Some basic properties of bulk elastic waves in anisotropic media

Wave Motion 40 (2004) 297–313

Some basic properties of bulk elastic waves in anisotropic media V.I. Alshits a,∗ , J. Lothe b b

a Institute of Crystallography RAS, Leninskii pr. 59, 119333 Moscow, Russia Institute of Physics, University of Oslo, P.O. Box 1048, Blindern, Oslo, Norway

Received 24 September 2003; received in revised form 11 February 2004; accepted 28 February 2004 Available online 22 July 2004

Abstract A short survey of some basic results in the theory of bulk elastic waves in anisotropic media is presented. A series of general properties of phase speed branches is deduced. Main features of the theory of acoustic axes are briefly discussed. Bulk wave modes in a semi-infinite medium with a free surface are considered: exceptional waves, the reflection problem and limiting inhomogeneous wave superposition are described. © 2004 Elsevier B.V. All rights reserved. Keywords: Elasticity; Anisotropy; Bulk waves; Acoustic axes

1. Introduction The theoretical description of bulk elastic waves in anisotropic bodies involves solving algebraic equations of degree 3 or higher, and obtaining explicit results is only possible for simplified situations due to symmetry or special relations between material parameters in model media. Nevertheless many fundamental properties of bulk waves in anisotropic media became understood just by way of explicit analysis of various particular illuminating situations admitting direct calculations (see, e.g. [1–6]). Such calculations together with appropriate computer studies created a mosaic picture containing many of the important features. However, this picture would be incomplete without more general theoretical results valid for unrestricted anisotropy and based on an approach, which can be defined in the following manner: analyze rather than solve, or, in other words, formulate the problems so that results might be obtained without explicitly finding all wave parameters. For instance, in order to find the number of some specific wave modes, conditions for their existence, or basic relations between their parameters, it is not necessary to know explicit solutions for these modes individually. The first realization of this approach in theoretical crystalloacoustics has been done by Fedorov in his book [7]. In this paper we shall present a survey of some general results obtained after Fedorov [7]. We stress that this is not a review but just an attempt of a systematic presentation of several important topics in the theory, which might be of interest for our colleagues. ∗

Corresponding author. Tel.: +7-95-330-82-74; fax: +7-95-135-10-11. E-mail address: [email protected] (V.I. Alshits). 0165-2125/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.wavemoti.2004.02.004

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2. Some basic concepts in the theory of bulk waves in crystals Consider an infinite medium with density ρ and elastic moduli tensor cijkl . A plane bulk wave of displacements u(x, t) propagating along the direction specified by a unit vector m: u(x, t) = u0 A exp[ik(m · x − ct)]

(1)

with scalar amplitude u0 , polarization A, wave vector k = km, and phase speed c = ω/k must, for u0 k  1, satisfy the Christoffel equation: (mm)A = ρc2 A,

(2)

where (mm) ≡ Q is the so-called acoustical tensor with components (mm)il = mj cijkl mk ≡ Qil . For a fixed direction m Eq. (2) determines three eigenvectors Aα (m), α = 1, 2, 3, and three corresponding eigenvalues λα (m) = ρc2α as roots of the cubic secular equation: det [Q − ρc2 I] = 0,

(3)

where I is the unit matrix. Q is symmetric, real and positive definite. Thus its eigenvalues λα are positive and the eigenvectors Aα can be chosen real and orthonormal. It is convenient to express Q as a spectral expansion in dyads: Q(m) = ρ

3
α=1

cα2 (m)Aα (m) ⊗ Aα (m).

(4)

Eq. (4) is valid for all directions of wave propagation except for acoustic axes, along which the phase speeds of a pair of isonormal waves coincide. For such a degeneracy direction m = md where, say c1 (md ) = c2 (md ), Eq. (4) can be transformed to Q(md ) = ρ{c12 (md )I + [c32 (md ) − c12 (md )]A3 (md ) ⊗ A3 (md )}.

(5)

It follows from Eq. (5) that any vector A ⊥ A3 (md ) is an eigenvector of the acoustical tensor Q(md ), i.e. is an allowed orientation for the polarization of a “degenerate” wave propagating along md with the phase speed c1 = c2 (see Fig. 1).

Fig. 1. The acoustic axis along m = md and the corresponding contour c1 (A) in the plane of degeneracy A = αA1 +βA2 , α2 + β2 = 1.

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3. General properties of phase-speed branches cα (m) 3.1. Invariance properties of the combination c12 + c22 + c32 on the sphere m2 = 1 Form the unit matrix I=n⊗n+s⊗s+t⊗t

(6)

built on the three mutually orthogonal unit vectors n, s, and t. The convolution of both sides of Eq. (6) with the tensor cijkl over the internal indices (j and k) may be written as V = (nn) + (ss) + (tt),

(7)

where V is a tensor of the second rank: Vil = cijjl .

(8)

Taking the trace on both sides of Eq. (7), making use of Eq. (4), one obtains [8–10]: 3
α=1

cα2 (n) +

3
α=1

cα2 (s) +

3
α=1

cα2 (t) =

tr V c11 + c22 + c33 + 2c44 + 2c55 + 2c66 = = const ρ ρ

(9)

irrespective of the orientation of the system {n, s, t}. Thus the following theorem is true: For any three mutually orthogonal directions of propagation, the sum of the squares of the phase speeds of the bulk elastic waves is constant for a given crystal and equal to the invariant combination: c11 + c22 + c33 + 2c44 + 2c55 + 2c66 . (10) ρ As an evident consequence of this theorem, one can also state that for any two coplanar pairs of orthogonal vectors n ⊥ s and p ⊥ q (i.e. n × s || p × q) the following identity [9] 3 
α=1

3  
 cα2 (n) + cα2 (s) = cα2 (p) + cα2 (q)

(11)

α=1

must be true. Let us now approach the problem from another side. It is clear from Eq. (4) that the sum c12 + c22 + c32 for one arbitrary direction m is also related to the tensors Q and V:   3
tr Q(m) V 2 cα (m) = =m· m. (12) ρ ρ α=1

Introduce, in the standard way, the characteristic ellipsoid of the tensors V/ρ [7]:   V r· r = 1. ρ Its radius-vector r = r(m)m has a length, which, by (12), may be expressed as −1/2  3
2 cα (m) . r(m) =

(13)

(14)

α=1

In these terms the geometrical meaning of theorem (10) becomes clear. Eq. (9) is equivalent to a known property of the ellipsoid for any orientation of the orthogonal system {n, s, t}: r−2 (n) + r −2 (s) + r −2 (t) = const.

(15)

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3.2. A general relation between max {c12 (m)} and min {c32 (m)} For m and n any two directions, due to the invariant property cijkl = cjilk it must be true that m · (nn)m = n · (mm)n.

(16)

m can be expanded in terms of the basis vectors Aα (n) and n correspondingly in terms of the Aα (m). Then obviously 3


[m · Aα (n)]2 = 1,

α=1

3


[n · Aα (m)]2 = 1.

(17)

α=1

Substituting the representation (4) of (mm) and the analogous representation of (nn) into (16) and numbering the speed branches according to the rule c1 (m) ≤ c2 (m) ≤ c3 (m), we have by (17) that ρc12 (n) ≤ m · (nn)m = n · (mm)n ≤ ρc32 (m).

(18)

Thus for any propagation directions m and n c1 (n) ≤ c3 (m).

(19)

From (19) we get the following relation between the top and bottom phase speed branches [10] The largest value of the phase speed in the bottom branch is not greater than the smallest value of the phase speed in the top branch.

(20)

In general, this theorem is valid irrespective of whether degeneracy is present or not. It leads to strict limitations on the possible configurations of the sheets of the phase velocity surface. In particular the whole inner sheet must be inside the sphere of radius c3min = min{c3 (m)} (Fig. 2). 3.3. Relations between the phase speeds of waves traveling along the directions m and A(m) Suppose that for the propagation direction m the polarization is A1 , so that by (2) A1 · (mm)A1 = ρc21 (m).

(21)

On the other hand, by (18): A1 · (mm)A1 = m · (A1 A1 )m ≥ ρc12 (A1 ),

(22)

where the inequality becomes an equality if the vector m happens to be the true polarization for the propagation direction A1 . Combining (21) and (22) we obtain [10] c1 (m) ≥ c1 (A1 ).

(23)

Fig. 2. Two schematic phase velocity curves in a plane through the origin intersecting the velocity surface: (a) permitted configuration; (b) forbidden configuration contradicting theorem (20).

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Fig. 3. Illustrations to Eq. (23) (a) and Eq. (25) (b).

Thus we arrive at the following important property of the bottom phase-speed branch (Fig. 3a): The polarization vector A1 of a wave propagating in the direction m at speed c1 (m) and belonging to the bottom branch points out another propagation direction where the wave speed c1 (A1 ) does not exceed c1 (m).

(24)

Pursuing this line of reasoning we obtain an algorithm for the search for a minimum, c1min = min {c1 (m)}. Of course, if the wave in question is pure longitudinal, this “other” direction simply coincides with the original direction. As is known, purely longitudinal polarizations indeed always correspond to extremal values of the speed cα (m). But on the bottom branch the minimum does not usually relate to longitudinal polarization. From (18) together with (22) follows another inequality [10] for the top branch: c3 (m) ≤ c3 (A3 )

(25)

(Fig. 3b), which enables us to formulate a new statement concerning the top velocity branch: The polarization vector A3 of a wave propagating in the direction m at speed c3 (m) and belonging to the top branch points out another propagation direction where the wave speed c3 (A3 ) is not less than c3 (m). (26) 3.4. Relations for degenerate phase speeds Now let us consider degeneracy between the bottom and middle branches in the direction md , c1 (md ) = c2 (md ) (Fig. 1). At the degeneracy point the polarization A can be any linear combination of the type A = αA1 + βA2 ,

α2 + β2 = 1,

(27)

where A1 and A2 are arbitrary orthogonal unit vectors in the plane of “degeneracy” perpendicular to the polarization A3 (md ) of the nondegenerate branch. Repeating the arguments which led to inequality (23), in this case we again get [10] c1 (m) ≥ c1 (A).

(28)

With all possible changes in α and β, the vector A (27) describes a unit circle in a plane containing A1 and A2 . From this viewpoint, the content of the inequality (28) may be expressed as a theorem The phase speed at a degeneracy point between the bottom and the middle branches is not less than the greatest speed on the bottom branch on the great circle of directions described by vector A (27) on the sphere m2 = 1. (29) See Fig. 1. Quite analogously one can prove the theorem [10] The phase speed at a degeneracy point between the top and the middle branch is not larger than the least speed on the top branch on the great circle of directions described by vector A (27) on the sphere m2 = 1.

(30)

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We note that at a point of triple degeneracy the vector A is in general directed in any arbitrary way. In this case the meaning of inequality (28) reduces to the following assertion [10]: Triple degeneracy can be realized only at a point where the speed on the bottom branch is a maximum and coincides with the minimum velocity on the top branch. (31) Incidentally, this assertion also follows directly from theorem (20).

4. Acoustic axes in crystals 4.1. Existence, non-existence and possible numbers of acoustic axes Let us denote the three eigenvalues of the acoustical tensor Q as λα, λβ and λγ , and suppose that along the direction m = md degeneracy between two of them occurs, say λα = λβ . Then, instead of (5), one can write Q(md ) = λα I + (λγ − λα )Aγ ⊗ Aγ .

(32)

In fact, this relation represents one of the forms of the equation for acoustic axes. Taking into account that Qij = Qji , we are dealing here with a system of six equations with six unknown values: λα , λγ and the directions of md and Aγ . Since each of these six equations is of the degree 2, the system may have at most 26 = 64 solutions. However, it is clear that solutions (md , Aγ ), (md , −Aγ ), (−md , Aγ ) and (−md , −Aγ ) are physically equivalent. Therefore among 64 solutions there are only 64/4 = 16 different solutions. This consideration does not work only for isotropic and transversely isotropic media where the above system of equations becomes degenerate and can provide a continuum of solutions. Thus, the following statement must be valid [11–13]: In crystals of unrestricted anisotropy, however not isotropic or transversely isotropic crystals, the total number of acoustic axes does not exceed 16.

(33)

Acoustic axes are very common in crystals. In fact, among real materials studied till now there are no examples of crystals free of acoustic axes. However, as a rule, the fastest phase-velocity branch remains nondegenerate. The only exception, found in 1972 by Ohmachi et al. [14], TeO2 crystals where all three wave branches are degenerate, has prevented attempts to prove that the empirical observation of a nondegenerate fast branch is a general property. In the same way, attempts to prove a theorem of obligatory existence of acoustic axes in crystals of unrestricted anisotropy were stopped after Alshits and Lothe [10] in 1979 introduced the example of a thermodynamically stable model crystal without acoustic axes. According to [10], any orthorhombic crystal with the elastic moduli c12 = c13 = c23 = 0 and 0 < c22 < c66 < c11 < C1 < C2 < c55 << c33 must be completely free of acoustic axes. In (34) C1,2 are constants defined by the relations



 c55 (2c44 − c22 ) c55 (2c44 − c22 ) C1 = min c44 , , C2 = max c44 , . c55 + 2c44 − c22 c55 + 2c44 − c22

(34)

(35)

Later an alternative example of an orthorhombic medium with nondegenerate phase speed branches was also found numerically [15,16]. The acoustic properties of a crystal without acoustic axes must be rather unusual. In particular, longitudinal wave normals will be obligatory in all three sheets of phase velocity, including so-called “quasi-transversal” sheets. And along these directions a purely transversal wave must propagate in the “quasi-longitudinal” fastest sheet. An example of a model medium free of acoustic axes would be impossible for systems of higher symmetry than orthorhombic, because any symmetry axis higher than the 2-fold axis must be an acoustic axis. Accordingly, in

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tetragonal crystals only one acoustic axis along a principal 4-fold axis is obligatory, the other acoustic axes may or may not exist (altogether there could be 1, 5 or 9 acoustic axes in 7 different combinations). In trigonal crystals there are only 2 possible variants: 4 (obligatory) or 10 acoustic axes. In cubic crystals 7 obligatory acoustic axes along 4- and 3-fold symmetry axes always exist, and no other degeneracies may occur in this symmetry system. In hexagonal crystals transversely isotropic with respect to elastic properties, apart from one obligatory acoustic axis along the principal axis, a cone of acoustic axes, formed by the intersection of two sheets of the phase velocity surface, will arise under the condition: (c66 − c44 )[(c11 − c66 )(c33 − c44 ) − (c44 + c13 )2 ] > 0.

(36)

The angle θ d between the acoustic axes of the cone and the principal axis is given by  2 1/2 −1 (c11 − c66 )(c33 − c44 ) − (c44 − c13 ) . θd = tan (c11 − c66 )(c66 − c44 )

(37)

A more detailed analysis of acoustic axes in crystals of particular symmetry can be found in the original papers [2,13,16–18]. 4.2. Invariant criterion for degeneracy Let us now consider the other forms of equations for acoustic axes. Consider the tensor F(m) = Q − λα I = (λβ − λα )Aβ ⊗ Aβ + (λγ − λα )Aγ ⊗ Aγ .

(38)

As any other planar tensor, F is characterized by a vanishing determinant, det F = 0, but does not vanish itself even at the degeneracy point (not triple). However the adjoint tensor to F: F¯ = (λβ − λα )(λγ − λα )Aα ⊗ Aα ,

(39)

clearly must vanish at any degeneracy point m = md where λα = λβ,γ : ¯ d ) = 0. F(m

(40)

As was noticed by Khatkevich [19], when Qij = 0 (i = j) Eq. (40) is automatically satisfied if F¯ 12 ≡ F13 F23 − F12 F33 = 0,

F¯ 13 ≡ F12 F23 − F13 F22 = 0,

F¯ 23 ≡ F12 F13 − F23 F11 = 0.

(41)

The system (41) determines both the directions of acoustic axes and the degenerate eigenvalues λα . Eliminating λα from (41), Khatkevich obtained the two conditions determining the direction of md : R1 = (Q11 − Q22 )Q13 Q23 − Q12 (Q213 − Q223 ) = 0,

(42)

R2 = (Q11 − Q33 )Q12 Q23 − Q13 (Q212 − Q223 ) = 0.

(43)

Alshits and Lothe [20] added to R1 and R2 five more components: R3 = (Q22 − Q33 )Q12 Q13 − Q23 (Q212 − Q213 ),

(44)

R4 = (Q11 − Q22 )(Q11 − Q33 )Q23 − (Q11 − Q33 )Q12 Q13 + Q23 (Q212 − Q223 ),

(45)

R5 = (Q22 − Q11 )(Q22 − Q33 )Q13 − (Q22 − Q33 )Q12 Q23 + Q13 (Q212 − Q213 ),

(46)

R6 = (Q33 − Q11 )(Q33 − Q22 )Q12 − (Q33 − Q22 )Q13 Q23 + Q12 (Q213 − Q212 ),

(47)

R7 = (Q11 −Q22 )(Q11 −Q33 )(Q22 −Q33 ) + (Q22 −Q33 )(Q213 − Q223 ) + (Q11 − Q22 )(Q213 − Q212 ).

(48)

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They formed the seven-component vector ␰ = {R1 , R2 , . . . , R7 } and proved that the equation: ξ=0

(49)

represents an invariant criterion of degeneracy valid in an arbitrary coordinate system independently of the condition Qij = 0 ( i = j). When Eq. (49) is solved and the direction m = md is found, the next steps in finding the wave characteristics along md are quite straightforward. In particular, in the coordinate system where Q23 = 0 the degenerate eigenvalue is given by λα =

Q11 − Q12 Q13 . Q23

(50)

For the parameters of the nondegenerate wave one has λγ =

det Q(md ) = tr Q(md ) − 2λα , λ2α

Aγ F(md )C,

(51)

where C is an arbitrary real vector. The polarizations of the degenerate waves may have any orientation in the plane orthogonal to Aγ . 4.3. Geometrical types of acoustic axes and polarization singularities near degeneracies Though orientations of acoustic axes are determined by the same Eq. (49) and the basic characteristics of the eigenwaves propagating along md are universal, degeneracy directions differ from each other by their neighborhood. They can be classified by geometrical types of contact along md of degenerate velocity sheets or/and by types of singularities in polarization fields Aα,β (m) around md . Geometrical characterization of acoustic axes has a long history. One can find mention of conical and tangent points of contact and lines of intersection of degeneracy sheets (the latter for transverse isotropy, see Eq. (37)) already in the papers by Herring [21,22] and Khatkevich [13,19], and in the books by Fedorov [7] and Musgrave [1]. Alshits and Lothe [20,23] (see also [24,25]) first noticed that geometrical features of degeneracies correlate with definite types of polarization singularities. This observation was elaborated in [26,27] where a complete classification of acoustic axes was constructed including all possible types of local geometry of the velocity sheets near the degeneracy, and the corresponding polarization singularities (see Fig. 4). The developed theory also provides the particular algebraic conditions for any type of degeneracy, without solving the wave equation, but using only appropriate convolutions of the elastic moduli tensor cijkl . As an example, let us construct the two vectors p and q: p = (S11 − S22 )md ,

q = (S12 + S21 )md ,

(52)

where the four matrices Sαβ are defined by Sαβ = (Aα Aβ )/2ρcd , with cd and A1,2 being the degenerate phase speed and the arbitrary orthogonal pair of unit vectors in the degeneracy plane, Eq. (27) and Fig. 1. It turns out that p and q determine the geometry of the contact of the velocity sheets. If the vector product p × q does not vanish, one has a degeneracy of the conical type. It is customary to characterize point singularities in plane distributions of vector fields by a topological “charge”—a Poincaré index n, which is defined as the angle (in 2π units) of aggregate rotation of vectors around a singular point. In these terms the topological charge of the conical degeneracy at md is given by n=

1 2

sgn[md (p × q)].

(53)

If p × q = 0 but p or q does not vanish, we have a wedge degeneracy at the point md or along the line passing through md . The case p = q = 0 corresponds to a degeneracy of the tangent type at a point or along a line. For distinguishing the degeneracy at a point from that along a line and for calculating the index n for degeneracies of

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305

Fig. 4. Geometrical and associated polarization types of degeneracies.

wedge-point (n = 0, ±1/2) or tangent-point (n = 0, ±1) types, the other convolutions of the tensor cijkl with md and Aα should be used [26,27]. It is essential that almost all types of acoustic axes exist in real crystals. Only the wedge-point and tangent-line degeneracies are known only for model crystals. Conical degeneracies (n = ±1/2) exist in practically all known crystals of the orthorhombic, monoclinic and triclinic symmetry systems. A 2-fold axis does not obligatorily provide a degeneracy; however when there happens to be an acoustic axis it must be of the tangent type, and its topological charge, depending on the material constants, may appear with any of the relevant values n = 0, ±1. Along a 3-fold axis, both in trigonal and in cubic crystals, the conical degeneracy with Poincaré index n = −1/2 always occurs. A 4-fold axis in tetragonal and cubic crystals is always a tangent acoustic axes with a Poincaré index n = ±1. In

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particular, in the cubic system, the choice of the sign of n is especially simple: n = sgn(c12 + c44 ). An ∞-fold axis in transverse isotropic crystals is also always a tangent acoustic axis, however its topological charge is definite: n = 1. In crystals of this symmetry one can also meet the wedge-line degeneracy, Eqs. (36) and (37). In the model crystal, where apart from (36) also the condition c44 = c66 is satisfied, the two symmetrical wedge lines must, in accordance with (37), coalesce into one tangent degeneracy line in the basal plane (θ d = π/2). The appearance of two crossing “meridian” wedge lines with tangent points at the places of their intersection on the “poles” is predicted [27] for the crystal Hg2 Cl2 at the temperature of the phase transition between the tetragonal and the orthorhombic symmetry states. In the same paper [27] the sporadic tangent degeneracy with Poicaré index n = 0 is predicted for the monoclinic phase of the crystal CsDSeO4 along the 2-fold axis. The above specific features must arise due to the coincidence c44 = c55 , which remains in the low symmetry phases of both crystals [28,29]. The extension of the theory [26] to piezoelectrics was given in [30,31]. It was found that the classification of acoustic axes remains, apart from renormalization of the algebraic conditions for particular types of degeneracies, due to contributions from piezoelectric moduli. However this contribution may qualitatively change the wave properties near a specific acoustic axis and even change the type of the degeneracy itself, see [31,32]. Piezoelectric coupling also causes a quasi-static electric field accompanying elastic waves. Its characteristics depend on the polarization of the elastic wave and therefore it will also have definite singularities near the directions md , see [30]. 4.4. Instabilities of degeneracies, conical and wedge refractions The different types of acoustic axes behave differently (disappearance, shifting, splitting) under small perturbations of the elastic properties. The analysis of this problem in [20,26] reveals that only acoustic axes of the conical type are always stable under perturbations, i.e. they can only shift orientation. Unstable points of degeneracy either split in accordance with the rule of conservation of topological charge (the sum of the Poincaré indices of the degeneracies after a perturbation must be equal to the index n of the initial degeneracy) or vanish, but only if n = 0. Such perturbations often arise due to different external influences, like electric fields, mechanical stresses or temperature variations in the vicinity of phase transitions. For instance, at a phase transition from a transverse isotropic to a trigonal crystal the ∞-fold axis (n = 1) is replaced by the 3-fold axis (n = −1/2). In accordance with the rule of index conservation, in addition to this degeneracy also three conical acoustic axes of index n = 1/2 must arise (Fig. 5). As was mentioned in [26] and studied in [33], even a conical degeneracy is unstable with respect to the “switching on” of a small damping, which is equivalent to a small imaginary perturbation of the phase speed, v → v−iδv. As a result of such a perturbation, the conical axis split into a pair of singular directions connected on the slowness surface and on the surface of damping by lines of intersection of corresponding sheets. The only two common inversely nonequivalent points of these lines on the unit sphere m2 = 1 correspond to new positions of acoustic axes. The polarizations of the degenerate waves along the two new singular directions are circular. In the vicinity of these points

Fig. 5. Splitting of the tangent degeneracy along the ∞-fold axis into four conical degeneracies at the phase transition from the hexagonal to the trigonal symmetry system.

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307

of degeneracy, polarizations are elliptic. The rotation of the large semi-axes of these ellipses around the degeneracy points corresponds to the Poincaré index n = ±1/4 in complete accordance with the rule of index conservation. The geometrical feature of the degenerate slowness sheets close to a conical point of their contact must create a corresponding cone of group velocities, which in turn will give rise to internal conical refraction. This phenomenon was first observed by Klerk and Musgrave [34] in Ni crystals along the 3-fold axis parallel to the direction [1 1 1]. Quite similarly in the vicinity of the wedge line one can expect wedge refraction, in a fan-like distribution of rays, instead of the usual cone. Such a sort of refraction theoretically predicted by Khatkevich [11,19] was later experimentally observed by Henneke and Green [35]. When the plane-wave approximation is replaced by packets of elastic waves, one has to answer the following natural question: how do such singular polarization fields in k space affect the wave fields in r space for beams propagating along directions of acoustic axes? In [36] the phenomenon of internal conical refraction was theoretically investigated from this viewpoint for acoustic and optical beams. The use of the Lorentzian profile made it possible to obtain results in an analytical form. The answer to the above question turned out to be quite non-trivial: the internal conical refraction creates a polarization disclination line in such a beam. The wave field vanishes at any point in such disclination and there are two orientational singularities n = 1 and n = −1 around these zeros in any cross-section orthogonal to the acoustic axis, z = const. > z0 .

5. Bulk waves in semi-infinite media 5.1. Some preliminary theoretical background Consider an anisotropic half-space with a free surface. Let us choose the coordinate system with the origin in the surface, the y-axis along the internal normal n to the surface and the x-axis along the arbitrary direction m of wave propagation, Fig. 6. In the chosen coordinate system the steady-state displacement field of the plane wave can be presented in the form of a superposition of partial waves:
u(x − vt, y) = A(y) exp[ik(x − vt)], A(y) = bα Aα exp(ikpα y), (54) α

which have equal x-components of the wave vector, kx ≡ k, and a common tracing speed v = ω/k, but different (α) y-components of the wave vector, ky = pα k, and polarization vectors Aα . It is convenient, in parallel with (54), to introduce the so-called traction field represented by a normal projection of the stress tensor:
n␴(x − vt, y) = −ikL(y) exp[ik(x − vt)], L(y) = bα Lα exp(ikpα y), (55) α

where Lα = [(nm) + pα (nn)]Aα and the orthogonality relation: Aα · Lβ + Lα · Aβ = δαβ

(56)

Fig. 6. Cross-section of slowness surface sheets by a sagittal plane.

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is fulfilled. In these terms the boundary condition of a free surface reduces to
bα Lα = 0.

(57)

α

As shown by Stroh [37], the six-vectors ␰α = (Aα , Lα ) and the parameters pα are eigenvectors and eigenvalues of some real 6 × 6 matrix. Therefore six functions pα (v) must be real or form complex conjugate pairs. In the range 0 <ν<ˆv known as a subsonic range all eigenvalues pα and eigenvectors ␰α occur in pairs of complex conjugates: p␣+3 = p∗α , ␰α+3 = ␰∗α , α = 1, 2, 3. At the first transsonic state v = vˆ one of the conjugate pairs coalesces into one degenerate eigenvalue p, ˆ which in the supersonic range v > vˆ splits into a pair of different real parameters. As is clear from Fig. 6, in the range v > vˆ there must exist at least two other transonic states, vˆ  and vˆ  , where the next pairs of the set pα , ␰α are similarly transformed from complex to real state, so that at ν > vˆ  all partial waves in the superposition (54) are homogeneous bulk waves. On the other hand, in the subsonic range ν < vˆ three of six inhomogeneous terms of this superposition contain infinitely increasing exponents as y → ∞ and the corresponding three amplitudes bα must vanish when describing surface localized phenomena. With the choice of numeration providing Im pα > 0 for α = 1, 2, 3, b4,5,6 = 0 describes a 3-partial field localized at the surface, i.e. a surface wave. Its amplitudes bα are to be found from the boundary condition of a free surface: b1 L1 + b2 L2 + b3 L3 = 0,

(58)

which will have non-trivial solutions only if the corresponding determinant of the 3 × 3 matrix {Lαi } vanishes, det{Lαi } = 0. This equation determines the velocity vR of the Rayleigh wave. Barnett and Lothe [38–40] have proved the theorem of existence of a unique solution of the above equation if the ¯ exp[ik(x + py one-partial limiting bulk wave at the first transonic state v = vˆ , u¯ = bA ˆ − vˆ t)], is not exceptional, i.e. does not satisfy the boundary condition of a free surface. Thus, it became important to find the conditions for the appearance of exceptional wave solutions and especially to establish the dimension of the sub-space occupied by such solutions in the 3D space of all possible orientations of the frame {m, n} specifying a surface wave geometry. The first detailed study of exceptional waves was accomplished in [23]. 5.2. Exceptional bulk waves Thus, the situations, when the limiting bulk wave at the first transonic speed vˆ is exceptional, play a crucial role in the theory of surface waves in anisotropic media, because in these particular situations closer analysis is needed for deciding the question of existence of a surface wave solution in the subsonic range v < vˆ . The other motivation for an interest in exceptional waves has purely applicational origin: the devices based on bulk waves are often preferable to their surface wave analogues. Formally, a bulk wave is defined as exceptional with respect to a family of parallel planes with normal n if the wave produces no tractions on these planes, σ ij nj = 0. With this definition, the energy flux of an exceptional wave, ˙ must be parallel to the surface, i.e. J · n = −u˙ i σij nj = 0. For the bulk wave (1) the stress field σ ij J = −u␴, = cijkl ∂uk /∂xl is σij (x, t) = iku0 µij exp[ik(m · x − ct)],

(59)

where µij = cijkl ml Ak . Strictly the real parts of (1) and (59) should be used in J. Linear polarizations allows more expedient procedures. Since cijkl = cjikl , µij is also symmetric. In these terms the Christoffel equation (2) can be rewritten as mi µij = ρc2 Aj .

(60)

Multiplying scalarly both sides of this equation by the vector n and taking into account that µij nj = 0, we obtain an important relation: A · n = 0.

(61)

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309

Thus, the following important property of exceptional waves is deduced [23,41]: An exceptional bulk wave is polarized in the plane with respect to which the wave is exceptional.

(62)

For a given surface wave geometry there are several possible exceptional waves. For instance, in Fig. 6 anyone of the three waves at speeds vˆ , vˆ  and vˆ  might be exceptional. The wave normal m of exceptional waves can be found from the condition for the occurrence of non-trivial solutions of the equation µij nj = 0 for the normal n to the exceptional plane: det ␮ = 0.

(63)

This equation determines the permitted directions m for propagation of exceptional waves. Before solving Eq. (63) one must find the polarization distribution A(m) for the wave branch considered from the Christoffel equation. The orientation of the unit vector m is completely specified by two spherical angles ϕm and θ m . The scalar equation (63) gives a relation between ϕm and θ m , F(ϕm , θ m ) = 0, which is a line on the unit sphere of all directions m2 = 1. For any m at this line the direction of the normal n to the corresponding traction free plane is specified by only one parameter—an angle ϕn in the plane orthogonal to m. This angle ϕn = f(ϕm , θ m ) may be found from the system µij nj = 0. Thus, in the 3D space {ϕm , θ m , ϕn } the exceptional wave orientations must generally occupy a 1D sub-space, i.e. lines on the sphere m2 = 1. The theorem of existence of such lines of exceptional wave solutions (EWS) for media of unrestricted anisotropy was proved in [23]. The intimate relationships between exceptional waves and acoustic axes were established. According to [23], the direction md of any acoustic axis determines the wave normal of at least one exceptional wave. And the properties of EWS lines on the unit sphere of directions depend on the type of degeneracy along md . For instance, the EWS lines emerge from and end on the points of conical degeneracy, passing in these points from one degenerate branch to another. On the other hand, the discussed lines necessarily pass through points of genuine tangent degeneracy (n = ±1) in one or in both degenerate branches. Only in the case of a sporadic tangent degeneracy (n = 0), it may happen that in the vicinity of md there are no other solutions for exceptional waves. However, even in this case a line of EWS necessarily exists in one of degenerate branches. And, finally, in media free of acoustic axes the lines of solutions for exceptional waves must be present in all velocity sheets. Thus, lines of EWS must be an important characteristic feature of the acoustic identity of most crystals. Indeed, such EWS lines were found explicitly and numerically for a series of crystals [4,42–44]. Some extension of the results [23] was accomplished in [43,45], where the equations for the orientation of the polarization of the exceptional waves propagating along the acoustic axes of different types and for the direction of EWS lines at md , were derived. Analysis of these equations allowed finding the number of exceptional waves that can propagate along different acoustic axes and correspondingly the number of the lines of solutions for exceptional waves going from degeneracy points or crossing them. 5.3. The reflection problem in the first transonic range In the first transonic range vˆ < v < vˆ  the given tracing speed v for the reflection problem specifies the incident and reflected waves. v−1 is then the projection on the surface of the inverse phase speeds of the incident and reflected waves (see Fig. 7), so that u(i) (r, t) = a(i) exp[ik(x + pi y − vt)], where

a(i ,r )

u(r) (r, t) = a(r) exp[ik(x + pr y − vt)],

(64)

= bi,r Ai,r and

pi = tan ψi ,

pr = tan ψr . a(i ,r )

(65) l(i ,r )

The vector amplitudes and the corresponding tractions = bi,r Li,r of the bulk waves (64) can be chosen real. Taking into account that the projections of the energy flux on the normal n: J(i,r) · n = − 21 kωa(i,r) · l(i,r)

(66)

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Fig. 7. Incident wave, reflected wave and limiting wave, related to the slowness surface.

for the incident and reflected waves must be negative and positive, respectively, one obtains a(i) · l(i) > 0,

a(r) · l(r) < 0.

(67)

Accordingly, with the normalization (56), 2Ai ·Li = 2Ar ·Lr = 1, the vectors Ai and Li must be purely real, and the vectors Ar and Lr must be purely imaginary. The boundary problem for the reflection at a free surface in terms of the Stroh formalism reduces to a requirement of linear dependence, similar to (57) and (58): L(0) = b1 L1 + b2 L2 + bi Li + br Lr = 0.

(68)

Certainly, four vectors are always linearly dependent. Thus, for any speed v, there must be a solution for the unknown amplitudes bβ . According to [46], the solution is given by bβ = C[Lβ L∗1 L∗2 ].

(69)

Here the constant C is determined by the known amplitude bi of the incident wave and [. . . ] denotes the triple scalar product. Quite similarly, one can find [46]: A(0) = b1 A1 + b2 A2 + bi Ai + br Ar = CL∗1 × L∗2 .

(70)

Multiplying (68) with (70) and making use of the orthogonality (56) we obtain the identity [46] b12 + b22 + bi2 + br2 = 0.

(71)

The solution (69) and the identity (71) may be generalized to cases of more complicated boundary problems [47–52]. In this section we have limited ourselves to an analysis of reflection only in the first transonic range vˆ < v < vˆ  , missing some features of reflection specific for “faster” regions ν > vˆ  , for instance the phenomenon of mode conversion, similar to a Brewster reflection in optics, where the incident and outgoing waves belong to different sheets of the slowness surface (see [53]). 5.4. Inhomogeneous limiting waves and their physical meaning ˆ and A particular and interesting situation arises in the limiting case vˆ ← v where ψi and ψr coalesce to ψ, ˆ accordingly, pi pr → p, ˆ ai , ar → aˆ and li , lr → l (the corresponding normalized Li and Lr diverge). This is the case of grazing incidence since the ray direction of the limiting wave is parallel with the surface (Fig. 7). The fourth

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partial solution required for satisfying the boundary condition can be obtained as follows: since u(i ) and u(r ) (64) are partial solutions, also u4 =

u(r) − u(i) , /ψ

/ψ = ψr − ψi

is a partial solution. With (65) in (64) going to the limit ∆ψ → 0 as vˆ ← v in (72), one obtains [46]   ˆ ∂a ikya(ψ) ˆ − vˆ t)], u4 (r, t) = + exp[ik(x + y tan ψ ˆ ∂ψ ψˆ cos2 ψ which is an inhomogeneous wave. In the surface y = 0, this wave exerts a force with amplitude   ∂l(ψ) . −␴4 n = ik ∂ψ ψˆ The condition of free surface can now be satisfied by a superposition   ∂l ˆ ˆ b1 L1 + b2 L2 + bl + b = 0. ∂ψ ψˆ

(72)

(73)

(74)

(75)

At grazing incidence, this requirement replaces (68). The solution of (75) could be obtained by limiting procedures starting with (69). We will not undertake such a discussion here. The construction leading to (73) is similar to the Achenbach–Gautesen [54] construction of a homogeneous surface wave solution. However one should notice that the combined solution containing the partial wave (73) in its literal form is divergent at infinity y → ∞ together with its derivatives. Nevertheless, this inhomogeneous solution has the definite physical sense as a limiting asymptotic form describing the wave field at grazing reflection (/ψ  1) not very far from the surface (ky  1//ψ), when indeed sin(/ψky)//ψ ≈ ky. Such particular solutions arise for the Lamb waves in elastic plates in the short-wavelength range when an infinite number of dispersion curves vn (k) for wave-guided modes tend to the level vˆ [55], independently of the “status” of the limiting wave, exceptional or non-exceptional: 1 πn 2 vn (k) ≈ vˆ + , n = 1, 2, 3, . . . (76) 2κ kd Here κ ∼ vˆ −1 is the radius of curvature of the external slowness sheet at ν−1 = vˆ −1 (Fig. 7), d is the thickness of the plate, and for each number n the wavelength is supposed to be short enough for the dispersion curve vn (k) to be close to the asymptotic level vˆ : νn− vˆ << vˆ , i.e. kd  πn. On the other hand, from elementary geometrical consideration in the vicinity of vˆ one has  2 /ψ 1 vn − vˆ ≈ , (77) ˆ 2κ 2 cos2 ψ which together with (76) provides the estimate /ψ ∼ πn/kd. Thus, the wave-guided solutions in the asymptotic range kd  πn contain the inhomogeneous component (73) in the pre-surface zones of thickness /y  1/k/ψ ∼ d/πn. Looking at Eq. (75) one would think that rather than the exceptional case ˆl = 0, the less restrictive situation b1 L1 + b2 L2 + ˆl = 0,

(78)

could also represent the limiting case of a surface wave of infinite penetration at vˆ . This would extend the definition of an exceptional limiting wave to a larger class of solutions than we studied above. However, as was rigorously proved in [46], Eq. (78) cannot be satisfied for ˆl = 0, and at ˆl = 0 there must be either b1 = b2 = 0 or b1 L1 + b2 L2 = 0. In the latter case a surface wave composed of only two partial waves coexists with the exceptional bulk wave at ν = vˆ , which is not ruled out by surface wave existence theorems [38–40].

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Quite similarly, at ˆl = 0 one cannot expect linear dependence   ∂l b1 L1 + b2 L2 + b = 0. ∂ψ ψˆ

(79)

As was proved in [46], the third term in (79) may not belong to the same plane as L1 and L2 . Thus, an extension of the concept of an exceptional wave in elastic half-space is not possible. On the other hand, extended exceptional waves representing a superposition of a single bulk mode with localized modes may arise in elastic plates [56] and in piezoelectric half-spaces [57,58].

Acknowledgements V.A. and J.L. are grateful for support received from the The Physics of Geological Processes group (PGP) at the University of Oslo. V.A. was also supported in part by The Russian Foundation for Fundamental Research (grant #01-02-16228) and by the Polish–Japanese Institute of Information Technology, Warsaw (research grant no. PJ/MKT/02/2003). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

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