b) vertical section. Note th a t obstacle boundaries are not required to vertical as shown in b).
will choose to arran g e th e basis functions into two sets as suggested in c h ap ter 3. Explicitly, we define an outgoing set (ip1)™ as
(jA\m _ I W n(kn, uj , z ) T™( r , 0) e ,wt, _ 1 = 1;
^ “ ' [{/„(£„,u>,z)R™(r,0) + K,(fcn,u ;,z )S ™ (r,ö )]e“ l" ‘1 1 = 2; (4.1)
where th e indices / , m , n refer to wavetype, azim uthal order and m o d a l order, re spectively. T h e functions Un, Vn, an d W n are th e displacem ent eigenfunctions for the modes which describe th e d ep th dependence and are readily d eterm in e d for b o th Love (/ = 1) an d Rayleigh (/ = 2) waves in plane-stratified e a r th models. T h e cylindrical harm onics R™,S™,T™ are defined as
R ” = z n r , S™ = k~'V\Y™, T j = - i x S ; , (4.2)
and are c o n stru cted using th e outgoing Ilankel function H m \ k nr) in th e definition of th e horizontal wave function Y™:
(4.3)
where th e N eu m an n factor em is equal to l / \ / 2 for m = 0 and 1 otherw ise. We em phasize t h a t this set of basis functions represents waves p ro p a g a tin g outw ards from r = 0, a fact which becomes obvious when we consider the a s y m p to tic form of Hm as v approaches infinity:
H £ \ k nr) « y C T - e x p [i(fc„r - | ( m + 1/2)] (4.4)
This ty p e of b ehaviour is expected of surface waves s c attered by obstacles in th e em b ed d in g stratified medium . We choose therefore to represent th e sc a tte re d field from a p a rtic u la r obstacle in term s of the basis functions from th e set (ip1)™ with respect to an origin of horizontal coordinates located w ithin t h a t obstacle. Note t h a t th e singularity due to the im aginary com ponent of poses no problem to us in this instan ce since th e scattered wave is only strictly defined outside the o b s ta c le ’s b o u n d in g surface. T h e second set of basis functions, which we designate
(ip1)™, is c o n stru cted by taking th e regular p a rt of (ip1)™ (an o p e ra tio n we den o te by an overhead caret) or, equivalently, s u b stitu tin g J m for in (4.3), t h a t is
(V)n = { W n( kn, u , z ) T ™( r , 0 ) e iut, j Un(kn, i o, z ) R™( r , 0) + Vn( kn, u, z ) S™( r , 0 ) > — \uit l =
1;
1
=2:
(4.5)This basis set is characterized by s tan d in g wave behaviour and is finite-valued at th e horizontal co o rd in a te origin (i.e. along th e z-axis). C onsequently, it is used to exp an d fields which we ex p ect will be well behaved at a given c o o rd in a te origin. 4.2.3 C om ponent Wavefields
T h e p ro b lem configuration is shown in figure 4.1 which illustrates a horizontal plan view of th e two scatterers in our em bedding medium . T h e surfaces Si and 52 are circular cylinders which enclose th e two scatterers and in which we have
located two c o o rd in a te origins 0 \ and O2. A global reference frame is centered at 0
somewhere between th e two obstacles and is associated again with a surface S which contains b o th scatterers. As in ch ap ter 3 we will consider th e to ta l displacem ent field u l w ithin th e stratified m edium outside S as th e sum of an unknow n scattered field us an d an incident field u ‘ which is taken as known an d defined to be th a t which would exist in th e absence of any heterogeneity,
U t = u i + us. (4.6)
T h e to tal sca tte re d field us can in tu rn be broken into two co n s titu e n t wavefields
usl and iT2 associated w ith th e individual fields scattered from e ith er of th e two
obstacles
us = u sl + us2. (4.7)
In ad d itio n , it is advantageous to introduce ‘ex citin g ’ fields uel a n d ue2 which represent th e to ta l wavefield impinging upon either obstacle and which give rise to
usl and us2:
= u 1 + 1T 2, (4.8)
= u 1 -f u s l. (4.9)
Note th a t in a single scatterin g approxim ation the second te rm in b o th (4.8) and (4.9) is ignored as a co n trib u tin g source to th e individual s c attered fields. Having defined th e various com ponent displacement fields of interest, we now wish to
represent these qu an tities in term s of th e basis function expansions in (4.1) and (4.5). T h e choice of ap p ro p ria te basis set for a given co m p o n en t of th e wavefield will d epend on th e n a tu re of the field and where it is to be ev aluated.
Since th e incident field u1is taken to originate ou tsid e th e surface S en co m p ass
ing b o th scatterers, it would, in the absence of any heterogeneity, be finite-valued th r o u g h o u t this volume. Hence it is ap p ro p ria te to ex p a n d u1 in term s of the
regular basis set ipa referred to th e global c o o rd in a te origin 0 . We indicate this explicitly by including the position vector r as an a rg u m e n t (see figure 4.1)
l m n
=£yr(r).
(4.10)a
Here, th e basis function coefficients for the incident field are d en o ted by a° where we have chosen to a b b rev iate the triple su m m a tio n over / , m , n by single su m m a tio n over th e com posite index o (c f . ch ap ter 3) for th e sake of brevity. In co n tra st, we ex p ect t h a t ou tsid e th e surface 5 th e total sc a tte re d field iT will behave as an o u tw ard p ro p ag atin g wavefield; thus we ex p an d u s in term s of ip1 referred (9,
iT = (4.11)
a
Now let us consider the displacem ent wavefields associated with th e individual scatterers explicitly. T h e two scattered fields u s l , us2 are n a tu ra lly expanded in
te rm s of o utgoing basis functions this time referred to th e origins 0 \ and O2,
respectively.
a
( 4 . 1 2 )
£ c ^ ( r 2 ). ( 4 . 1 3 )
a
Finally, th e exciting fields exhibit regular behaviour in th e vicinity of th e bounding surface of their respective obstacles (e.g. Si for u el ) and hence can be w ritten as
a
(4.14)
E ^ M - (4.15)
T h e m o tiv atio n for introducing these two la tte r q u an tities now becom es clear; it was shown in ch a p te r 3 th a t the coefficients of sc a tte re d wave series in th e form of (4.12), (4.13) could be related to those of a given exciting field as in (4.14), (4.15) by an infinite set of linear equations which when expressed in algebraic form c o n stitu te s a surface wave T -m atrix . We will make use of this p ro p e rty shortly, b u t tu r n first to an ex am ination of the tran slatio n o p e ra to rs for th e surface wave basis functions.
4.2.4 Translation Operators
Reconsider for a m om ent, th e basic objective of this section. We wish to derive a com posite T - m a tr ix relating th e coefficients of th e scattered field ca to th o se of th e incident field a °. Note t h a t our expansions for th e co rresponding fields are b o th referred to th e origin at 0 whereas, for exam p le th e individual s c attered fields from eith er obstacle are referred to origins a t 0 \ and O2. It is a p p a r e n t th e n t h a t
to apply th e T - m a tr ix formalism we must be able to express basis functions a t one c o o rd in a te origin to those of a different reference frame. To d e m o n s tr a te this p ro ced u re we will exam ine th e tran slatio n p roperties of th e scalar wave functions since th e corresponding properties for o u r vector basis functions follow alm ost trivially. Consider our two obstacle g eo m etry with th e following horizontal q u an itities defined: r = [r,Q,z) th e position vector of point P w ith respect to a reference fram e a t 0 , iq = ( r ^ ö ^ z ) th e same q u a n tity w ith respect to origin 0 1,
and d x ( = z) th e vector sep aratin g the two origins such t h a t r = iq + d ^ To establish th e a p p r o p r ia te translation operators for Tnm we will exploit th e following identity:
OO
Jm (knr)e imS = Xn- P( W i ) e ' (m- p> 0 ,,( C ,r 1)e,>’' \ (4.16)
p = — OO
where r = |r| etc. This expression and (19) below are generalizations of G r a f ’s addition th eo rem , see Erdelyi et a/., 1953. We n o te th a t th e expression in (4.3) in co rp o rates real-valued sinusoids. T hus to pose th e relation above in term s of our Tnm we m u st sep a ra te the complex exponentials in th e above expression into sine and cosine q uantities on either side of th e equality an d reduce th e sum from
^ra(r) = £ ^ mp(d1)V-n’’(r,)>
p = 0 (4.17) w h e r e Amp(dj) = 2 1 7---Cp Jm—p (hnd 1 ) ^ c o s ( m — p)(f) ^ s i n ( m - p)(j) - s i n ( m - p)cf) N c o s ( m - p)4> j + ( — 1 )P Jm+p(kndl ) ( c o s ( m T p)<f> { s i n ( m -f p)(f> s i n ( m + p)(f) - c o s ( m + p)(f> (4 .1 8 ) N o t e t h a t t h i s r e l a t i o n s h i p h o l d s for t h e w a v e f u n c t i o n Y™ sp e c if ic t o a g i v e n s u r f a c e w a v e ( L o v e o r R a y l e i g h ) m o d e o f i n d e x n a n d in v o lv e s a s u m m a t i o n o v e r a z i m u t h a l o r d e r p. W e c a n d e r i v e t h e s e c o n d o f t h e t w o r e q u i r e d t r a n s l a t i o n o p e r a t o r s w h ic h r e l a t e s t h e o u t g o i n g h o r i z o n t a l w a v e f u n c t i o n Y™ a t 0 \ t o t h e r e g u l a r w a v e f u n c t i o n s Knm a t 0 2 u s i n g t h e follo w in g i d e n t i t y ( h e r e w e h a v e p o s e d t h e r e l a t i o n in a f o r m e n c o u n t e r e d in t h e e n s u i n g a n a ly s i s ) O O H ^ ( k n r i ) e ^ = ] T + d 2))ei^ - P ) * J p( k nr i )ei(4 .1 9 ) p — — OO w h e r e </> is n o w t h e a n g l e o f t h e v e c t o r - d ! -f d 2. I t s h o u l d b e n o t e d t h a t t h i s r e l a t i o n is valid o n l y for ( - d i -f d 2) > r 2 o w i n g t o t h e s i n g u l a r i t y in • B y r e a r r a n g i n g t h i s e q u a t i o n in a s i m i l a r m a n n e r we o b t a i n ^nm ( r ) = £ ß r a p ( d 1)V'n'>( r i ) , (4 .2 0 ) P=0 w h e r e B mp is c o m p u t e d by s u b s t i t u t i n g t h e B e sse l f u n c t i o n s J m± p w i t h o u t g o i n g H a n k e l f u n c t i o n s //B]_p in ( 4 .1 8 ). T h e e x t e n s i o n o f t h e s e r e s u l t s t o t h o s e o f o u r s u r f a c e w a v e b a s i s f u n c t i o n s is r e a liz e d by i n s e r t i n g t h e e x p r e s s i o n s in (4 .1 7 ) a n d (4 .2 0 ) i n t o t h e r e l a t i o n s g iv en by (4 .2 ). O u r t a s k is m a d e e a s y w i t h t h e r e c o g n i t i o n t h a t t h e d i f f e r e n t i a l o p e r a t o r s in (4.2) a r e i n v a r i a n t a n d c a n n o t d e p e n d o n t h e c h o ic e o f c o o r d i n a t e s y s t e m . T h u s t h e t r a n s l a t i o n q u a n t i t i e s A 7np, i ? mp n e e d o n ly b e m o d i f i e d t o r e m a i n c o n s i s t e n t w i t h t h e c o n v e n t i o n s we h a v e e s t a b l i s h e d t h u s fa r. H e n c e w e s h a l l w r i t eW’On (r) = (V-')^(d, + r,) = £ ( / l' ) r ( d i) « - ' ) £ ( r i ) ,
(4 .2 1 )and
(r2) = M 2 + d, + r.) =
^ ( ß ' ) » ’’( - d2
+d,)(V>')S(n)
(4 22)P
T h e requirem ent th a t ( - d i + d 2) > r 2 for th e validity of (4.20) an d (4.22) places some restrictions on th e spatial relationship of th e two obstacles. Since th e rela tion in (4.22) will be required over th e surface of th e obstacle a t S\ a sufficient (th o u g h not necessary) condition for ( - d i - f d 2) > r 2 to hold is t h a t th e two cylin ders Si and S 2 circumscribing th e obstacles do not overlap. N ote th a t we have te m p o ra rily ex p an d ed th e co m p o u n d index <r to em phasize th a t th e s u m m a tio n s exist over azim u th a l order (p) only, hence th e q u an tities ( A l)™p, ( B l)™p are ‘d i a g o n a l1 with respect to wavetype and modal indices so t h a t a tra n s la tio n of our basis functions is indep en d en t of m ode and or w avetype an d couples a z im u th a l orders alone. We note th a t th e modified q uantities ( A l)™p, ( B l)™p will d ep en d on m o d e and w avetype insofar as we must include th e a p p r o p r ia te w avenum ber in th e calculation of each diagonal s u b m a trix element. To m a in tain an u n c lu tte re d p re sen tatio n for th e rem aining analysis we will in general refer to these q u an tities in te rm s of th e co m p o u n d indices e.g. A a,/, B (ri/. Finally, it m ay prove convenient in numerical applications where we are interested th e far-field pro p erties of th e s c a t tered field to use ap p ro x im ate expressions for th e tra n s la tio n o p e r a to r A ai/. T h is p ro ced u re is detailed in A ppendix B.