Chapter 3: Wave propagation fundamentals: From energy point of view, energy partitioning at interfaces

Transcription

1 Chapte 3: Wave popagation fundamentals: Fom enegy point of view, enegy patitioning at intefaces Befoe pusuing futhe on discussing specific topics in seismic exploation to a vaiety of applications, it is citical to undestand the basic wave popagation phenomena. In this chapte, we intoduce the fundamental solutions fo the wave equation we deived in Chapte, and discuss the fundamental popagation phenomena. Fist, we discuss the enegy consevation in tems of plane wave, cylindical wave, and spheical wave, coesponding to D, D, and 3D popagations. This phenomenon is known as geometic speading. Then, we discuss absoption, anothe main eason fo having seismic enegy dissipation. Finally, by discussing the Snell s law, the eflection, efaction and seismic phase convesion at a mateial contast inteface, as well as the enegy distibution, we discuss the appoximation by using the ay theoy, o geometic wave popagation. 3. Wave font and the Huygen s Pinciple It is clea that the wave field at a paticula moment is an extension and continuity of the field in the last moment. By using the concept of wavefont, which is defined as the suface on which the wave motion is in the same phase, this concept is used to descibe the Huygens Pinciple. Huygens pinciple states that evey point on a wavefont can be egaded as a new souce of waves. In Figue 3., the points P, P, P 3, P 4, and P i, on the wavefont AB can be egaded as the news souces. Afte an infinitesimal time of dt, the waves fom each of the new souce popagated to a distance of vdt, and foms a new wavefont A B. Be awae of that between A and B thee ae infinite numbe of points so the newly fomed wave font is a continuous phenomenon. A A P P P 3 P 4 P i B B vdt... Figue 3. Illustation of the Huygens pinciple.

2 Fom Figue 3., it is easy to see that the shape of the wave font depends on medium s velocity. Fo example, the wave geneated by a point souce popagates in a medium with constant velocity, the wavefont is in a shape of a spheical shell. 3. Wave enegy A simplified case fo the wave equation discussed in Chapte is the plane wave popagating in one diection, say the x-diection. In this case, the wave equation can be witten as u x v u t (3.) One solution fo a plane wave popagating in an unbounded, unifom medium can be expessed as u u exp[ i( ω t + kx)] u cos( ωt + kx) + iu sin( ωt + kx) (3.) This plane wave can be viewed as the wave geneated by a plane souce occupying the entie yz-plane (x) to geneate waves popagating in x-diection. Fo simplicity, let s only conside the eal pat of Eqn (3.). In this equation, u is the amplitude, ω is the angula fequency; k is called the wave numbe. We will show the elationship of k with espect to angula fequency ω, by demonstating Eqn (3.) does satisfy the -D wave equation (3.). Taking the seconday deivative of u with espect to space, hee the x- coodinate, is u x k u cos( ω t + kx) and putting the second deivative of u with espect to time on the ight hand side of Eqn (3.) gives u ω u cos( ωt + kx) v t v compaing the last equations leads to ω k v It is clea that the wave numbe k in space domain defines how many evolutions in a unit length scale, just like the angula fequency s (ω) function in time domain, to define how many evolutions in a unit time. They ae linked though the popagation velocity v, which is detemined by the physical popeties of the media the wave tavels though. In

3 the case discussed in this section, the wave does not expeience any lose of enegy. Clealy, it is not ealistic in the eal wold. We will discuss the enegy equilibium in the next section. 3.. Kinetic enegy At one paticula position with, the simplest wave displacement is not depending on the position anymoe and can be witten as u u cosωt the paticle velocity at this point then is u uωsinωt (and what is the expession fo acceleation?) the kinetic enegy density is then m Ek u ρu ω sin V ω t Please make sue you undestand the diffeence of wave popagation velocity as it passing though the media, and the paticle motion velocity that only vibating about an equilibium point of the paticle itself. Since the function of sin(ωt) has a ange of [-, ], then sin ωt vaies in the ange of [ ]. This gives the maximum kinetic enegy is E k max ρu ω 3.. Potential enegy: Stain Enegy In elasticity, the elastic defomation (stain) is caused by applying stess, the poduct of stess and stain, in analogue to wok which is the poduct of foce and displacement, in a macoscopic sense, is in a dimension of enegy. Since we ae taking the case in a micoscopic sense, the enegy actually is enegy density, i.e., the enegy in a unit volume of the medium. The wok done by stess is conveted to elastic stain enegy in the same amount and stoed in the medium. Elastic stain enegy is a kind of potential enegy due to some kind of ecovey foce (gavity is a ecovey foce, and thee is a gavity potential). E σ ε ( σ xxε xx + σ yyε [( λθδ + µε ) ε ] [( λε iiεii + µεε ] yy + σ ε + σ ε + σ ε + σ ε ) zz zz xy xy yz yz zx zx

4 since δ ε ε, then ii E ε ( σ ε ) σ ε This expession means that fo a given stain, if the stess applied is lage, the enegy stoed is also lage (This is due to the elastic modulus is lage). The stain enegy is a kind of the potential enegy that stoed in the elastic medium when it is defomed. Fom physical pinciple, the summation of the potential enegy and the kinetic enegy get to be a constant at any given moment. Let us examine the situation at two paticula moments: ωt, and ωt π/. At ωt, we have and u u cosω t u v u u ω sin At ωt π/ we have and u u cosω t π v u uω sin uω This is to say that at the moment the displacement is in its maximum u, the paticle velocity is zeo, and the kinetic enegy is zeo, all enegy has been stoed as the elastic stain enegy. In contast, when the displacement is zeo, the velocity eaches its maximum u ω, and the kinetic enegy is in its maximum, and all the elastic stain enegy has been eleased. The total enegy at this point is E k + E p Ek max ρu ω

5 T E k E p E kmax u E k E kmax E p t Figue 3., the balance of the kinetic enegy and elastic potential enegy at any moment Enegy Intensity I Enegy intensity is the total enegy flow though a unit aea in a unit time, so that it is the enegy density we have leaned above (times the volume, then divided by the aea and time). Imagine a cylinde, it happened have the wave enegy popagation diection coincides with the axis of the cylinde as shown in Figue 3.3 below k da vdt Figue 3.3 so that

6 E dadt E vdtda Ev ρω u v dadt I total whee v is the popagation velocity of the waves. It is clea that enegy intensity can also be called as enegy flow density. Fo the sake of enegy consevation, we should expect the total enegy in the entie domain of the media at any moment should be a constant; the total amount of enegy depends on the souce has adiated. Now we can discuss the waves with diffeent type of souces and thei elationship with enegy and enegy flow density. 3.3 Geometic speading: Spheical wave, cylindical wave and plana wave 3.3. Point souce a 3D case Fo the simplest case, i.e., a point souce in an infinite homogeneous medium, we should expect the following. Let s imagine two wave fonts, which make spheical shells with the same oigin (location of the souce). The adius to the oute shell is, which is geate than that of the adius of the inne shell. Thus, the suface aeas of the oute and inne shells ae 4π and 4π, espectively. By enegy consevation, the total enegy flowing though the oute shell and the inne shell at a given time should keep the same so that we have E I I S and E I 4π I S ( ρω u I 4π ) I ( ) ρω u u ( ) u to genealize this elation with the inne shell becoming a constant efeence shell close to the souce and the oute shell a geneic one we have u ( ) u This state that the amplitude is decaying against / fo the waves geneated by a pint souce, since the shape of the wavefont is spheical, this is geneally efeed as the geometic speading fo spheical waves Line souce a D case

7 Fo an infinitely long line souce, the shape of the wavefont is a cylinde; this is called the cylindical wave E E IS IS I πl I πl and I ( ) I ρω u ( ) ρω u u u to genealize this elation with the inne shell becoming a constant efeence shell close to the souce and the oute shell a geneic one we have u u This state that the amplitude is decaying against / fo waves geneated by a line souce, since the shape of the wavefont is cylindical, this is geneally efeed as the geometic speading fo cylindical waves Plane souce a D case If the souce occupies the entie x plane (as shown in the beginning of Section 3.), the shape of the wave font is plana, this wave is called the plane wave; thee is no amplitude decay fo plane wave. In summay we can view the enegy density flow of diffeent waves as: Point souce: spheical wave / decay; Line souce: cylindical wave / decay; Plane souce: plane wave no decay; In eality, the enegy decays by the wave field occupying lage and lage volume and enegy in the unit volume become less and less when the wave popagating fathe and fathe, this phenomenon is called geometic speading. The geometic speading alone can not lead to the complete dead-off of the seismic wave enegy. The ultimate dead-off of the kinetic enegy of seismic waves is due to the enegy absoption caused by the impefection of the eath mateials, i.e., the elastic enegy has been completely tansfeed to eath mantle.

8 3.4 Enegy dissipation caused by absoption (intinsic attenuation) Absoption is the enegy loss caused by the impefection o defect of the mateial, in the fom of enegy convesion fom mechanic to themal. This loss can be accounted fo by using the absoption coefficient α in the fom as A A e α Now we intoduce the concept of the quality facto Q, which is defined as the atio of the total elastic enegy and the enegy lost in one cycle, i.e., E Q π E Q can be thought as: afte how many cycles of vibation the elastic enegy can be dissipated, appaently lage Q means many cycles to dissipate the enegy so that the mateial tends to be moe close to pefect o moe puely elastic. In contast, if only afte vey few cycles the enegy is gone, the mateial is fa moe fom elastic. We have leaned that the kinetic enegy is popotional to the squae of the amplitude, i.e., we have E ρω A taking the efeence point at point, so we have π E A A ( A + A )( A A ) A ( A A ) Q E A A A ( A A ) A A ( ) ln( ) A A A Taylo expansion has been applied in the last step, since ln x +...( x > ) x On the othe hand, fom the oiginal definition of the absoption coefficient we have the amplitudes at points with only one cycle apat (one wavelength in space) can be expessed as then A Ae A A e α α ( + λ ) and A A A e A e α α ( + λ ) e α e α ( + λ ) e α + α + αλ e αλ

9 so we got that A ln( A αλ ) ln( ) αλ e αλ and Q πe E π π αλ This is the elation between the quality facto Q and the absoption coefficient α. Finally, afte conside the geometical speading and the absoption, a geneal fom of the solution of wave equation can be witten as: u α u e cos( ωt k) fo point souce (3D popagation) u α u e cos( ωt k) fo line souce (D popagation) α u u e cos( ωt k) fo plane souce (D popagation) Again, be make sue that the absoption is the mechanism esponsible fo complete deadoff of seismic vibations. Afte we discuss geometic speading and absoption, which occus even fo a unifom medium, we need discuss enegy patitioning at intefaces caused by heteogeneous medium. This is the basis fo diffaction, and scatteing. 3.5 Diffaction, and its kinetic appoximation: The ay theoy, o geometic wave popagation (Snell s law) 3.5. Geometic wave theoy, Snell s law The pocess of wave eflection may be defined as the etun of all o pat of a wave beam when it encountes the bounday between two media. The most impotant ule of eflection is that the angle of incidence is equal to the angle of eflection., which is known as the Snell s law (actually the simplest expession of it). Whee both these angles ae measued elative to an imaginay line which is nomal to the bounday. Figue 3.4 shows the situation of the incidence of a p-wave to a plana inteface sepaating medium and medium.

10 Figue 3.4. A plane p-wave impinging at the inteface with ρ v > ρ v. The ay diection of the eflected P- and S-wave, the tansmitted (efacted) P- and S- wave ae obeys the Snell s law: sinθ sinθ v v In the expession above the wave incidence and eflection/tansmission ae expessed by the diection of ays. Only conside geomety, and kinetics not conside the causes of the defomation o motion. This is the essence of the geometic wave theoy. It is an appoximation of the physical wave theoy. The pemise of this appoximation is that the fequency of the wavefoms is assumed to be infinitely high, o the wavelength is vey, vey shot compaed with the featues it studies. No diffaction phenomenon is consideed in the teatment hee, only eflections and efactions the Reflection Coefficient and tansmission coefficient Reflection is often quantified in tem of the eflection coefficient R. R is defined simply as the atio of the eflected and incident wave amplitudes. R A / Ai Whee Ai and A ae the incident and eflected wave amplitudes espectively. The value of the eflection coefficient elates to the magnitude of eflection fom the inteface between two media with diffeent physical popeties. The acoustic impedance Z of the two media involved dictates the magnitude of eflection fom a bounday. You will emembe fom ou discussion of 'acoustic paametes' that the acoustic impedance is simply the poduct of the density (ρ) and the sound speed (v) of the media.

11 Z ρv In acoustics, the acoustic impedance Z is measued in Rayles ( Rayle m/s.kg/m 3 kg/m /s); but it is not that popula in seismology. The full expession fo sound eflection coefficient vesus the incident angle α is: ( Z R ( Z / Z ) / Z ) + ( n ) tan α ( n ) tan α i i whee n (v /v ) and α i is the angle of incidence of the wave ay. Matlab Execise: Plot the Reflection coefficient against the incident angle fo a) ai-wate inteface; b) wate-ock (v3 m/s) inteface. Notice that since enegy is always conseved the emaining enegy that is not absobed must be tansmitted into the second medium wheeby it will undego efaction if the velocities in the two layes diffe. If absoption is negligible (a lossless medium) the total enegy of the eflected and the tansmitted waves should be equal to the oiginal enegy of the incident wave Nomal incidence If the waves ae nomally incident to the bounday the eflection equation can be simplified to: Z Z R α Z + Z fo i and the tansmission coefficient is T ZZ fo αi Z + Z Thus, eflection is a simple function of the impedance of the two media. If the two media have the same impedance thee will be no eflection. Since the impedance is the poduct of velocity and density it is possible fo example to have two media with diffeent densities o sound speed but the same acoustic impedance. In-class execise: Reflection and tansmission coefficient fo powe (enegy in a unit time) is the amplitude R and T squaed. What do you expect fo RR+TT? Explain why you get the esult.

12 Reflection coefficients have values that ange between - and +. Fom this ange we can identify 4 diffeent types of eflection: ) z >> z, R > (Rigid bounday), i.e. most of the acoustic enegy will be eflected without a change in phase. ) z << z, R > - (Soft o pessue elease bounday), i.e. most of the wave enegy is eflected with a 8 degee phase change. 3) z z, R, (No Reflection) 4) Simila acoustic impedance, - << R <<, some phase change. Pobably the most impotant thing to emembe is that eflection will be stong anywhee thee ae stong spatial gadients o contast in acoustic impedance. Some typical examples of impedance fo diffeent mateials and thei eflection coefficient in salt wate ae given below: Table 3. Acoustic impedance and eflection coefficient pf some mateials Mateial Z (Rayles) R ove salt wate Ai 4 - Fesh wate,48,.4 Salt wate,54, Wet fish flesh,6,. Wet fish bone,5,.4 Rubbe,8,.8 Ganite 6,,.8 Quatz 5,3,.8 Clay 7,7,.67 Sandstone 7,7,.66 Concete 8,,.68 Steel 47,,.94 Bass 4,,.9 Aluminum 7,,.83 In class execise: Ty and classify the above into igid soft and weakly eflective boundaies. An examination of the figues above you should tell you why militay submaines often have ubbe coatings in ode to minimize eflection and avoid detection (compae the eflection coefficients of steel and ubbe). Beaing in mind that fish flesh and bone have elatively low eflection coefficients why is do you think that we ae able to detect fish so effectively with fish finding echo soundes? (Hint conside how fish contol thei buoyancy)

13 The fact that diffeent mateials have diffeent acoustic impedance and eflection coefficients allows us to acoustically distinguish between diffeent tagets. Side-scan sona fo example uses this chaacteistic to distinguish between sand (pedominantly quatz), mud and ock. This is done by examining the diffeence in intensity of the acoustic etuns, often in conjunction with some sot of textual analysis. Execise Compute the magnitude of the eflection and tansmission coefficients at the bounday between a fesh uppe laye and a saline lowe laye of wate in a salt wedge estuay. Assume that the angle of incidence of the acoustic ay is 5 degees. The chaacteistics of the fesh and saline wate ae as follows: Fesh wate: c46m/s, density kg/m3 Saline wate: c59m/s, density 5kg/m