Inhomogeneous plane waves

Inhomogeneous Plane Waves

MICHAEL HAYES

Dedicated to J. L. Ericksen with gratitude and esteem on the occasion of his sixtieth birthday.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4I 2. The Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3. Bivectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4. Sums and Products of Bivectors . . . . . . . . . . . . . . . . . . . . . 51 5. Eigenbivectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6. Inhomogeneous Plane Waves Description . . . . . . . . . . . . . . . . . 62 7. Linear lsotropic Elastic Bodies . . . . . . . . . . . . . . . . . . . . . . 64 8. Linear Isotropic Viscoelastic Bodies . . . . . . . . . . . . . . . . . . . 6~ 9. Inhomogeneous Plane Waves in Linear Anisotropie Elastic Materials . . . . 72 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79'

w 1. Introduction

This paper deals with the propagation of elliptically polarised inhomogeneous t ime-harmonic plane waves. Such waves arise in many areas. Examples include Rayleigh, Love and Stoneley waves in classical linear isotropic elasticity theory, gravity waves in ideal fluids, TE and TM waves in electromagnetism, and viscoelastic waves. For the most part even though the applications given here are in the theory of isotropic and anisotropic elastic bodies, it should be apparent tha t the results have application in other areas, such as electromagnetism. The purpose of the paper is to show how the theory of complex vectors, or "bivectors" as Hamilton and Gibbs called them, may be used to give results on the polarisations of inhomogeneous plane waves. Also, the use of bivectors leads to a simple direct formulation of the eigenvalue problem for these waves.

The first part (w167 2-5) of this paper deals with bivectors. GIBBS [1] in his lecture notes gave an extensive development. Also, SYNGE [2], apparently unaware o f the work of GIBBS, derived further results and considered the eigenvalue problem for complex symmetric matrices. I have drawn extensively on these works o f GIBBS and SYNGE.

The link between bivectors and inhomogeneous plane waves is that the

42 M. HAYES

field, for example the displacement u or the electric field E, corresponding to a train of such waves is described in terms of two bivectors. Thus u = [A exp ico(S, x - - t)] +. Here o) is real and the amplitude bivector A determines the plane of polarisation. The slowness bivector S determines the planes of constant phase S + �9 x = constant, the planes of constant amplitude S - �9 x = constant, the phase slowness IS+ I and the attenuation factor IS- I.

GraBs showed that an ellipse may be associated with each bivector. If a bivector A = a + ib is given, where a and b are real vectors, then an ellipse may be drawn with centre at the common origin of a and b, and having a, b as a pair of conjugate semidiameters. In w 2 properties of conjugate diameters are recalled and a simple way of visualising the construction of the ellipse with given conjugate semi-diameters is sketched. Sums of bivectors are then considered (w 3) and proofs given of the basic result that if a' + ib' = ei~ + ib) where 0 is real, then a', b' is a pair of conjugate semi-diameters of the ellipse determined by a + ib.

Products of bivectors are defined in the usual way (w167 3, 4). It is seen for example that if the scalar product A �9 B of two bivectors is zero then the projection of the directional ellipse of one (A say) onto the plane of the directional ellipse of B is an ellipse whose major axis is parallel to the minor axis of the ellipse of B and whose aspect ratio is equal to that of the ellipse of B. Also, the plane of the ellipse of A may not be orthogonal to the plane of the ellipse of B. Further, if the vector product A A B of two bivectors is zero then A = o~B so that the ellipses of A and of B are similar and similarly situated.

The eigenvalue problem for complex symmetric 3 • 3 matrices Q is considered in w 5. It is proved that if Q possesses an isotropic eigenbivector A: Q A = 2 A ,

A . A = 0, then the secular equation det ( Q - 2 / ) = 0 has a double root. Conversely if the secular equation possesses a double root then Q possesses an isotropic eigenbivector. The significance of this lies in the fact that if the amplitude bivector A is isotropic (A. A = 0) then the corresponding elastic or electric field is circularly polarised.

The remainder of the paper (w167 6-9) is devoted to using these results on bivectors in the theory of elliptically polarised inhomogeneous plane waves.

The problem of how best to describe an inhomogeneous wave is considered in w 6. For homogeneous waves the field u (say) has the form u = [A exp i (k �9 x - - cot)] +, where k is the (real) wave-vector and the phase-speed is o9/[/e I. Typically, the direction of k is chosen and an eigenvalue problem solved to determine the phase speed and the corresponding amplitude A. For inhomogeneous waves, on the other hand, the directions of the normals to the planes of constant phase and to the planes of constant amplitude may not be chosen arbitra- rily. For example, for isotropic non-dissipative systems these planes must be at right angles to each other, whereas for isotropic dissipative systems these planes may not be at right angles to each other. It is shown that if the slowness bivector S is written S ---- Tei~(m + in) where T, 4~ are real, m, n are prescribed vectors: n is a unit vector and m is any vector lying in the plane normal to n, then T, 4~, A are determined from an eigenvalue problem. The directions of the normals to the planes of constant phase and the normals to the planes of constant amplitude lie along conjugate semi-diameters of the ellipse of S.

The familiar example of waves in a homogeneous isotropic elastic body is

Inhomogeneous Plane Waves 43

considered in w 7 and the results contrasted in w 8 with those of waves in homogeneous isotropic viscoelastic bodies.

Finally (w 9), waves propagating in homogeneous anisotropic elastic bodies are considered. Results on wave polarisations are obtained. Some universal relations are obtained connecting the acoustical tensors for inhomogeneous waves with acoustical tensors for homogeneous waves. These lead to results which are independent of the elastic coefficients. Finally, it is noted that the problem of determining directions in which circularly polarised homogeneous waves may propagate is readily resolved by examining the slowness surface and finding where the sheets intersect or touch.

In conclusion, the special case of an elastic cubic crystal is examined. It is seen that the criterion of double roots derived in w 5 is not empty and circularly polarised inhomogeneous plane waves may propagate for certain slownesses.

w 2. The Ellipse

Here those properties of an ellipse which are useful in the c[escription of bivectors are introduced. In particular, properties of conjugate diameters are detailed. A simple geometrical method is outlined for the construction of an ellipse with given conjugate semi-diameters and centre.

Two diameters of an ellipse are said to be conjugate if all chords parallel to one diameter are bisected by the other diameter. An equivalent statement is that the tangents at the extremities of one diameter are parallel to the other diameter. If an ellipse 6 ~ has major and minor semi-axes of lengths a and b respectively then referred to these rectangular axes the equation of the ellipse may be written

X 2 y2 1. (2.1)

An equivalent statement about two diameters being conjugate is that the product of their slopes with respect to the major axis is --b2/a 2.

With O, the centre of the ellipse, as centre draw a circle c~ of radius a. This is called the 'auxiliary circle' and has equation

c~: x 2 + y2 = a 2" (2.2)

Let Q, P be any two points on c~ such that QtgP -- zt/2 (see Fig. 1). If the projections of P and Q upon the major axis intersect the ellipse at R, H, respectively, then OR, OH is a pair of conjugate semi-diameters of 6 p. Indeed,

if PtgM ---- ~b (say) then the coordinates of P, R, Q, H are

P(a cos ~b, a sin ~b), R(a cos ~b, b sin ~b), (2.3)

Q(-a sin if, a cos if), H(--a sin ~b, b cos ~b),

and thus the product of the slopes of OR and OH relative to OM is --b2/a 2. If OM, ON are given to be a pair of conjugate semi-diameters of an ellipse,

then taking oblique axes x' and y ' along OM and ON, respectively, the equation

44 M. HAYES

H P

M

Fig. I

of the ellipse is: X '2 Y~2 2 a ' -~ + b' = 1, (2.4)

where O M = a', O N = b'. Given the conjugate semi-diameters there are a number of geometrical ways

of constructing the ellipse (see e.g. FRENCH & VIERCI~ [3]). Here I give a construction which is useful in visualising the role of the conjugate semi-diameters.

With O as centre draw two circles, ~ of radius a' and ~ of radius b' (see Fig. 2). Let Q be the angular distance 4, from O M on q-/. From Q drop the perpendicular Q R on O M and through R draw the line La parallel to ON. ~ has equation x' = a' cos ft. Similarly let P be the angular distance (zl/2 -- if) from O N on ~r From P drop the perpendicular P H on O N and through H draw the line

Fig. 2


J parallel to OM. The equation of ~ is y ' : b' sin 4,. Since the coordinates o f the point of intersection W of ~ and ./r are (a' cos 4,, b' sin 4,) it follows

that W is on the ellipse. Further, in the x ' - y ' plane, if O W = r, then

r : O~[cos 4, + O-~Nsin 4'. (2.5)

The tangent at 4, on the ellipse is parallel to

dr ~ ~-~ = - - O M sin 4, + ON cos 4,, (2.6)

which is parallel to the conjugate diameter at (4, + n/2) on the ellipse.

w 3. Bivectors

HAMILTON [4] used the term "bivector" to describe a complex vector A where A ---- c § i d and c, d are real vectors. With each bivector GraBs [1] associated an ellipse with a sense of direction and used the term "directional ellipse" to describe the ellipse. The pair (c, d) is a pair of conjugate semi-diameters of the directional ellipse of A if A = c § i d.

In this section bivectors and their associated directional ellipses are considered in detail. Bold face capital Latin letters A, B . . . . will be used to denote bivectors and bold face lower case Latin letters a, b . . . . will be used to denote real vectors.

GIBBS [1] showed that the pair (c' d') given by

c' q- i d' : e i ~ ~- i d) (3.1)

where 0 is given, is also a pair of conjugate semi-diameters of the directional ellipse of A : c ~- i d. Since this is a central result of importance in applications three separate proofs of it are given here.

Finally, in w 3.3 examples are given to show how a given bivector A may be written in the form A : ei~ -q- ib), where 4,, a, b are determined such that a �9 b : 0, so that a, b are the principal semi-axes of the directional ellipse of A. In each case the equation of the directional ellipse is written down.

3.1. The Directional Ellipse

Any pair of vectors c, d may be considered as conjugate semi-diameters ~f an ellipse. The ellipse is uniquely determined if the centre of the ellipse is at the common origin of c and d. Now GIBBS regarded A = c + i d as defining an ellipse--"the directional ellipse of A". Furthermore, if the angular direction is taken to be from imaginary to real, then the sense in which the ellipse is described is defined to be positive. For example if A ---- 2j + 3ik then the directional ellipse of A is described in the sense from positive z to positive y whilst the directional ellipse of B = 5k + 2ij is described in the sense from positive y to positive z. Also the directional ellipse of A and the directional ellipse of its

complex conjugate ,~ are described in opposite senses.

46 M. HAYES

In the following the adjective "directional" will be dropped and the term "ellipse of A" will be understood to mean the "directional ellipse of A".

If c, d is a pair of conjugate semi-diameters of the ellipse of A then c', d ' given by

c' + i d ' = (cos 0 + i sin 0) (c + i d), (3.2)

are conjugate semi-diameters of the ellipse of A. Three separate proofs are given of this:

(i) Analytically. (i/) Geometrically (GraBS). This gives a constructive method of finding

c ' + i d ' if c + i d and 0 are given. (iii) Using equation (2.5).

(i) Without loss of generality let c be along the x-axis and d be in the x-y plane. Then

C : C~, (3.3)

d = p i + qj, q :~ O, i . j ---- O.

The ellipse with these as conjugate semi-diameters is

r = c c o s @ + d s i n @ , z - - 0 , 0_< @=< 2z~,

o r

2 t, (c 2 + p2) X 2 - - - - x y - ~ - - y 2 : C 2. q q2

Now using (3.2) and (3.3), we have

c' = (c cos 0 -- p sin 0) i -- q sin Oj,

d' = (p cos 0 + c sin 0) i + q cos Oj.

(3.4)

(3.5)

(3.6)

It is easily checked that these are on the ellipse. Also the slopes of the tangents " a t " c' and d ' are, respectively,

q cos 0 - -q sin 0 p cos 0 + c sin O' c cos 0 -- p sin O' (3.7)

which are parallel to d ' and c', respectively.

(ii) Geometrically. If in Figure 3 O H ---- d, OR = c then the points Q, P determined by the point of intersection with the auxiliary circle of the perpendiculars

HK, R L on the major axis are such that QOP -- ~/2. Now rotate QOP rigidly clockwise through an angle 0 so that it occupies Q'OP'. Let the perpendiculars

'K' P 'L ' = ---- - - Q , intersect the ellipse at H' , R'. Then if PtgL @, P'tgL' @ 0

and Q()L = zc/2 + @, Q'(gL' = z~[2 + (@ -- 0) so that the coordinates of


P ' and Q' are

P'(a cos (4 - - 0), a sin (4 - - 0)), Q'(a cos (4 - 0 + z~/2), a sin (4 - 0 + z~/2))

= Q ' ( - a sin (4 - 0), a cos (4 - 0)),

where a, b are the major and minor semi-axes. Hence the coordinates o f R ' and H' are

R'(a cos (4 -- 0), b sin (4 - - 0)), H'( - -a sin (4 - - 0), b cos (4 - - 0)).

N o w O-~H' and O~R' is a pair o f conjugate semi-diameters since Q'OP' = ~/2, and

OH' = - -as in ( 4 - - 0) i + b cos (4 - - 0 ) j

= cos O(--a sin 4i + b cos 4j) + sin O(a cos 4i + b sin 4j)

= cos 0 d + sin Oc

and

OR' = a cos (4 - - 0) i + b sin (4 -- O)j

= cos Oc -- sin 0 d

C r"

This confirms that c ' and d ' is a pair o f conjugate semi-diameters. Incidentally this shows how c ' + i d ' are determined if c + i d and 0 are

given. Referring to figure 3, we see that OR and O H are given. Through R and H draw RP and HQ parallel to the minor axis. N o w rotate the or thogonal pair QOP clockwise th rough 0 to Q'OP'. Through Q' and P ' draw Q'H" and P'R' parallel

P

J

Fig. 3

48 M. HAYES

to the minor axis. Then OR' = c' and OH' = d'. In particular i f Q O P is rotated

clockwise through an angle ~b where (POL = qb) then c ' ---- a, d ' ---- b and

a + ib ---- ei~(c + i d) , (3.8)

<)r

c + i d = e-t~(a + ib), a - b ---- 0. (3.9)

This shows that any bivector B may be written in the form

B = ei~(r + is), r . s = O.

(iii) Direct proo f using equation (2.5). It was seen in w 2 that in the plane o f A the position vector r o f points on the ellipse o f A is given by

r = c cos q) + d sin % 0 ~ ~o ~ 2zr. (3.10)

This may be written

r = c ' cos (0 + ~0) + d ' sin (0 + 70, 0 ~ 0 -}- tP ~ 2~r, (3.11)

where

c ' = c cos 0 - - d sin 0,

d ' = c sin 0 + d c o s 0,

Thus c ' and d ' is a pair o f conjugate semi-diameters.

(3.12)

3.2. Principal Axes o f the Directional Ellipse

It has been seen that any bivector c + i d may be written in the form

c + i d = (cos q + i sin q) (a + ib), a . b = O, (3.13)

where a and b are the principal conjugate semi-diameters (GraBS). Analytically, if c + i d is given, then

a + ib = (cos q -- i s in q) (c + i d ) , (3.14)

a �9 a -- b . b = (cos 2 q - - i s in 2q) ( c . c - - d . d + 2 i c . d) .

Thus, assuming c �9 c + d �9 d, we have

2c" d tan 2q -- (3.15)

c . c - - d . d

and this is known. The quadrant which is chosen for 2q is the one for which sin 2q has the same sign as c . d and c o s 2 q h a s the same sign as c . c - - d . d ([1, p. 87]).

In the special case when c �9 c = d �9 d so that the conjugate semi-diameters


are equal in length, then

and

so tha t

(c + a ) . (c - a) = 0, (3.16)

c + i d = (1/1/2) (cos ~/4 - - i sin :r/4) [(c -- d) + i(c q- d)] , (3.17)

c - - d, c + d are a long the principal axes of the ellipse o f c + i d. Thus in every case c + i d m a y be writ ten in the fo rm (3.13) where a and b

are or thogonal . Here are some examples.

3.3. Examples

Example (3.1). Let

c=2j+3k, a = 4 j - s k .

Then

(c + i d ) . (c + i d ) = - -14(2 q- i) = ( a . a - - b . b) (cos 2q + i s in 2q).

Hence

1 tan 2q - - , sin 2q < 0, cos 2q < 0, tan q = - - 2 - - 1/5,

2

+(2 + r sin q - - 1/(10 + 4 1/5)'

- -1 cos q - - i/(10 + 4 I/5) '

a + ib = j (2 cos q + 4 sin q) + k(3 cos q - - 5 sin q)

q- i [ j ( - -2 sin q + 4 cos q) + k ( - - 5 cos q - - 3 sin q)]

- -1 i /5){J(--6 - - 4 I/5) + k(13 + 5 1/5) q- i[j(8 q- 2 1/5) + k(1 + 3 1/5)]}.

1/(10 + 4

The directional ellipse is

(2z - - 3y) 2 + (5y + 4z) 2 = (22) 2, x = 0.

Example (3.2). Let

c = 2i + 3j, d = - - 6 i q - j .

Then

( c q - i d ) . ( c + i d ) = - - 2 4 - - 1 8 i = ( a . a - - b . b ) ( c o s 2 q + i s i n 2 q ) .

50

Hence

M. HAYES

3 tan 2q = -~-, tan q = --3, sin 2q < 0,

--1 + 3 cos q = 1/10' sin q -- 1/10 ,

3/) a + ib ---- ~ I/T0 [2i + 3j 6- i ( - -6 i 6-j)]

= --1/10 [2i + ij].


(x + 6y) 2 + (2y -- 3x) 2 = (20) 2,

or

x 2 + 4 y 2 = 4 0 , z = 0 .

Example (3.3). Let

Then

Hence

cos 2 q < 0 ,

Z = 0 ,

c = 2 j - 3 k , a = 3 j - 2 k .

(c + i d ) . (c 6- i d ) = 24i = ( a . a -- b. b) (cos 2q 6- is in 2q).

t a n 2 q = o % s i n 2 q > 0 , q=--~- ,

(1 i) ( a 6 - i b ) = ~ ~-~ ( ( 2 + 3 i ) j - - ( 3 + 2 i ) k )

1 = ~ {(5) ( j -- k) 6- i ( j 6- k)}.


(2y + 3z) 2 6- (3y + 2z) 2 = 25, x = 0.

Thus, in conclusion, if a bivector A is given, there is associated with it a directional ellipse. However, each directional ellipse is associated with more than one bivector. Indeed all pairs of conjugate semi-diameters of the directional ellipse give "equivalent" bivectors, in the sense that they have a common directional ellipse. Analytically, bivectors B given by B = d~ 0 <= 0 < 2~ are equivalent. Geometrically, all pairs of conjugate semi-diameters of a directional ellipse give equivalent bivectors. Also any bivector A may be written

A = ei~ + ib), a . b = O,

where a and b are along the principal axes of the directional ellipse of A.


w 4. Sums and Products of Bivectors

In this section sums and products of bivectors are considered. In w 4.1 it is seen that in general A + B is not coplanar with either A or B. In w 4.2 the scalar product A �9 B of two bivectors is defined in the usual way. It is seen that in general the vanishing of A �9 B means (i) that the planes of A and B are not orthogonal, and as GIBBS showed (ii) the projection of the ellipse of A upon the plane of B is an ellipse whose major axis is orthogonal to the major axis of the ellipse of B and which is described in a sense similar to the ellipse of B. In w 4.3 'isotropic' or 'null' bivectors are considered and in w 4.4 the cross product A ^ B of two bivectors is considered. Finally, in w 4.5 orthonormal triads are considered.

4.1. Sums of bivectors

Let

be an arbitrary bivector and let

be an arbitrary scalar. Then

A----c + icl, (4.1)

: 1~1 ei~ (4.2)

o~A = I~l el~ c - k i d ) = t~1 (p + iq), (4.3)

where (p, q) is a pair of conjugate semi-diameters of the ellipse of A. Thus the effect of multiplication of a bivector by a complex scalar o~ is to rotate non-uni- formly the pair (c, d) of conjugate semi-diameters of the ellipse of A into another pair of conjugate semi-diameters of the ellipse of A and then each member of this pair if subjected to a uniform extension 1~1.

Two bivectors A and B are said to be parallel if there exists a scalar ~ such that

A = o~B. (4.4)

In this case the ellipse of A is similar to the ellipse of B so that the major and minor axes of the ellipse of A are parallel respectively to the major and minor axes of the ellipse of B and also the aspect ratio (i.e. the ratio of the length of major axis to the length of the minor axis) is the same for each ellipse. Further- more, the ellipses are described in the same sense. As GIBBS [1, p. 87] put it "Parallelism of bivectors signifies the similarity and similar position of their directional ellipses".

I f two bivectors are not parallel they are said to be linearly independent. Now let A and B be two bivectors:

Then

A : c + i d , B = e q - i f .

A + B = c + e + i(d + f ) ,

52 M. HAYES

and is thus a bivector whose ellipse has the pair (c + e, d + f ) as conjugate semi-diameters. In general the ellipses of A and B are not related in a simple manner to the ellipse of A + B. Indeed the ellipse of A + B is not in general coplanar with either' the ellipse of A or the ellipse of B.

There are a few special cases: (a) I f B = aA, so that A is parallel to B, then A + B is parallel to A and the ellipses of A, B, A + B are similar and similarly situated. (b) I f the major axis of the ellipses of A and B are both in the direction n then so also is the major axis of A § B.

m

If/3~, A~ (g = 1 . . . m) are m complex scalars and bivectors then ~] fl~A~ or

is a bivector whose ellipse has the vectors S(fl~A~)+ and S(/3~A~)- as conjugate semi-diameters. Of course if the A~ are all parallel to a bivector B then S/3~A~ is a bivector also parallel to B.

Three bivectors A, B, C are said to be coplanar if there exist scalars ~,/3, 7 not all zero such that o~A +/3B + ~,C = 0. The term 'coplanar ' is used in a very loose sense here for in general A + is not even coplanar with B+ and C+ since A+ : --(fiB/s)+ -- (7C!o0+ and this involves B - and C-.

4.2. The Scalar Product

The scalar product A �9 B of two bivectors A, B is defined by

A . B = (A -F + iA-) . (B + + iB-)

: (A+. B+ -- A - . B-) + i(A +. B- + A - . B+). (4.5)

Two bivectors are said to be perpendicular if their scalar product is zero. Now it is shown, following GIBBs [1] that if A . B = 0 then either A = 0 or B : 0 or the projection of the ellipse of A upon the plane of the ellipse of B is an ellipse whose major axis is orthogonal to the minor axis of the ellipse of B and is described in a sense similar to that of the ellipse of B. Also the plane of A may not be orthogonal to the plane of B.

First suppose that A and B are coplanar. Then they may be written

A : m(a + ib), where a . b : 0, (4.6)

B = n ( c + i d ) , where c . d : 0 ,

:and without loss of generality a is assumed to be along the major axis of A. Now :since c, d lie in the plane of a, b it follows that B may be written

B = n[~a +/3b + i(Ta + ~b)]

: n[(~ + i7) a + (/3 + i ~) b]

for some real scalars ~, 13, 7, 6- Thus A . B = 0 gives

~xa �9 a = Ob �9 b, ~,a �9 a q-/3b �9 b : O. (4.7)


Thus

B = n(6 - - ifl) ( [ ( b . b) a/(a. a) ] + ib}. (4.8)

Hence ( b . b ) a / ( a , a) and b are along the minor and major axes respectively of B, and also B is described in the same sense as A is described. See Figure 4. Indeed for A and B, omitting the scale factors m and n(6 - - i f l ) ,

a A: major axis a; minor axis b; aspect ratio -~-.

(4.9) a

B: major axis b; minor axis ( b . b)/a/(a �9 a); aspect ratio ~ - .

Thus if the bivector A : a + i b , a . b : 0 , is coplanar with B and if A . B = 0 t h e n B has the form

B ---- /t[(b �9 b) a + i(a. a) b], ( 4 . 1 0 )

where # is some scalar.

Fig. 4

Suppose now that A and B are not coplanar. First note that the planes of A and B may not be orthogonal. Since A �9 B = 0 it follows f rom (4.5) that

A+ . B+ --_ A - . B - , A + . B - = - - A - . B + . (4.11)

Now A+ A A - and B+ A B - are normal to the planes of A and B respectively. But using (4.11) gives

(A+ A A- ) " (B+ ^ e - ) = ( a +" e+) (A-" B-) - - (A+. B-) (A-" B+)

= (A+. B+)2 + (A+. B-) 2 =4= 0, (4.12)

so that the normals are not orthogonal. I f A ---- m(a + ib), a . b ---- 0, then B may be written as B = 0~a + fib + yc

where c . a = c . b = 0 . Then A . B = 0 implies o ~ a . a + i f l b . b = O . Thus

B : - - i f l { [ (b �9 b) a/(a. a) ] + ib} + yc, ( 4 . 1 3 )

is the form of B if A �9 B ---- 0 and A : m(a + ib). I t is clear that the term in chain brackets is the projection of B upon the plane of A. Hence as GIBBS [1, p. 89] puts it: " I f tWO bivectors are perpendicular the directional ellipse of either projected upon the plane of the other and rotated through a quadrant in that plane will be similar and similarly situated to the second".

54 M. HAYES

4.3. Isotropic Bivectors

A bivector A is said to be isotropic i f A �9 A ---- 0. N o w if A ----- m(a + ib), where a . b - - - - 0 then A . A = 0 implies t h a t a . a = b . b so tha t the ellipse o f A is a circle. The te rm " i so t rop ic" is used for a geometrical description and "nu l l " is used for an algebraic description.

Typical ly an isotropic bivector A may be writ ten

a = o~(i + ij), (4.14)

where o~ is a scalar. Every bivector B perpendicular to this has the fo rm

B =/5(i + ij) + •k. (4.15)

I f this bivector B is also isotropic then y = 0 f rom which it is concluded that two non-paral lel isotropic bivectors cannot be or thogonal ; or equivalently if two isotropic bivectors are or thogonal they are parallel (SYNGE [2, p. 14]).

Also if three non-zero l inearly independent bivectors A, B, C are mutual ly or thogonal none o f them is an isotropic bivector (SYNGE [2, p. 14]). For suppose A is isotropic. Then

A . A = A . B = A - C = B . C = 0. (4.16)

N o w A, B, C may be writ ten

A = o~(i + ij), n =/5( i + ij) + yk, C = O(i + ij) + ek, (4.17)

for some scalars 0c,/5, ~, ~, e, since A is isotropic and B and C are perpendicular to it. F r o m use of B �9 C ---- 0 it follows tha t ye = 0 so that ei ther y ---- 0 (in which case B is isotropic) or s = 0 (in which case C is isotropic). Thus either B or C is parallel to A, cont rary to hypothesis.

4.4. The Cross Product

The cross p roduc t A ^ B of two bivectors A and B is defined by

A ^ B = (A + + iA- ) A (B+ + iB-)

= (A+ ^ B+ -- A - A B--) + i(A + A B - + .4- ^ B+). (4.18)

As GIBBS [1] showed, if A ^ B ---- 0 then A is parallel to B. For, wi thout loss o f generality, A and B m a y be writ ten

A = o~(c + i d ) , c" d = 0, (4.19)

B = / 5 c + ?, d + ~h, h �9 c = h �9 d = 0 ,

for some (complex) scalars o~, /5, 7, ~. N o w if A ^ B = 0, then

7=//5, and

a = o (c + i a) , B = + i a ) ,

so that A and B are parallel .


Also, by direct expansion, for any two bivectors A and B,

A A B . A : O , A ^ B . B = O . (4.20)

Thus the projection of the ellipse of A upon the plane of the ellipse of A A B is an ellipse which is similar and similarly situated to the projection of the ellipse of B upon the plane of the ellipse of A A B. Both projected ellipses are similar and similarly situated to each other and are similar also to the ellipse of A A B, but the major axis of each of the projected ellipses is perpendicular to the major axis of the ellipse of A A B.

Example (4.1). Let

Then

ThiS is illustrated in the following example.

A = 2i + i j ,

B = k + 3ii. (4.21)

a A B = 1/13 l + ii, l" = ( - -2 j + 3k)/1/13. (4.22)

Now the unit normal to the plane of the ellipse of A A B is n given by

1 n = 7 i ~ (3j + 2k),

and the projection of A upon the plane of the ellipse of A A B is

A' = A -- ( A - n ) n = 2 i - - 2i

Similarly the projection of B is

(4.23)

B:

^

B' = B - - (B . n ) n = - ~ l + 3ii. (4.25)

Let x, y, z, k, denote coordinates along i, j , k, i. Then the various ellipses are (see Figure 5)

k2 A A B: x 2 + ]-~ = 1, 3y + 2z = 0; (4.26)

A: x z x 2 13k z -4" + y2 = 1, z = 0, projected into A': -~---~- - - 4 =

x 2 x z 13~ 2 + z 2 = 1, y = 0 projected into B ' : ~ + 9

Finally here is a case where A and B are coplanar.

- - 1 , 3 y + 2 z = 0 ;

(4.27)

= 1, 3y + 2z 0.

(4.28)

i r (4.24)

56 M. HAYES

^

B !

Fig. 5

Example (4.2). A , B coplanar. In this case A ^ B is a scalar multiple of the normal to the plane of A. For example if A = 0 c i + ~ j , B = T i q - ~ j then A A B = (~ ~ - / 3 7 ) k and the ellipse of A ^ B degenerates into a straight line along the z-axis. The 'plane ' of the ellipse of A ^ B is now any plane containing the z-axis and the projection of the ellipse of A or the ellipse of B onto any such plane is a straight line orthogonal to the z-axis in that plane.

4.5. Or thonormal Triads

Any bivector A may be written

A = o~(a + ib), a . b = 0, (4.29)

where 0~ may be complex. To obtain a unit bivector, A* (say), A * . A* ---- 1, f rom A, simply take

A * : (a + ib) / (a . a - - b . b) �89 (4.30)

This is well defined provided A is not isotropic in which case A �9 A = 0, or a �9 a -- b �9 b. Thus given a bivector A which is not isotropic one may obtain a


unit bivector A*. The ellipse of A* is similar and similarly situated to the ellipse of A.

Now an orthogonal triad of non-zero bivectors no two of which are collinear may not contain an isotropic bivector. Thus if an orthogonal triad of non-zero bivectors A, B, C is given and no two of them are collinear then A*, B*, C* form an orthonormal triad:

A * . A * = B * . B * = C * . C * = 1, (4.31)

A * . B * = B * - C * = C * . A * = 0 .

If A*, B*, C* form an orthonormal triad then the 3 • matrix T given by

T = B * , (4.32)

C*

is orthogonal: T r = T --1, since, by (4.31)

rr = B* (A* c * ) = 10101 . (4.33)

c* [0011

Further details may be found in SYNGr [2, p. 15].

w 5. Eigenbiveetors

In this section the standard eigenvalue problem for complex symmetric 3 • 3 matrices is considered. In particular the possibility of having isotropic eigenbivectors is examined in detail. The main result is that a necessary and sufficient condition that a complex symmetric 3 • 3 matrix have an isotropic eigenbivector is that the matrix have a double eigenvalue.

Let Q be a complex symmetric 3 • 3 matrix. Then its eigenbivectors A are given by

QA ----- ;tA, (5.1)

and the eigenvalues ;t by

d e t l Q - 211 = 0. (5.2)

SYNGE [2] has shown that if A and B are eigenbivectors corresponding to eigenvalues ;t and # respectively, and if ;t =4= # then A and B are linearly independent (A ~ o~B) and orthogonal: A �9 B = 0. The method of proof is standard. Also SYNGE [2] has shown that if ;t and # are distinct eigenvalues and A and B are the eigenbivectors corresponding to them respectively, then A and B cannot both be isotropic for A and B are orthogonal and linearly independent. But, as shown in w 4 if A and B are both isotropic and orthogonal then A is parallel to B which contradicts the statement that A and B are linearly independent.

58 M. HAYES

SYNGE [2] also discusses complex orthogonal matrices T : T r = T -1. Such a matrix transforms the bivectors A, B into the bivectors A', B ' : A ' = TA, B ' = TB. Also A ' r B ' : ArB, so that an orthonormal triad transforms into an orthonormal triad and an isotropic bivector transforms into an isotropic bivector.

lsotropic Eigenbivectors

If Q has three distinct eigenvalues 2', 2", 4 '" (say)then the corresponding eigenbivectors are linearly independent and mutually orthogonal and none of them is isotropic, since an orthogonal triad of linearly independent bivectors cannot contain an isotropic bivector.

Theorem. I f Q has an isotropic eigenbivector then the corresponding eigenvalue must be double. I f an eigenvalue is double then Q possesses an isotropic eigenbivector.

Proof. Given that O = O r and that

Q A = 2 A , A . A = 0 , (5.3)

we are to show that 2 must be a double root of det I Q - 211 = 0. Write

A = r + is, r . r = s . s = l, r . s = 0 , (5.4)

without loss of generality. Let t be chosen so that

t . t = l , t . r = t . s = O , (5.5)

and then r, s, t form an orthonormal triad of vectors. Referred to this triad

let O have the form Q given by

6 = 0

Referred to this triad A has the form' ( l , i, 0). Now by (5.3)

Thus

= 4 .

0

(5.6)

(5.7)

c ~ + i ~ = 2 , ~ + i ~ = 2 i , r + i O = O , (5.8)


and hence

I:i~ io I Q = ~ + i ~ 0

t - i o o 9, J

(5.9)

and (~ has eigenvalues ;t (double) and ?. Suppose now that Q has a double eigenvalue ;; (say), to show that Q possesses

an isotropic eigenbivector. There are two cases: (i) two eigenvalues equal and not equal to the third

(ii) all three eigenvalues equal. (The treatment here runs parallel to the treatment in SVNGE [2] for a problem on eigenbivectors.)

Case (i). Let the eigenvalues be 2, 2 and /~ (=1= ;t). Let A and B be eigenbivectors corresponding to ;t and/~ respectively. Then, since Q is symmetric and 2 # / ~ A and B are orthogonal and not parallel. There are three possibilities:

i (a ) a . a # O , B . B ~ O ;

i (b) A- A = 0, B . B # 0;

i(c) A " A =t= O, B . B ~- O.

The possibility A . A = B . B = 0 is not allowed since A . B = 0 and A is not parallel to B.

It will be shown that for possibility i (a) it is possible to choose an isotropic eigenbivector, and that possibility i (c) is not allowable. It is also seen that possibility i (b) does not lead to any inconsistency.

Possibility i (a). Now

QA = 2A, QB = #B, A . B = 0, ;t :4=/z. (5.10)

Without loss of generality, assume that A, B are normalised and complete an orthonormal triad with C. Then

/AI T = B , (5.11)

tc! is an orthogonal matrix: T r = T -1 and it transforms Q into Q' given by

I!~176 Q ' = T Q T T = iz 0 . (5.12)

0 CTQC

60 M. HAYES

But 2 is a double root. Therefore it = C r Q G

Hence D, given by value it.

and

Q'__- /~ .

0

(5.13)

D = A + iC is an isotropic eigenbivector of Q with eigen-

Possibili ty i (b). Here

QA = 2/1, QB =/zB, A �9 B = 0, A �9 A = 0, B �9 B =]= 0,

it double root.

B is not isotropic. Thus complete an orthonormal triad B, C, D with bivectors C, D. Since A . B -----0, and since A is isotropic, i t m a y be written

a = C q- i D . (5.14)

Also

Thus if we write

it follows that

Take

Then

and

Q C q- i q D -= itC q- i2 D,

C r Q C + i c r q O = it,

D r q C q- i D r O D = lit.

(5.15)

c, TO D = D T o e = or (5.16)

C r Q C = it - ion,

D r Q D = 2 + ict. (5.17)

T = ;i ol

(5.18)

I~ 0

Q' : - T Q TT = 2 - - ior

a

o i O~

i t + i a

(5.19)

l! ~ O~

0r = , ~ , ,

i t + i x

(5.20)


which simply states that C + iD is an isotropic eigenbivector of Q. Also Q has double roots 4. Thus this Possibility i (b) does not lead to any inconsistency.

Possibi l i ty i (c). Here

Q A = h A , Q B = # B , A . B = 0 , B - B = 0 , A . A 4 = 0 ,

2 double root, 2 4=/z.

Since A is not isotropic, complete an orthonormal triad A, C, D with bivectors C, D. Since B �9 A = 0 and B is isotropic, B may be written

B = C @ i D .

Proceeding as in Possibility i (b), conclude with T r = (A , C, D) that Q' = T Q T r is given by

[4 0 0 ]

Q ' = 10 / ~ - - i ~ ~ 1. (5.21)

/

to # + i o r

Thus O has eigenvalues 2 and ,u (double). But 2 is the double eigenvalue. Hence this Possibility i (c) is not allowed.

Case (ii). All three eigenvalues equal. Here

o a = 2A, (5.22)

and 2 is a triple root of (5.2). Suppose A is not isotropic: A �9 A =4= 0. Then choose B, C to complete an orthonormal triad A, B, C.

Let

[cj Then

Q ' = T Q T r = o~ , (5.24)

fl where ~ = B Q B r, fl = B Q C r, ), = C Q C r. Now the eigenvalues are all equal to 2. Hence Q' has the structure o/.

O' = 2 + fl . ( 5 . 2 5 )

fl 2 - i f l Now B' - - iC' = T B - - i T C = (0, 1, O) -- i(0, O, 1) is an isotropic eigenbivector of Q' and hence B - - iC is an isotropic eigenbivector of O.

Thus it follows from Cases (i) and (ii) that if Q has a double eigenvalue then it possesses an isotropic eigenbivector.

62 M. HAYES

w 6. Inhomogeneous Plane Waves. Description

The results of the previous sections are now applied in the consideration o f inhomogeneous plane waves. Such waves arise in many areas. Examples include Rayleigh, Love and Stonely waves in classical linear isotropic elasticity theory, gravity waves in ideal fluids, TE and TM waves in electromagnetic theory, viscoelastic plane waves and generalised Rayleigh waves in anisotropic elasticity theory. To fix ideas attention will be restricted for the most part to elasticity and visco- elasticity theory.

The displacement u corresponding to a single train of inhomogeneous plane waves may be written

u = A exp i c o ( S , x - - t ) . (6.1)

Here A and S are bivectors. A is called the amplitude bivector and S is the slowness bivector. Also co is real and is called the angular frequency. The real part of u is

u + = {.4+ cos co(S +" x t) A- sin co(S +. x -- t)} exp (--coS-. x). (6.2)

The planes S+ �9 x ----- constant are planes of constant phase and S- �9 x = constant are planes of constant amplitude. Equation (6.2) represents an infinite train of elliptically polarised plane waves. The waves travel in the direction of S + with slowness IS+ I. They are attenuated in the direction of S-. The period is 2~r[co. For fixed x = (x* say) the displacement vector u+ lies on an ellipse which is similar and similarly situated to the ellipse of A, namely the ellipse whose conjugate semi-diameters are A+ exp (--coS-- x*) and A- exp (--coS- �9 x*). At any given time the displacement vector is along one semi-diameter of this ellipse and the particle velocity is parallel to its conjugate semi-diameter. As t increases the sense in which the particle moves along the ellipse is from A+ to A-.

To illustrate the way in which homogeneous and inhomogeneous waves axe described, such waves are now briefly discussed within the context of linear homogeneous anisotropic elasticity theory.

For a homogeneous elastic body the basic equations are [5]

tij = COklekt, 2ekl = l'lk,1 DI- Ul,k, tij, j = 9 OZUi/~tZ, (6.3)

CijklUk,lj = Q ~2Ui/~t2.

Here the stresses are denoted by to, the infinitesimal strains by e 0, the elastic constants by CUk l, density by ~ and " , / " means differentiation with respect to xl. The elastic constants are assumed to have the symmetries

Cijkl = Cjikt = CU1k = Cklij. (6.4)

Suppose that the wave train (6.1) is to propagate in this body. Then inserting (6.1) into (6.3)4. gives the propagation condition

( C , : k l S l S j - - Q ~,k) Ak = 0 . (6.5)


Then if A is not identically zero the secular equation

l coktS1Sj - - ~ t~ik I = 0, (6.6)

must be satisfied. Now if S is a vector, S = n/(c) (say) where n is a unit vector and c is the phase

speed, the secular equation (6.6) is the cubic in c2:

l cimnln~ - - ~c 2 a;~ l = 0. (6.7)

Thus if n is specified, the phase speeds c~ (0~ = 1, 2, 3) (say) are determined by solving (6.7) and then the corresponding amplitude vectors A (~) (say) are obtained to within a scalar multiple from (6.5). As the direction of n is varied, all the possible solutions are obtained.

However, for inhomogeneous waves there is a far greater variety of solutions possible. For inhomogeneous plane waves there are two basic directions, namely the directions of the normals to the planes of constant phase and to the planes of constant amplitude. To obtain a systematic development I propose to write the slowness bivector

S = Te 'r + in), m . n = 0, n �9 n ---- 1, T real. (6.8)

The direction of the unit vector n is specified and the magnitude and direction of m, which is orthogonal to n is also specified. Then both T and 4, are obtained by solving the secular equation (6.6):

I CijklTZeE'*(mt + in1) (my + inj) - - e bikl = O. (6.9)

Knowledge of T and 4, ensures that the planes of constant phase and constant amplitude are known, and that the phase speed and attenuation factor are determined.

Specifically, suppose n ---- i, the unit vector along the x-axis and m is parallel to j the unit vector along the y-axis. Now n is held fixed and m is allowed to vary in magnitude. The corresponding T 's and 4,'s are found in each case. Next n = i, as before, but m is rotated out of the xy-plane. Let the new m = a j + ilk. With n held fixed the magnitude of m is allowed to vary. Again the corresponding T 's and 4,'s are obtained from (6.9) in each case. Then systematically the vector m takes any magnitude and any direction in the y - - z plane, the corresponding T 's and 4,'s being found in each case. Now n is varied in direction and m allowed to have any magnitude and any direction at right angles to n. In this way all the possible solutions are obtained.

Returning to (6.8), we see clearly that m and n are along the principal axes of the ellipse of S, or put another way, the normals to the planes of constant phase and planes of constant amplitude are along conjugate semi-diameters of the ellipse of m + in. Also

S+ = T ( cos 4, m - - s in 4, n ) , IS+ 1 = T ( c o s 2 4 , m 2 + s i n 2 4 , ) �89 (6.10)

S - ---- T (sin 4' m + cos 4, n), [ S - I = T (sin 2 4, m 2 + cos 2 4,)�89

64 M. HAYES

The angle 0 between the planes of constant phase and planes of constant amplitude is given by

m

tan 0 = (m 2 _ 1) cos ~b sin ~" (6.11)

It is clear from (6.8) that these planes are orthogonal if exp (i~) is purely real or purely imaginary, so that either sin ~ = 0 or cos ff ---- 0, and also when m 2 : 1, for then the ellipse of m + in becomes a circle and all conjugate semi- diameters of a circle are orthogonal. The case rn --~ 0 (and m --~ co) corresponds to the case when normals to the planes of constant phase and planes of constant amplitude are both parallel to n (and m).

Also note that the wave train described by equation (6.1) is [6, 7]

circularly polarised if A �9 A : 0, (6.12)

linearly polarised if A/x A = 0.

Finally note that any linear combination of elliptically polarised inhomogeneous plane waves with common angular frequency co,

u : y~ A(~) exp io~(S(~), x - - t) , (6.13) o~

is also elliptically polarised. For given x the particle displacement is elliptical:

u : exp (--loot) [a q- ib], (6.14)

where the conjugate semi-diameters a and b are given by

a + ib : y , A(~) exp [iw(S(~) �9 x)]. (6.15) cr

w 7. Linear Isotropic Elastic Bodies

In this section trains of inhomogeneous plane waves are considered in the context of classical linear isotropic elasticity theory. As usual, the secular equation may be factored, the single root corresponding to a longitudinal wave and the double root corresponding to the transverse wave. The transverse wave is ,considered in detail (w 7.1) and specific examples are given. It is shown (w 7.1.1.) how the amplitude bivector may be chosen so that circularly polarised plane waves may propagate in every direction. This is to be expected in view of the Theorem in w 5. For the transverse waves the planes of the amplitude bivector and the slowness bivector may not be orthogonal. Also, the projection of the ellipse of the amplitude bivector upon the plane of the slowness bivector is an ellipse, similar to that of the slowness bivectoI but whose major axis is orthogonal to the major axis of the ellipse of the slowness bivector. For the longitudinal wave the slowness and amplitude bivectors are parallel. Finally, in w for given m q- in the superposition of the longitudinal and transverse waves is examined.

In a homogeneous isotropic elastic body the stresses are given by

tii : 2#ely q- 2ek k (~ij, (7.1)


where 2 and # are constants. In the absence of body forces the equations o f mot ion are

/*12i,jj + (2 + fl) lik,ki = ~ O2Ui/Ot2. (7.2)

Let

u i : A i e x p k o [ S ' x - - t } , S : T e i * ( m + i n ) , n . n : 1, n . m - - O . (7.3)

Then f rom (7.2)

[/*SjSj r ~- (• -~ /*) SiSk - - ~ (~ik] Ak = 0, (7.4)

which leads to the secular equat ion

det [ (/*SjSj -- e) 0ik -[- (2 + #) S~Sk[ = 0. (7.5)

Equat ion (7.5) has the double root

/ i S . S : ~, A . S = 0, (7.6)

where A is the ampli tude bivector. This corresponds to the transverse wave. The other roo t of (7.5) is

(2 + 2/,) S . S = e, B = 0~S, (7.7)

where B is the ampli tude bivector and o~ is some scalar. This corresponds to the longitudinal wave.

7.1. Transverse Waves

From (7.3)2 and (7.6),, it follows that

/*T2e 2~r (m 2 - - 1) = e,

and hence, since T is real,

if m 2 > 1, 4 ~ = 0 ,

(7.8)

)�89 (m + in), =/~TZ(m 2-I), S=4- /~(m--- F I)

if m 2 < 1, 4' = ~/2, 9 = #T2( 1 -- m2), S = =~ (1 -- m 2 (ira -- n),

I S+[ -- (/*(1 e_ m2 ))} . (7.10)

Clearly m 2 ~ 1 is not possible since 9 ,# , T =I= 0. Also the planes of constant phase are or thogonal to the planes of constant amplitude, since exp i$ is real. This is also obvious by taking the imaginary part of(7.6)1 which gives S + �9 S - = 0.

66 M. HAYES

For these solutions A is any bivector satisfying A �9 S ---- 0. Hence the displacements may be written

u -- [3(m-lrh + in + 710) expi t~ :~ f z ( m i L ~) r e . x - t x

e x p ( ~ ( ~ ~�89 x) m 2 ) 1 (7.11) " '

where fi, 7 are scalars and p is a vector orthogonal to the plane of the ellipse of S. Similarly for m 2 < 1

u -~ f l ' (m- l l h -t- in + 9,'p) exp i~o (1 - m 2 n �9 x -- t x

exp

where fl', 9,' are scalars. For example, let

Then

n ---- i , m = 3 ] . ( 7 . 1 3 )

1 1 T = (e/8,@, S = • + i,)(e/8/0 ~,

i, + 9,k) exp io~(~3(9/8/~0�89 -- t) exp (T~o(9/8~0 �89 x).

(7.14)

_1 This is a train of waves, propagating with phase speed (8t~I9)2/3 along the y

1 axis and attenuated along the x axis with attenuation factor ~(~2/8,u) ~-. If 9, is taken to be real then the polarisation ellipse has a principal axis along i and the other principal axis along ( j /3 + 9,k). If 9' is purely imaginary and equal to i (say) then the polarisation ellipse has a principal axis along j and the other principal axis along i + ~k~ Ifg, is complex, 9, --~ ~ + / 7 * (say) then the polarisation ellipse has ( j /3 + ~k) and (i + y ' k ) as a pair of conjugate semi-diameters.

As another example, let

Then

n = i, m = j / 4 .

1 ( j ) 1 T = (169/15~0 '2, S = • t -~--- i (169/15/0 e, (7.15)

�9 e �89 t} e �89

InhomogeneousPlaneWaves 67

Finally, note that in equation (7.11) if m tends to infinity then the expression for u becomes

u : fl(in q- ~p) expico{(~//z) �89 rh . x -- t}, (7.16)

where th is a unit vector in the direction of m. This is the displacement corresponding to the usual unattenuated wave.

7.1.1 Circularly Polarised Waves. Returning to the expressions (7.11) and (7.12) for the displacement u, we noted that ~ and ~' were arbitrary. These may be chosen to give a circularly polarised wave. Thus if p is a unit vector, 7 is chosen so that the amplitude vector is isotropic. Thus with m 2 ~ 1,

(m- ' rh q- in q- 7P) " ( m- l rh q- in + 7P) : O,

leads to

[rn 2 - 1\�89 a : m-l + in + p j ,

(7.17)

[m 2 - 1~�89 7 : ~ = ~ m 2 ] , (7.18)

s - - " +

In every case, as m varies, the plane of polarisation contains n. The circles of polarisation are obtained by rotating circles about n.

Similarly with m 2 < 1,

, [ 1 - - m 2 ] � 8 9 7 =:J=i \ m2 ] ,

{ [l--m2'�89 ~ (q) �89 a = f l ' rn-'rh q- in =[= i ~ - - - ~ - ) P I ' S :- q- # ( l _ m 2 )

(7.19)

In every case, as m varies, the plane of polarisation contains m. Circles of polarisation are obtained by rotating circles about m.

Thus the planes of polarisation are quite different in the two cases. Note however that in each case the plane of polarisation contains the normal to the plane of constant amplitude.

7.1.2 Note on polarisation. For the transverse waves solution (7.6), the amplitude bivector A and the slowness bivector S must satisfy A �9 S = 0. Thus, by use of the results of w 4.2, it follows that the planes of A and S may not be orthogonal. Also the ellipse of S projected on to the plane of A is an ellipse whose aspect ratio is the same as the aspect ratio of the ellipse of A and whose major axis is perpendicular to the major axis of the ellipse of A.

68 M. HAYES

7.2. Longitudinal Waves

Using (7.3)2 and (7.7)1 we obtain

(2 + 2t0 T 2 e 2 i ~ ( m 2 - - 1) ---- e,

and hence

(7.20)

if m 2 > l, ~b = O, 0 = (2 q- 2#) T2(m 2 -- 1),

( ~ i ) � 89 + in), S = 4 - ( 2 + 2 # ) ( m 2 - 1 (7.21)

if m 2 ~ 1, ~b = Jr/2, ~ = (2 -}- 2#) T2(1 -- m2),

' + 2#)9(1 -- m 2))�89 S ---- 4- ((2 (ira -- n) .

Since the amplitude bivector B = 0~S, it follows that for the longitudinal wave the polarisation ellipse is similar and similarly situated to the ellipse of S. Also for the same reason the maximum displacement is either along the normal to the planes of constant amplitude or along the normal to the planes of constant phase.

Further note that for given m and n, the amplitude bivector A corresponding to the transverse wave and the amplitude bivector B corresponding to the longitudinal wave satisfy A . B = 0. Thus the particle paths corresponding to the two waves are ellipses whose planes may not be at right angles to each other. Also the projection of the ellipse of B upon the plane of the ellipse of A is an ellipse which has the same aspect ratio as the ellipse of A but whose major axis is perpendicular to the major axis of the ellipse of A. Also the ellipse of the slowness bivector S is similar and similarly situated to the ellipse of A.

Also note that in equation (7.21) if m tends to infinity then S tends to 1

rh[9/(2 § 2#)] ~ and the ordinary unattenuated longitudinal wave is recovered.

7.3. Superposition o f Two Wave Trains

Finally for given fixed m and n, consider the motion in any plane parallel to the plane of B due to the superposition of the two wave trains. The displacement of a particle arises as a linear combination of the two basic displacements and hence in general will be also elliptical. The ellipses for different particles will generally be different from each other.

The total displacement in the plane of B may be written

u ---- [fl(m-~rh + in) exp ( i toTxm. x -- ~oTtn. x} + e~(m q- in) exp ( io)Tzm. x

, -- toT2n, x}] exp (--iogt), (7.22)

where ~, fl are taken to be real constants, and assuming m 2 > 1, we have

tzT2(m 2 -- 1) = e, (2 + 2#) T2(m 2 -- 1) = e. (7.23)


The displacement (7.22) may be written

u = (3 exp ( - -coTln . x) [m-Xrh + in] + o~ exp (--roT2n" x

+ ico(T2 -- T1) m . x) [m + in]} exp iw(T~m, x -- t) . (7.24)

To fix ideas consider points on the line n �9 x = e in the plane of B. Let

/3* = / 3 exp (--o~T:) , 0~* = o~ exp (--~oT2e). (7.25)

Then the displacement (7.24) may be written

u ---- [/3*(m-lrh + in) + ~* exp (i~o(T2 -- 7"1) m . x) (m + in)]

• exp i~o(Tlm, x -- t). (7.26)

Now the ellipse of u will have principal axes along m and n if exp (ito(T2 -- T) • m �9 x) is purely real. There are two possibilities. Particles initially at the intersection of n . x - - - - c and the lines

r - - Ti) m - x = 2z~q, q = 0, q-1, + 2 . . . . (7.27)

will move on identical ellipses 8 (say) in the place of B, and the principal axes of these ellipses will be parallel to the principal axes of S. For the second possibility particles initially at the intersection of the lines n �9 x ---- c and

~o(T2 - - 7"1) m . x = zr(2r + l), r = 0, 4-1, ~ 2 . . . . (7.28)

will move on identical ellipses 8 ' (say) in the plane of B, and the principal axes of these ellipses will be parallel to the principal axes of S. In general the ellipses 8 and o ~' will not be the same. The only similarity between them is that their principal axes are parallel. Their major axes need not be parallel.

I t has been assumed that o~ and fl in (7.22) are real. I f they are complex, only minor modifications have to be made to the above.

w 8. Linear Isotropic Viscoelastic Bodies

The theory of plane inhomogeneous waves propagating in linear viscoelastic bodies runs parallel to the corresponding treatment for elastic bodies. For isotropic bodies the constants 2,/z used in w 7 are now replaced by complex functions 2(o~),/z(~o) of ~o. For simplicity the explicit dependence on o~ is dropped and in this section it is understood that 2, /z are complex function of co.

The secular equation factors as in w 7 leading to the transverse and longitudinal waves:

/ z s . s = q , a . s = 0 ; (8.1)

(2 + 2#) S . S = O, B = aS. (8.2)

To fix ideas it is assumed that

# + > 0 , 2+ + 2/Z+ > 0, /z- < 0, ).- + 2/Z- < 0. (8.3)

70 M. HAYES

8.1. Transverse Waves

Using (7.3)2 and (8.1)1 gives

#T2(m 2 - - 1) = Qe -210,

and hence

(8.4)

cos 2~b = #+T2(m 2 -- I), Q sin 24, = --#-TZ(m 2 -- 1). (8.5)

Thus, by use o f (6.11), the angle 0 between the planes o f constant phase and planes o f constant amplitude is given by

2m I/~1 tan 0 ---- (1 - - m 2) # - " (8.6)

It is clear f rom (8.5) that m z 4= 1. Hence, since # - 4= 0, m 4= 0, it follows that 0 4 = 0, 0 4= z~/2 whereas for elastic isotropic bodies the only possibility allowed was 0 = z~/2.

Assume m 2 > 1 and use (8.3); it follows that

tan 2~ = - -# - /#+ , zt/4 > 4~ > 0,

O = Z2( m2 - - 1)1/~], (8.7)

S = ~ e" (.( ~ ) �89 ( rn + in ) . m 2 - - 1)I~1

Similarly if m 2 < 1, on use o f (8.3), it follows that

- - # - 3z~ zt tan 2~ = #+ , P = Tz( 1 - - mZ) I~1, -T > ~ > - Y '

(8.8) (.( e ) (re+in). S = q-e i* 1-- m z) ]#l �89

N o w for these solutions A is any bivector satisfying A �9 S = 0. Hence, for example f rom (8.7),

u=f l (m- l rh+in -q -Yp ) e x p i ~ ) �89 } m2 - 1)IN [cosrbrn- -s inrbn] .x - t

• exp --e) - (cos4 ,n + sin4, . x m 2 1) I~1 m) , (8.9)

where fl, ~, are scalars and p is a unit vector or thogonal to the plane o f the ellipse o f S.

Suppose, for example, that

r 1 #+ = -~-, /z- = - - -~-, n = i, m = 3j. (8.10)


Then

I/~ ~ 3 T---- T , ~ = ] ~ - , t a n 0 - - 2 '

u = fl (~ + ii + yk) expiog{(~2 ) [3cos~2J-- sin-~2 i] " x t}

• exp --co-~-- 3 sin ~'~j-t- cos' i '~i �9 x . (8.11)

Also, as in w 7.1.1, y may be chosen so that the wave is circularly polarised. The remark on polarisation (w 7.1.2), is valid here also.

8.2. Longitudinal Waves

Use of (7.3)2 and (8.2) leads to

cos 2ff ---- (2 + 2/,) + T2(m 2 -- 1), (8.12)

sin 2~ = --(2 + 2/,)- T2(m 2 -- 1),

2m[2 + 2/,l tan 0 ----- (1 -- m 2) (2 + 2/,)-" (8.13)

We assume rn 2 > 1 and use (8.3). Then it follows that

tan 25 = --(2 + 2/,)-/(2 + 2/,) +, - ~ > $ > 0,

= T2(m 2 -- 1)]2 + 2/,], (8.14)

( 0 )�89 S = - r (m 2 _ 1 ) [ 2 + 2 / , [

Similarly for m 2 < 1,

3z~ zr -~->r e=T2(l--m2)[2+2/,[,

( 0 )�89 (8.15) S : q-e t'r (1 - - m 2 ) 1 2 -~- 2/, I

Since the amplitude bivector B : aS it follows that the polarisation ellipse of the displacement is similar and similarly situated to the ellipse of S. However, in contrast to the elastic case, the maximum displacement is neither along the normal to the planes of constant phase nor along the normal to the planes of constant amplitude, since in this case these normals are not along the principal axes of the ellipse of S.

72 M. HAYES

8.3. Superposition o f Two Wave Trains

Now, as in w 7.3, consider the superposition of two wave trains each with the same given m + in. Attention is restricted to motion parallel to the plane of B. The purpose here is to show how the situation discussed in w 7.3 for isotropic elastic bodies is modified for an isotropic viscoelastic body.

The total displacement in the plane of B may be written

u = f l (m- lrh + in) exp ( io[Txei~'(m + i n ) . x - - t]}

q- a ( m -k i n ) e x p {i~[T2ei*'(m -F in)" x - - t]}, (8.16)

where 0~, fl are taken to be real constants, and assuming m 2 > 1 we have

~ T ~ ( m ~ - - I) = qe-2i% (2 + 2#) T g ( m : - - 1) = ee -2i*~. (8.17)

Now (8.16) may be written

u = [fl(m-lrh -k in) -k o~ exp [+o~(b -- d ) . x] (m -k in) exp {ico(c - - a ) . x}]

• exp ioJ{(a + ib ) . x - - t}, (8.18)

where

a q- ib = T~d~'(m q- in), (8.19)

c q- i d ---- Tzei#2(m q- in).

Hence, in the plane of B, the polarisation ellipse will be similarly situated to the ellipse of S for particles initially on the lines

(c -- a ) . x = 2zrq, q ---- 0, - l-1, . . . (8.20)

In particular, the ellipses will be similarly situated to the ellipse of S and similarly situated with respect to each other for particles intially at the intersections of a line (b -- d ) . s = constant and the lines (8.20).

Thus a difference between the motions in the elastic and viscoelastic media is that the 'grids' are rectangles in the elastic case and parallelograms in the viscoelastic case.

w 9. Inhomogeneous Plane Waves in Linear Anisotropie Elastic Materials

The theory of plane waves in homogeneous anisotropic linearly elastic materials was briefly touched upon in w 6. In this section the theory is developed further. First of all the basic equations are recalled. Then the secular equation is considered and the consequences for the polarisation of the waves examined. Next in w 9.2 the mean energy flux vector for a wave train is seen to have a simple geometrical relation with the ellipse of the slowness bivector of the train. In w 9.3 some universal relations are set out. In w 9.4 the possibility of having circularly polarised waves in the particular case of a cubic crystal is briefly explored.


9.1. Wave Polarisations

The propagation condition and the secular equation governing the propagation of inhomogeneous plane waves in an anisotropic homogeneous elastic body are given by equations (6.5) and (6.6). These may be written, on using equation (6.8), as

(Qik -- ~ T-2e-2ir C~ik) Ak = 0, (9.1)

det I Qtk - 0 T-2e-2ir Oikl = 0, (9.2)

where the acoustical tensor Q is given by

Qik(m + in) = Cokt(m t + int) (mj + inj). (9.3)

Using equation (6.4) we see that the acoustical tensor is symmetric.

Let the roots of (9.2) be denoted by T~e ir 0~ = 1, 2, 3. Let the corresponding eigenbivectors be denoted by A ~). Then, in view of the symmetry of Q, it follows that the A's are orthogonal:

A(~)- A~) = 0 (o~, fl =1= ; ~x, fl = 1,2,3) . (9.4)

If no two of the roots are equal, then none of the A ~) may be isotropic so that for the given m + in, circularly polarised waves may not propagate. Also, the plane of the ellipse of A ~) may not be perpendicular to the plane of the ellipse of A ~ and the ellipse of A ~) projected onto the plane of A t~) is an ellipse which is similar to the ellipse of A ~) but whose major axis is perpendicular to the ellipse of A (~). Thus the ellipses of A ~) and A (v) (o~, fl, y ~ ) when projected onto the plane of the ellipse of A (~) are similar and similarly situated ellipses and similar also to the ellipse of A ~) but with major axes orthogonal to the major axis of the ellipse of A (~).

To illustrate this, suppose the ellipse of A ~ has principal semi-axes p and q. Then A (~) are of the form

A ~ p ' q = O ,

A(2)= kei~(~5~+ iqq~.q)+g r, r . p = r . q = O , (9.5)

A (3)= l e i~ (p~ + iq -~q)+ tr,

where h, k, /, g, t, 2, #, ~ are scalars related by

p l 15) d 0'+~) + g t r . r = 0 , (9.6) kl 7p q.

to ensure that A (2) �9 A (3) : 0.

74 M . HAYES

The total displacement in the plane of A O~ may be written

u = {h(p + iq) + ( A + i q - ~ ) [ + k exp i[(# -- 2) + 0)(c -- a) " x]

• exp (--0)(d -- b) . x} + I exp i[(u -- 2) + 0)(e -- a ) . x]

• exp [--0)(f -- b) . x}] )exp ([i0)[a + ib] ) . x -- t} + it], d /

where

(9.7)

a + ib = Tleit~(m + in),

c + i d : T2eir + in), (9.8)

e + if : Taeir + in).

At the points of intersection of the lines

0)(c -- a ) . x + (# -- 2) : 2z~q~, ql : 0, =[= 1 . . . . . (9.9)

0)(e -- a ) - x + (u -- 2) : 27~q2 , q2 = 0, ~= 1 . . . . .

the particle paths will be ellipses whose principal axes are the principal axes of A ~ In general these ellipses will have just that one feature in common.

A similar statement may be made about motion parallel to the plane of A <2) or A ~

9.2. Mean Energy Flux

The energy flux vector r and its mean } are defined by

r t - -~ _ _ t y i V j ,

2nl~o

r i = (0)/27/7) f r i tit. (9.10)

o

It has been shown [8, 9] for a train of inhomogeneous plane waves propagating with slowness S in a homogeneous elastic body that

~. S - = 0, ~. S + = E, (9.11)

where /~ is the mean energy (kinetic plus stored) density. Now S + and S - are conjugate diameters of the ellipse of S and equation (9.11)t expresses the fact that the component of ~ in the plane of S is along the normal to the ellipse "a t " S +. For given (m + in) the three slownesses S(~) have similar and similarly situated ellipses. On each of these at the "points" S(+) in the plane of So), ~o) is along the normal, l~ say. Thus

r~ - - (l~. S+-------~ + 2~(S+ A S~-), or = 1, 2, 3, (no sum on or (9.12)

where 2~ are scalar multipliers. Of course for an isotropic body 2~ = 0.


Also using (9.11)

T(~)?(~) �9 m = (cos ~ ) / ~ ,

T(~)~(~). n = --(sin $~)/~,

( ~ . m) 2 + (r~" n) 2 = j~2T~-2.

For isotropic materials either ~ ---- 0, in which case ~. m = ff~/T, or $~=~r/2 , in which case r . m = 0 , r . n - - - - - -E /T .

(9.13)

[ " n = O,

9.3. Universal Relations

In this section some universal relations involving the phase speeds and attenuation factors are obtained.

On expanding equation (9.3), the acoustical tensor may be written

Oik(mrh q- in) = Cijk t{m21"t l j f i l l - - t l j f l t ~- im(rhtft j + fillhj)} = aik( -mrh - fi),

(9.14)

where the unit vector n has been denoted by h for emphasis. Now the unit vectors (rh 4- h)t/2 are orthogonal and in terms of them

2(filth j + filjnl) = (fil I ~- nl) (filj -~ nj) -- ( f i l l-- nl) (filj - - flj), (9.15)

and hence

Q(mrh + iri) = m2Q(rh) - Q(h) + imQ \ 1/2 ] ~ (9.16)

where the acoustical tensors on the right correspond to homogeneous plane waves. Thus it follows that

Q(rnrh q- ifi) q- Q(mrh - ifi) = 2mZQ(rh) - 2Q(h), (9.17)

f l + h l ,\ - h Q(mrh q- ih) -- Q(mth - i n ) = 2 in /Q [ ~ ) - Q [ " - ~ - ) t " (9. 18)

3

The trace of Q(h)is equal to ~ ~c2(h), the density times the sum of the squares a = l

of the three phase speeds c~,(h) of the homogeneous waves which may propagate along h. Thus taking the trace in equation (9.2) and using (9.16) gives the universal relation

3 ~,~ T~2e-2ic'~[m + in] ~x=l

t l,a + h i t h - - h + - ,mc ,9,9,

76 M. HAYES

Similarly, from (9.17)

3 3 2 - - 2 i # ~ r " {Tg e t m m + ifi] + T~-2e-2 'oC ' [mth - - ifi]} ---- 2 '~ { m 2 c ] ( t n ) - - c ] ( h ) } ,

c~=l or

(9.20)

and from (9.18)

3 TZ2e - 2i~c' mrh -- {TZ2e-2i*~[mrh q- iti] -- [ ih]}

= 2im ~] c 2 2 (9.21) o=1 -Tr-I - co - -Tr -

Now if ~a is any triad of mutually orthogonal unit vectors the sum of the 3 3

squared speeds • ~] c2(~a) is invariant and equal to cool9. Now, using (9.20), by a = l / ~ = 1

replacing rh, h in turn by qa, tle, and using a common value r for m we have

3 3 �9 ~ - - 2 --2iq~ar ^ Y~ {TZEe-2'*~'[rq# -Jr iqe] q- l~, e trqa -- i~e]}

f l w = l e ~ = l

3 3 ---- 2(r 2 -- 1) ~ Y~ c~(~a)

~x=l 8 = 1

----- 2(r 2 -- 1) cuu/p. (9.22)

Finally, if A is known to be an eigenbivector of q corresponding to the root Te +i~, then from (9.1) and (9.16),

Q T - 2 COS 2qbA'/1 = [ m 2 Q i k ( t h ) - - aik(~l)] [ .4 iAk]+ , (9.23)

= - - m [Qik (rh q- h] rh -- h ~T -z sin 2~bA. t

9.4. Circularly Polarised Waves

It was shown in w 5 that a necessary and sufficient condition that a complex symmetric 3 • 3 matrix have an isotropic eigenbivector is that the matrix have a double root. Now if the eigenbivector is isotropic it means that the corresponding wave displacement is circularly polarised. Thus a necessary and sufficient condition for circularly polarised waves in anisotropic media, which are not subject to inter- nal constraints such as inextensibility or incompressibility, is that the secular equation has a double root. This is valid whether we are dealing with homogeneous or inhomogeneous plane waves.

Hence circularly polarised plane homogeneous waves may propagate in the direction n in an anisotropic body if for that direction n two sheets of the slowness surface intersect or touch. MUSGRAVE [5] presents many examples of sections of slowness surfaces. One can see at a glance for which directions the sheets intersect


or touch. Thus for example, for beryl [5, p. 99] circularly polarised homogeneous waves may propagate along the axis of symmetry and along directions which lie in a cone making an angle of (45 ~ with the axis of symmetry. For zinc [5, p. 100] such waves are possible only along the axis of symmetry.

Turning now to inhomogeneous waves consider propagation in an elastic cubic crystal. There are three elastic constants ell , c**, c12 and the equations of motion in standard notation are [5]

c02u 1

r 4- C44Ul,22 4- C44Ul,33 4- (C12 -[- C44 ) (U2,12 "At- U3,13) = Q t3t 2 '

~2U 2 (C12 -[- r Ul,12 -[- C44U2,11 -{- r -]- C44U2,33 4- (C12 At- C44) U3,23 = e "8t 2 ,

~2U 3 (cl~ + c44) (ul,13 + u2,23) + c44u3.H + c44u3.2~ + c11u3,33 : Q --b-i-c-.

Let

m --- m(ml i + m2J),

m 2 + m 2 2 : n 2 4 - n 2 - - 1,

Then the components of Qo[m + in] are

n = n t i + n2j ,

mini 4- mzn2 = O.

Q,1 = m:(c,~ m2 + c44 m2) + 2im(c,~m,n~ 4- com2nz) - (c,~n 2 4- c44n2),

Qlz = [m2mlmz 4- im(mlna + mznO - nln2] (cx2 + c4,,) - - Q21,

022 : mZ(co m2 + c~ m2) 4- 2im(cltm2n2 + c44mln~) - (cHn~ + c44n2),

Q33 : ( m 2 - - 1) c**,

Qta = Qz3 = Qa, = Q32 = 0.

The roots T-Ze -2ir of the secular equation are given by

~ T - 2 e -2 i~ = (m 2 - - 1) c44,

and

( Q T - 2 e - 2 i r 2 - - (9T-2e -2i~) (m 2 - - 1) (Cll ~- c44) 4- 0r : 0 ,

(9.25)

(9.26)

where

0r : m4[CllC44 -~- ((Cll - - r 2 - - (r -~- C44) 2} mlm2 ] 2 2 _+_

+ 2im3[(Cll - - c44) 2 - - (c12 -[- c44) 2] (m 2 -- m E) mln I +

+ m2[(elz 4- e44) z -- (c~2t + e~4) + 6((el~ -- c44) z - - (c,2 4- e44) z) m~n~] +

+ 2tm[(cll -- e44) z - - (c12 + c44) 2] (n~ -- n22) mini +

~_ [CllC44 _]_ ((Ct I - - e44)2 (c12 _]_ C44)2} 2 2 - - nln2] . (9.30)

Equation (9.29) has a double root provided

(m 2 -- 1) 2 (cll 4- c**) 2 = 40r (9.31)

(9.29)

(9.28)

(9.27)

78 M. HAYES

N o w m is real, so that in view of the expression (9.30) for o~, this leads to two equations for m. In general these will not be consistent. However they may be so in certain cases. Here are two such cases.

Case (i), ml : n2 ---- 1, nl : m 2 = 0. Then (9.31) becomes

and hence

[(rn 2 + 1) ( c ~ - - c**)] 2 : 4m 2 ( c , 2 + c**) 2, (9.32)

Thus

m2(cx2 + c**) -4- 2m(c11 - - c44) + (c12 + c**) : 0. (9.38)

The discr iminant is (--/5) given by (9.34). Thus these circularly polarised waves are possible only in the second class o f materials (potass ium fluoride, etc.).

Thus in the cases where /5 < 0 there are four possible real values for m. Corres- ponding to each a circularly polarised wave with

2 e T - 2 e - 2 i * = ( m 2 - - 1) (cn + c44), (9.39)

m2(cl l - - c44) -~- 2m(c12 q- c44) q- ( C l l - - C44) = 0 . (9.33)

The condit ion that the roots for m be real is that the discr iminant /5 (say) given by

fl ---- 4(cl l + c12) (c12 -~ 2C44 - - C11), (9.34)

be non-negative. Referr ing to the table on pages 278-9 of MUSGRAVE'S Crystal Acoustics we see that /5 > 0 for thir teen o f the materials (including a luminium, iron, nickel, silicon, indium, ant imonide) a n d / 5 < 0 for the othei thirteen mater ia ls cited (including potass ium fluoride, sodium bromide, silver bromide) . Incidentally, 15 = 0 if c~2 + 2c4, ----- ct~, in which case the body is isotropic.

Thus in the cases for which /5 > 0, there are four possible real values for m. Corresponding to each a circularly polar ised wave with

2 e T - 2 e -2ir : (m 2 - - 1)(el l "~- C44), (9.35)

where m is given by one of the four roo ts in (9.33), m a y p ropaga te in the mater ia l Typically, the corresponding displacements are of the fo rm

ul = a cos co(Tmx - - t) exp - - toTy,

u2 = a sin co(Tmx -- t) exp - - e~Ty,

u3 ---- 0, (9.36)

where a is a constant , and m satisfies one of the equat ions (9.33).

Case (ii), ml : nl = m2 : --n2 = (2) -�89 Then (9.31) becomes

(m 2 + I) 2 (c12 + c , , ) 2 = 4m2(cl l - - c**) 2. (9.37)


where rn is a root of (9.38) may propagate. Typically the corresponding displacements are of the form

( m ) ( 1 ) ul = a c o s c o T ~ - ~ ( x + y ) - - t e x p - - t o T ~ ( x - - y ) ,

m u2 = a sin o~

u3 ----0. (9.40)

Acknowledgment. I am greatly indebted to my colleague T. J. LAFFEY for many valuable discussions about complex matrices. I thank D. MCCREA for referring me to SYNGE'S paper, D. JUDGE and J. KENNEDY for their interest and help and D. HURLEY for drawing the diagrams.

This work was supported under Grant 19/79 with the National Board for Science & Technology.

References

1. GraBs, J. W., Elements of Vector Analysis, 1881, 1884 (privately printed) ~ pp. 17- 90, Vol. 2, Part 2 Scientific Papers, Dover Publications, New York 1961.

2. SYNGE, J. L., The Petrov Classification of Gravitational Fields. Comm. Dublin Inst. for Adv. Studies, A, No. 15, Dublin, 1966.

3. FRENCH, T. E., & C. J. VIERCK, Engineering Drawing, 9 th ed., McGraw-Hill, New York, 1960.

4. HAMILTON, W. R., Lectures on Quaternions, Hodges & Smith, Dublin, 1853. 5. MUSGRAVE, M. J. P., Crystal Acoustics, Holden-Day, San Francisco, 1970. 6. LUNEaURG, R. K., Mathematical Theory of Optics, Univ. of California Press, Berke-

ley, 1966. 7. RIVLIN, R. S., & R.A. Torn'IN, Electro-magneto-optical Effects, Arch. Rational

Mech. Anal. 7, 434 (1961). 8. SYNG~, J. L., Flux of Energy for Elastic Waves in Anisotropic Media, Proc. R. Ir.

Acad. 58 A, 13 (1956). 9. HAYES, M., Energy flux for trains of inhomogeneous plane waves, Proc. R. Soc.

Lond. A 370, 417 (1980).

Department of Mathematical Physics University College

Dublin

(Received May 24, 1983)

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Inhomogeneous plane waves