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Inhomogeneous Plane Waves in Incompressible Elastic Materials
Wave Phenomena: Modern Theory and Applications
C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Hollaml), 1984
175
INHOMOGENEOUS PLANE WAVES I N INCOMPRESSIBLE ELASTIC MATERIALS Michael Hayes Department of Mathematical Physics University College Dub1 i n
Gibbs b i v e c t o r s [l] a r e used t o g i v e a d e s c r i p t i o n of inhomogeneous plane waves i n a n i s o t r o p i c homogeneous
I t i s shown
incompressible l i n e a r e l a s t i c m a t e r i a l s .
t h a t i f t h e slowness b i v e c t o r i s n o t i s o t r o p i c then t h e a c o u s t i c a l t e n s o r has double r o o t s i f a c i r c u l a r l y polari s e d wave propagates, and conversely, i f t h e a c o u s t i c a l t e n s o r has double r o o t s then t h e corresponding wave i s c i r c u l a r l y polarised. INTRODUCTION
1.
This i s a sequel t o a previous paper
121 where t h e corresponding problem
of inhomogeneous plane waves i n compressible a n i s o t r o p i c e l a s t i c and compressi b l e i s o t r o p i c v i s c o e l a s t i c m a t e r i a l s was considered. t h a t instead of specifying a d i r e c t i o n
a s i n d e a l i n g with homogeneous
p l a n e waves, f o r inhomogeneous plane waves a p a i r
n
where
i s a u n i t v e c t o r and
m
The e s s e n t i a l i d e a i s
@, E)
is perpendicular t o it.
is specified,
Associated with
1. + IIJ. Thus i n s t e a d o f s p e c i f y i n g a d i r e c t i o n 3, a d i r e c t i o n a l e l l i p s e i s s p e c i f i e d . t h i s p a i r i s an e l l i p s e , t h e d i r e c t i o n a l e l l i p s e o f t h e b i v e c t o r
The corresponding slowness b i v e c t o r and
4
I t i s seen t h a t
are real.
is written Te"
S = Te [m
+
11.)where T
i s determined from t h e s e c u l a r
equation and t h e corresponding amplitude b i v e c t o r i s determined as an eigenb i v e c t o r of t h e a c o u s t i c a l t e n s o r . I t i s seen, i n t h e u s u a l way t h a t t h e r e a r e i n general j u s t two waves which may propagate f o r a given choice of t h e d i r e c t i o n a l e l l i p s e of t h e slowness bivector.
The corresponding amplitude b i v e c t o r s
arthogonal:
A.B =
0.
&,
(say) a r e mutually
Hence f o r a given d i r e c t i o n a l e l l i p s e of t h e slowness
b i v e c t o r t h e p a r t i c l e displacements corresponding t o t h e two waves l i e on p l a n e s which may not be orthogonal.
Also t h e e l l i p s e s corresponding t o
e i t h e r displacement when p r o j e c t e d onto t h e p l a n e of t h e d i r e c t i o n a l e l l i p s e
M. Hayes
of t h e slowness b i v e c t o r a r e s i m i l a r and s i m i l a r l y s i t u a t e d with major a x i s perpendicular t o t h e major a x i s of t h e e l l i p s e of t h e slowness b i v e c t o r . On t h e assumption t h a t t h e slowness b i v e c t o r i s not i s o t r o p i c , it i s shown
t h a t t h e a c o u s t i c a l t e n s o r has double r o o t s i f a c i r c u l a r l y p o l a r i s e d wave propagates.
Conversely, i f t h e a c o u s t i c a l t e n s o r has double r o o t s then t h e
corresponding wave i s c i r c u l a r l y p o l a r i s e d .
Thus a s t h e r o o t s coalesce t h e
amplitude b i v e c t o r s merge and t h e o r t h o g o n a l i t y condition A.A- = -
0, so t h a t t h e wave i s c i r c u l a r l y p o l a r i s e d .
A.B
= 0
becomes
The condition t h a t t h e
r o o t s be double leads t o a q u a r t i c with complex c o e f f i c i e n t s .
If t h i s q u a r t i c
has a r e a l r o o t then a c i r c u l a r l y p o l a r i s e d inhomogeneous plane wave may propagate with a p p r o p r i a t e slowness.
The superposition of t h e displacements corresponding t o t h e p a i r of waves with given d i r e c t i o n a l e l l i p s e i s a l s o considered.
For motion p a r a l l e l t o t h e
plane o f t h e amplitude b i v e c t o r of e i t h e r wave it i s seen t h a t f o r e q u i d i s t a n t p o i n t s on c e r t a i n l i n e s t h e t o t a l displacement e l l i p s e s a r e similar and s i m i l a r l y s i t u a t e d with r e s p e c t t o each o t h e r and with r e s p e c t t o t h e p r o j e c t i o n of t h e slowness b i v e c t o r onto t h e plane of t h e amplitude b i v e c t o r .
52. -
EQUATIONS OF MOTION The c o n s t i t u t i v e equations a r e taken t o b e
=
u . .
1,1
where
djik,t
Here t h e s t r e s s e s a r e denoted by u.
(2.11
'
0 ,
dijk9. =
displacements by
' d i j k e Uk,,t
-p 6 . ij
t.. = 11
dk,tij
=
=
dij,tk
*
t i j , t h e e l a s t i c c o n s t a n t s by
and t h e comma denotes d i f f e r e n t i a t i o n : u
The summation convention i s used throughout.
Also
p
dijka.
,
the
~ E, auk/axk ~
.
is a scalar, the
h y d r o s t a t i c p r e s s u r e which i s t o be determined from t h e equations of motion and t h e boundary conditions.
Equation ( 2 . 2 ) expresses t h e f a c t t h a t a l l deformations
i n t h e body a r e i s o c h o r i c .
The equations of motion a r e given by 2 a u
at.. *
=
I
P
y
i
at
i n t h e absence o f body f o r c e s , where
(2.41
>
p
is t h e material density.
Inserting
177
lnhomogeneous Plane Waves
(2.1) i n t o ( 2 . 4 ) g i v e s dijki
13. -
a 2u i / a t
P
k , a j - P , ~=
.
2
ISHOMOGEiiEOUS I'LAiiE WAVES
Now i t i s assumed t h a t t h e displacements
a r i s e due t o t h e propagation ui of an i n f i n i t e t r a i n of inhomogeneous p l a n e waves i n t h e body. Thus
Here
A i s a bivector:
&
z-.~
&+ +
=
The planes of constant phase a r e amplitude a r e = constant.
I&-
2,
and so i s
s+.x=
t h e slowness b i v e c t o r .
c o n s t a n t , and t h e planes of constant
The period of t h e wave is
Equation
27r/w.
(3.1) d e s c r i b e s an i n f i n i t e t r a i n of e l l i p t i c a l l y p o l a r i s e d inhomogeneous
plane waves.
For f i x e d
x(= x*
say) t h e displacement v e c t o r
’1
A+ exp(- w
A-expc-
uZ-.f).
&,
z-.~*) and
e l l i p s e which i s s i m i l a r and s i m i l a r l y s i t u a t e d t o t h e e l l i p s e of t h e e l l i p s e whose conjugate semi-diameters a r e
l i e s on an namely
A t any given time t h e displacement v e c t o r i s along one semi-
diameter of t h e e l l i p s e and t h e p a r t i c l e v e l o c i t y i s p a r a l l e l t o i t s conjugate semi-diameter.
t
As
t h e e l l i p s e i s from
i n c r e a s e s t h e sense i n which t h e p a r t i c l e moves along
&+ t o
A-.
The p l a n e of p o l a r i s a t i o n i s determined by
t h e plane of t h e e l l i p s e of t h e amplitude b i v e c t o r Circularly polarised i f Linearly p o l a r i s e d i f
Te"
S = where
5
The slowness b i v e c t o r
T
vector.
g
angles t o problem.
and
$
(5+,):I
are real.
and t a k i n g
m.:
.
The wave i s
-A.A _
=
0 ;
(3.2)
&A&
=
0.
(3.33
When equation ( 3 . 2 ) is s a t i s f i e d t h e b i v e c t o r "nullt'.
A -
A -
i s s a i d t o be " i s o t r o p i c "
or
may be w r i t t e n
__
= 0, n.n = 1
,
(3.4)
By f i x i n g on a p a r t i c u l a r choice of t h e u n i t
5 t o be o f a r b i t r a r y [chosen) magnitude and a t r i g h t
1 , i t will be seen t h a t T and 4 a r e determined by an eigenvalue 5 and may be regarded a s t h e p r i n c i p a l axes of an e l l i p s e .
I 78
M. Hayes
Then
S/T
(z+,g-) i s
and
13i s
el$@ +
=
a p a i r of conjugate semi-diameters of t h i s e l l i p s e
t h u s a p a i r of conjugate semi-diameters of a similar and s i m i l a r l y
s i t u a t e d e l l i p s e whose major and minor axes a r e
(m .+):1
axes of t h e e l l i p s e of
times t h e major and minor
Also
S+
= T(cos$m
-
sin$n-),
S-
= T(sin@
+
cos@),
The angle
T
2 3 ; = T(cos2$m2 + s i n $)
12-1 =
,
z f .
T(sin2$m2 + cos $)
(3.5)
between t h e planes o f constant phase and t h e planes of
8
constant amplitude i s given by tan 8
=
[Z]
m
(3.6)
(m 2 - l ) c o s $ s i n $
The condition (2.2) gives
A.
2
=
0.
(3.7)
This means i n general t h a t t h e e l l i p s e of t h e amplitude b i v e c t o r
2
e l l i p s e o f t h e slowness b i v e c t o r l a r t o each o t h e r .
S
A
and t h e
may not l i e on p l a n e s which a r e perpendicu-
Also, t h e p r o j e c t i o n of e i t h e r
(5 say)
upon t h e plane o f
i s an e l l i p s e whose aspect r a t i o ( r a t i o of major t o minor a x i s ) i s equal t o
t h e aspect r a t i o of t h e e l l i p s e of
2.
t h e minor a x i s o f t h e e l l i p s e of
5
and whose major a x i s i s perpendicular t o
A A. A =
Exceptionally it may happen that
i s a s c a l a r and a l s o t h a t polarised.
84. -
I t i s seen i n
2. 55
=
0
i s p a r a l l e l t o 2: A = as, where so t h a t t h e wave i s c i r c u l a r l y
a
that t h i s i s a p o s s i b i l i t y f o r i s o t r o p i c bodies.
THE PROPAGATION CONDITION
Here it i s assumed t h a t t h e d i r e c t i o n a l e l l i p s e of eigenvalue equation f o r t h e determination of that t h e r e a r e j u s t two non-zero eigenvalues.
Te"
5
i s given.
i s obtained.
The
I t i s seen
I f t h e s e a r e n o t equal it i s
shown t h a t t h e corresponding eigenbivectors a r e orthogonal. Now i f follows t h a t
p
has t h e form (3.1), t h e n from the equations o f motion (2.5) must have a s i m i l a r form.
Thus write
it
179
lnhomogeneous Plane Waves
p
=
P expiw
(5. x
.
-t)
(4.1)
I n s e r t i n g t h e expressions (3.1) and (4.1) i n t o t h e equations of motion and P
using c3.5) t o e l i m i n a t e
2. 2 SO,
and assuming
leads t o
Using (3.4) t h i s may be w r i t t e n (4.3) where
Q.
lk
*
m,
1 ~ k R ks j
%i
=
'
=m+ip,m .n = O , n .n = l .
S Thus, i f
* *
= d..
(4.41
a r e given, equation (4.3) i s an eigenvalue problem f o r t h e
determination of
and t h e corresponding eigenbivector
T, $
A.
The propagation condition (4.3) l e a d s t o t h e s e c u l a r equation d e t (pT-' f o r the determination o f det(Q)=0
zero.
e-21'
-
6ik
[Te'$)
= 0,
f o r given
*
*
s .s - * *
since det
6.i k)
<.
) = 0,
rns m
eigenbivectors be
T-2e-21' and
T 2 e210
be denoted by Alp-', respectively.
I t i s c l e a r from (4.4)
and hence one r o o t of (4.5)
Thus equation (4.5) i s a q u a d r a t i c i n Let t h e r o o t s
(4.5)
Then
that
is
.
h2p
-1
.
Let the corresponding
(4.6) (4.7) I t i s now shown t h a t if h l
11.B
=
A2
0.
,
then (4.8)
180
M. Hayes
Now s i n c e
A. S* = B. BiQikAk
4
since
*
-
i s symmetric.
= 0, i t follows from equation (4.4) t h a t
A.Q. 1 ik Bk
X1 f
e l l i p s e of
B
A. g
(4.9)
(4.10)
= 0.
A,
and
The e l l i p s e s of
A
S
a r e on planes no two of which may be
S
and o f
when p r o j e c t e d on t h e plane of t h e
a r e s i m i l a r and s i m i l a r l y s i t u a t e d .
r o t a t e d through a quadrant. t i o n s of t h e e l l i p s e s o f t i o n s of t h e e l l i p s e s o f
B &
Examples o f such t r i a d s [3],
The p r o j e c t i o n s a r e s i m i l a r
i n t h e p l a n e of
and s i m i l a r l y s i t u a t e d t o t h e e l l i p s e of
For example
,
X2,
Hence t h e e l l i p s e s of orthogonal.
= 0
Hence from (4.6) and (4.7)
(A1, - X2) Thus, f o r
- AiQikBk
BiQikAk
3
when they a r e
A similar statement may be made about t h e projec-
and
5
and
upon t h e plane o f upon t h e plane of
A, B,
A
and about t h e projec-
2.
s a t i s f y i n g (4.11) a r e e a s i l y constructed.
take
c4.12) a, 6, 6
where
are scalars, satisfying
ci
2
-
B2
=
Returning t o equation (4.3) i t i s seen t h a t i f eigenbivector f o r given of
Te'+
i s known t o be an
and i s not i s o t r o p i c , then t h e corresponding value
i s given by p ~ - 2 e-21'
since
*
1.
A.5=
0.
If
&
A1 . A .1 = Q i k ~ i ~ =k Qi kA iA k '
i s i s o t r o p i c , then it w i l l be shown i n
corresponding eigenvalues a r e double and hence t h e v a l u e of half t he t race of
(4.13)
Te"
86
that the
i s given by
Q/p:
2pT
-2 -214
e
=
A
Qii*
(4.14)
181
Inhomogeneous Plane Waves
55. -
ISOTROPIC MATERIALS
In t h i s s e c t i o n t h e propagation of inhomogeneous plane waves i n i s o t r o p i c homogeneous incompressible e l a s t i c bodies i s considered. o f waves.
There a r e two c l a s s e s
In t h e f i r s t t h e slowness b i v e c t o r i s i s o t r o p i c , t h e waves a r e
c i r c u l a r l y p o l a r i s e d and t h e s o l u t i o n i s u n i v e r s a l i n t h e sense t h a t t h e slowness does not depend upon t h e shear modulus which d e s c r i b e s t h e e l a s t i c response of
In t h e second c l a s s , t h e p r e s s u r e term i s zero, t h e waves a r e
the material.
t r a n s v e r s e , and t h e amplitude b i v e c t o r may be chosen so t h a t t h e wave is c i r c u l a r l y polarised. The c o n s t i t u t i v e equation (2.1) now reads t.
=
Ij
p
where
i s a constant.
-
p6ij
+
p(ui,j
,
+ uj,i)
(5.1)
Using t h i s and equation (4.1) i n equation (2.4)
leads
to
Now
AiSi
=
0
and hence
either
ci) (5.3)
where
a
is a scalar
Cii) =
P
In
c a s e Ci),
usj sj
0 ,
p
s.s-
=
aSi expiw@. 5 - t )
=
-
=
0
ipwaexpwC2.
,
a
This i s a c i r c u l a r l y p o l a r i s e d wave. p l a n e s of constant amplitude:
5'. 5-
r e s t r i c t i o n on t h e magnitude of case ( i i ) ,
.
(5.4)
we have t h e s o l u t i o n ui
In
= p
the solution is
2.
,
-t) , arbitrary.
Planes of constant phase a r e orthogonal t o = 0.
Also
Is+( =
15-1
.
There i s no
I t i s independent of t h e shear modulus P
.
M. Hayes
182
A
where
where
From (5.6)2
0.
m2 < 1 ,
Thus, i f
and i f
A. 5 =
i s any b i v e c t o r s a t i s f y i n g
m2 > 1
y, y'
,
a r e a r b i t r a r y and 2.2 = 1,
1.11=
r.fl= 0 .
I n both c a s e s t h e
planes o f constant phase a r e orthogonal t o t h e planes of constant amplitude.
m
5
c Cp/p) homogeneous t r a n s v e r s e wave i s recovered. As
-+
a,
then from ( 5 . 9 ) ,
For given " +
15,and m
-+
a, m, A
s a t i s f y i n g (S.?),
the waves are c i r c u l a r l y p o l a r i s e d .
-f
y
12 + or
and t h e usual
y'f,
y'
may be chosen s o t h a t
Thus, from equation (5.8),
take
y
given
bY (5.10)
Then (5.11) The displacement v e c t o r lies on a
corresponds t o a c i r c u l a r l y p o l a r i s e d wave. c i r c l e i n t h e p l a n e spanned by
m
2
> 1,
take
yt
given by
m
and ;{
+- (-2-1) 1 4E} m
.
Similarly, f o r
(5.12)
Then (5.13)
183
Inhomogeneous Plane Waves
corresponds t o a c i r c u l a r l y p o l a r i s e d wave, t h e c i r c l e of p o l a r i s a t i o n l y i n g i n a plane spanned by
86.
m m-m
1 and
1 4 IS}
{ -i (1 - -2)
m
STRUCTURE OF THE ACOUSTICAL TENSOR.
.
CIRCULARLY POLARISED WAVES.
Here t h e s t r u c t u r e o f t h e a c o u s t i c a l t e n s o r i s considered more f u l l y .
In
i s not i s o t r o p i c , it i s shown t h a t i f t h e s e c u l a r
particular,assuming t h a t
equation has a double r o o t then a c i r c u l a r l y p o l a r i s e d wave may propagate i n t h e material.
Also, it i s shown, assuming
S
.S 9
0, t h a t i f a c i r c u l a r l y p o l a r i s e d
wave propagates then t h e s e c u l a r equation has a double r o o t .
i s n o t i s o t r o p i c i s made, f o r , i n g e n e r a l , i n
The assumption t h a t
d e r i v i n g t h e form of t h e a c o u s t i c a l t e n s o r expressions o f t h e form enter.
I t was seen i n
o b t a i n a s o l u t i o n with
35
[s@SJ/(s.s)
f o r an i s o t r o p i c m a t e r i a l t h a t i t i s p o s s i b l e t o
5.2 =
0.
However, i n g e n e r a l , i f t h e d e t a i l a d s t r u c t u r e
of t h e e l a s t i c c o e f f i c i e n t s i s n o t given then it is e s s e n t i a l t o assume t h a t
S
i s not i s o t r o p i c . Now s i n c e any b i v e c t o r
where
a
may be w r i t t e n [l]
5
=
+
ib), a.b =
0,
i s a s c a l a r , i n general complex, we may, without l o s s of g e n e r a l i t y ,
write
where
T,
0,
respectively.
m
a r e r e a l and Then, assuming,
defined by equation (4.4),
i,
2.S
a r e u n i t v e c t o r s along t h e x and y axes 2 0 so that m $: 1, it i s seen t h a t Q,
may b e w r i t t e n * *
and has components given by
184
M. Hayes
Thus t h e matrix
4
has t h e form
where
One eigenvalue of
X1, AZ,
(0)
191
i s zero s i n c e
and t h e o t h e r two, denoted by
a r e given by
A,
t h e corresponding e i g e n b i v e c t o r s being
where
= 0
given by
tl, t2 a r e given by
Now tl t2 =
and hence
m2
-
1
,
(6.9)
185
Inhomogeneous Plane Waves
=
A . B-
(6.10)
0.
A1so
A.2 -
B.5
0,
=
=
(6.11)
0.
In t h e s p e c i a l case when (6.12) it follows from equation (6.4) t h a t
(:i ; Q3i) 6
Cii)
=
and t h e non-zero eigenvalues a r e bivectors
CO,
0, y)
Q,,
(1, m, 0)
and
and
(6.13)
,
a + imB
where
w i t h corresponding eigen-
is arbitrary.
y
The f i r s t of
t h e s e corresponds t o a l i n e a r l y p o l a r i s e d wave, t h e second t o an e l l i p t i c a l l y p o l a r i s e d wave.
(Recall
m
2
9 1).
CIRCULARLY POLARISED WAVES
F i n a l l y , the p o s s i b i l i t y of equa
eigenva ies i s considered.
The eigenvalues a r e equal provided L
a
(6.14)
so that (6.15) Then A1
=
From equation (6.4),
A2
=
(a + imB + Q3,)/2
(Q) i s now given by
.
(6.16)
186
M. Hayes
where the upper and lower s i g n s correspond t o t h e upper and lower s i g n s i n (6.15).
C(say) i s given by
The corresponding eigenbivector
(6.18) It is clear that
5.C =
0,
and thus the wave corresponding t o t h e double r o o t
is c i r c u l a r l y polarised. I t may be noted t h a t t h e c i r c l e s of p o l a r i s a t i o n corresponding t o t h e d i f f e r e n t s i g n s i n equation (6.15) and t h e r e f o r e a l s o i n (6.18), a r e g r e a t c i r c l e s i n t h e u n i t sphere.
They a r e described i n opposite senses.
most e a s i l y seen i n t h e s p e c i a l c a s e
_C = _i r i-k
are unit circles i n the
-
xz
if it i s p o s s i b l e
-
when m = 0.
This i s Then
plane, described i n opposite senses.
Also, r e t u r n i n g t o equation (.6.14), which i s t h e condition f o r double r o o t s ,
this condition may be w r i t t e n , using equation (6.5),
CQ,,
+
2 1 m Q12
-
m
2
Q2,
-
-
(1
2
as 2
m 1 Q33}
(6.19) This is a q u a r t i c i n real root for
m
m
with complex c o e f f i c i e n t s .
I f t h i s equation has a
then t h e r e a r e two corresponding c i r c u l a r l y p o l a r i s e d waves.
In general it w i l l not possess r e a l r o o t s f o r
m.
Assuming t h a t 5 i s not i s o t r o p i c it has been shown t h a t i f t h e a c o u s t i c a l tensor Q) has double r o o t s then t h e corresponding eigenbivector i s i s o t r o p i c and accordingly a c i r c u l a r l y p o l a r i s e d inhomogeneous wave may propagate. shown, assuming t h a t possesses an
Now, it i s
S
i s not i s o t r o p i c , t h a t i f t h e a c o u s t i c a l t e n s o r (Q) i s o t r o p i c eigenbivector then 9 has a double eigenvalue.
Now from (4.4) t h e a c o u s t i c a l t e n s o r
9 n
has t h e form (6.20)
cij
=
6ij
-
SiSj
‘mSm
(6.21)
187
Inhomogeneous Plane Waves
and (6.22)
0
Without l o s s of g e n e r a l i t y , l e t
be given by
(6.23)
i s t o be an i s o t r o p i c eigenbivector of
Now
@)
so t h a t it s a t i s f i e s (6.24)
C6.25)
(6.26)
without l o s s of g e n e r a l i t y .
_A.S_
5
= 0,
I t i s assumed that
2.5
0
and since
may be assumed t o have t h e form (6.27)
where
i s some s c a l a r .
6
6
2.2
Now
i’ 6 -1
2
=
-1
-6
and
=
(
(6 -1)a - t b -6c ( 6 * + l ) b -la -16c
-:(a
62,
=
and
&:)
-1
62+1 -16
0
2 (6 -1)b -if -6g
- t b +(6’
+tb)
,
(6.28)
2
(6 - 1 ) ~- t g -6h
+ l ) f -16g
-6(b + t f )
-6(c
+tg)
(6.29)
Then from (6.24) it follows t h a t
,
2tb
=
f - a
61
=
6(a + tb) - cc + 18)
,
(6.301
188
M. Hayes
and (6
-1)a - t b -6c
(S2 + l ) b - i a -16c
62G =
[-&:a On expanding r o o t Csince
17. -
+
-1
a -6g
2 6 (a+2tb)+Ca+ib)-t6g
= 0,
(6
2
-1)c-Ig-dh
-tc+(6
-
-t6Ca + ib)
tb)
IQ - e r l
191
( ~ 5+ l~) b
2
6(c + 1g)
i t i s seen t h a t t h e s e c u l a r equation has a zero
X
= 0 ) , and a double r o o t
given by ( 6 . 3 0 ) 2 .
SUPERPOSITION OF WAVE TRAINS WITH COMMON DIRECTIONAL ELLIPSE
In t h i s s e c t i o n t h e s u p e r p o s i t i o n of two wave t r a i n s with common d i r e c t i o n a1 e l l i p s e i s considered.
S
For given f i x e d
p a r a l l e l t o t h e plane of t h e e l l i p s e of
*
t h e motion i n any plane
is examined.
The displacement o f
a p a r t i c l e a r i s e s a s a l i n e a r combination of t h e two b a s i c displacements and (The sum of two b i v e c t o r s i s
hence, i n general, w i l l a l s o be e l l i p t i c a l . also a bivector.)
The e l l i p s e s f o r d i f f e r e n t p a r t i c l e s w i l l g e n e r a l l y d i f f e r .
However, it i s seen t h a t t h e e l l i p s e s a t c e r t a i n p o i n t s on a c e r t a i n l i n e are i d e n t i c a l both i n o r i e n t a t i o n a n d i n t e r m s of t h e lengths of t h e i r p r i n c i p a l axes. I t i s assumed t h a t
where
a,,
B,
a, b
A.B=
5, A,
a r e given by
I t i s e a s i l y checked that
a r e assumed t o be r e a l .
A.S-1 = B.S-2 = 0. Also Tle 161 , T2e1" of t h e s e c u l a r equation ( 4 . 5 ) f o r given 2 = corresponding eigenbivectors a r e The t o t a l displacement
E* =
A
and
a r e assumed t o be t h e s o l u t i o n s (ol/B)
u_"Csay) in t h e p l a n e of
A expiw C Tle'"(crx
+ iBy]
+ {B - (B.n) - - n) expiu{T2e"2(ax
+
ii,
and t h e
respectively.
-
&
may be w r i t t e n
t 1 + ’By)
-
t)
,
(7.21
189
lnhomogeneous Plane Waves
A
2 is t h e u n i t normal t o t h e p l a n e of
where
given by
- s i n d i + acossk
z =
2 2 2 l ( s i n S+a cos 6)
(7.3)
’
Now
-
bfisind[ii
=
Bcos6
(1+B c o s S )
IacosSj + sin8k) ]
,
(7.41
and
-s* -
=
(s”.fl)” 5 6
l[lL
-
’c0s6 ~ a c o s +~ si i n & & )I . 2 2 (1+B cos 6)
Notice t h a t t h e e l l i p s e s of t h e p r o j e c t i o n s of
and
5
(7.5)
upon t h e plane of
are s i m i l a r and s i m i l a r l y s i t u a t e d . Now
u*
may be w r i t t e n
+ a-’b@sin6{iL
-
BCOSG
(1+6 cos 6)
(acossi
+
sin6k) 1 exp
IWP]
,
(7.61
where (7.71 I t i s c l e a r t h a t f o r given
x, y,
t h e displacement
%* l i e s on an e l l i p s e .
The axes of t h e e l l i p s e w i l l v a r y from p o i n t t o p o i n t f o r p o i n t t o point. the ellipse a t
However, if exp(iwb) M
has p r i n c i p a l axes along
t h e ellipses at the points axes of t h e e l l i p s e of
5
M
u v a r i e s from
i s purely r e a l a t a point
i
M(say)
and (ctcosdi + sin@).
then
Thus
have p r i n c i p a l axes p a r a l l e l t o t h e p r i n c i p a l
and a l s o of course t o t h e p r i n c i p a l axes of t h e
e l l i p s e of t h e p r o j e c t i o n of
2.
Of course t h e s e e l l i p s e s a t t h e p o i n t s
M
will not be i d e n t i c a l but they do have t h a t one f e a t u r e i n common, t h a t t h e i r p r i n c i p a l axes a r e p a r a l l e l .
The p o i n t s
M
a r e any p o i n t s o n t h e e q u i d i s t a n t
190
M. Hayes
d
parallel lines
2
:
where
k,
:
wk[a[cos$)x $
-
p(sin$)y]
qs,
=
q
0,
...
+1, f 2 ,
(7.8)
a r e given by T e 2
‘42
- Tle
Consider t h e p o i n t s o f i n t e r s e c t i o h
32 with t h e l i n e s
:
.
(7.9) M*(say)
(asin+)x + ( ~ c o s $ ] y = The term
exp(iwu)
p o i n t s and has t h e same value a t each.
of a l i n e
7
constant
(7.10)
i s p u r e l y r e a l a t each of t h e s e
Thus f o r t h e s e p o i n t s
M
*
the
displacement e l l i p s e s a r e a l l i d e n t i c a l and are s i m i l a r and s i m i l a r l y s i t u a t e d t o t h e e l l i p s e of t h e p r o j e c t i o n of
ACKNOWLEDGMENT.
5
upon t h e p l a n e of
A.
This work was supported by t h e National Board f o r Science
6 Technology under Grant 19/79.
191
Inhomogeneous Plane Waves
REFERENCES [l]
Gibbs, J.W. Elements of Yector Analysis, 1881, 1884 ( p r i v a t e l y p r i n t e d ) E pp 17 - 90, Vol. 2 , p a r t 2 , S c i e n t i f i c Papers, Dover Publications, New York, 1961.
[Z]
Hayes, M. Inhomogeneous Plane Waves. appear)
[3]
Synge, J . L . The Petrov C l a s s i f i c a t i o n of G r a v i t a t i o n a l F i e l d s . Dublin I n s t . f o r Adv. S t u d i e s , No. 15, Dublin, 1964.
Arch. R a t ' l Mech. Anal. 1984 ( t o
A,
Corn.
C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Hollaml), 1984
175
INHOMOGENEOUS PLANE WAVES I N INCOMPRESSIBLE ELASTIC MATERIALS Michael Hayes Department of Mathematical Physics University College Dub1 i n
Gibbs b i v e c t o r s [l] a r e used t o g i v e a d e s c r i p t i o n of inhomogeneous plane waves i n a n i s o t r o p i c homogeneous
I t i s shown
incompressible l i n e a r e l a s t i c m a t e r i a l s .
t h a t i f t h e slowness b i v e c t o r i s n o t i s o t r o p i c then t h e a c o u s t i c a l t e n s o r has double r o o t s i f a c i r c u l a r l y polari s e d wave propagates, and conversely, i f t h e a c o u s t i c a l t e n s o r has double r o o t s then t h e corresponding wave i s c i r c u l a r l y polarised. INTRODUCTION
1.
This i s a sequel t o a previous paper
121 where t h e corresponding problem
of inhomogeneous plane waves i n compressible a n i s o t r o p i c e l a s t i c and compressi b l e i s o t r o p i c v i s c o e l a s t i c m a t e r i a l s was considered. t h a t instead of specifying a d i r e c t i o n
a s i n d e a l i n g with homogeneous
p l a n e waves, f o r inhomogeneous plane waves a p a i r
n
where
i s a u n i t v e c t o r and
m
The e s s e n t i a l i d e a i s
@, E)
is perpendicular t o it.
is specified,
Associated with
1. + IIJ. Thus i n s t e a d o f s p e c i f y i n g a d i r e c t i o n 3, a d i r e c t i o n a l e l l i p s e i s s p e c i f i e d . t h i s p a i r i s an e l l i p s e , t h e d i r e c t i o n a l e l l i p s e o f t h e b i v e c t o r
The corresponding slowness b i v e c t o r and
4
I t i s seen t h a t
are real.
is written Te"
S = Te [m
+
11.)where T
i s determined from t h e s e c u l a r
equation and t h e corresponding amplitude b i v e c t o r i s determined as an eigenb i v e c t o r of t h e a c o u s t i c a l t e n s o r . I t i s seen, i n t h e u s u a l way t h a t t h e r e a r e i n general j u s t two waves which may propagate f o r a given choice of t h e d i r e c t i o n a l e l l i p s e of t h e slowness bivector.
The corresponding amplitude b i v e c t o r s
arthogonal:
A.B =
0.
&,
(say) a r e mutually
Hence f o r a given d i r e c t i o n a l e l l i p s e of t h e slowness
b i v e c t o r t h e p a r t i c l e displacements corresponding t o t h e two waves l i e on p l a n e s which may not be orthogonal.
Also t h e e l l i p s e s corresponding t o
e i t h e r displacement when p r o j e c t e d onto t h e p l a n e of t h e d i r e c t i o n a l e l l i p s e
M. Hayes
of t h e slowness b i v e c t o r a r e s i m i l a r and s i m i l a r l y s i t u a t e d with major a x i s perpendicular t o t h e major a x i s of t h e e l l i p s e of t h e slowness b i v e c t o r . On t h e assumption t h a t t h e slowness b i v e c t o r i s not i s o t r o p i c , it i s shown
t h a t t h e a c o u s t i c a l t e n s o r has double r o o t s i f a c i r c u l a r l y p o l a r i s e d wave propagates.
Conversely, i f t h e a c o u s t i c a l t e n s o r has double r o o t s then t h e
corresponding wave i s c i r c u l a r l y p o l a r i s e d .
Thus a s t h e r o o t s coalesce t h e
amplitude b i v e c t o r s merge and t h e o r t h o g o n a l i t y condition A.A- = -
0, so t h a t t h e wave i s c i r c u l a r l y p o l a r i s e d .
A.B
= 0
becomes
The condition t h a t t h e
r o o t s be double leads t o a q u a r t i c with complex c o e f f i c i e n t s .
If t h i s q u a r t i c
has a r e a l r o o t then a c i r c u l a r l y p o l a r i s e d inhomogeneous plane wave may propagate with a p p r o p r i a t e slowness.
The superposition of t h e displacements corresponding t o t h e p a i r of waves with given d i r e c t i o n a l e l l i p s e i s a l s o considered.
For motion p a r a l l e l t o t h e
plane o f t h e amplitude b i v e c t o r of e i t h e r wave it i s seen t h a t f o r e q u i d i s t a n t p o i n t s on c e r t a i n l i n e s t h e t o t a l displacement e l l i p s e s a r e similar and s i m i l a r l y s i t u a t e d with r e s p e c t t o each o t h e r and with r e s p e c t t o t h e p r o j e c t i o n of t h e slowness b i v e c t o r onto t h e plane of t h e amplitude b i v e c t o r .
52. -
EQUATIONS OF MOTION The c o n s t i t u t i v e equations a r e taken t o b e
=
u . .
1,1
where
djik,t
Here t h e s t r e s s e s a r e denoted by u.
(2.11
'
0 ,
dijk9. =
displacements by
' d i j k e Uk,,t
-p 6 . ij
t.. = 11
dk,tij
=
=
dij,tk
*
t i j , t h e e l a s t i c c o n s t a n t s by
and t h e comma denotes d i f f e r e n t i a t i o n : u
The summation convention i s used throughout.
Also
p
dijka.
,
the
~ E, auk/axk ~
.
is a scalar, the
h y d r o s t a t i c p r e s s u r e which i s t o be determined from t h e equations of motion and t h e boundary conditions.
Equation ( 2 . 2 ) expresses t h e f a c t t h a t a l l deformations
i n t h e body a r e i s o c h o r i c .
The equations of motion a r e given by 2 a u
at.. *
=
I
P
y
i
at
i n t h e absence o f body f o r c e s , where
(2.41
>
p
is t h e material density.
Inserting
177
lnhomogeneous Plane Waves
(2.1) i n t o ( 2 . 4 ) g i v e s dijki
13. -
a 2u i / a t
P
k , a j - P , ~=
.
2
ISHOMOGEiiEOUS I'LAiiE WAVES
Now i t i s assumed t h a t t h e displacements
a r i s e due t o t h e propagation ui of an i n f i n i t e t r a i n of inhomogeneous p l a n e waves i n t h e body. Thus
Here
A i s a bivector:
&
z-.~
&+ +
=
The planes of constant phase a r e amplitude a r e = constant.
I&-
2,
and so i s
s+.x=
t h e slowness b i v e c t o r .
c o n s t a n t , and t h e planes of constant
The period of t h e wave is
Equation
27r/w.
(3.1) d e s c r i b e s an i n f i n i t e t r a i n of e l l i p t i c a l l y p o l a r i s e d inhomogeneous
plane waves.
For f i x e d
x(= x*
say) t h e displacement v e c t o r
’1
A+ exp(- w
A-expc-
uZ-.f).
&,
z-.~*) and
e l l i p s e which i s s i m i l a r and s i m i l a r l y s i t u a t e d t o t h e e l l i p s e of t h e e l l i p s e whose conjugate semi-diameters a r e
l i e s on an namely
A t any given time t h e displacement v e c t o r i s along one semi-
diameter of t h e e l l i p s e and t h e p a r t i c l e v e l o c i t y i s p a r a l l e l t o i t s conjugate semi-diameter.
t
As
t h e e l l i p s e i s from
i n c r e a s e s t h e sense i n which t h e p a r t i c l e moves along
&+ t o
A-.
The p l a n e of p o l a r i s a t i o n i s determined by
t h e plane of t h e e l l i p s e of t h e amplitude b i v e c t o r Circularly polarised i f Linearly p o l a r i s e d i f
Te"
S = where
5
The slowness b i v e c t o r
T
vector.
g
angles t o problem.
and
$
(5+,):I
are real.
and t a k i n g
m.:
.
The wave i s
-A.A _
=
0 ;
(3.2)
&A&
=
0.
(3.33
When equation ( 3 . 2 ) is s a t i s f i e d t h e b i v e c t o r "nullt'.
A -
A -
i s s a i d t o be " i s o t r o p i c "
or
may be w r i t t e n
__
= 0, n.n = 1
,
(3.4)
By f i x i n g on a p a r t i c u l a r choice of t h e u n i t
5 t o be o f a r b i t r a r y [chosen) magnitude and a t r i g h t
1 , i t will be seen t h a t T and 4 a r e determined by an eigenvalue 5 and may be regarded a s t h e p r i n c i p a l axes of an e l l i p s e .
I 78
M. Hayes
Then
S/T
(z+,g-) i s
and
13i s
el$@ +
=
a p a i r of conjugate semi-diameters of t h i s e l l i p s e
t h u s a p a i r of conjugate semi-diameters of a similar and s i m i l a r l y
s i t u a t e d e l l i p s e whose major and minor axes a r e
(m .+):1
axes of t h e e l l i p s e of
times t h e major and minor
Also
S+
= T(cos$m
-
sin$n-),
S-
= T(sin@
+
cos@),
The angle
T
2 3 ; = T(cos2$m2 + s i n $)
12-1 =
,
z f .
T(sin2$m2 + cos $)
(3.5)
between t h e planes o f constant phase and t h e planes of
8
constant amplitude i s given by tan 8
=
[Z]
m
(3.6)
(m 2 - l ) c o s $ s i n $
The condition (2.2) gives
A.
2
=
0.
(3.7)
This means i n general t h a t t h e e l l i p s e of t h e amplitude b i v e c t o r
2
e l l i p s e o f t h e slowness b i v e c t o r l a r t o each o t h e r .
S
A
and t h e
may not l i e on p l a n e s which a r e perpendicu-
Also, t h e p r o j e c t i o n of e i t h e r
(5 say)
upon t h e plane o f
i s an e l l i p s e whose aspect r a t i o ( r a t i o of major t o minor a x i s ) i s equal t o
t h e aspect r a t i o of t h e e l l i p s e of
2.
t h e minor a x i s o f t h e e l l i p s e of
5
and whose major a x i s i s perpendicular t o
A A. A =
Exceptionally it may happen that
i s a s c a l a r and a l s o t h a t polarised.
84. -
I t i s seen i n
2. 55
=
0
i s p a r a l l e l t o 2: A = as, where so t h a t t h e wave i s c i r c u l a r l y
a
that t h i s i s a p o s s i b i l i t y f o r i s o t r o p i c bodies.
THE PROPAGATION CONDITION
Here it i s assumed t h a t t h e d i r e c t i o n a l e l l i p s e of eigenvalue equation f o r t h e determination of that t h e r e a r e j u s t two non-zero eigenvalues.
Te"
5
i s given.
i s obtained.
The
I t i s seen
I f t h e s e a r e n o t equal it i s
shown t h a t t h e corresponding eigenbivectors a r e orthogonal. Now i f follows t h a t
p
has t h e form (3.1), t h e n from the equations o f motion (2.5) must have a s i m i l a r form.
Thus write
it
179
lnhomogeneous Plane Waves
p
=
P expiw
(5. x
.
-t)
(4.1)
I n s e r t i n g t h e expressions (3.1) and (4.1) i n t o t h e equations of motion and P
using c3.5) t o e l i m i n a t e
2. 2 SO,
and assuming
leads t o
Using (3.4) t h i s may be w r i t t e n (4.3) where
Q.
lk
*
m,
1 ~ k R ks j
%i
=
'
=m+ip,m .n = O , n .n = l .
S Thus, i f
* *
= d..
(4.41
a r e given, equation (4.3) i s an eigenvalue problem f o r t h e
determination of
and t h e corresponding eigenbivector
T, $
A.
The propagation condition (4.3) l e a d s t o t h e s e c u l a r equation d e t (pT-' f o r the determination o f det(Q)=0
zero.
e-21'
-
6ik
[Te'$)
= 0,
f o r given
*
*
s .s - * *
since det
6.i k)
<.
) = 0,
rns m
eigenbivectors be
T-2e-21' and
T 2 e210
be denoted by Alp-', respectively.
I t i s c l e a r from (4.4)
and hence one r o o t of (4.5)
Thus equation (4.5) i s a q u a d r a t i c i n Let t h e r o o t s
(4.5)
Then
that
is
.
h2p
-1
.
Let the corresponding
(4.6) (4.7) I t i s now shown t h a t if h l
11.B
=
A2
0.
,
then (4.8)
180
M. Hayes
Now s i n c e
A. S* = B. BiQikAk
4
since
*
-
i s symmetric.
= 0, i t follows from equation (4.4) t h a t
A.Q. 1 ik Bk
X1 f
e l l i p s e of
B
A. g
(4.9)
(4.10)
= 0.
A,
and
The e l l i p s e s of
A
S
a r e on planes no two of which may be
S
and o f
when p r o j e c t e d on t h e plane of t h e
a r e s i m i l a r and s i m i l a r l y s i t u a t e d .
r o t a t e d through a quadrant. t i o n s of t h e e l l i p s e s o f t i o n s of t h e e l l i p s e s o f
B &
Examples o f such t r i a d s [3],
The p r o j e c t i o n s a r e s i m i l a r
i n t h e p l a n e of
and s i m i l a r l y s i t u a t e d t o t h e e l l i p s e of
For example
,
X2,
Hence t h e e l l i p s e s of orthogonal.
= 0
Hence from (4.6) and (4.7)
(A1, - X2) Thus, f o r
- AiQikBk
BiQikAk
3
when they a r e
A similar statement may be made about t h e projec-
and
5
and
upon t h e plane o f upon t h e plane of
A, B,
A
and about t h e projec-
2.
s a t i s f y i n g (4.11) a r e e a s i l y constructed.
take
c4.12) a, 6, 6
where
are scalars, satisfying
ci
2
-
B2
=
Returning t o equation (4.3) i t i s seen t h a t i f eigenbivector f o r given of
Te'+
i s known t o be an
and i s not i s o t r o p i c , then t h e corresponding value
i s given by p ~ - 2 e-21'
since
*
1.
A.5=
0.
If
&
A1 . A .1 = Q i k ~ i ~ =k Qi kA iA k '
i s i s o t r o p i c , then it w i l l be shown i n
corresponding eigenvalues a r e double and hence t h e v a l u e of half t he t race of
(4.13)
Te"
86
that the
i s given by
Q/p:
2pT
-2 -214
e
=
A
Qii*
(4.14)
181
Inhomogeneous Plane Waves
55. -
ISOTROPIC MATERIALS
In t h i s s e c t i o n t h e propagation of inhomogeneous plane waves i n i s o t r o p i c homogeneous incompressible e l a s t i c bodies i s considered. o f waves.
There a r e two c l a s s e s
In t h e f i r s t t h e slowness b i v e c t o r i s i s o t r o p i c , t h e waves a r e
c i r c u l a r l y p o l a r i s e d and t h e s o l u t i o n i s u n i v e r s a l i n t h e sense t h a t t h e slowness does not depend upon t h e shear modulus which d e s c r i b e s t h e e l a s t i c response of
In t h e second c l a s s , t h e p r e s s u r e term i s zero, t h e waves a r e
the material.
t r a n s v e r s e , and t h e amplitude b i v e c t o r may be chosen so t h a t t h e wave is c i r c u l a r l y polarised. The c o n s t i t u t i v e equation (2.1) now reads t.
=
Ij
p
where
i s a constant.
-
p6ij
+
p(ui,j
,
+ uj,i)
(5.1)
Using t h i s and equation (4.1) i n equation (2.4)
leads
to
Now
AiSi
=
0
and hence
either
ci) (5.3)
where
a
is a scalar
Cii) =
P
In
c a s e Ci),
usj sj
0 ,
p
s.s-
=
aSi expiw@. 5 - t )
=
-
=
0
ipwaexpwC2.
,
a
This i s a c i r c u l a r l y p o l a r i s e d wave. p l a n e s of constant amplitude:
5'. 5-
r e s t r i c t i o n on t h e magnitude of case ( i i ) ,
.
(5.4)
we have t h e s o l u t i o n ui
In
= p
the solution is
2.
,
-t) , arbitrary.
Planes of constant phase a r e orthogonal t o = 0.
Also
Is+( =
15-1
.
There i s no
I t i s independent of t h e shear modulus P
.
M. Hayes
182
A
where
where
From (5.6)2
0.
m2 < 1 ,
Thus, i f
and i f
A. 5 =
i s any b i v e c t o r s a t i s f y i n g
m2 > 1
y, y'
,
a r e a r b i t r a r y and 2.2 = 1,
1.11=
r.fl= 0 .
I n both c a s e s t h e
planes o f constant phase a r e orthogonal t o t h e planes of constant amplitude.
m
5
c Cp/p) homogeneous t r a n s v e r s e wave i s recovered. As
-+
a,
then from ( 5 . 9 ) ,
For given " +
15,and m
-+
a, m, A
s a t i s f y i n g (S.?),
the waves are c i r c u l a r l y p o l a r i s e d .
-f
y
12 + or
and t h e usual
y'f,
y'
may be chosen s o t h a t
Thus, from equation (5.8),
take
y
given
bY (5.10)
Then (5.11) The displacement v e c t o r lies on a
corresponds t o a c i r c u l a r l y p o l a r i s e d wave. c i r c l e i n t h e p l a n e spanned by
m
2
> 1,
take
yt
given by
m
and ;{
+- (-2-1) 1 4E} m
.
Similarly, f o r
(5.12)
Then (5.13)
183
Inhomogeneous Plane Waves
corresponds t o a c i r c u l a r l y p o l a r i s e d wave, t h e c i r c l e of p o l a r i s a t i o n l y i n g i n a plane spanned by
86.
m m-m
1 and
1 4 IS}
{ -i (1 - -2)
m
STRUCTURE OF THE ACOUSTICAL TENSOR.
.
CIRCULARLY POLARISED WAVES.
Here t h e s t r u c t u r e o f t h e a c o u s t i c a l t e n s o r i s considered more f u l l y .
In
i s not i s o t r o p i c , it i s shown t h a t i f t h e s e c u l a r
particular,assuming t h a t
equation has a double r o o t then a c i r c u l a r l y p o l a r i s e d wave may propagate i n t h e material.
Also, it i s shown, assuming
S
.S 9
0, t h a t i f a c i r c u l a r l y p o l a r i s e d
wave propagates then t h e s e c u l a r equation has a double r o o t .
i s n o t i s o t r o p i c i s made, f o r , i n g e n e r a l , i n
The assumption t h a t
d e r i v i n g t h e form of t h e a c o u s t i c a l t e n s o r expressions o f t h e form enter.
I t was seen i n
o b t a i n a s o l u t i o n with
35
[s@SJ/(s.s)
f o r an i s o t r o p i c m a t e r i a l t h a t i t i s p o s s i b l e t o
5.2 =
0.
However, i n g e n e r a l , i f t h e d e t a i l a d s t r u c t u r e
of t h e e l a s t i c c o e f f i c i e n t s i s n o t given then it is e s s e n t i a l t o assume t h a t
S
i s not i s o t r o p i c . Now s i n c e any b i v e c t o r
where
a
may be w r i t t e n [l]
5
=
+
ib), a.b =
0,
i s a s c a l a r , i n general complex, we may, without l o s s of g e n e r a l i t y ,
write
where
T,
0,
respectively.
m
a r e r e a l and Then, assuming,
defined by equation (4.4),
i,
2.S
a r e u n i t v e c t o r s along t h e x and y axes 2 0 so that m $: 1, it i s seen t h a t Q,
may b e w r i t t e n * *
and has components given by
184
M. Hayes
Thus t h e matrix
4
has t h e form
where
One eigenvalue of
X1, AZ,
(0)
191
i s zero s i n c e
and t h e o t h e r two, denoted by
a r e given by
A,
t h e corresponding e i g e n b i v e c t o r s being
where
= 0
given by
tl, t2 a r e given by
Now tl t2 =
and hence
m2
-
1
,
(6.9)
185
Inhomogeneous Plane Waves
=
A . B-
(6.10)
0.
A1so
A.2 -
B.5
0,
=
=
(6.11)
0.
In t h e s p e c i a l case when (6.12) it follows from equation (6.4) t h a t
(:i ; Q3i) 6
Cii)
=
and t h e non-zero eigenvalues a r e bivectors
CO,
0, y)
Q,,
(1, m, 0)
and
and
(6.13)
,
a + imB
where
w i t h corresponding eigen-
is arbitrary.
y
The f i r s t of
t h e s e corresponds t o a l i n e a r l y p o l a r i s e d wave, t h e second t o an e l l i p t i c a l l y p o l a r i s e d wave.
(Recall
m
2
9 1).
CIRCULARLY POLARISED WAVES
F i n a l l y , the p o s s i b i l i t y of equa
eigenva ies i s considered.
The eigenvalues a r e equal provided L
a
(6.14)
so that (6.15) Then A1
=
From equation (6.4),
A2
=
(a + imB + Q3,)/2
(Q) i s now given by
.
(6.16)
186
M. Hayes
where the upper and lower s i g n s correspond t o t h e upper and lower s i g n s i n (6.15).
C(say) i s given by
The corresponding eigenbivector
(6.18) It is clear that
5.C =
0,
and thus the wave corresponding t o t h e double r o o t
is c i r c u l a r l y polarised. I t may be noted t h a t t h e c i r c l e s of p o l a r i s a t i o n corresponding t o t h e d i f f e r e n t s i g n s i n equation (6.15) and t h e r e f o r e a l s o i n (6.18), a r e g r e a t c i r c l e s i n t h e u n i t sphere.
They a r e described i n opposite senses.
most e a s i l y seen i n t h e s p e c i a l c a s e
_C = _i r i-k
are unit circles i n the
-
xz
if it i s p o s s i b l e
-
when m = 0.
This i s Then
plane, described i n opposite senses.
Also, r e t u r n i n g t o equation (.6.14), which i s t h e condition f o r double r o o t s ,
this condition may be w r i t t e n , using equation (6.5),
CQ,,
+
2 1 m Q12
-
m
2
Q2,
-
-
(1
2
as 2
m 1 Q33}
(6.19) This is a q u a r t i c i n real root for
m
m
with complex c o e f f i c i e n t s .
I f t h i s equation has a
then t h e r e a r e two corresponding c i r c u l a r l y p o l a r i s e d waves.
In general it w i l l not possess r e a l r o o t s f o r
m.
Assuming t h a t 5 i s not i s o t r o p i c it has been shown t h a t i f t h e a c o u s t i c a l tensor Q) has double r o o t s then t h e corresponding eigenbivector i s i s o t r o p i c and accordingly a c i r c u l a r l y p o l a r i s e d inhomogeneous wave may propagate. shown, assuming t h a t possesses an
Now, it i s
S
i s not i s o t r o p i c , t h a t i f t h e a c o u s t i c a l t e n s o r (Q) i s o t r o p i c eigenbivector then 9 has a double eigenvalue.
Now from (4.4) t h e a c o u s t i c a l t e n s o r
9 n
has t h e form (6.20)
cij
=
6ij
-
SiSj
‘mSm
(6.21)
187
Inhomogeneous Plane Waves
and (6.22)
0
Without l o s s of g e n e r a l i t y , l e t
be given by
(6.23)
i s t o be an i s o t r o p i c eigenbivector of
Now
@)
so t h a t it s a t i s f i e s (6.24)
C6.25)
(6.26)
without l o s s of g e n e r a l i t y .
_A.S_
5
= 0,
I t i s assumed that
2.5
0
and since
may be assumed t o have t h e form (6.27)
where
i s some s c a l a r .
6
6
2.2
Now
i’ 6 -1
2
=
-1
-6
and
=
(
(6 -1)a - t b -6c ( 6 * + l ) b -la -16c
-:(a
62,
=
and
&:)
-1
62+1 -16
0
2 (6 -1)b -if -6g
- t b +(6’
+tb)
,
(6.28)
2
(6 - 1 ) ~- t g -6h
+ l ) f -16g
-6(b + t f )
-6(c
+tg)
(6.29)
Then from (6.24) it follows t h a t
,
2tb
=
f - a
61
=
6(a + tb) - cc + 18)
,
(6.301
188
M. Hayes
and (6
-1)a - t b -6c
(S2 + l ) b - i a -16c
62G =
[-&:a On expanding r o o t Csince
17. -
+
-1
a -6g
2 6 (a+2tb)+Ca+ib)-t6g
= 0,
(6
2
-1)c-Ig-dh
-tc+(6
-
-t6Ca + ib)
tb)
IQ - e r l
191
( ~ 5+ l~) b
2
6(c + 1g)
i t i s seen t h a t t h e s e c u l a r equation has a zero
X
= 0 ) , and a double r o o t
given by ( 6 . 3 0 ) 2 .
SUPERPOSITION OF WAVE TRAINS WITH COMMON DIRECTIONAL ELLIPSE
In t h i s s e c t i o n t h e s u p e r p o s i t i o n of two wave t r a i n s with common d i r e c t i o n a1 e l l i p s e i s considered.
S
For given f i x e d
p a r a l l e l t o t h e plane of t h e e l l i p s e of
*
t h e motion i n any plane
is examined.
The displacement o f
a p a r t i c l e a r i s e s a s a l i n e a r combination of t h e two b a s i c displacements and (The sum of two b i v e c t o r s i s
hence, i n general, w i l l a l s o be e l l i p t i c a l . also a bivector.)
The e l l i p s e s f o r d i f f e r e n t p a r t i c l e s w i l l g e n e r a l l y d i f f e r .
However, it i s seen t h a t t h e e l l i p s e s a t c e r t a i n p o i n t s on a c e r t a i n l i n e are i d e n t i c a l both i n o r i e n t a t i o n a n d i n t e r m s of t h e lengths of t h e i r p r i n c i p a l axes. I t i s assumed t h a t
where
a,,
B,
a, b
A.B=
5, A,
a r e given by
I t i s e a s i l y checked that
a r e assumed t o be r e a l .
A.S-1 = B.S-2 = 0. Also Tle 161 , T2e1" of t h e s e c u l a r equation ( 4 . 5 ) f o r given 2 = corresponding eigenbivectors a r e The t o t a l displacement
E* =
A
and
a r e assumed t o be t h e s o l u t i o n s (ol/B)
u_"Csay) in t h e p l a n e of
A expiw C Tle'"(crx
+ iBy]
+ {B - (B.n) - - n) expiu{T2e"2(ax
+
ii,
and t h e
respectively.
-
&
may be w r i t t e n
t 1 + ’By)
-
t)
,
(7.21
189
lnhomogeneous Plane Waves
A
2 is t h e u n i t normal t o t h e p l a n e of
where
given by
- s i n d i + acossk
z =
2 2 2 l ( s i n S+a cos 6)
(7.3)
’
Now
-
bfisind[ii
=
Bcos6
(1+B c o s S )
IacosSj + sin8k) ]
,
(7.41
and
-s* -
=
(s”.fl)” 5 6
l[lL
-
’c0s6 ~ a c o s +~ si i n & & )I . 2 2 (1+B cos 6)
Notice t h a t t h e e l l i p s e s of t h e p r o j e c t i o n s of
and
5
(7.5)
upon t h e plane of
are s i m i l a r and s i m i l a r l y s i t u a t e d . Now
u*
may be w r i t t e n
+ a-’b@sin6{iL
-
BCOSG
(1+6 cos 6)
(acossi
+
sin6k) 1 exp
IWP]
,
(7.61
where (7.71 I t i s c l e a r t h a t f o r given
x, y,
t h e displacement
%* l i e s on an e l l i p s e .
The axes of t h e e l l i p s e w i l l v a r y from p o i n t t o p o i n t f o r p o i n t t o point. the ellipse a t
However, if exp(iwb) M
has p r i n c i p a l axes along
t h e ellipses at the points axes of t h e e l l i p s e of
5
M
u v a r i e s from
i s purely r e a l a t a point
i
M(say)
and (ctcosdi + sin@).
then
Thus
have p r i n c i p a l axes p a r a l l e l t o t h e p r i n c i p a l
and a l s o of course t o t h e p r i n c i p a l axes of t h e
e l l i p s e of t h e p r o j e c t i o n of
2.
Of course t h e s e e l l i p s e s a t t h e p o i n t s
M
will not be i d e n t i c a l but they do have t h a t one f e a t u r e i n common, t h a t t h e i r p r i n c i p a l axes a r e p a r a l l e l .
The p o i n t s
M
a r e any p o i n t s o n t h e e q u i d i s t a n t
190
M. Hayes
d
parallel lines
2
:
where
k,
:
wk[a[cos$)x $
-
p(sin$)y]
qs,
=
q
0,
...
+1, f 2 ,
(7.8)
a r e given by T e 2
‘42
- Tle
Consider t h e p o i n t s o f i n t e r s e c t i o h
32 with t h e l i n e s
:
.
(7.9) M*(say)
(asin+)x + ( ~ c o s $ ] y = The term
exp(iwu)
p o i n t s and has t h e same value a t each.
of a l i n e
7
constant
(7.10)
i s p u r e l y r e a l a t each of t h e s e
Thus f o r t h e s e p o i n t s
M
*
the
displacement e l l i p s e s a r e a l l i d e n t i c a l and are s i m i l a r and s i m i l a r l y s i t u a t e d t o t h e e l l i p s e of t h e p r o j e c t i o n of
ACKNOWLEDGMENT.
5
upon t h e p l a n e of
A.
This work was supported by t h e National Board f o r Science
6 Technology under Grant 19/79.
191
Inhomogeneous Plane Waves
REFERENCES [l]
Gibbs, J.W. Elements of Yector Analysis, 1881, 1884 ( p r i v a t e l y p r i n t e d ) E pp 17 - 90, Vol. 2 , p a r t 2 , S c i e n t i f i c Papers, Dover Publications, New York, 1961.
[Z]
Hayes, M. Inhomogeneous Plane Waves. appear)
[3]
Synge, J . L . The Petrov C l a s s i f i c a t i o n of G r a v i t a t i o n a l F i e l d s . Dublin I n s t . f o r Adv. S t u d i e s , No. 15, Dublin, 1964.
Arch. R a t ' l Mech. Anal. 1984 ( t o
A,
Corn.