The Elastic Wave Equation. The elastic wave equation

Transcription

1 The Elasc Wave Eqaon Elasc waves n nfne homogeneos soropc meda Nmercal smlaons for smple sorces Plane wave propagaon n nfne meda Freqency, wavenmber, wavelengh Condons a maeral dsconnes nell s Law Reflecon coeffcens Free srface Reflecon esmology: an example from he Glf of Mexco

2 Eqaons of moon ρ f + j σ j Wha are he solons o hs eqaon? A frs we look a nfne homogeneos soropc meda, hen: σ j λθδ j + µε j σ j λ δ + µ k k j ( j + j ) ρ f + j k k j µ ( λ δ + ( + )) ρ f + λ + µ + µ k k j j j j j

3 Eqaons of moon homogeneos meda ρ f + λ + µ + µ k k j j j We can now smplfy hs eqaon sng he crl and dv operaors and x + - ρ f+ ( λ + µ ) - µ y + z hs holds n any coordnae sysem Ths eqaon can be frher smplfed, separang he wavefeld no crl free and dv free pars

4 Eqaons of moon P waves ρ ( λ + µ ) - µ Le s apply he dv operaor o hs eqaon, we oban ρ θ ( λ + µ ) θ Acosc wave eqaon where θ P wave velocy or θ α 1 θ α λ + µ ρ

5 Eqaons of moon shear waves ρ ( λ + µ ) - µ Le s apply he crl operaor o hs eqaon, we oban ρ ( λ + µ ) θ + µ ( ) we now make se of Wave eqaon for shear waves θ ϕ and defne o oban hear wave velocy ϕ β 1 ϕ β µ ρ

6 Elasodynamc Poenals Any vecor may be separaed no scalar and vecor poenals Φ + Ψ where P s he poenal for Φ waves and Ψ he poenal for shear waves θ Φ ϕ Ψ Ψ P-waves have no roaon hear waves have no change n volme θ α θ ϕ β ϕ

7 esmc Veloces Maeral and orce P-wave velocy (m/s) shear wave velocy (m/s) Waer 15 Loose sand 18 5 Clay 11-5 andsone Anhydre, Glf Coas 41 Conglomerae 4 Lmesone Grane Granodore Dore Basal 64 3 Dne Gabbro

8 olons o he wave eqaon - general Le s consder a regon who sorces η c η Where n cold be eher dlaaon or he vecor poenal and c s eher P- or shear-wave velocy. The general solon o hs eqaon s: η( x, ) G( a x c) j j ± Le s ake a look a a 1-D example

9 olons o he wave eqaon - harmonc Le s consder a regon who sorces η c η The mos approprae choce for G s of corse he se of harmonc fncons: ( x, ) A exp[ k( a x c)] j j

10 olons o he wave eqaon - harmonc akng only he real par and consderng only 1D we oban ( x, ) Acos[ k( x c)] k x c kx kc π π π ( ) x ω x λ λ T c k λ T ω A wave speed wavenmber wavelengh perod freqency amplde

11 phercal Waves η c η Le s assme ha η s a fncon of he dsance from he sorce r η 1 rη + rη r c η where we sed he defnon of he Laplace operaor n sphercal coordnaes le s defne o oban η η r η c η wh he known solon η f ( r α)

12 Geomercal spreadng so a dsrbance propagang away wh sphercal wavefrons decays lke r 1 η f ( r α) η r 1 r... hs s he geomercal spreadng for sphercal waves, he amplde decays proporonal o 1/r. If we had looked a cylndrcal waves he resl wold have been ha he waves decay as (e.g. srface waves) η 1 r

13 Plane waves... wha can we say abo he drecon of dsplacemen, he polarzaon of sesmc waves? Φ + Ψ P Φ P + Ψ Φ... we now assme ha he poenals have he well known form of plane harmonc waves Aexp( k x ω) Ψ B exp( k x ω) P Akexp( k x ω) k B exp( k x ω) P waves are longdnal as P s parallel o k shear waves are ransverse becase s normal o he wave vecor k

14 Heerogenees.. Wha happens f we have heerogenees? Dependng on he knd of reflecon par or all of he sgnal s refleced or ransmed. Wha happens a a free srface? Can a P wave be convered n an wave or vce versa? How bg are he ampldes of he refleced waves?

15 Bondary Condons... wha happens when he maeral parameers change? ρ 1 v 1 welded nerface ρ v A a maeral nerface we reqre conny of dsplacemen and racon A specal case s he free srface condon, where he srface racons are zero.

16 Reflecon and Transmsson nell s Law Wha happens a a (fla) maeral dsconny? Medm 1: v 1 1 sn sn 1 v v 1 Medm : v B how mch s refleced, how mch ransmed?

17 Reflecon and Transmsson coeffcens Le s ake he mos smple example: P-waves wh normal ncdence on a maeral nerface Medm 1: r1,v1 A R R A ρα ρ α ρ1α 1 + ρ α 1 1 Medm : r,v T T A ρ1α 1 ρ α + ρ α 1 1 A oblqe angles conversons from -P, P- have o be consdered.

18 Reflecon and Transmsson Ansaz How can we calclae he amon of energy ha s ransmed or refleced a a maeral dsconny? We know ha n homogeneos meda he dsplacemen can be descrbed by he correspondng poenals esmology and he Earh s Deep Ineror n -D hs yelds Φ + Ψ x y z x z z Φ Ψ x Φ + y x Ψ x Ψ an ncomng P wave has he form Φ z Ψ ω A exp ( a j x j α) α y z

19 Reflecon and Transmsson Ansaz... here a are he componens of he vecor normal o he wavefron : a (cos e,, -sn e), where e s he angle beween srface and ray drecon, so ha for he free srface Φ Ψ A expk( x B expk'( x x x 3 3 an e c) + an f c' ) Aexpk( x 1 + x 3 an e c) where c k α cos e ω cos e α ω c c' k' β cos f ω cos β f P e f P r V r wha we know s ha σ σ xz zz

20 Reflecon and Transmsson Coeffcens... png he eqaons for he poenals (dsplacemens) no hese eqaons leads o a relaon beween ncden and refleced (ransmed) ampldes R PP A A 4 an e an 4 an e an f f (1 + (1 an an f f ) ) R P A A 4 an e (1 an f ) 4 an e an f + (1 an f ) These are he reflecon coeffcens for a plane P wave ncden on a free srface, and refleced P and V waves.

21 Case 1: Reflecons a a free srface A P wave s ncden a he free srface... j P V P The refleced ampldes can be descrbed by he scaerng marx P P P d d P d d

22 Case : H waves For layered meda H waves are compleely decopled from P and V waves H There s no converson only H waves are refleced or ransmed d d d d

23 Case 3: old-sold nerface P V r P r V P To accon for all possble reflecons and ransmssons we need 16 coeffcens, descrbed by a 4x4 scaerng marx.

24 Case 4: old-fld nerface P V r P r P A a sold-fld nerface here s no converson o V n he lower medm.

25 Reflecon coeffcens - example

26 Reflecon coeffcens - example

27 Refracons waveform effecs

28 caerng and Aenaon Propagang sesmc waves loose energy de o geomercal spreadng e.g. he energy of sphercal wavefron emanang from a pon sorce s dsrbed over a sphercal srface of ever ncreasng sze nrnsc aenaon elasc wave propagaon consss of a permanen exchange beween poenal (dsplacemen) and knec (velocy) energy. Ths process s no compleely reversble. There s energy loss de o shear heang a gran bondares, mneral dslocaons ec. scaerng aenaon whenever here are maeral changes he energy of a wavefeld s scaered n dfferen phases. Dependng on he maeral properes hs wll lead o amplde decay and dspersve effecs.

29 Inrnsc aenaon How can we descrbe nrnsc aenaon? Le s ry a sprng model: The eqaon of moon for a damped harmonc oscllaor s mx + γx γ x + x + m x + εϖ + k m kx x x + ω x ϖ ε γ mϖ k m 1/ where ε s he frcon coeffcen.

30 Q The solon o hs sysem s x( ) A e εϖ sn( ϖ 1 ε ) so we have a me-dependen amplde of A ( ) A e εϖ A e ϖ Q and defnng 1 A1 ε δ ln Q A Q π δ Q s he energy loss per cycle. Inrnsc aenaon n he Earh s n general descrbed by Q.

31 Dsperson effecs Wha happens f we have freqency ndependen Q,.e. each freqency looses he same amon of energy per cycle? A( x) A e ( fπqv) x hgh freqences more oscllaons more aenaon low freqences less oscllaons less aenaon Conseqences: - hgh freqences decay very rapdly - plse broadenng In he Earh we observe ha Q p s large han Q. Ths s de o he fac ha nrnsc aenaon s predomnanly cased by shear lace effecs a gran bondares.

32 Q n he Earh Rock Type Q p Q hale 3 1 andsone Grane Perdoe Mdmanle Lowermanle Oer Core

33 caerng