u z = UiJje'mS, an d V T = ( - l J - W j — J ' e - “'“’ X J vt = - ( - 1 ) n Wj Jj e~m6 (A 3 ) v z = 0.
H ere we have suppressed th e harm onic tim e d ep endence e -1^ com m on to all phys ical fields an d ad opted th e n o ta tio n of K e n n e tt (1983) w here Vi, I/t ,and Wj are th e
eigenfunctions for th e it h Rayleigh m o d e an d j t h Love m o d e (in th e n o ta tio n of Aki a n d Richards [1980] W = 11, V = ri an d U = — r 2). T h e square root of - 1 is
d en o ted by a small roman i to distinguish it from th e italic index i which represents m o d al order. We have employed ab b rev iatio n s of th e following form
= k { r
J { — J m ( T )
(-4.4)
J j — J n i k j T )
J'i
=dXiJi.
We will consider these quantities and their respective trac tio n s on a circular cylinder of radius r = a centered a t th e origin and ex ten d in g to infinite dep th . Since trac tio n s are zero at th e free surface and b o th trac tio n a,nd displacem ent te n d to zero as z approaches infinity we need only consider the co m ponents of trac tio n over th e vertical surface of the cylinder in evaluating ( A .l) . T h ese are
t r(u) = [2 tikiViJ! + A J, (djJi - k t Vt)} eim<?
t f f ( U ) = 2 [L p i r n ö
t!(u) = fi(dy, + k,ui) j ; e " ' \
and
(-4.5)
U(v) = -(-1 )» 2 M-
jH / > ( j ; - ± J j )
e~'n0
4 (v ) = - ( - 1 YtikjWj (2JV + J;) e-i’*9
(A6)
‘.(v) = ( - l
X jWe n o te in applying ( A .l) over the specified geom etry t h a t th e azim uthal d e p e n dence co n trib u tes a common integral factor f ^ d O e*(™-n)0_ por m ^ n this fa c tor vanishes and B e t ti’s identity is satisfied trivially. We therefore consider th e case m = n where this integral introduces a factor of 2ir. After som e algebraic m an ip u -
lation
I
d S [t(u) • v - t(v)- u] J ( r = a )-
2 n a ( - \ ) m
( - i m )J~Az[W j
[2/<(A//'./,
-+ 2 '1 ( 1 - ^ 4 - |
J .'
+ 2 '‘ ( | v ; - | f - w ; ) - / ' A- / . -
a! ; . / , / .
-n U i d .W j — JiJj
+XWjdJJi
— J , J , } . X j X j JN ote t h a t th e term s involving the p ro d u cts
J[J'-
cancel and by employing Bessel’s eq u atio n (e.g.J"
-fj-J[ + (I — -p-)Jt
= 0) for th e ap p ro p ria te values ofi
andj
this m ay be fu rth e r reduced to r oo 27ra( —l ) m ( - i m )J
dz
(A.7)
- (A + 2/i)
Vj Wj —-
l ( A 8 ) + —(XWjdJJi - pUid,Wj)\
J,=
0. XjSince this expression m ust not be dep en d en t on th e choice of radius
a
we have (after m ultiplying by—k^kja)
r Az
[(A + 2 /0ViW,kf - ßV.WjIc] + k,(ßU, dzW, - XW,dJJ,)\
= 0 . (A.9) which establishes a functional o rthogonality relation between Love and Rayleigh wave eigenfunctions.A.3 RAYLEIGH WAVE ORTHOGONALITY
In deriving th e Rayleigh wave orth o g o n ality relation in cylindrical coordinates we ap p ly th e sam e arg u m e n t used in the previous section and consider u as before, b u t in this case choose v and th e corresponding tractio n over a circular cylindrical surface to be
vr = ( - l ) mVjJ' e- lm9
ve =
(A10)
X jvz
= ( - 1)mUJJJe-'m\
an d <,(v) = ( - 1 )m2nkjVjJ'f + XJ,
- M*.(V) = (•/' - A / , ) ' . —i m #
<,(v) = ( - 1 ) " > [d2V, + kj Uj ) J ‘Je - i"'>,
( A l l )
N ote th a t satisfaction of B e t ti’s identity for two Fourier-Bessel co m ponents of dif ferent azim u th al order in A n will follow trivially as in th e previous section. We recognize th a t th e integrals over th e horizontal surfaces (z = 0 ,z —► oo) in ( A .l) vanish so th a t we m ay write ( A .l) as
f dS [t(u ) v - t (v) • u] = 0
J { r = a )
= 2 j r a ( - l ) m H d z [2 nV,V, J j - k j j v j ( ) - \ V ,V } (fc.J, J] - kJ.JyJ'i)
+A (VjdJLJiJiJ; - V, dJJjJjJ',) + fiUiUj (k,J[J, - k j J ’J,) ( A12) +fc ( U j d y i j f j j - U i d y j ' j J i ) + — 2ßV,Vj U ,j;j] - v p . )
X i X j
where th e ab b rev iated variables in ( A 12) are evaluated at r — a after differen tiatio n . T h e first te rm involving second derivatives of Bessel functions can be expressed as two sep a ra te term s using Bessel’s eq uation, b o th containing Bessel functions and their first derivatives and one of which exhibits dependence on az im u th al order. After some algebraic m a n ip u latio n , those term s in ( A .12) d epending on azim u th a l order can be m ade to cancel yielding
2 ) r a ( - l ) m
rd z [(A +
2n)V,VJ ( k j j j j f - k, J\J ; )
+fiU,UJ ( ki Jj J- - kjJ' jJi)
-
A (VidJJj Jj J[-
V jdJJiJ.J^)+ it {U jdyjjJ i - u . d y r y , ) .
All term s exhibit a som ew hat similar s tr u c tu re with regard to horizontal d e p e n dence an d we can rearrange (A. 12) such th a t th e d ep th dep en d en ce on either side of th e eq u atio n is in th e form of a single factor
r oo
/
d z[(A + 2
n W iV jk j+
itUiUjk,- XVidJJj+
fiU jdtVi\Jo
= r
d ; [(A +
2it)V,V,k,+
itUiU.k,- A
VjdJJi +nUid,Vj)( A 1 3 )
Jo
where we have shown the horizontal dependence explicitly. Since th e horizontal and vertical dependencies have been isolated we may write this as
F ( k i, k J) X l ( a , k i, k J) = F ( k j , k i) X 2( a , k i, k j ). (A .14) If we fix k{,kj with i ^ j this is equivalent to C \ X \ ( a ) - C2X 2(a) = 0 where C\ and C2 are constants. This equation will hold in two distinct situations: i) X \ and X 2 are linearly dependent functions and hence C\ and C2 may be non-zero; or ii) X \ and X 2 are linearly independent functions so th a t we require C\ = C2 — 0. To investigate the relationship between X \ and X 2 we note th a t the result in (13) m ust be independent of the choice of radius a. We may therefore simplify our task by choosing a non-zero such th a t k xa is a zero of J m(;r). Equation (A .14) thereby reduces to
F(ki, k j ) Xi ( a ) = 0. (A .15)
In this case J^n (kla) will not be zero and since equation (A .l) m ust hold for all perm issible values of i.,j and is independent of the choice of earth model which governs th e values of k{,kj\ A'i(a) will not, in general, be zero. Therefore we m ust have
ro o
/ d* [(A + 2n)ViVjkj + nUiUjki - XVidJJj + fiUjdzVt) = 0. (A.16) Jo
Also note th a t the stan d ard expression of Rayleigh wave eigenfunction orthogonal ity may be retrieved by simply adding (or subtracting) the expression in (A. 16) to itself w ith the indices i , j interchanged.