INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS J. Phys. D: Appl. Phys. 5 00 007 04 PII: S00-770867-6 Anisotropic reference edia and the possible linearized approxiations for phase velocities of qs waves in weakly anisotropic edia Lin-Ping Song and Manfred Koch Departent of Geohydraulics and Engineering Hydrology School of Civil Engineering University of Kassel Kurt-Wolters-Strasse 4 D-409 Kassel Gerany Received 5 June 00 Published 4 Noveber 00 Online at stacks.iop.org/jphysd/5/007 Abstract Using perturbation theory we propose possible first-order approxiations of qs-wave phase velocities for a weakly anisotropic ediu under anisotropic reference edia. The chosen reference edia have the features of non-degenerate qs waves. It is found that the usually arbitrarily chosen zeroth-order eigenvector basis associated with the degenerate shear waves in an isotropic reference ediu are none other than the shear-wave polarization vectors in one such anisotropic reference ediu. It follows that the qs-wave phase velocities are approxiated as linear functions of the elastic constants of a weakly anisotropic ediu. The expressions illustrate that the qs-wave speeds have the ost probably sensitive coefficients C C C 66 [C +C 66 C + C ] [C +C C + C ] [C +C C + C ] and those quantifying the deviation fro orthorhobic syetry. Nuerical exaples in an orthorhobic ediu show that the present sipler approxiations are sufficiently applicable to propagation directions being close to syetry planes or around syetry axes and achieve in these cases the sae order accuracy of approxiation as the quadratic equations derived fro degenerate perturbation theory. In light of the used anisotropic reference edia the validity of the approxiations presented should be liited to anisotropic solids that possess at least one plane of syetry.. Introduction In anisotropic edia there are three bulk odes that propagate with distinct phase velocities and the particle displaceents for these wave odes are generally neither along nor perpendicular to the wave noral. These odes are usually called qp and qs waves and becoe pure P and S waves only in isotropic edia. The phase velocities and the polarization vectors of these three waves governed by the well-known Christoffel equation are generally very coplicated functions of the wave noral and the elastic constant tensor. On syetry planes of orthorhobic or higher syetry systes the coplexity of these expressions is reduced to a certain extent. Yet fro a theoretical point of view as well as for practical applications in fields as diverse as seisology and non-destructive evaluation it is quite useful to siplify such coplicated functions under soe reasonable assuptions or conditions. This ay not only offer better physical insight into the dependence of wave propagation but also substantially facilitate the solution of forward and inverse probles. Motivated by these actual needs a nuber of researchers have ade efforts to seek approxiations of the wave phase velocities in anisotropic edia with higher syetry e.g. transverse isotropy and arbitrary syetry by eploying linearizing ethods [ 9]. Aong these ethods the perturbation approach is a powerful tool. Recently a review on the various linearized approxiations of phase velocities 00-77/0/007+08$0.00 00 IOP Publishing Ltd Printed in the UK 007
L-P Song and M Koch was given by Song et al [0]. Farra [] further discussed the use of the higher order perturbations on the approxiations of phase velocities and polarizations of qp and qs waves. So far the use of perturbation theory in the area of wave propagation is usually carried out by assuing an isotropic reference ediu which has the siplest analytical solution. In this approach the anisotropy is naturally treated as a perturbation fro isotropy. It follows that the first-order approxiation of the qp phase velocity along an arbitrary direction of propagation is explicitly expressed as a linear function of the elastic constants of weakly anisotropic edia. It has been deonstrated that the qp approxiation works very well even for fairly large degrees of anisotropy [ 7]. For the qp wave Mensch and Farra [] developed a Hailtonian ray tracing approach for a weakly orthorhobic ediu using an ellipsoidal reference ediu whose principal axes are parallel to the principal axes of orthorhobic syetry. Ettrich et al [] presented an averaging technique to copute a bestfitting reference ellipsoidal ediu to an arbitrary anisotropic ediu. For qs waves the degenerate case occurs for reference isotropic edia. For the general syetry case the derived first-order approxiations for qs-wave phase velocities along an arbitrary direction are coplicated non-linear functions of the elastic constants. Song et al [0] suggested qs-wave phase velocity expressions which are linear functions of the elastic constants assuing that wave propagation has a slight deviation fro a syetry plane of the orthorhobic ediu. Mensch and Rasolofosaon [6] presented the approxiations of qs-wave phase velocities by a coordinate-rotational technique which also take the sae for as those fro the firstorder degenerate perturbation theory. Notably they used a special anisotropic reference ediu for their derivation. In particular this reference ediu has characteristics that are isotropic with respect to a qp wave but ay have two different shear waves. Therefore such characteristics naturally spur us to apply in a different way fro Mensch and Rasolofosaon [6] nondegenerate perturbation theory to qs waves in such kinds of reference edia. As a result first-order approxiations of qs-wave phase velocities are obtained that are now expressed in the for of linear functions of elastic constants of an anisotropic ediu as given below. In the following sections we first describe the fundaental equations being used in the developent of the perturbation theory. Second the choices of special reference anisotropic edia are discussed in detail. Then the explicit linear expansions of the phase velocities are presented for general and orthorhobic edia. Finally the accuracy and liitation of the approxiations are illustrated by nuerical exaples.. Perturbation expansions of Christoffel equations: non-degenerate case Elastic wave propagation in hoogeneous anisotropic edia is described by the Christoffel equations [4]: Ɣa = σ a. In this equation Ɣ is the syetric Christoffel tensor whose eleents are Ɣ = C kl n k n l ; C kl is the fourth-rank tensor of elastic stiffnesses of the ediu with at ost independent constants and n j the coponent of wave noral n; a is the eigenvector or unit polarization vector of the equation; σ = ρv is the corresponding eigenvalue where ρ is the density of the ediu and v is the phase velocity. The characteristic equation of is Ɣ σi =0 where I is an identity atrix. This equation being cubic in σ generally has three distinct roots corresponding to a qp and two qs waves with utually orthogonal polarization vectors. Closed for expressions for the three roots can be obtained for an arbitrary anisotropic ediu and these siplify for certain special cases [5]. Now let us consider perturbation of the Christoffel equation. Assuing that we have a reference ediu with the elastic constants kl then the original elastic constant atrix C kl of the considered ediu can be decoposed into [ 5] C kl = kl + εc kl where εc kl is regarded as a perturbation and ε is a sall paraeter used to develop the perturbation series [6]. The perturbation of εc kl of the reference ediu leads to the perturbed Christoffel tensor Ɣ = +εɣ. So the perturbed Christoffel equation is + εɣ a = σ a 4 where the eigenvalue and eigenvector have expansions in powers of ε σ = a = a 0 + εσ + εa + 5a + 5b where =. Substituting equations 5a and 5b into equation 4 and equating the coefficients of powers of ε we obtain for the non-degenerate case the first-order eigenvalue corrections and the first-order eigenvector corrections [5 ]: a σ = q = a0t Ɣ a 0 a 0T Ɣ a 0 q σ 0 σ q 0 6a a 0 q 6b where q = and the superscript T eans vector transpose. One observes that the eigenvalue and eigenvector corrections are dependent on the unperturbed eigenvectors defined in a reference ediu [6]. Here the purpose of presenting equation 6b is to show that the first-order eigenvalue correction equation 6a is iplicitly validated under the condition σ 0 σ q 0 i.e. there is no singularity in a direction of propagation. Moreover we would like to ephasize the constraints that not only the perturbation of the eigenvalue should be sall equation 5a but the perturbation of the polarization as well equation 5b if an anisotropic reference ediu is used. This is why the 008
Approxiations for phase velocities of qs waves use of an ellipsoidal anisotropic reference ediu should be interesting for directions close to a syetry plane as described below. In an isotropic reference ediu where the degenerate S-wave eigenvectors are undeterined the associated zeroth-order polarization vectors having to be constructed by the linear cobination of the two chosen vectors in a plane perpendicular to a wave propagation direction depend on the perturbed elastic paraeters and therefore the constraint on the polarization can be satisfied autoatically [].. Anisotropic reference ediu When using perturbation theory one has the freedo to choose a reference ediu. As a general rule the physical properties of the chosen reference ediu should be as close to those of the studied ediu as possible and at the sae tie a siple analytical solution should exist. Obviously a straightforward choice is to use an isotropic reference ediu and assue weak anisotropy of the ediu under study. However since an isotropic ediu has the two degenerate shear waves degenerate perturbation theory has to be used in such a case in order to develop the first-order approxiations for qs waves. As a result the associated approxiations then have to be expressed intricately as the square roots of a quadratic function and are thus highly nonlinear in the elastic constants of the ediu. Next we consider the other possible reference ediu having non-degenerate shear waves. One for of a reference ediu that has such desired properties was proposed by Mensch and Rasolofosaon [6]. Nevertheless in their developent the wave propagation characteristics were ignored for such a special reference ediu. Now we start with a reference ediu that possesses orthorhobic syetry. Using the Voigt atrix notation C αβ α β =...6; C αβ = C βα for the elastic constants C kl the possible reference elastic atrix is αβ = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 66. 7 This atrix has seven independent elastic constants. Following Mensch and Rasolofosaon [6] the independent nuber in this atrix could be further reduced to four by iposing the conditions: = C0 C0 8a = C0 C0 8b = C0 C0 66. 8c Given the wave noral vector n the coponents of the corresponding Christoffel atrix are = C0 n + C0 66 n + C0 n = C0 66 n + C0 n + C0 n = C0 n + C0 n + C0 n = Ɣ0 = C0 = Ɣ0 = C0 = Ɣ0 = C0 66 n n n n n n. Now the reference ediu defined by the elastic atrix 7 and the conditions 8 is isotropic for the qp wave whose square phase velocity is in any direction and whose polarization vector is along the wave noral n. Denoting the polarization vectors of the two qs waves as ξ and η let us consider possible analytical solutions of the qs waves in the reference ediu. For the oent it suffices to write the following relations for the above elastic atrix: n in j = Ɣ0 ξ iξ j = η + C0 η + C0 66 η η iη j = ξ + C0 ξ + C0 66 ξ n iξ j = n iη j = 0 ξ iη j = ξ η + ξ η + 66 ξ η. 9 0 Note that the above derivations have used the cross-product relationships aong the three orthogonal eigenvectors n ξ and η. Fro 0 it is interesting to observe that the vectors n and ξ and n and η are orthogonal with respect to the reference Christoffel tensor. We call this property conjugate orthogonality by analogy with the conjugate concept in nuerical atheatics. To ake the two shear polarization vectors satisfy the conjugate orthogonal condition too we need to ipose new constraints together with conditions 8 on the elastic atrix. First let us set ξ iη j = ξ η + ξ η + 66 ξ η = 0. Then in connection with the vector orthogonal relation ξ η + ξ η + ξ η = 0 we have C0 ξ η + C0 66 ξ η = 0 a C0 ξ η + C0 66 ξ η = 0 b C0 66 ξ η + C0 66 ξ η = 0. c The relations iply the following four possible choices or cases: = C0 C0 66 ; where ξ = 0 and ξ η + ξ η = 0. = C0 66 C0 ; where ξ = 0 and ξ η + ξ η = 0. C0 = C0 66 ; where ξ = 0 and ξ η + ξ η = 0. 4 = C0 = C0 66 ; where ξ and η are arbitrarily chosen in a plane perpendicular to n. 009
L-P Song and M Koch Note that for the last 4 and the second possible choices the elastic atrix 7 under conditions 8 corresponds to a well-known isotropic reference ediu that will not be considered further here and the reference ediu of Mensch and Rasolofosaon [6] see their equation 7 respectively. For the first three cases we can now easily write down the eigenvalues and eigenvectors of their corresponding Christoffel equation : Case = n = n n n a = 66 n + n + C 0 n ξ = n n 0 n n b = η = n n n n n c n n where n = n + n /. Case = n = n n n a = n + n + C 0 n ξ = n 0 n n n b = η = n n n n n c n n where n = n + n /. Case = n = n n n 4a = n + n + C 0 n ξ = 0 n n n n 4b = η = n n n n n 4c n n where n = n + n /. For all three cases we can specify the eigenvalues associated with the P wave and the two S waves as P = S = ax = in S respectively by definition the subscript S represents the fast shear wave and S the slow shear wave. We observe that the solutions are siple and analytical. They represent one pure longitudinal wave and two pure nondegenerate shear waves. The polarization vector of one of the pure shear waves always lies in one of the syetry planes for exaple in the x y plane for case. In particular we can give a physical interpretation of the arbitrarily chosen unit vectors for the degenerate transverse waves coonly used in an isotropic reference ediu [7 0 ]. Indeed they are exactly the shear polarization vectors for case. Siilarly one ay choose one of the other two sets of shear eigenvectors for the cases and for the degenerate perturbation case. The solution characteristics of the three anisotropic reference edia suggest that a reasonable choice of the depend on that syetry plane which is the closest to the region of interest. Chadwick and Norris [7] studied the conditions under which the slowness surfaces of an anisotropic elastic aterial consist of aligned ellipsoids. Interestingly by adding an elastic constant constraint C = C +C to the conditions C discussed by Chadwick and Norris [7] their results for the eigenvalues S and eigenvectors P of the Christoffel equation could be siplified as those of case. The set of eigenvalues for each of the cases here describes the slowness surfaces with two spherical sheets and one ellipsoidal sheet. Following Chadwick and Norris [7] a ediu defined by the elastic atrix 7 and subjected to both the conditions 8 and one of the first three further possible choices on the reference shear oduli ay in a generalized sense be regarded as one class of ellipsoidal reference ediu. 4. Specification of the reference elastic constants In this section we would like to specify the reference elastic constants in ters of a weakly anisotropic ediu. Like Mensch and Rasolofosaon [6] we ay siply set the qp and qs wave oduli along a specially interested direction of wave propagation e.g. the vertical or the horizontal direction as reference constants. They ay be set as the averaging quantities for instance = C + C + C = C0 = C + C 66 = C 66 for case. More rigorously we can use Fedorov s ethod [8] to derive the reference elastic constants by iniizing a function F that is the su of the squared difference between the Christoffel atrices Ɣ and of the anisotropic and reference edia respectively: F = Ɣ = ij= ij= Ɣ + ij= ij= Ɣ 5 where the sybol denotes the averaging operation over all directions of n in order to obtain a relation that is solely dependent on the aterial properties. Through this iniization procedure and soe algebraic anipulations we obtain the reference elastic constants of the elastic atrix under the conditions 8 5 = C + C + C +C +C 66 +C +C +C +C 6a 70 = 4C +4C +4C 66 C +8C +8C +C +C 8C 6b 00
Approxiations for phase velocities of qs waves 70 = 4C +4C +4C 66 +8C C +8C +C 8C +C 6c 70 66 = 4C +4C +4C 66 +8C +8C C 8C +C +C. 6d Equation 6a is just the longitudinal odulus already given in an isotropic reference ediu [8]. Then for case averaging any two e.g. and C0 of three equations 6b 6d would produce the other reference shear coponent required: 70 = 70 = 9C +9C +4C 66 +C +C +8C +C 8C 8C 7 where the sybol denotes the arithetic average. Of course equation 7 can also be obtained by setting = a priori in the iniization of function F 5. The derivation for the cases = 66 and = 66 is siilar. In addition the averaging of C0 and C0 66 in equations 6b 6d yields the shear odulus in an isotropic reference ediu [8]: 5 = C + C + C 66 + C + C + C C + C + C. 8 The reference ediu specified by such a iniization procedure is called an isotropic/anisotropic replaceent ediu IRM/ARM [6 8 8]. To quantify the intrinsic deviation of the considered ediu fro a reference ediu and taking at the sae tie the effects of wave propagation into account we ay use Ɣ 9 Ɣ where denotes the nor and Ɣ and are the Christoffel atrices of the considered ediu and a reference ediu respectively. Mensch and Rasolofosaon [6] directly used the elastic constant atrices to do this. 5. Expansions for the phase velocities of qs waves Using the eigenvalues in one of the anisotropic reference edia and their non-degeneracy as discussed in section we can directly write down the phase velocities to first-order: ρvqp = P + Ɣ n i n j = Ɣ n i n j 0a ρv = axɣ ξ i ξ j Ɣ η i η j 0b ρv qs = inɣ ξ i ξ j Ɣ η i η j 0c where the corrected shear-wave eigenvalues follow fro the relationships + Ɣ ξ i ξ j = Ɣ ξ i ξ j + Ɣ η i η j = Ɣ η i η j. For coparisons the first-order expressions for the qs-wave phase velocities under a degenerate isotropic reference ediu are also listed as [0 ] [ ρvqs = Ɣ ξ i ξ j + Ɣ η i η j ] ± Ɣ ξ i ξ j Ɣ η i η j +4Ɣ ξ i η j where + and before the square root ter correspond to and qs waves respectively. Substituting the expanded expressions for the coponent Ɣ and the corresponding unperturbed polarization vectors in one of the anisotropic reference edia e.g. in equation for case into equation 0 we get the following explicit firstorder approxiate forulae for the phase velocities in general anisotropy: ρvqp = { C n 4 + C n 4 + C n 4 +C +C 66 n n +C +C n n +C +C n } n + { 4C 6 n n +4C 5 n n +4C 6 n n +4C 4 n n +4C 5 n n +4C 4 n n +4C 4 +C 56 n n n +4C 5 +C 46 n n n +4C 6 +C 45 n n } n a ρv = axɣ ξ i ξ j Ɣ η i η j b ρv qs = inɣ ξ i ξ j Ɣ η i η j c where Ɣ ξ i ξ j = n { C66 n 4 + [ C + C C +C 66 ] n n + C n n + C n n +[ n n C 6 n n C 6 n n + C 46 n n + C 56 n n + n n n C 4 n + C 5 n C 45 n ]} Ɣ η i η j = C n + C n n + [ C +C 66 C + C ] n n n n + { [C + C C +C ]n +[C + C C +C ]n } n +4n n n [ n C6 n + C 6n C 45 + C 6 n ] +C45 n n n +n n n n n n [C4 +C 56 n + C 5 +C 46 n ]+n n n C5 n + C 4 n +n n n C 5 n + C 4n. It can be seen that the qp wave in equation a is ost sensitive to the 5 elastic constants or cobinations C C C C +C C +C 66 C +C and C 5 C 6 C 4 C 6 C 4 C 5 C 4 +C 56 C 5 +C 46 and C 6 + C 45 [ 6 8] the latter 9 paraeters of which involve the elastic constants quantifying the deviation fro orthorhobic syetry. The expressions in equations b c illustrate that the qs-wave speeds are ost probably sensitive to the elastic constants or cobinations C C C 66 [C +C 66 C + C ] [C +C C + C ] [C +C C + C ] and those elastic constants appearing in the latter 9 qp-wave sensitive paraeters. 0
L-P Song and M Koch For higher order syetry edia naely orthorhobic edia the equations b c for qs waves can be further reduced to ρvqs = axɣ ξ i ξ j Ɣ η i η j a ρvqs = inɣ ξ i ξ j Ɣ η i η j b where Ɣ ξ i ξ j = n [ C66 n 4 + [ C + C C +C 66 ] n n + C n n + C n ] n Ɣ η i η j = C n + C n n + [ C +C 66 C + C ] n n n n + {[ n C + C C +C ] n + [C + C C +C ] n }. Equations a b have been suggested by Song et al [0] who considered that the ter Ɣ ξ i η j in equation ay be sufficiently sall to be negligible when the wave vector deviates only slightly fro the x y syetry plane because its expansion has the leading factor n n n as shown below: Ɣ ξ i η j = n n n n { C n C n + C C + C +C 66 n n +[C +C C +C ]n }. 4 Note that the above expansions are based on the eigenvectors ξ and η in case. Under the reference ediu in case the approxiate expressions of qs-wave phase velocities in a weakly orthorhobic ediu will take the for ρvqs = axɣ ξ i ξ j Ɣ η i η j 5a ρvqs = inɣ ξ i ξ j Ɣ η i η j 5b where Ɣ ξ i ξ j = n [ C n 4 + [C + C C +C ] n n + C n n + C 66n ] n Ɣ η i η j = C 66 n + C n n + [ C +C C + C ] n n n n + { n [C + C C +C 66 ]n +[C + C C +C ]n }. As stated previously we ay choose the eigenvectors ξ and η in case as a zeroth-order eigenvector basis associated with the degenerate shear waves in an isotropic reference ediu. With this choice the expansion of the ter Ɣ ξ i η j in equation now is Ɣ ξ i η j = n n n n { C n C n + C 66 C + C +C n [ n + C +C C +C 66 ] n } 6 and also has the leading factor n n n. This again illustrates that the ter Ɣ ξ i η j in equation ay be sufficiently sall to be negligible when the wave vector is near the x z syetry plane. These two sets of approxiations and 5 for weakly orthorhobic edia assuing that the polarization of one of the qs waves is not too uch perturbed fro that of a pure ode for the reference edia of cases and show that the qs-wave phase velocities propagating near the x y plane or the x z plane will be ostly influenced by the different elastic constants or cobinations. 6. Nuerical exaples We test the validity of the linearized approxiations for the qs waves using the orthorhobic ediu of Farra [] which has the density-noralized elastic constants in units of k s : C = 0.8 C =. C = 8.5 C =.6C =.9 C 66 = 4.C =.C =.9 and C =.7. The corresponding IRM ediu to be used for coparison in the coputation has elastic constants = 0.04 C0 = 4.0 as calculated by equations 6a and 8. The corresponding ARM ediu coputed by equations 6a 7 and 6d has elastic constants = 0.04 = =.85 66 = 4.4. The intrinsic deviations of the orthorhobic ediu fro the IRM and ARM edia are 9.5% and 9.% respectively. Here the intrinsic IRM deviation of 9.5% is saller that of 0.6% in Farra [] who used the elastic tensor for the calculation. First let us inspect the plot at the polar angle easured fro the vertical z-axis of 60. In figure a both the linear equation and the quadratic approxiations equation for fast quasi-shear wave are good with average relative errors of only 0.% and 0.05% respectively. One also notices that the quadratic equation is a better approxiation in the iddle aziuthal range than the linear one. Copared to the fast quasi-shear wave the approxiations for the qs wave slow quasi-shear wave have higher errors with.09% for the linear and 0.80% for the quadratic equation. The approxiate phase velocities of the qs wave are larger than the exact ones in the aziuthal range 0 90. This iplies the need for soe aount of corrections fro the higher order perturbations see details in Farra []. Here we only focus on the issues of the first-order approxiation of qs-wave phase velocities. When the polar angle is gradually increased towards the x y plane the above described differences between the approxiate and the exact phase velocities are reduced significantly and the approxiate curves get closer and closer to the exact ones as shown in figures b d. In particular the sipler linear equations could yield satisfactory approxiations like the quadratic equations. For the linear approxiations the errors at a polar angle of 70 are now decreased to 0.% and 0.66% for and qs waves respectively; and for a polar angle of 80 the errors are further reduced to 0.07% and 0.0% for the two qs waves respectively. Even within the polar angle range 80 90 the errors in the linear approxiations of waves are saller than those in the quadratic ones as depicted in table. It is observed that the direction factor n contained in the linear and quadratic equations has an iportant effect on the accuracy of their approxiations. When the wave propagates near the z-axis it ay cause nuerical instabilities. For exaple at a polar angle of 0 the linear equation equation cannot properly approxiate qs waves possibly due to the relatively saller values of n and the large deviation of the polarization vector fro that of a pure ode see equation. However using the linear equations 5a 5b which now contain the direction factor n and correspond to one reference shear polarization in the x z plane see equation the qs waves can be well approxiated like using the quadratic equations as illustrated in figure a. At this polar angle the average errors in 0
Approxiations for phase velocities of qs waves a Velocity k/s...0 Polar angle at 60 qs b Velocity k/s...0 Polar angle at 70 qs c Velocity k/s.9 0 0 0 0 40 50 60 70 80 90 Aziuth deg...0 Polar angle at 80 d Velocity k/s.9 0 0 0 0 40 50 60 70 80 90....0.9 Aziuth deg Polar angle at 90 qs qs.9 0 0 0 0 40 50 60 70 80 90 Aziuth deg.8 0 0 0 0 40 50 60 70 80 90 Aziuth deg Figure. The exact and approxiate linear equations a and b and quadratic phase velocity curves of qs waves in the orthorhoboic aterial for the aziuths of 0 90 at the fixed polar angles a 60 b 70 c 80 and d 90. Table. Averaged relative errors as a percentage between the exact and approxiate linear equations a and b and quadratic phase velocities of qs waves. For each polar angle the calculations are carried out at the aziuths of 0 90. Polar angle - - -qs -qs 60 0. 0.05.09 0.80 70 0. 0.0 0.66 0.50 80 0.07 0.09 0.0 0.5 8 0.0 0.09 0.6 0. 85 0.0 0.07 0.05 0.04 90 0.000 0.06 the linear approxiations are 0.0% and 0.05% for and qs waves respectively being exactly the sae as those of the quadratic approxiations. For the polar angles of 0 and 40 the results in figures b c fro both the linear 5a 5b and the quadratic equations have siilar orders of approxiation accuracy; however they are less satisfactory due to the singularity between this polar angle interval [] than those at the polar angle of 0 shown in figure a. 7. Conclusions This paper discussing the choices of a set of ellipsoidal reference edia explored the possible first-order approxiations of qs-wave phase velocities for a weakly anisotropic ediu by non-degenerate perturbation theory. The derived approxiations appear as linear functions of the elastic constants of an anisotropic ediu. The corresponding expressions show that the qs-wave speeds are ost probably sensitive to the elastic coefficients C C C 66 [C +C 66 C + C ] [C +C C + C ] [C +C C + C ] and those constants quantifying the deviation fro orthorhobic syetry. In addition we could give a physical interpretation that the usually arbitrarily chosen zeroth-order eigenvector basis associated with the degenerate shear waves in an isotropic reference ediu are none other than the shear-wave polarization vectors in one of anisotropic reference edia discussed here. In for the present expressions are different fro the usual first-order degenerate approxiation of qs-wave phase velocities which are the square roots of a quadratic function of the ediu elastic constants. We see that the perturbed approxiations ay be siple or coplex depending on the reference ediu used. Of course the validity of the associated approxiations ight be liited to a certain range. The nuerical tests in the orthorhobic ediu deonstrate that the sipler linear approxiation is sufficiently applicable to propagation directions close to syetry planes or around syetry axes and does in these cases achieve the sae order of accuracy of approxiation as 0
L-P Song and M Koch a Velocity k/s.05.00 Polar angle at 0 b Velocity k/s.04.00.96 Polar angle at 0 qs.95 qs.9.90 0 0 0 0 40 50 60 70 80 90 Aziuth deg 0 0 0 0 40 50 60 70 80 90 Aziuth deg c.04.00 Polar angle at 40 Velocity k/s.96.9 qs 0 0 0 0 40 50 60 70 80 90 Aziuth deg Figure. The exact and approxiate linear equations 5a and 5b and quadratic phase velocity curves of qs waves in the orthorhoboic aterial for the aziuths of 0 90 at the fixed polar angles a 0 b 0 and c 40. the quadratic equations. However in view of the anisotropic reference edia used the valid use of these approxiations should be liited to aterials having at least one plane of syetry. Note that the quadratic approxiations can be used in a relatively large range of angles as they have the property of rotational invariance in a plane perpendicular to the wave noral [0]. In suary this study provides a theoretical and practical basis to possibly siplify inverse probles involving aterial characterization in non-destructive testing or seisology under qs-wave arrival easureent slightly deviating fro syetry planes/axes. Acknowledgents We are very grateful to V Farra for any valuable discussions and coents in the course of this work. We used her progra to calculate the exact phase velocities of qs waves in an orthorhobic ediu. We express our sincere thanks to A G Every I Psencik and the two anonyous reviewers whose coents and suggestions helped to iprove the anuscript and to clarify certain issues. LPS is thankful to T B Guttea for his help with soe issues regarding wording. References [] Backus G E 965 J. Geophys. Res. 70 49 [] Crapin S 98Wave Motion 4 [] Every A G and Sachse W 99 Ultrasonics 0 4 [4] Gassann F 964 PAGEOPH 58 6 [5] Jech J and Psencik I 989 Geophys. J. Int. 99 69 [6] Mensch T and Rasolofosaon P 997 Geophys. J. Int. 8 4 [7] Psencik I and Gajewski D 998 Geophys. 6 754 [8] Sayers C M 994 Geophys. J. Int. 6 799 [9] Thosen L 986 Geophysics 5 954 [0] Song L-P Every A G and Wright C J. Phys. D: Appl. Phys. 4 05 [] Farra V 00 Geophys. J. Int. 47 9 [] Mensch T and Farra V 999 Geophys. J. Int. 8 [] Ettrich N Gajewski D and Kashtan B 00 Geophys. Prospecting 49 [4] Auld B A 990 Acoustic Fields and Waves in Solids vol Malabar FL: Krieger [5] Every A G 980 Phys. Rev. B 746 [6] Butkov E 968 Matheatical Phys. Reading MA: Addison-Wesley chapter 5 [7] Chadwick P and Norris A N 990 Q. J. Mech. Appl. Math. 4 589 [8] Fedorov F I 968 Theory of Elastic Waves in Crystals New York: Plenu 04