ACOUSTIC WAVE EQUATION. Contents INTRODUCTION BULK MODULUS AND LAMÉ S PARAMETERS

Transcription

1 ACOUSTIC WAE EQUATION Contnts INTRODUCTION BULK MODULUS AND LAMÉ S PARAMETERS

2 INTRODUCTION As w try to vsualz th arth ssmcally w mak crtan physcal smplfcatons that mak t asr to mak and xplan our obsrvatons. Lnr (4, p.33 consdrs thr typs of smplfd dscrptons for th arth matrals: flud, porous and sold. Fluds,.g. gass and lquds charactrstcally can dform only by comprsson and dcomprsson. Only acoustc, or sound, or P-wav or comprssonal body wavs ar th natural wav vbratons that can propagat through flud arth matrals. In solds w can smplfy P-wavs va an acoustc modl so as to gnor th ffct of couplng to shar strsss (p.55, Ilk and Amundsn. Not that n ths cas th shar modulus stll rmans as on of th Lamé s paramtrs. For lquds th shar modulus s and for solds t s non-zro. In a flud, ssmc data can b collctd by hydrophons n th form of prssur masurmnts. Th prssur fld s a scalar fld, whch s smplr to dal wth mathmatcally, than a tnsor fld. From th followng mathmatcal drvatons w can rach svral accurat concpts to hlp us vsualz partcl stran durng acoustc wav transmsson n a flud. Frst, wthn a flud, and at a gvn pont, partcl moton ncrass as th prssur gradnt ncrass. Scond, th largr th partcl dnsty th slowr th partcl acclraton. Not that an unwathrd pc of grant s dnsr than a pc of slat from a blackboard and so basd ON ONLY th proprty of ts dnsty partcl acclraton wll b smallr n grant than n slat. [Q] Thrd, th mor ncomprssbl th flud th fastr th partcl moton. A fast partcl moton s rlatd wth a fast transmsson of mchancal vbraton through th flud. If w us th ball and sprng modl ths mpls that as w substtut stffr and stffr sprngs n th pctur partcl moton s fastr and th vbraton or wav s abl to cross btwn balls at a fastr rat ( spd of sound. On th Acoustc Wav Equaton Most popl would say th spd of sound ncrass as th dnsty of th matral that sounds travls through. For xampl, sound wavs travl fastr n th watr of th swmmng pool than by shoutng abov th watr. Yt, th acoustc quaton of moton mpls th oppost. Explan why th spd of sound s so much gratr at th cntr of th arth than n th crust nar th surfac of th arth n trms of th acoustc wav quaton and ts physcal mplcatons. On Hydrophons vrsus Gophons Q. 1 On land ssmc survys data s collctd by dvcs that ar abl to convrt ground vlocty or acclraton nto a voltag. In marn sttngs ssmc stramrs tow only hydrophons whch ar arrays of prssur transducrs. Snc acoustc wavs produc partcl moton n fluds shouldn t w b abl to us 3-componnt gophons n fluds as wll as prssur transducrs? Thnk through and xplan why you thnk hydrophons ar th choc for marn acquston work.

3 Q. A ssmc acquston company s currntly marktng dgtal 3-compnnt acclromtrs as a substtut for 3-componnt gophons. What s th advantag of usng an acclromtr ovr a gophon for a ssmc land-basd survy? Mathmatcal Drvaton W saw n th scton on tnsors that T = n σ Th total forc F = ( F1, F, F3 surfac gvn ara da s xrtd by th mdum on to th volum through th small F = σ n da whr σ n s forc pr unt ara (prssur (1 For xampl n a flud: σ Pδ. ( whr P s prssur and whr comprsson s by convnton ngatv. Exprsson ( can also b xprssd as

4 σ 11 σ P σ 33 P P P δ P 1 P 11 δ δ Not that w can vw th Kronkr dlta as a scond ordr tnsor whr thr ar NO offman-dagonal componnts bcaus thr s no shar,.. Formattd: Hghlght Formattd: Hghlght σ = σ So now combnng (1 and ( w hav F Pδ n da Pn da that s, n vctoral notaton, F Pnˆ da. If thr s no gradnt n th prssur thr s no nt forc actng on t. For xampl a nutrally buoyant sphr wll nthr rs nor fall mmrsd n a flud. BULK MODULUS AND LAMÉ S PARAMETERS W am to show that how th bulk modulus s rlatd to Lamé s paramtrs Prvously, n dalng wth th lastc wav quaton w saw that Hook s law for th cas of an sotropc, htrognous mdum took on th form Formattd: Hghlght Formattd: Hghlght

5 σ = λδ and th scalar xprssons that rlatd th stran fld to th gradnt of th dsplacmnt fld or th dlataton wr as: = u k, k = u = (Th volumtrc stran s th sam as th sum of th lnar strans n ach of th prncpal drctons f th dformaton s small, as n th cas of dal lastcty. (Proof of ths rlaton s gvn to th radr to vrfy, untl w dmonstrat ths approxmaton n class Formattd: Undrln For a hyrdrostatc prssur fld whr σ substtutng ( to obtan : = σ = σ P P δ = λδ (3 w can rphras Hook s Law. W contract th ndcs, makng = th sum: n ordr to consdr only th non-zro contrbutons to Pδ = λδ P 3 = P 3 = λ 3 ( λ 3 (Not that By rplacng trms wth othr quvalnt xprssons, notd mmdatly abov, P 3 = ( λ 3 λ 3 P 3 P λ + µ 3 P = λ + µ 3 P In ths form, w can show that th bulk modulus ( k can b xprssd n trms of Lamé s paramtrs: k = λ + µ 3

6 In th acoustc cas, w hav that µ = and k = λ, so that quaton (3 can b r-xprssd as Pδ = κδ P = κ = k u = k (4 from whch w s that th dvrgnc of th dsplacmnt fld s proportonal to th prssur. Partcl acclraton and ts rlaton to dnsty th prssur fld and wav vlocty W can prdct th dffrnt paramtrs n th quaton of moton: ρu = σ = ( Pδ, ( P δ + Pδ,,,, P δ P δ, P P, x (non-zro componnts only xst for = In vctoral notaton w can also xprss ths as ρu P P (5 u ρ In ths form th quaton of moton tlls us that th partcl acclraton n a body ncrass wth largr gradnts n th prssur fld but dcrass as th matral bcoms dnsr and rqurs mor nrgy to mov. W can also stmat how prssur changs n spac can affct th partcl acclraton, by takng th dvrgnc of th abov xprsson. ( P = ( ρu P = ρ( u + u ρ = ρ ( u + u ρ t

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