ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM


 Lynne Anderson
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1 ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look on ust one pont n space. Ths s the case of fxng geophones or sesmometers n the feld, or Euleran descrpton. We wll begn by a smple case, assumng that we have (1 an sotropc medum, that s that the elastc propertes or wave velocty, or not drectonally dependent and that ( our medum s contnuous. By examnng a balance of forces across an elemental volume and relatng the forces on the volume to an deal elastc response of the volume usng Hooke s Law we wll derve one form of the elastc wave equaton. Let us begn by examnng the balance of forces and mass (Newton's Second Law for a very small elemental volume. The effect of tracton forces and addtonal body forces ( f s to generate an acceleraton ( u per unt volume of mass or densty ( : u σ f, (1 >To Acoustc Wave Equaton, where the doubledot above u,the denotes the second partal dervatve wth respect to tme u (. The deformaton n the body s acheved by dsplacng ndvdual partcles about ther central restng pont. Because we consder that the behavor s essentally elastc, the partcles wll eventually come to rest at ther orgnal pont of rest. Dsplacement for each pont n space s descrbed by a vector wth a tal at that pont. u ( u1, u, u3 Each component of the dsplacement, u depends on the locaton wthn the body and at what stage of the wave propagaton we are consderng. Densty ( s a scalar property that depends on what pont n 3D space we consder: ( x, x, x 3 or, n other words x, x, x ( ( 1 3 x Body forces such as the effect of gravty are dscarded. The homogeneous equaton for moton (a partal dfferental equaton states that the acceleraton a partcle of rock undergoes whle under the nfluence of tracton forces s proportonal to the stress gradents across ts surface: (for 1,,3 Each bass vector component of the acceleraton as for example 1 s expressed as σ11 σ1 σ13 u ˆ ˆ 1x1 x1 x1 x x 3 Fnally, n complete ndcal notaton: u σ,, for a gven elemental volume.
2 Remember from the chapter on stran that the nfntesmal deformaton at each pont depends on the gradents n the dsplacement feld: 1 e ( u p, q uq, p (substtutng, p for and q for 1 u p uq ( q p 5 Emprcally, t has been shown that for small strans ( 10, and over short perods of tme (Lay, Wallace rocks behave as deal elastc solds. The most general form of Hooke s Law for an deal elastc sold s: σ ce (4 where c s a fourthorder tensor contanng de elastc constants or matrx components that defne the elastc propertes of the materal n the an ansotropc and nhomogeneous medum. Each component c or elastc constant has dmensons of pressure. Each component c s ndependent of the stran e and for ths reason s called a constant although elastc constants vary throughout space as a functon of poston. We can reduce the number of constants to two n varous steps. Frst we can reduce the number to 36 because t follows that snceσ y e are symmetrc: c c and qp c c. that Through thermodynamc consderatons (beyond the scope of ths course we can demonstrate c c so that even n the case of ansotropy the number of constants can be reduced to 1. However, t s possble to often solve many geologcal problems by consderng that rocks have sotropc elastc propertes. The assumpton of sotropy reduces the number of ndependent elastc constants to ust. In summary for an sotropc, contnuous medum we can reduce the elastc constant tensor to the followng: c λδ δ µ ( δ δ δ δ p (5 p q q where λ y µ are known as the Lamé elastc parameters or propertes. Lamé parameters λ y µ can be expressed n terms of other famlar elastc parameters such as Young s modulus E and Posson s rato σ : Eσ E λ ; µ ( 1 σ (1 σ (1 σ
3 Other elastc parameters can also be expressed n terms of λ y µ ( also known as Lamé s frst and second parameters. For example, ncompressblty K relates the change n pressure surroundng a body to the correspondng relatve change n volume of the body: P E K V λ µ (7 V 3 3(1 σ Substtuton of equaton (5 nto equaton (4 shows that tracton forces and stran are related for an sotropc medum n the followng manner: If we add over repeated subndces: [ λδ δ µ ( δ p δ q δ q δ p ] e σ λδ λδ δ e µδ δ e µδ δ e µδ p q ( δ e δ e δ e ( p 1,,3 δ q ( e1 q eq e3q δ ( p 1,,3( e1 q eq e3q µδ q q p From the defnton of a Kronecker delta, the only terms that wll be nonzero and contrbute to the stress tensor wll be those whch have ther subndces equal to each other. That s for the second term on the rght of the equals sgn, values exst f p and q. Smlarly, for the thrd term on the rght of the equals sgn values exst f p and q. Wth ths smplfcaton we arrve at: λδ e µδ δ e µδ δ e kk Because the deformaton tensor s symmetrc e e leadng to the result that Equaton σ λδ e µ e (8a >To Acoustc Wave kk In experments we observe dsplacement, ground velocty and acceleraton so t makes sense to express the stresses n terms dsplacements, σ u λδ k k u µ u or, n complete ndcal notaton: σ λδ u µ u u (8b k, k (,, snce
4 ( u u 1 and ekk e11 e e33 uk, k u e,, Note too, that deformatons V u, ( where V s the relatve change n volume, for nfntesmal V We obtan the wave equaton for dsplacements n a general sotropc medum by substtutng (8b nto the equaton of moton u f (1 σ, usng the product rule. [ λδ u k k µ( u u ] f,,,, ( λδ u µ (u u µ (u u f, k,k,,,,,, λ δ u λδ u µ (u u µ (u u f after expanson, k,k k,k,,,,,, Let us take each of the terms on the rght hand sde separately to demonstrate the applcaton of ndcal notaton. For each term I, only the case where can contrbute n the Kronecker delta, so λ δ u, k, k λ δ u, λ u, k, k k, k λ u,, because we can nterchange the repeated k s by repeated s because they both sgnfy summaton over the range of values for ;.e., 1 through 3. λ u λ u ( µ u µ u µ ( u u f,, k,k,,,,,, I II III IV V (9 After some algebra we show that an alternatve expresson can be obtaned by addng (9 vectorally from 1,, 3 to arrve at: u µ (10 ( λ µ ( u µ u λ u µ u ( u f Two fundamental body wave types: P waves and S waves From the equaton of moton (10 n vectoral form, we can demonstrate (Posson, 18.., that n an nfnte, elastc, and sotropc, homogeneous medum two types of partcle moton assocated wth travelng trans of deformaton can be predcted.
5 Snce λ and µ are constant n a homogeneous medum, we have that λ and µ because there are no spatal changes n ther values. Ths leaves: u ( λ µ ( u µ u both equal zero But, we can use the dentty: µ u ( u µ ( u ( µ u, (dentty 1 so that ( λ µ ( u µ ( u (11 Through the Helmholtz theorem (aka fundamental theorem of vector calculus, the dsplacement vector feld can be decomposed nto two ndependent component felds: a scalar feld together wth a vectoral feld. u Θ Ω The frst term on the rght (scalar s nonrotaton and the second term (a vectoral feld s rotatonal. The scalar feld of dsplacement can not experence rotatons and the vectoral feld can not experence a dvergence. In other words ( " vector " 0 (dentty ( " 0 " scalar, (dentty 3 Now, f we take the dvergence of (11 we can smplfy the whole expresson. The second term on the rght becomes zero because u s a vector quantty, and ts rotatonal s zero (dentty : u ( u [( λ µ ( u ] ( u( λ µ We can change varable names by defnng a new scalar feld varable Θ u so that the mmedately precedng expresson looks lke: Θ Θ ( λ µ Θ Θ Θ, where Θ λ µ
6 In order to propagate ths type of deformaton through the medum the body must expand and contract (dvergence s nonzero. Ths porton of the dsplacement feld can not have a rotaton component; only a dvergence component. One soluton to ths scalar wave equaton s to make Θ V P λ µ Now, f we take the rotatonal of the general equaton of moton (11.e., Because u s a scalar feld and the rotatonal of the gradent of ths feld s zero (dentty 3. So, the frst term on the rght of the equaton goes to zero: ( u ( u ( λ µ ( u µ ( u ( u µ ( u We can now change varable names by defnng a new vector feld varable Ω u so that the mmedately precedng expresson looks lke: Ω µ µ µ Ω [ ] ( λ µ ( u µ ( u ( Ω { ( [( Ω ]} Ω because the frst term goes to zero snce the dvergence of the rotatonal of Ω s zero (dentty. One soluton to ths vector wave equaton s to make Ω V S µ The separaton nto dfferent wavefelds s useful to know for practcal reasons. If a full 3 component, D array of geophones can record a wavefeld, then the wavefeld can be broken out by separatng the feld nto the transverse waves and the compressonal waves. The dvergence of the full 3C, D array data set would leave behnd only transverse waves. The curl of the same data set would leave behnd only compressonal waves. Of course the wavefeld can not be separated out completely because our 3D experments collect data over a D grd so that the partal dervatves n the z drecton may have to be consdered as constant.
7 In Summary, we have seen that (1 the stran s related to dsplacement feld by the followng relatonshp 1 u u e ( ( and that Newton s second law provdes the followng relatonshp between partcle acceleraton and stress: u σ, (3 Also, Hooke s Law can be wrtten as: σ λδ e kk µ e (4 Fnally, when we substtute Hooke s Law nto Newton s Law through the expresson of stress as a functon of dsplacements we can eventually fnd the soluton to two wave equatons that gve us the speed of propagaton of two waves through elastc, sotropc materal: Ω V S µ and Θ VP λ µ
8 Appendx for secton on the wave equaton >Back to text In ths secton we state that vectoral manpulaton of the expresson: u ( λ µ u, µ u, λ, u, µ, ( u, u, I II III IV f (9 for 1,, 3 leads to the alternatve expresson: u µ ( λ µ ( u µ u λ u µ u ( u f In order to show the steps n detal, let us examne each of the terms I through IV on the rght hand sde of equaton (9. Startng wth I : e.g., when k1,,3, for commutatve partal dervatves u k,k u1 u u3 1 3 u u u u u 1 k,k, For II : u, u 1 ( u 1 u u 3 3
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