived from the harmoi futio of time. ( ) ( ) v t u t p t u p t v p p t p ( ) ( ) ( ) ( ) ( ),,,, t i i e t p ω ω

13 Noie Egieerig / Aouti Ra Theor A more rigorou derivatio of Ra theor Ra are defied to be urve whih are everwhere ormal to the wavefrot From the derivative of ra erie, ubtitutio of the orret form for the wave equatio lead to grad grad, d dy d dx ( ) ( ) i i i ω ω ω

14 Noie Egieerig / Aouti Ra Theor A more rigorou derivatio of Ra theor Firt, the equatio i to be true for all value of ω The oeffiiet of ω mut alo vaih, ra ( ) ( ) ( ) Y X, grad wavefrot Eikoal Equatio

15 Noie Egieerig / Aouti Ra Theor A more rigorou derivatio of Ra theor Uig the hai rule of diretio with regard to diretio, eikoal equatio lead to the Sell law The iteit,, mut atif the d equatio. θ i, d dy d d d dy d d

16 Noie Egieerig / Aouti A more rigorou derivatio of Ra theor The olutio to thi equatio i Ra Theor ( ) ( ) ep ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) A A δ A grad δ ( ) ( ) ( ) ( ) otat alog the ra tube A Eerg Flu Coerved

17 Uderwater oud propagatio Soud wave of uderwater Ra Theor Water tramit oud wave far better tha it doe optial, radio or mageti wave, ad o oud i ued eteivel for uderwater ommuiatio The temperature ear the urfae i warm, ad temperature i dereaed alo to m. t mea that peed of oud i alo dereae Uder the m, however, the ireae of preure lead to a ireae of oud peed Soud Speed C Depth Noie Egieerig / Aouti - 7 -

18 Uderwater oud propagatio Ra Theor The ra propagatig dowward will evetuall reahe the regio whih peed of oud i haged iverel, ad the ra are propagatig upward. Fiall the ra are trapped withi the regio. Thi regio are alled oud hael 48 Soud Speed m A Soud Chael Depth Noie Egieerig / Aouti - 8 -

19 Uderwater oud propagatio Ra Theor The refratio of oud ra b aomalie of the oud peed profile i the oea a lead to the formatio of hadow zoe the ae of a oud oure i the oea ear a poitio of maimum i the oud veloit, ra propagatig upward move ito regio where the oud peed dereae ad are refrated upward, ad vie vera. Soud Speed Soure Shadow zoe Depth Noie Egieerig / Aouti - 9 -

20 Soud propagatio i the atmophere Soud refratio o a da time Ra Theor Durig the datime the air temperature ted to dereae with height above earth, ad hee the oud peed dereae upward. Ra are the bet up, ad mot of the oud will pa over the head of a ditat oberver dereae upward Soud propagatio o a tpial da Noie Egieerig / Aouti - -

21 Soud propagatio i the atmophere Soud refratio o a ight time Ra Theor Sometime o a lear ight, the groud ool more quikl tha the air, the ra are bet bak to the groud are heard more dititl b a oberver ireae upward Soud propagatio o a lear ight followig a warm da Noie Egieerig / Aouti - -