THE NATURE OF THE SOUND WAVE POLARIZATION FIELD FOR ANISOTROPIC ELASTIC MEDIA WITHOUT ACOUSTIC AXES. THE CRITERION FOR THE PRESENCE OF ACOUSTIC AXES

§ 1. Introduction. Statement of Problem.

A sound wave in direction n, U

=

Aexp i(k<n,x> - wt) must satisfy the eigenvalue equation

( i) MA = ).A,

where the acoustic operator M

=

M(n) is defined by ( ii)

are the elastic coefficients. The eigenvalue is ).. = pc2 '

where p is the density and c the sound velocity along the

propagation vector kn . M is a symmetric matrix. As function of n it possesses antipodal symmetry,

(iii) M(n)

=

M( -n) •

·Thus, for every direction n, there are typically three sound waves with different velocities and mutually orthogonal polarizations A1 , A2 , A3, corresponding to the three solutions of the eigenvalue problem. A direction for which two sound velo- cities are equal is called an acoustic axis, cf. [2].

In the case of no acoustic axes, i.e. no degeneracy, the triple vector field, (A_{1 ,} A_{2 ,} A3 ), with the A's chosen as unit vectors, and with some initial sign convention, gives a continuous (in fact analytic, or even algebraic) field as function of n.

Thus, we arrive at the notion of a continuous polarization frame field on the surface of the unit sphere

When acoustic axes are present, discontinuities in the polarization field will result.

sound propagation in triclinic crystals. However, some problems remain.

In [1] it was shown, by an example, that linear anisotropic media without acoustic axes are possible. Topologically it turns out that there are two classes of possible polarization fields agreeing with the symmetry (iii).In this paper a complete deriva- tion of the two classes will be given. Further, it will be shown that only one class occurs as possible solution for actual acoustic operators (ii).

The usual case includes acoustic axes. Alshits and Lothe [1] generalized the Khatkevich condition for presence of acoustic axes to an invariant form that covers all cases. However, the relation between these conditions and the discriminant of the eigenvalue problem (i) is not clear. This problem will also be

dea~with in the present paper.

In order to give a smooth presentation of the main points, we delay to § 6 some of the proofs. This section also contains summary and conclusion.

§

2. Basic Theory and Notation. Flags, Frames and Spinors.

In the following sections some elementary topological con- structions play a central role. A brief introduction before the actual analysis, may be in order.

Let S

=

Sym(3) be the vector space of real symmetric 3x3-matrices and ~ c S the subset of matrices with multiple eigenvalues. Then ~ is a codimension 2 algebraic variety, and S- ~ is a connected open subset of S (§

5).

To every symmetric matrix M in S is associated its eigenvalues A1 ^~ A2 ^~ A3

(always in this order) and its eigenspaces e1 , e 2 , e 3 (in the corresponding order). If M is in S- b., then e = (e1 ,e 2 ,e^{3 )} is an ordered orthogonal system of lines through the origin in

~3, or briefly a flag.

The set of all flags in ~³ will be denoted Fl(3). It can be parametrized as follows. Each flag line is an element of the projective plane p2

'

determined by a set of homogeneous coordi- nates. If A _{1. -}- (a ) A - (a ) and A₃ = (ai₃ .. ) are sets of

i l ' 2 - i2

homogeneous coordinates of a flag (e ,e ,e ), then by orthogona-

l 2 3

lity

(1)

I

a. a. = 0 •

l l l2

Thus Fl ( 3) is the subset of P 2 x P 2 x P 2 whose mul tihomogeneous coordinates (A₁,A 2 ,A 3 ) satisfy the conditions (1). In particu- lar Fl(3) is a projective algebraic variety.

The matrix A= (A₁,A 2 ,A3 ) whose columns are A₁ ,A 2 ,A3

is not uniquely determined by the flag e = (e₁,e 2 ,e3 ). However, if we require the Ai to be of unit length and positive orienta- tion (i.e. det A= +1), then there are only four choices of A.

More precisely, the canonical mapping p: S0(3) + Fl(3), which to a rotation matrix (or frame) associates the flag determined by its column vectors, is a 4-sheeted covering of Fl(3). To see this let Il ^E S0(3) be the matrix

r1 0 ^I

-~J

Il =

i~

^{-1 0}

and define I2' I3 similarly (by cyclic permutation of the diago- nal elements). Let I a = I be the unit matrix. Then

R

=

{I₀,I₁,I₂,I_{3 }} is an abelian subgroup of S0(3), and two matrices A,A' project onto the same flag if and only if

A'

=

Aik for some k. Thus Fl(3) can be identified with the homogeneous space S0(3)/R and p with the canonical projection S0(3) ~ S0(3)/R. Note that R is not a normal subgroup of S0(3), so that Fl(3) does not inherit a group structure.

Let p : Spin(3) ~ S0(3) be the connected double covering

of S0(3), realized by the group of unit quaternions Spin(3).

Recall that the quaternions are the four-dimensional system of hypercomplex numbers

q

=

a + bi ₁ + ci ₂ + di ₃

where a,b,c,d are real, and the pure quaternionic units ik satisfy the multiplication rules

=

^-1

( 2 )

...

We denote by

-

q the conjugate of q,

( 3) _q

- ₌

_{a - bi1 - ci2 - di3 ,}

and by lql its norm or absolute value,

( 4)

Linearly the quaternions form a 4-dimensional real vector space H with Euclidean metric (4). This is analogous to the complex numbers, which form a 2-dimensional space C with the same type of metric. Also the set of unit quaternions

s

^{3 c [} is a group under (quaternion) multiplication. This group is Spin(3).

its real and pure (or vector) part. The subset of pure quaternions (a

=

0) is a 3-dimensional linear subspace of

fi,

which will be identified with B.-^{3 -}_- ^R3 ₌ (b,c,d)" Note that q is pure if and only

The quaternions act linearly on B_3 by

(5)

^{v,... v'}

₌

_qvq

-

If q E 8pin(3), then lv'

I = I

^{vi and so (5)} is an orthogonal operation,

Moreover, path from The mapping

v' = A ( v).

q

is a rotation, since det to q on 83.

cr(q) = A q

Then det

A

=

^{1. (Let} q(t) be a q

Aq ( t) :: det A q ( O) = 1 ) •

is the required covering projection 8pin(3) + 80(3). Note that A ^-1.

and A- = q q

We shall need an explicit expression for A

q (as a matrix) in terms of the spin variables a,b,c,d of q. Up to common change of sign for

e

and r we may write

q

=

^cos

e

+ sin

e

r , r ² = -1 •

Then r gives the axis and 28 the angles of rotation for the orthogonal transformation Aq. We have

cos 8 = a ,

and r

=

b'i 1 + c'i 2 + d'i 3 with

. . .

Freezing r, we see that _{A q}

=

A

e

is simply the 1-parameter group in 80(3) with infinitesimal generator 288,

-d' 0 b'

c'}

-b I

0

Thus Aq

=

^{exp(26S). Since S}

=

- I , expansion in power series reduces to

A

=

I + sin2eS + (1-cos2e)s^{2 •} q

Substitution for e and b', c', d' now yields the expression

( 6) {

2(a²+b²)-1 A

=

2(bc+ad)

q

2(bd-ac)

2(bc-ad) 2(a²+c²)-1 2\cd+ab)

2(bd+ac) } 2(cd-ab) 2(a²+d²)-1 The formula shows that cr(ik)

=

cr(-ik)

=

Ik'

course that cr(1)

=

cr(-1)

=

^I.

k

=

^1,2,3, and of

Since Spin(3) = S³ is simply connected, pcr : Spin(3) + Fl(3) is the universal covering of Fl(3) and cr : Spin(3) +S0(3) the

universal covering of S0(3). We can therefore read off the funda- mental groups of Fl(3) and SO( 3) as the counterimages cr ^-1 R and cr ^-1 {I}. By (6) they are the quaternion group

( 7 )

and its subgroup Q₀

=

^{±1}, respectively. Thus nFl(3) ~ Q.

There is a simple relation between S-~ and Fl(3). Let s³ be the set of all triples of real numbers (A 1 ,A 2 ,A 3 ) with

A1 ^< A2 ^< A3 • This is an open contractible subset of a3.

Let a : S-~ + s³ x Fl(3) be the mapping ( 8 )

where A1 ^< A2 ^< A3 are the eigenvalues of M and e

=

(e 1 ,e 2 ,e 3 ) the corresponding flag of eigenspaces. This mapping is easily

seen to be a (birational) homeomorphism. Since s³ is contractible, the flag mapping a_{2 :} S-~ + Fl(3) is a homotopy equivalence,

( 9)

Consequently no homotopy invariant information is lost by passage from S-A to Fl(3).

§

3 Topologically Possible Polarization Fields.

We turn to the eigenvalue problem for an acoustic operator.

Let M

=

g(x) be a field of symmetric 3 x3-matrices on

antipodal invariant and with simple eigenvalues, i.e. a mapping g :

s2

+S-A such that g(-x)

=

g(x).

*)

For the present we assume only that g is continuous, and study what are then the possible eigenspaces. The further question of which of these eigenspaces are also possible eigenspaces for an actual acoustic operator with its specific analytic properties ((ii),in §1), will be treated in

§ 4.

From g we derive the flag field e

=

a₂(M)

=

a₂(g(x)), which to each x associates the flag of eigenspaces of M at x.

Evidently this is also antipodal invariant.

More generally we may consider arbitrary flag fields

f:

s2

+ Fl(3), not a priori derived from symmetric matrices. Since

s

² is simply connected, f lifts to S0(3) and even to Spin(3), i.e. there is a frame field f' and a spinor f" such that

f(x) = p(f'(x)) = p(cr(f"(x))), x E S^{2 •} Moreover, f' and f" are unique up to the actions of R and Q, respectively. I.e. besides f' the lifts to S0(3) are f ' • I

1 ' f' • I

2 and f' • I _{3 •} Similarly, besides ±f" the lifts to Spin(3) are +f"·l·

- 1 ' ±f"•i₂ and +f"·l· _{- 3 •} This much results from the general theory of covering spaces.

More specifically we get

*) From now on we denote by x rather than n the variable point on S2 •

Lemma 1. Any lift to S0(3) of an invariant flag field on

s

² is invariant.

Before proving this lemma, let us underline that it states a real property of the flag-field itself which is best recognized in its lift. Thus, although the physical eigenspaces form a flag- field (no particular sign of polarization), it is important to introduce lifts to bring out all properties of flag field. **)

The pro-of ·of lemm-ac·1 proceeds as follows. Consider the diagram of f and its lifts

( 1 0)

f ^I f ¹¹

'

From the identity f(-x) = f(x) follows p(f¹(-x)) = p(f¹(x)), i.e. f ¹(-x) = f' (x)Ik for some Ik in R at each point x. By the connectivity of S² a relation

( 11 ) f ' ( -x)

=

^{f 1 ( x)} Ik

then holds identically in x for a fixed k. We claim this requires k= 0, i.e. Ik is the identity matrix. In fact from

( 11) follows

( 12) f ¹¹(-x) = f ¹¹(x)q

**) This is also the reason why in ref [ 1 ] it was found conve- nient to assign a direction to polarization and to discuss in terms of vector fields of polarization rather than the physical flag

fields.

up in Spin(3), where Replacing x by -x or q2

=

1. Thus q This ends the proof.

q is one of the two spinors above Ik.

in this identity gives f"(x)

=

f"(x)q 2 is either 1 or ^-1 and so covers I _{0 •}

Consider the now invariant frame field f' and its spinor lift f". By (12) we have either

( 13) f"(-x)

=

^{f 11(x)}

or

(14) f"(-x)

= -

f"(x).

Thus we have still two possibilities: f" invariant or f"

equivariant. Of course f" is not the only lift of f', but the only other is -f", which is of the same variance as f". In fact all eight spinor lifts of the flag field f are of the same variance, as is immediate from the previous discussion.

In contrast to the situation in lemma 1 we cannot here rule out either possibility. Call f' twisted if it has equivariant spinor lifts and non-twisted if it has invariant lifts. Thus an invariant flag field has invariant frame lifts which may be twisted or non-twisted.

Because of lemma 1 emphasis will now be put on frame fields rather than flag fields. Consequently we un-prime our notation one notch and write f,g, ••• for frame fields and f' ,g', ••• for their spinor lifts.

Two invariant frame fields f,g on

s

² are invariantlx homotopic if there is a continuous one-parameter family of invari- ant frame fields H(x,t)

=

Ht(x), 0 ~ t ~ 1, such that H₀

=

f and H₁

=

g. We require H to be a continuous map of both vari- ables (x,t) E

s

^{2 x} [0,1 ]. Invariant homotopy is an equivalence relation.

Proposision 2. All twisted frame fields on S² are (mutually) invariantly homotopic, all non-twisted frame fields are invariantly homotopic, and no twisted frame field is in- variantly homotopic to a non-twisted.

Thus there are two prototypes of invariant frame fields on S² (up to invariant deformation). Clearly the constant field ( 1 5) f(x) - I

represents the non-twisted type. At the end of this section we give a representative of twisted type, thus exhibiting both cases.

For proof of the second claim in proposition 2 it suffices to show that any invariant mapping f ^{1 :} S² + Spin(3) is invariantly homotopic to the constant 1 (i.e. the mapping with value 1).

This also implies the first claim, because if f ¹ ,g^{1 :} S² + Spin(3) are equivariant, then ^{f I • g I -1} is invariant and by the second claim homotopic to

connecting f ' . g' ^-1 1 .

to

But if H' _t

1 ' then homotopy connecting f' to g'.

is an invariant homotopy K' _t

=

^H'•g_t ¹ is an equivariant Projecting down to S0(3) we see that the twisted invariant fields f and g are connected by a (twisted) invariant homotopy Kt.

Consider the diagram

Spin(3)

( 16 )

y...,

⁺

s

² ___[_>

s

^{0 (} 3 )

II ;1 p2

7

of consistent mappings f,f' and f" and double coverings. It is immediate that f is untwisted if and only if f" lifts to

Spin(3). But any mapping f"' : P² ~ Spin(3) is homotopic to 1 since Spin( 3) =

s

³ is 2-connected. And any homotopic _{H "'}

t from f "' to 1 gives an invariant homotopy H'

t ^{from f' to 1 .} Finally, if f and g are invariantly homotopic on

s

^{2 ,} then f" and g" are homotopic on P2 • By the homotopy lifting pro- perty of covering spaces, either both f" and g" lift to Spin(3) or neither lift. Thus f and g are of the same type.

Proposition 2 may be viewed as a statement about the set [P²,S0(3)] of (ordinary) homotopy classes of frame fields on p2

'

namely that this consists of two elements. It is easily seen that restriction to "equator" P¹ c: P² sets up a bijective correspon- dence [P²,S0(3)] ~ [P¹,S0(3)]. (Replace S² ~ P² with S¹ ~ P¹ in the proof in§ 6). Thus restriction to equator already deter- mines the spin type of a field f. We use this for constructing

an invariant frame field of twisted spin type on

s

^{2 •}

We need a diagram of consistent mappings s¹ ~> f so(3)

P! ^fy/'

p l /

such that f" is not homotopic to a constant. We may identify P¹ with a copy of

s

¹ and parametrize it by its angular argument

v (0 < v ~ 2n). Then the mapping

( 17) f "(") v

= f

-s1n ^{co. s v ,} v, ^sin cos ^{v ,} v ,

^o

⁰

j

0 ' 0 '

is homotopically non-trivial, as is well known. Now if v

=

^p(~)

is the double covering of ^pl

'

^{then p} is multiplication by 2, v

=

2q>.

Substitution in (17 gives the twisted invariant frame field f f(lP)

=

f"(2tp) on S^{1 •} As it is homotopic to a constant (it factorizes through p), it can be extended over the upper half sphere S~ and by invariance over the rest of the sphere S 2 . The result is a twisted invariant frame field on S2 . Such an

extension, in spherical coordinates lJ),l)J (0 ~ lP ~ 2TI ,-TI/2 ~ 1jJ ~ TI/2) is given by

( 18) f(tp, 1jJ)

=

T(tp)S(21jJ)T(tp)

where

ros

^{!.P, sin lj),}

_~}

^{! r1}

^'

⁰

^' _s~ntJ

T(tp)

=

-s~n <.p, cos lj), S(l)J)

=

1~

^{cos 1jJ}

^'

'

⁰ , -sin 1jJ , cos 1jJ

For 1jJ

=

0 we get f(tp,O)

=

f(<.p), so that f(<.p,ljJ) is in fact an extension. Moreover, f(<.p,TI/2)

=

f(lJ),-TI/2)

=

^I _' ^{so that f}

is well defined on the sphere. Finally f(<.p+TI,-l)J)

=

f(lP,l)J), showing that f is invariant.

§

4.

The polarization Field of Anisotropic Media without Acoustic Axes.

When no acoustic axes are present, the polarization field

must belong to either the non-twisted or the twisted type, described in the preceeding section.

The constant field (15),the prototype of the non-twisted field, can be thought of as derived from a field of symmetric matrices M

=

^g(x) of the form

gll (x) 0 0 0 g22( x) 0

0 0 g. (x)

33

Acoustic operators close to this diagonal form are possible. In fact Alshits and Lathe [1] have recently shown that a stable orthorombic material without acoustic axis is possible, and this gives an example of non-twisted polarization.

The question remains whether cases of no acoustic axis and twisted polarization field may exist.

The acoustic operator is of the form g .. _{l ]}

= r:

_{k 1} B .. xk x 1 kl _{l ]}

with

'

kl lk lk

B .. = B .. = B . . .

lJ J l l]

(See eq. (ii), §1) In terms of the elastic coefficients

B~~ = ~(C.k

l.+C.l k.) lJ l ' J l ' J

We shall be able to prove the following theorem.

Theorem 3. Let M

=

g(x) be a field of symmetric matrices on with simple eigenvalues and entries which are quadratic forms. Let A

=

f(x) be a frame field of eigenvectors for g . Then f lS non-twisted.

Thus, the conclusion lS that cases of no acoustic axes and twisted polarization field cannot occur.

lift q when a ln the

For a proof of Theorem 3, several steps are required.

Consider an

=

^{f' ( x)}

column of particular

invariant Thus A

=

A ( i . e . a coordinate

0 0 z

frame field A

=

^{f(x) and its}

A ln the notation of ^§ 2. ~!Je

q

unit eigenvector of M

=

^{g(x) )}

direction

spinor ask

lies

In terms of the spln variables a,b,c,d the condition is that some one of the six equations

2(bd-ac) = ±1

( 1 9) 2(cd+ab) = ±1

2 (a ²+d²)-1 = ± 1

must be satisfied. This follows from the expression (6).

Applying the identity a²+b²+c²+d²

=

¹ the equations (18) appear in the form

b-d

=

^a+c

=

^{0 b+d}

=

^a-c

=

⁰

( 2 0 ) c-d

=

^a-b

=

^{0 c+d}

=

^a+b

=

⁰

b

=

^c

=

^{0 a}

=

^d

=

⁰

Given linear forms h,l, ... ln the variables a,b,c,d we denote by

s

h' l ' . . . the intersection of Spin(3)

= s

³ with the linear subspace h = l = • • • = 0 ln H Clearly z-polarization occurs whenever the spinor lift f' hits one of the six circles

s

b-d,a+c ' b+d,a-c '

s ' s

_a, ^d From now on we assume f and therefore f' smooth.

Lemma 4. Suppose f' lS equivariant. Then the solution set to any equations h(f'(x))

=

l(f'(x)

=

0 is non-empty.

We defer the proof to §6.

Applying lemma 4 to the pairs of linear forms occuring ln (20) gives six different pairs of antipodal points on where z-polarization takes place when f:S ^{2 -+} 80(3) lS a twisted frame field. On the other hand it follows from the eigenvector equation

MA. =/....A.

l l l (1<i<3)

that M13

=

_M23

=

0 whenever A.

l falls in the z-direction.

Thus the quadratic forms g 13 (x) and g 23 (x) are simultanously zero along 6 different directions. This is impossible unless

g 13 and g 23 are linearly dependent, which means that the matrix field g:S 2 ⁺ S-~ maps into a proper linear subspace of S

=

Sym(31

Call the matrix field g special if it maps into a proper linear subspace of S and ordinary otherwise. We have just proved that an ordinary invariant matrix field g with entries which are quadratic forms cannot have a twisted eigenvector frame field f .

with B

=

gl Bt = flag

Suppose g is special. We have g .. (x) =

l ]

L B .. xk x 1 kl k,l l ]

( 1 ~ i ' J < 3)

B~~ _l]

=

^B~~ _l]

=

^B~~ _Jl

.

We may perturb the set of constants

{B~~} l] to a set Bl such that the corresponding matrix field lS ordinary, and such that the homotopy gt with constants

(1-t)B +tB 1 maps into ^s-~

.

Then a2gt lS an invariant field homotopy from a2g to a2gl

'

^{where a2} is the flag mapplng ( 9 ) . Since a2g lifts to f the whole homotopy lifts to an invariant homotopy ft from f to some f 1 covering

Since g 1 is ordinary, f 1 is non-twisted and therefore so is f (proposition 2). This completes the proof of theorem 3.

§5. The Criterion for Presence of Acoustic Axis

The case of no acoustic axis discussed in the preceding, lS exceptional. Usually acoustic axes are present.

In the mathematical model the inclusion of acoustic axes means that there are directions x where the matrix M

=

^g(x)

has multiple eigenvalues, i.e. the mapping g into S

=

Sym(3) will meet the discriminant variety ~ . The structure of ~

therefore plays an important role in the study of g or the

polarization frame fields of g ' as shown ln ^{[ 1} J • Below we glve an approach to ^/::, different from [ 1], which better displays its geometry, and we settle some points left open ln [ 1 J •

Let M

=

^{{M .. }} be the general symmetric 3 x 3-matrix. It

l ]

will be convenient to denote the SlX free parameters of M by Mil = ul M23 = M3 2 = Vl

M22 = u2 M3l = Ml3 = v2 M3 3 = u3 M12 = M21 = v3

This permits us to identify M with the six-vector (u,v) = (ul ,u2 ,u3 ,vi ,v2 ,v3) and

s

itself with ^R6 = ^R6

- =u,v

The subset ^/::, c R6 of matrices with multiple eigenvalues lS glven by a homogenous polynomial equation (27), and so i t lS a conical affine variety. It contains as subset the line r of matrices with triple eigenvalues (all matrices AI, A Eli).

A real number A is eigenvalue for M if the matrix M- A I has rank < 3 or, equivalently, if its determinant D

=

^D(A,M)

vanishes. Recall that

( 21 ) D -

where c.

=

^c.(M) are the characteristic invariants of M,

l l

cl - ul +u2 +u3

(22) 2 2 . 2

c 2 - Cu 2u3-v 1 ) + Cu3u 1 -v2) + (ul u2 -v3)

2 2 2

c 3 - u1Cu2urv1) +u2Cu3u1-v2) +u3Cu1u2-v3) + 2Cv1v2vru1u2u3) Since c 1 , c 2 , c 3 are homogenous polynomials of degree ^{1 ,} 2 , 3 , respectively, D is a cubic form in the seven variables

Let

s

be the eigenvalue variety in R xR⁶ =R⁷

=A. =u, v =A. , u, v defined by the equation ' D(A.,M)

=

0 and ~ the subset defined by D(A.,M)

=

^DA.(A.,M)

=

^{0 Thus (A.,M) lS ln S when A. is an}

eigenvalue of M and in ~ when A. lS a multiple eigenvalue.

In particular ~ contains as subset the line r of pairs (A.,A.I), A. E R • Let ~=S + =u,v

R

be the restriction to

s

of the linear projection R X R⁶ + R⁶

=A. =u,v =u,v

( 2 3 ) ~(A,M) = M

The eigenvalue variety S has a physical interpretation as well as a geometric. Geometrically S is a 3-sheeted ramified coverlng space of with ramification locus The pro- jection ~ sends ~ to ~ and r to r , and lS two-to-one

on 15. - r and one-to-one on r . It should be noticed that ~

disconnects S In fact S -~ is a union of three disjoint

"' "'

open sets OI, 0_{2 ,} 0 3 ln S, each of which projects isomorphically onto R ⁶ - ~ by ~ .

=u,v The inverse of ~10. lS cr.(M) = (A..(M),M)

l l l

"'

If we take the closure of 0.

l ln S , we obtain the sheet

s . .

l

Since cr. _l is defined and continous on all of _=u,v R^{6 "'} S. = a. R ⁶ l l=u,v

"'

Thus S = S I U S ₂ U S _{3 ,} ~ = (

S

^I

n S

_{2 )} ^{U (}

S

2

n S

_{3 )} and r =

s

^I n

s

2 n

s

3 •

The three sets S , ~ and r are conical affine varieties in R⁷ given by homogenous polynomials in the variables A.,u_{1 , •••} ,v_{3 •}

Physically

S

may be considered the "universal slowness cone"

for elastic wave propagation. The ordinary slowness cone of any particular elasticity problem M

=

^g(x) lS just the counterimage

in

(24)

by the mapping

g -

{A=

^T2

M= g(x) ,

R7 =A.,u,v

i.e. the solution set to D(T 2 ,g(x))= 0. The cut between s~

g and ^T = 1 in R⁴ 1s the usual slowness surface.

=T,X

From the discussion below follows that ^~ S , !J. and ^{~ ~} r are subvarieties of _g_⁷ of codimension 1 , 3 and 6 , respectively,

,... ~ ,...

and with singularity s·ets singS = ^{:, and sing t:, = r

.

Consequently,

"' ^{,... ,... ~}

when g 18 transvers·e to

s

(i.e. to the smooth strata S-!J.

"'

LX-r

,... ^and "'

r

₎

_'

i t hits the ramification locus ^~ LX 1n a finite

'

number o~ directions in R⁴ only, and the higher ramification locus r not at all. Thus, in the transverse case we have:

(a) The slowness cone

S-g

is a 3-dimensional conical variety in with a finite number of acoustic axes (possibly none), which form the singularity set of ^s~.

g

(b) At an accoustic axis only double degeneracy occurs.

(c) The local normal structure of an acoustic axis in ^s~

g (outside the origin) is isomorphic to the local normal structure

,...

of !J. 1n S along any direction different from r .

Furthermore (a), (b), (c) presents the stable situation for an anisotropic crystal, since transversality is persistent under small perturbations.

In view of property (c) i t is interesting to explicitize the

~ ,... ~

normal space to !J. in S locally at a point 1n !J.-r .

Proposition 6. At any point of ^~ !J. not in r there 1s an

R

⁷ such that

=~,u,v

~ l"oJ f"t,J l"oJ

( s n B , !J. n B) is ana- open neighbourhood B 1n

lytically isomorphic to ⁽ C x

B ,

⁴ ox

! ) ,

⁴ where C c R ³ is an elliptic cone through the origin o .

Thus, let HcR⁶ be an affine plane meeting !J. transversally at M say' (not 1n r ) and let ^,... H

=

^{1r -1} H be the l i f t to

~

-

This implies that H meets ~ transversally at (~,M) and S transversally at (v,M) , where v is the simple and ~ the double eigenvalu~ of M . Then locally at (~,M) H meets ~

1n (~,M) only and S 1n a small cone with vertex (~,M) , and

~

-

locally at (v,M) H meets S in a small disk.

(~M)

•

• M

The proof of proposition 6 is deferred to § 6.

"' "'

Explicitly· .. · the equations for

s

^{and !J. are}

( 2 5) -A 3 + c A 2 -c2A +c3 ₁

=

⁰

and

-A 3 +clA2 -c2A + c3

=

⁰

( 2 6)

-3A 2 +2c₁A - c2

=

⁰

Elimination of A ln ( 2 6) g1ves (27)

which lS a defining equation for !J. • Being homogenous (of degree 6) ln the variables u_{1 , •••} ,v 3 it defines a conical affine variety 1n as claimed. The polynomial d

=

^d(M)

is of course the discriminant of the characteristic polynomial (23).

The discriminant equation (27) does not reveal much of the geometry of !J. , nor does (25) or (26) tell much about !J. or S To gain better insight we proceed as follows. For an arbitrary member (A,M) of S we have a unique decomposition

O,M) = (O,Ma) + O,A.I)

with Ma = M - A I . Moreover A. lS a double (triple) eigenvalue for M if and only i f 0 lS a double (triple) eigenvalue for Ma Thus we have a direct sum decomposition

"'

s

= _{Sa +}

r

( 2 8)

!J. = _{!J.a +}

r

"' "'

where Sa ^{::> !J.} a is the intersection of

s

^{::> !J.} with the hyperplane I.

=

^{0 1n} Identifying this with we have

(the set of symmetric matrices with 0 as eigenvalue) and

'K

a = !J. a

"' "'

( o o o with 0 as multiple eigenvalue). In other words S ::> !J. ::> r is

the skew cylinder over S 0 ^{:::> t:,} 0 => { 0} ln the direction (1, 1,1,1,0,0,0) ln

To obtain equations for S 0 and ^t:,0 we set A

=

0 in (25) and (26), getting

( 2 9) (for s 0 )

(30)

=

^{0 (for}

The second set of equations can be replaced by homogenous equations of 2.degree only. For the matrix M belongs to _/:,0 if and only if i t is of rank ^{< 1}

'

which means that all 2.order determinants -

vanish. Thus let c .. =

l] c .. (u,v) be the

l ] 2.order determinants of

e l l - u2u3 - v2 ₁ c22 ^- 2 2

u3ul - v 2 c33 ^- ulu2 - v 3 c23 ^- v2v3 - ul v 1 c 3 1 - v3vl - u2 v 2 c12 ^- vlv2 - u3 v 3 Then ( 3 0) lS equivalent to

( 31 )

We may subdivide s 0 into s 0e and s 0h given by

c 3 = 0, c 2 ;::: ⁰ and c 3 = 0, c 2 ^{< 0 ,} respectively (the elliptic and the hyperbolic part of S 0 ) .

Since c 2 is a quadratic form in the variables ui,vj , the semialgebraic sets S 0e and S 0h are again conical.

Further subdivision by means of the remainig invariant c 1 (c 1 ;::: ^0, c 1

s.

⁰⁾ only gives splitting into positive and negative half-cones /:,0

=

^{/:,0+ u /:,o- ' soe}

=

^{soe+ u} soe- ' ... E.g.

is the set of symmetric matrices with at most one non-zero eigen- value and this positive.

of the points in then (31) gives well known

M,

equations for the Veronese surface V2 c p⁵ The Veronese surface is the image of P² by an imbedding P² ~ P⁵ obtained as follows.

A symmetric matrix M = (u,v) has rank 1 precisely if M= ±t*t

for some non-zero ln This yields

u1 = +t2 - 1 u2 = +t2 - 2 (32)

u3 = +t2 - 3 v1 = ±t2 t3 v2 = ±t3 t1 v3 = ±t 1 t2

which lS a parametrization of ~0 ^• More precisely the sign + gives a parametrization of ~O+ and the sign If we regard (t 1 ,t 2 ,t 3 ) as homogenous coordinates of a point ln P² and (u 1 ,u 2 ,u 3 ,v 1 ,v 2 ,v 3 ) as homogenous coordinates of a point

in P5 , then the signs are irrelevant and (32) defines the imbed-

ding of P² ln called the Veronese surface. Elimination of the parameters t 1 ,t 2 ,t 3 glves back the equation (31). Thus ~0 is simply the affine cone ln R

=u,v over the imbedded copy

Similarly one can glve parametrizations of S₀e and soh ' from which i t follows that _{· oe}

s

and are affine cones over projective semialgebraic sets homeomorphic to S⁴ and

P4 , respectively. Finally s₀

a projective variety

is the affine cone ln R⁶

=u,v over given by the single determinant equation (29). It is easily checked that

w

⁴ lS a projective hypersurfade in P⁵ w~th singular locus V2 .

It follows that _~0 is a 3-dimensional cone ln R⁶ which

=U,V

lS smooth (i.e. a manifold) except the origin, and that S₀ lS a 5-dimensional cone which is smooth except along ^~₀ ^. And so ~

is a 4-dimensional cone in ^R7

=A.,u,v which is smooth except along ,...

the generatrice r , and S is 6-dimensional and smooth except along

'K .

Since

n:S

⁺ R⁶ is a finite algebraic map, the

u,v

projection !J. =

n'K

ls also 4-dimensional. Thus

'K

and t:,. are of codimension ² ln

S

^and _~u,v ^R6

From (32) and (28) we get the parametrization (A., t) ^... (A.,u,v) of ^{,... /::,.}

'

^where

ul = ±tl+A. 2 Uz = ±t~+A.

u3 = ±t~+A.

( 3 3)

vl = ±tzt3 Vz = ±t3tl v3 = ±tlt2

and (33), as i t stands, is a parametrization of t:,.

=

t:,.+ U!J.

Thus we have the members of t:,. (or

'K )

parametrized by their unique multiple eigenvalue and their one unique eigendirection, i.e. to (A.,t₁,t 2 ,t 3 ) corresponds the matrix M

=

(u,v) with multiple eigenvalue A. and corresponding eigenspace x .L t ,

(and the remalnlng eigenvalue either >A.

or <A. depending on the slgn in (33)).

From (33) we obtain

describing the matrices of t:,. with no off diagonal zeroes. To capture the rest of ^t:,. we consider the corresponding expressions for the higher order terms ( uk- u . )( u . - u . )

l l . J (three equations) and

(u2-u3)(u3-ul)(ul-u2) form they are

(one equation). Written out in homogenous

(u2-u3)v 2v 3 2 2 + vl (v2-v3)

=

0 (34a) Cu3-u1)v3v 1 2 2

+v2Cv3-v1)

=

0 (u 1-u 2 )v 1v 2 2 2

+v3(vl-v2)

=

⁰

(u 3-ul) (ul -u2 )vl + (u 1-u 2 )v 2v3 2 2

0 + v 1 ( v 1 -v 2)

=

(34b) (u 1-u 2 )(u 2 -u 3 )v 2 + (u2-u3 )v3v1 2 2 + v 2 ( v 2 -v 3 )

=

0 (u 2-u3)(u 3-ul)v3 + (u3-ul )vl v2 + v3 v3-vl) ( 2 2

=

⁰

(34c)

Apart from linear rearrangement these are the seven conditions put up by Alshits and Lothe as an invariant generalization of the two Khatkevich conditions. Locally at any particular M at most two of these are independant, the others being redundant or col- lapsed to the trivial identity 0

=

⁰

set of matrices with at least one v.

l

E.g. let V1 be the sub- zero. Then (34b), (34c) and one of (34a) are redundant on ~-V1 , and one is left with the two Khatkevich conditions. On the other hand (34a) collapses on V1 , and then the remaining conditions are important, discussed as special cases by Khatkevich.

To exhibit the structure of ~ completely (as an algebraic variety) the seven conditions (34) are necessary, as well as sufficient. We shall make this claim precise.

For this purpose i t is convenient to symmetrize somewhat the expressions in (34b). This can be done by a simple linear rear- rangement. Consider

Cu 2 -u3)v2v3 2 2 Rl - +v 1 (v2-v3)

=

0 R2 Cu3-u1)v3v1 2 2

(35a) ^- +v2(v3-vl)

=

0

R3 Cu 1-u 2 )v 1v 2 2 2

- +v 3 Cvl-v2)

=

0

(35b) Rs ^- R6 ^- (35c) R7 ^-

Then (35) is equivalent to (34), since (35a) = (34a), (35b) = (34a) +2(34b), and (35c) = (34c).

Let P = R[u,v] be the polynomial algebra in the slx

The set of polynomials vanishing on ~ form an ideal I = I(~) ln P , which is in fact generated by R1 ,R2, ... ,R7 .

Theorem 5. Every polynomial Q vanishing on ~ can be

written Q=q 1R1 +•••+q 7R7 forsuitable q 1 , ... ,q 7 ln R[u,v].

The proof is deferred to §6.

For any nonnegative integer n , let P be the real vector n

space of polynomials homogenous of degree n . Then p

=

:L p

n:::o n (direct sum), and for multiplication we have P m" P n c P m+n . This splitting is inherited on the ideal I so that I = :L I and

n:::o n P •I m n c I m n + By theorem 5 there are no non-zero polynomials vanishing on ~ below degree 3 . For F = q 1 R1 + • • • + ^q7R7 homo- genous of degree 3 the coefficients q. must be constants.

l

Hence R1 , ... ,R 7 form a set of generators for the real vector space I _{3 •} As they are linearly independant (over

R),

they even

b . . I 3 ~ R7

glve a asls, l.e.

Corollary 6. R1 , ... ,R 7 lS a linear basis for ^I3 ^(~) and a minimal generator s~t fo~ the full ideal I(~) .

It should perhaps be mentioned that theorem 5 and corollary 6 holds equally well over the complex numbers, i.e. with f[u,v] ln stead of R[u,v] and in stead of t:, , provided ~:,c c

c

⁶

= =u,v is defined by the same seven equations (35). Over the complex numbers, however, (35) lS no longer equivalent to (27). The equi- valence between (35) and (27) is a real phenomenon and is explained ln proposition 8 below.

The natural operation of the rotation group S0(3) on Sym( 3) = R ⁶

=u,v

M ,.,. M' _(M'

=

CMC*)

can be extended to P

=

^{B[u,v] by}

F ,.,. F' (F' (M) = F(M'))

This operation is linear and multiplicative, preserves degree and leaves I(t:,) invariant. In this way we get a sequence of repre-

of SO ( 3) . an inner product in P such that ^p = l:P

n

Moreover we can choose is an orthogonal decom- position and S0(3) operates orthogonally on P .

Corollary 7. I ₃(t:,) is the irreducible ?-dimensional repre- sentation of SO ( 3) .

Since we know that R_{1 , •••} ,R₇ span the invariant space I ₃(t:,), it suffices to check on the 1-parameter subgroup S0(2) c S0(3) describing rotations around the z-axls ln By standard methods we find a set of unitary eigenfunctions

This shows that I ₃(t:,) lS the unlque ?-dimensional irreducible orthogonal representation of 80(3) .

Since the discriminant variety ~ lS g1ven by the equations R1 = • • • = R7 = 0 and also by the single equation d = 0 ( 27), one suspects a simple relation between d and R1 , ... ,R 7 Of course, by theorem 7 for sui table q. , homogenous of

l

degree 3. But in addition d is nonnegative and vanishes only when all R. vanish.

l

Proposition 8. The discriminant d lS a positive definite quadratic form in the polynomials R1 , ... ,R 7 ,

(36) d = l: a .. R. ² + 2 l: a .. R.R.

l l l l i<j lJ l J

The following coefficients are uniquely determined a44 = ass = a66 = a77 = 1

(37)

a47 = as? = a67 = 0

.

The remalnlng are only determined up to relations 1 /3 a23 = az6 = as 6 ^~ - a3 s = -l/4al7 1/3 a1 3 = a3 4 = a46 = - a1 6 = - 1/4a27 1 /3 a12 = a1s = a4s = -a24 = - 1 /4 a 3 7 (38) a14 + az s + a3 6 = 0

a 11 - 2az s + 2a3 6 = 1 5 azz· - 2a3 6 + 2a14 = 1 5 a33 - 2a14 + 2az s = 1 5

However, only one of th~ expressions (36) lS invariant under rotation:

( 3 9 ) d

= 15R~

⁺

15R~

^{+ 15R; +}

R~

^{+ • • •}

Proposition 8 (once i t lS stated) can be proved by comparing coefficients in (36), using (35) and (27). The condition

ad/ae = 0 (cf. sec 1) singles out the unlque diagonal form (37).

Since this is positive definite all the expressions (36) are posi- tive definite. We give no further details.

By (38) there are precisely 5 degrees of freedom in the expression for d . This gives rise to interesting relations between the Ri's. Namely set a 23 = w1 , a 13 = w2 , a 12 = w3 ,

with

Then (36) takes for form

P 1 = 3R 2R3 + R2R6 - R3R5 + R5R6 - 4R 1R7 P2 = 3R3Rl + R3R4 - R1R6 + R5R4 - 4R2R7 (40) P 3 = 3R1R2 +R 1R5 -R 2R4 +R 4R5 -4R 3R7

P 4 = 3Rt - 3R~ - 2R 3R6 + R1R4 + R2R5

P5 = 3Rf- 3R~ + 2R 2R5 - R3R6 - R1R4

But by (39) w1P1 + • • • +w₅P₅ must be zero for all values of This means that all P. vanish identically.

l

Corollary 9. The generators R1 , ... ,R 7 satisfy the quadratic relations P 1 = • • • = P₅ = 0 .

It can be checked that P_{1 , •••} ,P₅ span an invariant subspace of I 6 ( ^{11) ,} the 5-dimens ional irreducible representation of SO ( 3)

An in~eresting consequence of corollary 9 is that not all equations R1 = ••• =R 7 = 0 are needed to define ¹¹ set theoreti- cally. Requiring the four conditions

one deduces from corollary 9 and the expressions (40) that Thus (41) is sufficient to give

The number of equations cannot be further reduced and s t i l l glve all of 6. • On the other hand i t should be borne in mind that while (41) and even (27) glve 6., neither can give the full set of polynomials vanishing on 6. • The "paradox" disappear i f we pass to the complexification

c

⁶ since here the equations (27),

=u,v ' (35) and (41) are no longer equivalent.

§ 6. Delayed Proofs. Summary and Conclusion

We return to lemma 4. Consider again the frame field

f:S 2 + S0(3) and its spin l i f t f' :S 2 + Spin(3) . For the proof we need

Lemma 10. Suppose f' is equivariant and transverse to sh Then the solution set to the equation h(f'(x))

=

0 consists of an odd number of simple closed smooth curves in S2 , of which pre- cisely one is antipodal invariant.

The solution set Shf' lS the counterimage and trans- versality (which means that the gradient of hf' never vanishes on shf' ) ensures that shf' lS either empty or a finite number of simple closed curves ( [ 3] P· ^{208). Since hf'} lS equivariant, i t must be 0 somewhere on s2

'

^{so shf'} is not empty. The set is stable under antipodal involution on S 2 , and therefore so lS the complement s ² - shf' This means the complement con- sists of a number of antipodal connected open zones on which hf' has constant sign. Since hf' is equivariant, there is a negative zone for each positive and vice versa. Thus there is an even

number of zones and an odd number of curves. But then at least one curve C is mapped to itself by the antipodal mapping. Its

complement S2 - C consists of two antipodal connected sets 0+ , 0 and therefore the remaining curves must lie either in 0+ or ln 0 But evidently no curve in 0 + or 0

invariant. This proves lemma 10.

can be antipodal

To prove lemma 4 we have to show that the equations h(f'(x)) =l(f'(x)) = 0 have a common solution when f'

variant. Suppose there is no solution. Then a suitable small perturbation of f will yield a twisted invariant frame field f ₁ with a spinor lift f'

1 transversal to h(fi(x)) = l(f~(x)) = 0 has no solution.

because by lemma 10 the sets and

and such that But this ^lS impossible,

contain antipodal invariant curves C and D , which necessarily meet in antipodal points x+ , x_ . Clearly x+ ,x_ are solutions to the perturbed equation. The contradiction proves lemma 4. The perturbation to f ₁ is best performed at the level of f" (cf. diagram (16)).

First project sh,l to submanifolds ph 'pl and Ph ₁ ' ln SL(3) . Then perturb f" into a homotopic mapping f"

1 which

lS transverse to Ph and P₁ and avoids This defines a twisted invariant mapping f ₁ on and an invariant homotopy from f to f ₁

defining f i .

Since f lifts, so does the homotopy, thereby

Next we turn to proposition 6. To prove proposition 6 consider the action of the rotation group S0(3) on S

=

R ⁶ ,

=u,v

M ... CMC* C E S0(3)

The orbit of M has a tangent space TM at M , which as linear subspace of S lS simply [o(3),M]

TM = [o(3),M].

[o(3),M] is the set of all Lie products [L,M] with L 1n o(3).

Define an inner product on S by <A,B> = trace(AB) (This gives a metric on S

=

=u,v

R

⁶ equivalent to the euclidean metric.) With respect to this inner product the orthogonal complement NM to TM lS seen to be the centralizer to M 1n S ,

NM = {A € sIAM = ^{MA} •}

Suppose M has eigenvalues A.₁ = A.₂ = ~, A.₃ = v (~<v). Since the orbit of M pass through the diagonalization of M , we may as well suppose M is diagonalized,

M =

{~

^:

~

^:

~ l

o,o,vj

The general matrix of the centralizer is then

{

uv 1 ' v ' ⁰₀ ^} A = , u_{2 ,}

0 , 0 , u₃

hence NA lS 4-dimensional (and the orbit 2-dimensional). By adding the normal field N

=

^N(u1,u₂ ,u₃,v) to M we get an ex- pression for the matrices M' in a normal slice to the orbit de- pending on four parameters (the minimal number),

We use this to determine the local structure of

S

^at (~,M) . Since (S,~) is a cylinder over Cs0 ^,~

0

^), we may consen- trate on the particular case

ture of

s

0 at (O,M) , i.e.

~

=

⁰ and determine the local struc- at M. The general matrix

I

I

from the normal slice (in S ) now looks like

The intersection of S₀ with the normal slice 1s clearly given by the equation c₃(M')

=

^{0, or}

For u

=

(u₁,u₂,u_{3 )} small, this 1s equivalent to

which exhibits an elliptic cone C in E~ and a cylinder on that cone,

c

^{X R} 1n R³ X R

=u =v Thus the trace of 1n a

=v

normal slice to the orbit is analytically of the form Cx R, and so

s

0 is, locally at M of the form C x

E

³ (the orbit being 2-dimensional). Thus S is, locally at (O,M) , of the form

The singularity stratum o

xE

⁴ corresponds to ~ and

shows that ~ is smooth locally around M ~ as already established.

Finally we turn to the proof of theorem 5. Consider a poly-

and let F = F(O) + • •• + F(r) be its decomposition in homogenous components.

It is easily verified that F vanishes on ~ if and only if all F(i) vanishes. Therefore it suffices to prove the theorem for homogenous polynomials Q . We proceed by induction. Obviously the claim holds for polynomials of degree 0 . Assume it has been verified for homogenous polynomials of degree :k-1 , and let

Q

=

Q(u,v) be homogenous of degree k > 1 .

We make a linear change of coordinates 1n More pre- cisely change the u-coordinates to

-I

and leave the v. unchanged. Also, for convenience, introduce

l

w 3 = - ( w l +w 2 ) = u l - u 2 . In the new coordinates the equations.

(35) take the form R.(w,v) = 0, and the lack of the

l z-coordinate

shows that ~ ^c R6 is a cylinder over

""'Z 'w' ^{v ~} n R⁵ along the

=w,v z-axis. Expanding Q = Q(z,w,v) in powers of z ,

Q

=

Q 0 + Q l z + ^{0 0 0} + Qk z k where Q. = Q. (w,v)

l l does not depend on z , i t follows that Q.

l

vanishes on ~ for 0 < i < k . By our induction assumption Q_{1 , •••} ,Qk can be expressed in terms of R_{1 , •••} ,R 7 .

to see that Q₀ can be so expressed.

It remains

When vl =v2 =v3 = Thus Q₀(w,O) vanishes Q₀(w,O) = w1w2w3h1 (w)

(with new f. ) .

l

0 on for

Rl, ... ,R6 collapse and R7 = wlw2w3 {w₁w2w3 = ^0} ln R2

=w ' showing that some polynomial hl It follows that

collapse and

R7 = (w1w2+v 3 )w 3 . Thus v 3f 3 (w,O,O,v 3 ) vanishes on {w

1

^w2 ^{-v~ = 0}}

ln R=w,v 3 3 and we would like to conclude that f 3 (w,O,O,v 3 ) =

2 2

(w₁w₂+v 3 )h(w,v 3 ) for some h. In fact w₁w₂ +v 3 = ⁰ defines a non-degenerate quadric ~' ln R

=w,v and i t is easily seen that its vanishing ideal is the prime ideal I' generated by the

single polynomial is ln I'

slnce v₃f ₃(w,O,O,v_{3 )} lS and v₃ is not. Therefore f ₃(w,O,O,v_{3 )} is a multiple of R' and so v₃f ₃(w,O,O,v_{3 )} lS a multiple of v₃R'

=

~(R

6

^+viv

3

^+v~v

3

^+(w

1

^-w

2

^)v

1

^v

2

⁾

can be written

Similarly by taking restrictions to v _{3 -}- v₁ = 0 and v₂ = v₃ = 0 we find

so that altogether

This term is made up of monomials con-

taining or But by (35)

each of these are congruent (mod R_{1 , •••} ,R_{6 )} to a multiple of v₃ Thus we have

It follows that q vanish on ^~ . Since q lS homogenous of degree k-1 , by assumption it can be expressed in terms of the R. l Hence so can Q 0 . This completes the proof of theorem 5.

Surrirriary and Conclusion: Crystals without acoustic axes always have a nontwisted polarization field. The twisted type, although topologically possible, cannot be a solution for sound propagation ln anisotropic materials.

Usually crystals have acoustic axes. The relation between the Khathevich condition for acoustic axis and the discriminant of the

eigenvalue problem, has been el~cidated. Also the geometry of acoustic operator space, in the vicinity of acoustic axes, has been derived in a general way.

References

[1] V.I. Alshits and J. Lathe, "Elastic Waves in Triklinic Crystals", accepted for publication in Kristallografiya.

[2] F.I. Fedorov, "Theory of Elastic Waves in Crystals"

Plenum Press. New York, 1968.

[3] Guillemin and Pollack, "Differential Topology", Prentice-Hall, 1974.

[41 A.G. Khatkevich, "Acoustic Axes ln Crystals", Kristallografiya '}_, 742 (1962).