Burridge and Knopoff 1964 BSSA BodyForceEquivalentsForSeismicDislocations

7/22/2019 Burridge and Knopoff 1964 BSSA BodyForceEquivalentsForSeismicDislocations

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Bul let in of the Seismological Society of Am er ica. Vol. 54 No. 6~ pp . 1875-1888 . D e c e m b e r 1 9 6 4

B O D Y F O R C E E Q U I V A L E N T S F O R S E I S M I C D I S L O C A T I O N SB Y P ~ BURRIDG E AND L I ~ N O P O F F

A B S T R A C TAn explicit expression is derive d for the bod y force to be applied in th e absence of a dislocation,which produces radia t ion ident ical to th at of the dislocation. This eq uivalent force dependson ly upon the source and the e la s ti c p roper t i e s o f the med ium in the imm edia te v ic in i ty o f thesource and no t up on the prox im ity of any ref lect ing surfaces . The the ory is developed for d is-locat ions in an an isotropic inhomogeneous m edium; in the examples iso tropy is assumed. Fo rdisplacemen t dis locat ion faul ts , th e double couple is an exa ct equivalen t bod y force .

1 INTRODUCTIONT h e i n t e r e s t i n t h e f o r c e e q u i v a l e n t p r o b l e m a n t e d a t e s i ts s o l u ti o n b y a c o n -

s i d e ra b l e n u m b e r o f y e a r s . T h e d e b a t e s o v e r s i n g le c o u p l e a n d d o u b l e c o u p l ee a r t h q u a k e f o ca l m e c h a n i s m s h a v e b y n o w b e e n w e ll v i e w e d a n d r e vi e w e d . F o r ad i s cu s s io n o f t h e p r o b l e m o f e a r t h q u a k e s o u r c e m e c h a n i s m s s ee S t a u d e r ( 19 6 2 ).

I t h a s b e e n p o i n t e d o u t b y s e v e r a l a u t h o r s ( K n o p o f f a n d G i l b e r t , 1 9 6 0; B a l a k i n a ,S h i r o k o v a , a n d V v e d e n s k a y a , 1 9 6 0) t h a t t h e s o l u t i o n t o t h e p r o b l e m o f t h e s e i sm i cr a d i a t i o n f r o m a s u d d e n l y o c c u r r i n g e a r t h q u a k e i n t h e e a r t h ' s i n t e r i o r is l i k e ly t ob e c o n n e c t e d w i t h th e s o lu t i o n t o a d i s l o c a t i o n p r o b l e m , or , i n t h e t e r m i n o l o g yo f B a k e r a n d C o p s o n ( 1 9 5 0 ) , t o a s a l t u s p r o b l e m . I n th e s e p r o b l e m s t h e d i sp l a ce -m e n t f i e ld o r s t r e ss f ie l d u n d e r g o e s t h e m o r e o r le s s s u d d e n c r e a t i o n o f a d i s c o n -t i n u i t y a c r o ss t h e f a u l t s u r f a ce .

P e r h a p s t h e m o s t c o m p l e t e s o l u t i o n t o t h e d i s l o c a t i o n p r o b l e m h a s b e e n g i v e nb y K n o p o f f a n d G i l b e rt . T h e s e a u t h o r s s h o w e d t h a t a m o d e l o f a n e a r t h q u a k e c a nb e r e p r e s e n t e d a s a l i n e a r c o m b i n a t i o n o f t h e s o l u t io n s to a n u m b e r o f f u n d a m e n t a lp r o b le m s . T h e r e m a i n i n g p r o b l e m , t h a t o f d e t e r m i n i n g t h e a p p r o p r i a t e l in e a rc o m b i n a t io n s , w a s n o t a t t e m p t e d .

A l t h o u g h K n o p o f f a n d G i l b e r t w e re p r i m a r i l y i n te r e s t ed i n t h e r a d i a ti o n p a t t e r n sf r o m t h e i r v a r i o u s d i s l o c a t i o n m o d e l s t h e y a l s o p r o v i d e d t h e f o r c e e q u i v a l e n t s t h a tw o u l d p r o d u c e t h e s a m e f i r s t m o t i o n s a s e a c h o f t h e i r so u r c es . I n t h i s p a p e r w e fi n dt h e f o r c e e q u i v a l e n t s f o r p h y s i c a l l y a n d m a t h e m a t i c a l l y r e a s o n a b l e d i s lo c a t io nm o d e l s .

N a b a r r o ( 1 9 5 1 ) o b t a i n e d t h e r a d i a t i o n f r o m a s p r e a d i n g d i s l o c at i o n b y e s s e n t ia l lyt h e s a m e m e t h o d a s K n o p o f f a n d G i l b e r t b u t i n a l es s e x p l ic i t f o r m . H e h a d i nm i n d m a i n l y m e t a l lu r g ic a l a p p li c at io n s . R e c e n t l y M a r u y a m a ( 1 9 63 ) h a s a t t e m p t e dt o d e r i v e t h e f o r c e e q u i v a l e n t s b u t h i s r e s u l t s a r e o b s c u r e d b y a l g e b r a i c d e t a i l a s-s o c i a t e d w i t h t h e e x p l ic i t e x p r e s si o n o f a c e r t a i n G r e e n ' s f u n c t i o n .

I n t h i s p a p e r w e s p e c if i ca l ly e x c l u d e c o n s i d e r a t i o n o f t h e r a d i a t i o n p a t t e r n . W ea r e c o n c e r n e d w i t h t h e b o d y f o r ce w h i c h w o u l d h a v e t o b e a p p l i e d i n t h e a b s e n c eo f t h e f a u l t t o p r o d u c e t h e s a m e r a d i a t i o n ( i n a ll r e s p e ct s , n o t o n l y fi rs t m o t i o n s )a s a g i v e n d is l o c at io n . W e f i n d t h a t t h e f o r c e e q u i v a l e n t d e p e n d s o n l y u p o n t h es o u r c e m e c h a n i s m a n d t h e e l a s ti c p r o p e r t i e s o f t h e m e d i u m i n t h e i m m e d i a t ev i c i n i t y o f t h e f a u l t a n d n o t u p o n a n y r e f l e c t i n g s u r f a c e s o r o t h e r i n h o m o g e n e i t i e sw h i c h m a y b e p r e s e n t i n t h e m e d i u m . T h e t h e o r y i s d e v e l o p e d f o r d i s l o c a t i o n s i n

1875


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1 8 7 6 BULLETI N OF THE SEISMOLOGIC L SOCIETY OF MERICa n i n h o m o g e n e o u s , a n i s o t r o p i c m e d i u m a l t h o u g h i n t h e s p e c i f i c e x a m p l e s g i v e n a tt h e e n d o f t h e p a p e r w e h a v e a s s u m e d i s o t ro p y .

W e a s s u m e f r o m t h e o u t s e t t h a t t h e G r e e n ' s f u n c t i o n e x i s t s . T h i s i s p h y s i c a l l yr e a s o n a b l e s i n c e i t i s t h e r e s p o n s e t o a n i n s t a n t a n e o u s b o d y f o r c e c o n c e n t r a t e d a ta p o i n t . W e t h e n d e r i v e a r e p r e s e n t a t i o n t h e o r e m i n t e r m s o f t h i s G r e e n ' s f u n c t i o na n d i n t e r p r e t t h e s u r f a c e in t e g r a l s a s r e p r e s e n t i n g c e r t a i n s u r f a c e d i s t r i b u t io n s o fb o d y f o rc e s a n d c o u p l e s . F i n a l l y , s p e c if ic e x a m p l e s a r e e x h i b i te d .

2 . A REPRESENTATION THEOREM AND RECIPROCITY RELATIONT h e e q u a t i o n s o f m o t i o n f o r a n i n h o m o g e n e o u s a n i s o t ro p i c e l a st ic s o li d a r e

c , i , q x ) u ~ , , ~ x , t ) ) , j - p x ) a d x , t ) = - f , , , , t ) ( 1 )w h e r e u d x , t ) i s t h e / - c o m p o n e n t o f t h e d i s p l a c e m e n t v e c t o r , c ijp q(X ) a r e t h ee la s ti c c o n s ta n t s o f t h e m e d i u m a n d f i( x , t ) t h e / - c o m p o n e n t o f t h e b o d y f o r ce a tt h e p o i n t x = ( x ~ , x ~ , X a) a n d t i m e t. T h e c o o r d i n a t e s y s t e m i s r e c t a n g u l a r c a r -t e s i a n . W e h a v e u s e d t h e n o t a t i o n F , s = O F / O x ~ a n d F = O F ~ O r . T h e s u m m a t i onc o n v e n t i o n a p p l i e s t o l e t t e r s u b s c r i p t s , c s j p q = c i v q = C p q ~ i , a n d f o r a n i s o t r o p i csol id cs~.pq = ),~sSvq + g(SspS~q + &qSjv) w he re h a nd ~ ar e th e La in 6 c o ns ta n ts oft h e m e d i u m a n d ~ t h e K r o n e c k e r d e l ta .

S u p p o s e v d x , t ) i s a n o t h e r m o t i o n d u e t o b o d y f o rc e s g i (x , t ) . T h e n( C S j p q ~ } a , q ) , j - - P ~ S = - - g S . 2 )

E q u a t i o n s ( 1 ) a n d ( 2 ) a r e s at is fi e d i n a v o l u m e V b o u n d e d b y a s u r f a c e S , a n d f o ra ll t i m e . W e a s s u m e t h a t f i a n d g s v a n i s h f o r t < - T , T a c o n s t a n t , a n d t h a t u sa n d vs a ls o v a n i s h f o r t < - T . T h i s l a s t i s a c a u s a l i t y c o n d i ti o n w h i c h g u a r a n t e e st h a t t h e d i s t u r b a n c e d o e s n o t s t a r t b e f o r e t h e f o r c e w h i c h c a u s e s it .

O n r e p la c i ng t b y - t i n e q u a t i o n ( 2 ) w e h a v e

w h e r e

a n d

[csjpq~p,q] . i - vs = - a ~

~ ( x , t ) = v ~( x, - - t ) , g ' ( x , t ) = g d x , - - t )

vv = O, t > T .

( 3 )

W e n o w m u l t i p l y ( 1 ) b y O s, ( 3 ) b y u ~ , s u b t r a c t , i n t e g r a t e o v e r al l o f V , a n df u r t h e r i n t e g r a t e w i t h r e s p e c t t o t f r o m - oo t o ~ . T h i s y ie l d s

.o _ e ~ _ u ~ )

- - c l j ~ ,~ ~ i .~ u ~ ,q - - u s . i ~ , , q ) = ~ d t d V u i ~ s - ~ i f l ) .( 4 )


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BODY FORCE EQUIVALENTS 877

T h e i n t e g r a ls o v e r t a r e i n f a c t f i n it e si n ce t h e i n t e g r a n d s v a n i s h o u t s i d e t h e i n t e r v a l( - T , T ) .

T h e l a s t t e r m o n t h e l e f t - h a n d s i d e o f ( 4 ) v a n i s h e s s i n c e c ~ j~ q = % ~ . S in ce pi s i n d e p e n d e n t o f t w e m a y i n t e g r a t e t h e s e c o n d t e r m w i t h r e s p e c t t o t a n d f i n d t h a tt h e r e s u lt v a n i s h e s s in c e u~ a n d ~ v a n i s h f o r t < - T a n d ~ a n d ~ v a n i s h f o rt > T . T h e f i r st t e r m o n t h e l e f t- h a n d s id e m a y b e t r a n s f o r m e d b y m e a n s o f t h ed i v er g e n c e th e o r e m s o th a t w e h a v e t h e i d e n t i t y

f _ : d t f v d V ( u ~ - O ~ f ~ ) = - I : d t f s d S n j( O ~ c ~ j ~ ,q u ~ ,q - u ~ c ~ i~ q O ~ ,q ), 5 )w h e r e n~ i s a n o u t w a r d d r a w n u n i t v e c t o r n o r m a l t o S .

I f we no w set g i (x , t ) = ~ 5(x , t ; y , - s ) w her e ~(x, t ; y , - s ) - - 5 (x1 - y l ) ( x 2 -y 2 ) 5 ( x 3 - y 3 ) ~ ( t -t- s ) , y i s a p o i n t i n V , a n d 5 i s t h e D i r a c d e l t a f u n c t i o n , t h e n~ i (x , t ) = ~ ( x , t ; y , s ) a n d ( 5 ) b e c o m e su , ( y , s ) = ~ d t G ~ , ( x , - t ; y , - - s ) f ~ ( x , t ) d V ~

- t - f _ : d t f s n j { G , ~ ( x , - t ; y , - s ) c ~ i ~ q ( x ) u ~ , q ( x , t ) 6 )

u i ( x , t ) c ~ j ~ q ( x ) G ~ n , q ( x , - t ; y , - s ) } d S ~ ,w h e r e G ~ n ( x ; t ; y , s ) i s t h e d i s p l a c e m e n t i n t h e / - d i r e c t i o n a t ( x , t ) d u e t o a n i n -s t a n t a n e o u s p o i n t f o r ce o f u n i t i m p u l s e in t h e n - d i r e c t i o n a t ( y , s ) .

I f , in ( 6 ) , u i a n d G ~ s a t i s f y th e s a m e h o m o g e n e o u s b o u n d a r y c o n d i t i o n s o n St h e n t h e s u r f a c e i n t e g r a l w i l l v a n i s h . I f , i n a d d i t i o n , f i ( x , t ) = & m ~ (x , t ; y ' , s ' ) ,t h e n ( 6 ) g i v e s

G ,~ m (y , s ; y ' , s ' ) = G ~ ,~ ( y', - s ' ; y , - s ) ( 7 )where G~m(x , t ; y ' , s ' ) s a ti s fi e s t h e s a m e b o u n d a r y c o n d i ti o n s a s G , , ,( x , t ; y , s ) .T h e r e f o r e ( 6 ) m a y b e re w r i t t e n a s

u n ( y , s ) - - d t G , i ( y , s ; x , t ) f i ( x , t ) d V x

+ d t n~,{G,i(y , s; x , t ) c ~ p q ( x ) U p . q ( X , t ) 8 )

- u~(x, t ) c is p q ( x ) G , ~ p , q , ( y , s ; x , t ) } d S x .H e r e

0 G ~ p (y , s ; x , t ), ,~ ,~ , y , 8 ; x , t ) =


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1878 BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICAw h e r e q i n d i c a t e s t h a t t h e s u b s c r i p t r e f e r s t o t h e s e c o n d s e t o f a r g u m e n t s o f G nva n d t h e s u m m a t i o n c o n v e n t i o n s ti l l a p p li e s.

E q u a t i o n ( 7 ) i s t h e r e c ip r o c al t h e o r e m . I t w a s e x a m i n e d b y K n o p o f f a n d G a n g i( 1 9 5 9 ) . I t i s a s p e c i a l c as e o f H e l m h o l t z s r e c i p r o c a l t h e o r e m i n g e n e r a l i z e d m e -c h a n i c s ( s e e W h i t t a k e r , 1 9 44 ; L a m b , 1 8 8 9 ) . E q u a t i o n ( 5 ) i s B e t t i s r e c i p r o c a lt h e o r e m a p p l i e d t o u~ a n d ~ a n d i n t e g r a t e d w i t h r e s p e c t t o t . ( L o v e , 1 94 4, p p . 1 7 3 -1 74 ) E q u a t i o n ( 8 ) i s t h e r e p r e s e n t a t i o n t h e o r e m w h i c h w e s h a ll u s e in t h e n e x ts e c t i o n . D e H o o p ( 1 9 58 ) d e r i v e s a s i m i l a r t h e o r e m f o r a n i s o t r o p i c h o m o g e n e o u sm e d i u m a n d o u r n o t a t i o n h a s f o l lo w e d h i s. T h e f i rs t r e p r e s e n t a t i o n t h e o r e m o f t h i st y p e w a s d e r i v e d b y K n o p o f f ( 1 9 5 6 ) .

L e t i t b e a s s u m e d t h a t w e w i sh t o f in d t h e r a d i a t i o n f r o m p r e s cr i b ed d i s c o n t i n u i -t i e s i n t h e d i s p l a c e m e n t a n d i t s d e r i v a t i v e s a c r o s s a s u r f a c e Y~ i m b e d d e d i n V ( F i g u r e1 ) . L e t v b e th e u n i t n o rm al to ~ a n d l e t [u~ ] x , t ) an d [u p.q ]( x , t) b e t h e d i s c o n t i n u i -t i e s i n u i a n d up.q a c r o s s ]~ a n d i n t h e d i r e c t i o n o f ~ a t t h e p o i n t x a t t i m e t . A s s u m eu l a n d G ~- s a t i s f y t h e s a m e h o m o g e n e o u s b o u n d a r y c o n d i t i o n s o n S a n d a p p l y

FIo 1 Schematic diagram of surfaces and volume used in the integrat ion

e q u a t i o n ( 8 ) t o t h e r e g i o n b o u n d e d i n t e r n a l l y b y ~ a n d e x t e r n a l l y b y S . G mi d o e sn o t h a v e p r e s cr i b ed d i s c o n t i n u i ti e s on Z . T h e n t h e s u r f a c e i n t e g r a l o v e r S v a n i s h e sa n d w e a r e l e f t w i t h t h e s u r f a c e in t e g r a l o v e r Z o n l y :

u ,~ (y , s ) = ~ d t G m ~ (y , ; x , t )f~ (x , t ) d V ~

9)- G ~i(y , s ; ~ , t)cljpq ~)[U~,q] ~, t)}dY,~.

E q u a t i o n ( 9 ) i s a r e p r e s e n t a t i o n o f u m i n t e r m s o f t h e p r e s c r ib e d d i s c o n t i n u it i e s i nu a n d i t s d e r i v a t i v e s a c r o s s Z .

3 . EQUIVALENT FORCESB y m e a n s o f t h e p r o p e r t i e s o f t h e d e l t a f u n c t i o n a n d i t s d e r i v a t i v e s w e s h a l l i n -

t r o d u c e v o l u m e i n t e g r a l s i n t o t h e s e c o n d t e r m o n t h e r i g h t h a n d s id e o f (9 ).


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BODY FORCE EQUIVALENTS 1 8 7 9W e n o t e t h a t

a n d

w h e r e

a n d

G ~ ( y , s ; ~ ,t ) = f ~ ~ (x , [ ) G ~ ( y , s ; x , t ) d V ~ .

- G , , ~ , q , ( y , s ; ~ , t ) = f v ~ q (x ; ~ ) G ~ ( y , s ; x , t ) d V , ,

~ ( x ; ~ ) = ~ ( x ~ - ~ ) ~ ( x ~ - ~ ) ~ ( z ~ - ~ )

~ q ( x ; ~ ) = ~ x ~ ~ ( x ; ~ ) .

I f t h e s e e x p re s si o n s a r e s u b s t i t u t e d i n ( 9 ) w e g e tu m ( y , s ) = i : d t f v G m p ( y , s ; x , t ) [ t p ( x , t ) - f ~ d 2 ~ . { [ u i ] ( ~ , t ) c ~ , q ( ~ ) ~ q ( x , ~ . )

~- [u~ ,q](~, t) ~.~q(~)8(x; ~ ) } ] d V ~ .B u t f ~ ( x , t ) r e p r e s e n t s a b o d y f o r c e ; s i n c e t h e s u r f a c e i n t e g r a l w i t h i n t h e s q u a r eb r a c k e t is i n v o l v e d i n t h e s a m e w a y a s f p , w e c o n c l u d e t h a t t h e e f f e ct o f t h e p r e -s c r i b e d d i s c o n t i n u i t i e s a c r o s s Z i s t h e s a m e a s t h e e f f e c t o f i n t r o d u c i n g e x t r a b o d yf o r c e s e p ( x , t ) , g i v e n b y

t _- - f d Z ~ { [ u ~ ] ( ~ , t )c l sp q ( ~ ) 8 q ( x ; ~ ) + [u~.q](~, t)cp~.~q(~)~(x; ~) } , ( l O )i n t o a n u n f a u l t e d m e d i u m .E q u a t i o n ( 1 0 ) h o l ds fo r a n y i n h o m o g e n e i t y a n d a n i s o t r o p y o f t h e e l a st ic m e d i u mp r o v i d e d o n l y t h a t t h e e q u a t i o n s o f m o t i o n a r e g i v e n b y ( 1 ) , a n d c iip q = c ~q ~i a te a c h p o i n t o f V .

4 MATHEMATICALLY CONSISTENT DISCONTINUITIESW e c a n n o t a s s ig n v a l u e s t o [u ~] a n d t o [U p,q ] c o m p l e t e l y a r b i t r a r i l y . F o r i n s t a n c e ,

i f 2; i s p a r t o f t h e p l a n e x 3 = 0 , t h e n [ u~ ,l] = [ u p ]a a n d [ up ,2 ] = [ up ],~ s o t h a t t h ed i s c o n t i n u i t i e s i n t h e s e t a n g e n t i a l d e r i v a t i v e s a r e d e t e r m i n e d w h e n [ up ] i s s p e c if ie d .T h e d i s c o n t i n u i t i e s i n t h e n o r m a l d e r i v a t i v e s [u ~,3 ] h o w e v e r , m a y b e s p e c if i e d i n -d e p e n d e n t l y . T h i s f e a t u r e w a s n o t c o n s id e r e d b y K n o p o f f a n d G i l b e r t.

T h e d i s c o n t i n u i t y i n t h e n o r m a l t r a c t i o n a c r o ss x3 = 0 is g i v e n b y[T~] = Iv31] = c3~pq[up,q] = [3~1 + c 3 i ~ 3 [ u ~ , 3 ]

w h e r e r~.~ i s t h e s t r e s s t e n s o r a n d [ 3 i] i s t h e p a r t o f t h e d i s c o n t i n u i t y [ T~ ] t h a t i s d e -


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1 8 8 0 BUL L E T IN OF T HE SE ISMOL OGI CAL SOC I E T Y OF AME RI CAt e r m i n e d w h e n [ u~ ] i s s p e c if ie d . C l e a r l y [T ~] m a y b e g i v e n a r b i t r a r y v a l u e s b ys p e c i f y i n g [u p.3 ] s u i t a b l y i f d e t c3~p3 ~ 0 . T h i s c o n d i t i o n i s c e r t a i n l y s a t i s f ie d b y a ni s o t r o p i c s o l i d s i n c e o n l y t h e d i a g o n a l t e r m s a r e n o n - z e r o , a n d c 3m - - c3223 = /~,c3333 = ~ - F 2 # w h e r e X a n d / ~ a r e t h e L a m d c o n s t a n t s o f t h e m e d i u m . I t i s n o ts a t i s fi e d , h o w e v e r , b y a l i q u i d s i n c e u = 0 ,F r o m n o w o n w e s h a ll a s s u m e t h a t w e m a y c h o o s e t h e s ix q u a n t i t i e s [u ~] a n d[ T i ] = ~ .[r~ .] i n d e p e n d e n t l y . W e r e w r i t e ( 1 0 ) , a c c o r d i n g l y , a s

e~(x, t) = -- f~ {[u,](~, t ) ~ c , j p q ~ ) 6 q x , ~ ) J c [Tp](~, t ) 6 x , ~ ) } d Y , ~ . ( 1 1 )T h e q u a n t i t i e s [T ~] m e a s u r e t h e f a i l u r e o f t h e t w o s id e s o f t h e d i s c o n t i n u i t y t o o b e yt h e l a w o f e q u a l a n d o p p o s i t e a c t i o n a n d r e a c t i o n , t h a t i s t h a t s o u r c e s o f s t r e s s a r ei n t r o d u c e d o n Z .

W e n o t e t h a t t h e e q u i v a l e n t f o r c e d e p e n d s o n l y o n t h e lo c a l p r o p e r t i e s o f t h em e d i u m i n t h e i m m e d i a t e v i c i n i ty o f t h e s u r f a c e ~ .5 . E Q U I V A L E N T T O T A L F o R c E S W H E N T H E T R A C T I O N S A R E C O N T IN U O US

I n t h i s c a s e t h e t o t a l e q u i v a l e n t f o rc e o b t a i n e d b y i n t e g r a t in g o v e r V is g i v e n b y

f v e , x , t ) d V = - f = t { f v 6 ~ (x , ~ ) d V , } d 2 ;~ .I t i s s e e n t o b e z e r o s i n c e

f v 6 q x , ~ ) d V = = fs n4$(x, ~ ) d S = = 0f o r ~ o n ~, s i n c e S a n d ~ h a v e n o c o m m o n p o i n t .A l s o t h e t o t a l m o m e n t a b o u t a n y c o o r d i n a t e a x i s i s z e ro si n c e t h e m o m e n t Ma b o u t a n y a x is m a y b e w r i t t e n a s

fM - - . Iv { x , e p x , t ) - xper(x, t ) } d V .T h i s i s

M = - - f = v~ [u ~](~ , t ) { c ~ , q ( ~ ) f v x , ~ q x , ~ ) d V x -B u t

f f X ~r ( Xj ~ ) d V l r

C l j r q ~ ) i v X p ~ q ( X ) d V = d Y ,

s i n c e c i i ~ r = c l j r ~ .

= - c ~ i p ~ ~ , ) * , q + c ~ s ,q S ~ ,,

= - - c i j ~ r + C ~ i r~ = 0


7/14

BODY FORC E EQUIVALENTS 1881H e n c e , i n a n a n i so t r o p ic , i n h o m o g e n e o u s m e d i u m t h e a p p l i c a t i o n o f a d i s pl a ce -

m e n t d i s lo c a t io n l e a ds t o f o r ce e q u i v a le n t s t h a t h a v e z e r o t o t a l m o m e n t a n d z e ror e s u l t a n t f o r c e a t a l l i n s t a n t s o f t i m e .

6 . S O M E S P E C IF I C EX MPLESW e n o w c o n s i d e r o n l y f a u l t i n g i n a n e i g h b o r h o o d i n w h i c h t h e m e d i u m i s i so -t r o p i c . F i r s t w e c o n s i d e r s o m e c a s e s i n w h i c h t h e n o r m a l t r a c t i o n s a r e c o n t i n u o u s ,i .e . [T p] = 0 . L e t x3 = 0 b e t h e s u r f a c e Z a c r o s s w h i c h t h e d i s p l a c e m e n t s a r e d i s -co n t i nu ou s . T h u s ~ = 0 , ~2 = 0 , ~3 = 1 .

N o n - p r o p a g a t i n g f a u l ts( i ) [ u ~ ] = ~ x~ ) 8 x~ )H t ), u 2 ] = 0 , [u ~ ] = 0 .

3 , H t )~

~ 5 . H t )

Fro. 2 . a) Disp lacem ent d is locat ion a t the focus for mo del i . b) Eq uiv alen t double couplein an unfau l ted med ium fo r model i .I n t h i s m o d e l a t a n g e n t i a l d i s p l a c e m e n t o f t h e m a t e r i a l i n x a > 0 o c c u r s r e l a t i v e

t o t h a t i n x3 < 0 . T h e d i s p l a c e m e n t h a s a s u d d e n o n s e t a t t = 0 ; t h e r e a f t e r i t r e -m a i n s c o n s t a n t . T h e f o r m a l s t a t e m e n t a b o v e i m p l ie s t h a t t h e a r e a o v e r w h i c h t h ef a u l t i n g o c c u r s h a s b e e n s h r u n k i n t h e l i m i t t o t h e o r i g i n .

I n ( 1 1 ) t h e o n l y n o n - ze r o t e r m s a r e t h o s e f or w h i c h ( i j p q ) i s (1 , 3 , 1 , 3 ) and(1 , 3 , 3 , 1 ) .

T h u s= ~e l ( x , t ) - - ~ ( x l ) ( x2 )~ ( x 3 ) H ( t )

e ~ x , t ) = 0

e3(x, t) = - p 8 ( x l ) 8 ( x 2 ) ~ ( x 3 ) H ( t ) .T h i s i s t h e f a m i l i a r d o u b l e c o u p l e f o r c e e q u i v a l e n t w h i c h b e g i n s t o a c t a t t = 0a n d t h e r e a f t e r i s h e ld a t a c o n s t a n t l ev e l. I t m a y b e r e p r e s e n te d s c h e m a t i c a l l y a ss h o w n i n F i g u r e 2 .(i i) [u~] = 0, [u2] = a ( x l ) a ( x ~ ) H ( t ) , [m] = 0 .T h i s i s s i m i l a r t o ( i ) . T h e x 2 - a xi s r e p l a c e s t h e x l - a x i s.


8/14

88 2 BULL ETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA

e ~ x , t ) = 0

e : ( , , , t ) = - ~ ( x ~ ) ~ ( ~ ) ~ ( x ~ ) H ( t )e 3( x, t ) = - ~ ( x l ) ~ ' ( x ~ ) ~ ( x ~ ) H ( t ) .

A g a i n t h e r e s u l t i s a d o u b l e c o u p l e .i i i ) [ u ~] = 0 , [ u 2 ] = 0 , [m] = 8 ( x l ) ~ ( x 2 ) Y ( t ) .

T h i s r e p r e s e n t s a s u d d e n s e p a r a t i o n o f t h e t w o s id e s o f t h e f a u l t a s f o r e x a m p l e ,a n e x p l o s i on i n a p l a n e c r a c k . F o r t h i s c a s e th e n o n - z e r o t e r m s i n ( i , j , p , q ) a r e

3 , 3 , 1 , 1 ) , 3 , 3 , 2 , 2 ) a n d 3 , 3 , 3 , 3 ) .

oH t ) 3 . H e ,

F I G . 3 . a ) D i s p l a c e m e n t d i s l o c a t i o n a t t h e f o c u s f o r m o d e l 3. b ) E q u i v a l e n t d o u b l e t s i n a nu n f a u l t e d m e d i u m f o r m o d e l 3 .

F r o m 1 1 ) w e g e te ~ (x , t ) = - ~ ' ( x l ) ~ ( x 2 ) ~ ( x 3 ) H ( t )e 2( x, t ) = - ~ ( x ~ ) ~ ' ( x 2 ) 8 ( x 3 ) H ( t )e 3 (x , t ) = - ( h + 2 ~ ) ~ ( x ~ ) ~ ( x 2 ) ~ ' ( x 3 ) H ( t ) .

T h e f o r c e e q u i v a l e n t c o n s is t s o f t h r e e l i n e a r d o u b l e t s a l ig n e d a l o n g t h e t h r e e c o o r -d i n a t e a x e s F i g u r e 3 ) .U n i l a t er a l m o v i n g f a u l t s

i v ) l u l l = H ( v t - x l ) { H ( x l ) - H ( x l - a ) } 6 x 2 ) , [ u 2 ] = 0 , [u s] = 0 .I n t h i s c a s e m a t t e r i n x 3 > 0 b e t w e e n x l = 0 a n d x l = v t o n t h e l i n e x 2 = x 3 - 0h a s a p e r m a n e n t t a n g e n t i a l d i s p l a c e m e n t i n t h e x l - d i r e c ti o n r e l a t i v e to th e m a t t e ra d j a c e n t t o i t i n x3 < 0 . T h e d i s lo c a t i o n s t o p s a t x ~ = a w h e r e a is th e l e n g t h o ft h e f a u l t .

A s i n 1 ) t h e o n l y t w o t e r m s i n 1 1 ) a r e ( i , j , p , q ) = 1 , 3 , 1 , 3 ) a n d 1 , 3 , 3 , 1 ) .


9/14

B O DY F O R C E E Q U I V A L E N T S 883

H e n c e 1 1 ) g i ve s

O x , t ) = - -t L f f H ( v t - } ) { H } ) - - H } - - a ) } 6 7 ) 6 x ~ - } ) 6 x 2 - - 7 )

. 6 ( x 3 ) d } d 7

e2 x , t ) - -- - 0

e 3 x , ) = -~ , f f H v t - - } ) { H ( } ) - - H ( } - - a ) } ~ ( 7 ) 8 ( x l - - } ) 8 ( x 2 - - 7 )d d 8 x 3 ) d } d n .

W ]. ,I , 3

2~ ' ~ ~e ' ~ r ~ r ~ r ~ t '~ J

\ o H ( v t - x , )

FIG. 4. a) Pro pag a t i ng d i sp l acement d i s l oca t i on i n t he se ismic source for mo de l i v . b )E q u i v a l e n t p r o p a g a t i n g d o u b l e c ou p l e i n a n u n f a u l t e d m e d i u m f o r m o d e l iv .T h i s s h o w s t h a t i n t h i s c as e w e h a v e a n e x t e n d i n g l in e o f d o u b l e c o u p l e s s t a r t i n ga t t = 0 a n d e x t e n d i n g b e t w e e n x ~ 0 a n d s t u n t i l v t = a , w h e n n o fu r t h e r p r o p a g a -t i o n t a k e s p l a c e .

I t f o l l o w s t h a tO x , t ) = - t ~ H ( v t - X l ) { H ( X l ) - - H X l - - a ) } ~ x 2 ) ~ t x ~ )e 2 x , t ) = 0e ~ x , t ) = - t ~ { 5 x l ) - ~ x l - m i n a , v t ) ) } ~ ( x 2 ) 5 ( x ~ ) H ( t ) .

T h i s is s h o w n s c h e m a t i c a l l y i n f i g u re 4. W e n o t e t h a t t h e l in e o f d o u b l e c o u p l e sg i v e s a li n e o f s i n g le c o u p l e s e l t o g e t h e r w i t h t w o i s o l a t e d s i n g l e f o r c e s e3 w h e n t h ei n t e g r a t i o n s a r e c a r r i e d o u t .

v ) [Ul] = 0 , [u2] = H ( v t - x l ) { H ( x l ) - H ( x l - a ) } ~ x 2 ) , [ u 3 ] = 0 .T h i s r e p r e s e n t s a r e la t i v e t a n g e n t i a l d i s p l a c e m e n t o v e r t h e s a m e r e g i o n a s i n

m o d e l i v ) b u t in t h e x ~ - d i r ec t io nel X, t ) = 0


10/14

1 8 8 4 BULLETI N OF THE SEISMOLOGIC L SOCIETY OF MERICe~ x, t ) = - - u H ( v t - x l ) { H ( x l ) - - H ( x l - a ) } 6 ( x 2 ) 6 ( x a )e~ x, t ) = - - u H ( v t - - x l ) { H ( x l ) - - H ( x ~ - - a ) } 6 ( x 2 ) 6 ( x 3 ) .

T h i s is s h o w n s c h e m a t i c a l l y i n f ig u r e 5 a s a n e x t e n d i n g l in e o f d o u b l e c o u p l es .v i ) In1] = 0 , [u~] = 0, [ua] = H ( v t - x l ) { H ( x l ) - - U ( x ~ - - a ) } 6 x 2 ) .

T h i s r e p r e se n t s t h e o p e n i n g o f a g r o w i n g l e n t ic u l a r c a v i t y . F o r m u l a 1 1 ) n o wg i v e s

e l x , t ) = - ~ { 6 x l ) - ~ x l - m i n v t , a ) ) } 6 ( x 2 ) 8 ( x a ) H ( t )e2 x , t ) = - - X H ( v t - x l ) { H ( x l ) - H ( x l - - a ) } 8 ( x 2 ) ~ ( x a )e3 x , t ) = - ( ~ q - 2 ~ ) H ( v t - x ~ ) { H ( x , ) - - H ( x l - - a ) } ~ ( x = ) 6 ( x a ) .

L 3 * H v t - x 0

2

~'J~ ~'~'? '~v1

5 H v t - x =)

v t ~

FIG. 5. a Propaga t i ng d i sp l acement d i s l oca t i on on t he se i smi c source for mode l v . b )E q u i v a l e n t p r o p a g a t i n g d o u b l e c o u p le in a n u n f a u l t e d m e d i u m f o r m o d e l v .T h i s m a y b e r e p r e s e n t e d a s in F i g u r e 6 . T h e s o l u t i o n c o n s i s ts o f a s et o f p r o p a g a t i n gd o u b l e t s i n th e t h r e e c a r d i n a l d i r e c ti o n s . T h e d o u b l e t s i n t h e x l - d ir e c t io n i n t e g r a t ei n t o t w o s i n g le f o r c es a t t h e e n d s o f t h e f a u l t .

B e f o r e c o n s i d e r i n g t h e d i s c o n t i n u o u s t r a c t i o n s w e c a l c u l a t e t h e e q u i v a l e n t f o r c e si n c a s e s i ), i i) a n d i ii ) w h e n t h e a x e s a r e r o t a t e d .

L e t u s t a k e t h e f a u l t p l a n e a s x s = x.~ t a n 4 ; h e n c e v l = 0 , v2 = - s i n 4 , v3 =c o s 4 . L e t } a n d v b e c o o r d i n a t e s i n th e f a u l t p l a n e , d E = d } d n a n d

~ x , 6 ) = ~ x ~ - ~ ) ~ x 2 - n c o s 4 ) ~ x 3 - ~ s i n 4 ) .

v i i ) [ u l] = 6 } ) 6 n ) H t ) , [ u 2 ] = 0 , [ us ] = 0 .T h i s is l ik e c a s e i ) , b u t a r o t a t i o n t h r o u g h a n a n g l e 4 a b o u t t h e x l - ax i s h a s b e e np e r f o r m e d .

F o r m u l a 1 1 ) is , i n t h i s c a s e ,

t = - f { 8 ( } ) 6 ( n ) H ( t ) v f l p q S q ( x , 4 } d d n= - - ~ c i v ~ 6 ~ ( x ) H ( t )


11/14

BODY FORCE EQUIVALENTS 885

T h e n o n - z e r o t e r m s a r e t h o se f o r w h i c h ( p , j , q ) a r e 1 , 2 , 2 ) , 1 , 3 , 3 ) , 2 , 2 , 1 )a n d 3 , 3 , 1 ) . C o l l e c ti n g t h e s e r e s u l t s w e h a v ee , x , t ) \ i - - p , ~ l X l ) ~ x 2 ) ~ t x 3 ) H , ) \ t - - p , ~ X l ) ~ l x 2 ) ~ x , 3 ) H t )

e3 x, t ) l \ - -t L 6 ( x l )~ ( x 2 ) ~ ( x 3 ) H ( t ) IT h u s , i n th i s c a s e, t h e e q u i v a l e n t f o r c e a p p e a r s a s a l in e a r c o m b i n a t i o n o f t h e f o r c ee q u i v a l e n t s f o r a f a u l t i n th e p l a n e x 8 = 0 a n d o n e i n t h e p l a n e x2 = 0 . T h e c o e f -f i c ie n t s a r e t h e x 2 a n d x8 c o m p o n e n t s o f t h e n o r m a l t o t h e f a u l t p l a n e .

J

15

2l l l l

. H . t )

5

. H v t - x , )

[ [ ~ [ l . 8 v t - x , )

FIG. 6 . a) Pro pag ating displacem ent d is locat ion on th e se ismic sou rce for model v i . b ~}Equ iva len t p ropaga t ing doub le t s in an unfau l ted med ium fo r mode l v i .v i i i ) [u~ ] = 0 , [ u s] = 6 } ) 6 v ) H t ) c o s , [u 3] = 6 } ) 6 ~ ) H t ) s i n .

T h i s i s a t a n g e n t i a l d i s p l a c e m e n t p e r p e n d i c u l a r t o t h a t i n v i i) a n d r e p r e s e n ts ar o t a t i o n o f m o d e l i i) a b o u t t h e x ~- ax is t h r o u g h a n a n g l e . F o r m u l a 1 1 ) g i v e s

e , x . t ) = - - f ~ d~d~F o r a n o n - z e r o c o n t r i b u t i o n i ~ 1 , j ~ 1 .C o l l e c ti n g t h e c o n t r i b u t i o n s f r o m t h e v a r i o u s t e r m s , w e h a v e

u 2 x , t ) = c o s 2 - - g ~ ( x l ) ( f ( x 2 ) ~ ( x 3 ) H ( t ) + s i n 2 g ~ ( x l ) ~ ( x ~ ) ~ ( x 3 ) H ( t )u s x , t ) \ - - ~ ( x l )( f ( x2 ) ~ ( x~ ) H ( t ) / \ - - t ~ ( x l ) ~ ( x2 ) ~ (.c~)H (t) /

T h e d i a g r a m i n f i g ur e 7 s h o w s t h e f o r c e e q u i v a l e n t f o r = 4 5 i n w h i c h c a s ec o s 2 = 0 a n d s i n 2 = 1 .

F o r = 0 t h i s f o rc e e q u i v a l e n t l o ok s s o m e w h a t d i ff e re n t f r o m t h a t o f e x a m p l ei i ) , b u t b y r o t a t i n g t h e a x e s t h r o u g h a n a n g l e , re s o l v i n g t h e f o r c e s a l o n g t h e


12/14

1886 B U L L E T I N O F T H E S E IS M O L O G I C A L S O C I ET Y O F A M E R I C An e w a x e s, a n d t h e n u s i n g r e l a ti o n s l ik e

0 t t, z ~ ~ ( x ~ ) + ~ ( x ~ ) 0 ~ ~ ( x ~ )~(x~)~ (x~) = ~( xJ ) ox~ ox~w h e r e x ~ i s o n e o f t h e o l d c o o r d i n a t e s a n d x ~' a n d x ~' a r e n e w o n e s , i t c a n b e s e e n t h a t( i i) a n d ( v i i i) a r e b u t d i f f e r e n t d e s e r i p t i o n s o f t h e s a m e s e t, o f f o r c e e q u i v a l e n t s .T h e s a m e r e m a r k m a y b e m a d e f o r m o d e l s ( i) a n d ( v i i) .( i x ) lu g] = 0 , [u 2] = - - s in $ ( } )5 (n )H ( t ) , [u a] = co s 6 (} ) ~(n )H ( t ) .

@ ~52

....

3.H t)$ 7 2

F r o , 7, a ) D i s p l a c e m e n t d i s l o c a t i o n a t t h e f o c u s f o r m o d e l ( v i ii ), b ) E q u i v a l e n t d o u b l e t si n a n u n f a u l t e d m e d i u m f o r m o d e l ( v i i i) .T h i s c o r r e s p o nd s t o m o d e l ( i i i) w i t h a r o t a t i o n o f a x e s. T h e r e s u lt s , w h i c h a r e n o ta s e l e g a n t a s t h o s e f o r ( v i i ) a n d ( v i i i) a r e

e l (x , t ) )e~(x, t)e3(x, t)

= ~--X~(xl)~ (x2)~(x3)H(t))\--XS(xl) ~(x2)~ (x3)H(t) /

+ 0 t-2t~ sin 2 ~(xl)~ (x2)~(x3)H(t)- 2 g c o s 2 ~(xl)~(x2)6 (xa)H(t) /

+: o )

s in 2 / ~ (x l ) /~ (x 2 )~ ' (x , )H ( t ) .sin 2~(xl)5 (x2)~(x3) H (t) /

Equivalent forces when the displacements are continuous but the tractions are not.T h e b a s i c e q u a t i o n ( 1 1 ) r e d u c e s i n t h ~ s c a s e t o

e~ ,,, t) = - [T~] ~, t)~ , ,, ~ )d ~ .I f w e a g a in sp ec ia l i ze to x s = 0 fo r ~ , t h en [ T ~] = [rp3 ] w h ere r~j i s t h e s t r e s s t e n so r .( x ) . N o n - p r o p a g a t i n g f a u l t

[T~] = t~(xl)6(x2)H(t) .


13/14

BODY FORCE EQUIVALENTS 1887w h e r e t v i s a c o n s t a n t v e c t o r

e~ x , t ) = - t p S x l ) ~ x 2 ) ~ x 3 ) H t ) .T h i s i s a n i s o l a t e d f o r c e a c t i n g a t t h e p o i n t 0 , 0 , 0 ) i n t h e o p p o s i t e d i r e c t i o n t ot h a t o f th e d i s c o n t i n u i t y i n t r a c t i o n a n d w i t h t h e s a m e m a g n i t u d e .. x i) . U n i l a t e r a l p r o p a g a t i n g f a u l t

[Tp] = t p H v t - x l ) { H x l ) - - H x ~ - - a )}~ x2)ep x , t ) = - t v H v t - x l ) I H x l ) - H x l - a ) } ~ x 2 ) ~ x 3 ) .

H e r e , t h e e q u i v a l e n t f o r c e i s a l in e d i s t r i b u t i o n o f f o r c e p r o p a g a t i n g a l o n g t h e l i n eo f f a u l ti n g . T h e f o rc e a l w a y s a g re e s i n m a g n i t u d e a n d d i r e c t io n w i t h - [ T v ] , t h en e g a t i v e o f t h e d i s c o n t i n u i t y i n t h e t r a c t i o n .

x i i ) . B i l a t e r a l f a u l t sB i l a t e r a l f a u l t in g c a n b e c o n s i d e r e d f o r a ll t h e s e m o d e l s. T h e m a t h e m a t i c s i n a ll

c a s e s i s r e l a t i v e l y s i m p l e s i n c e t h e s o l u t i o n s f o r t h e u n i l a t e r a l p r o t o t y p e s h a v e a l -r e a d y b e e n g i v e n . W e r e p l a c e t h e u n i l a t e r a l p r o p a g a t o r H v t - x i ) H t ) b y t h eb i l a t e r a l p r o p a g a t o r l H v t ~ - Xl) - H - v t Jr- x ~ ) l H t ) . N o n e w f e a t u r e s a r e i n -t r o d u c e d i n t o t h e s o l u t i o n s a l r e a d y o b t a i n e d . T h e r e s u l t s o b t a i n e d i n t h e e a r l i e rm o d e l s a r e s im p l y m o d i f ie d to t a k e i n t o a c c o u n t t h a t t h e d i s l o c a ti o n s a ls o p r o p a g a t ei n t h e o p p o s i t e d i r e c t i o n .

7 GENERAL REMARKSA l t h o u g h t h e b o d y f o r c e s e q u i v a l e n t t o a g i v e n d i s l o c a ti o n a re s u p p o s e d t o a c t i n

a n u n f a u l t e d m e d i u m a n d t h e r e f o r e c a n n o t i n a n y s e n s e r e p r e s e n t r e a l f o rc e s a c t i n go n t h e r e a l m e d i u m , t h e s e f o rc e e q u i v a l e n t s m a y n e v e r t h e l e s s p r o v i d e a u s e f u l t h e -o r e t i c a l t o o l . T h e i r u s e f u l n e s s l i e s i n t h e f a c t t h a t i f t w o d i s l o c a t i o n s h a v e t h e s a m ee q u i v a l e n t f o r c e t h e y a ls o e m i t t h e s a m e r a d i a t i o n . T h u s , w h e n w e w i s h t o fi n d t h er a d i a t i o n f r o m a g i v e n d i s l o c a t i o n , i t i s s o m e t i m e s p o s s i b l e t o f i n d , b y m e a n s o f t h ee q u i v a l e n t f o r c e s , a s e c o n d d i s l o c a t i o n g i v i n g t h e s a m e r a d i a t i o n a s t h e o r i g i n a ls o u r c e ; t h e r a d i a t i o n f r o m t h e s e c o n d s o u r c e m a y b e s i m p l e r t o t r e a t a n a l y t i c a l l y .B m T i d g e , L a p w o o d , a n d K n o p o f f 1 9 64 ) h a v e u s e d t h i s t e c h n i q u e to c o m p u t e t h er a d i a t i o n f r o m a d i s l o c a t io n i n t h e p r e s e n c e o f a p l a n e f r e e s u r fa c e .

A s a n o t h e r e x a m p l e w e m e n t i o n a m e t h o d o f f in d i n g t h e s o u r c e f u n c t i o n ins p h e r i c a l p o l a r c o o r d i n a t e s c o r r e s p o n d i n g t o a n i s o la t e d p o i n t b o d y f o r c e a c t in g a tt h e p o i n t b , 0, ) . I n t h i s m e t h o d t h e p o i n t f o r c e i s e x p r e s s e d a s a s e r ie s o f v e c t o rh a r m o n i c s u r f a c e d i s t r i b u t io n s o v e r t h e s p h e r e r = b M o r s e a n d F e s h b a c h , 1 95 3 ,p . 1 8 9 8 ) . T h e r a d i a t i o n c o r r e s p o n d i n g t o e a c h te r m o f t h e s e ri es m a y t h e n b e f o u n db y i m p o s i n g d i s c o n ti n u i ti e s in t r a c t i o n a c r o ss r = b i n a c c o r d a n c e w i t h o u r fo r m u l a

1 1 ) w h e n [u i] = 0 . T h i s t y p e o f s i n m l a t i o n o f p o i n t s o u r c e s w a s a l s o u s e d b yL a m b 1 9 0 4 ) f o r t h e h a l f s p ac e .

T h e r e s u l t s o f t h e p r e s e n t c a l c u l a ti o n s s ho w t h a t f o r p r o p a g a t i n g o r n o n - p r o p a g a t -i n g d i s p la c e m e n t d i s l o c at i o n f a u lt s , t h e p r o p a g a t i n g o r n o n - p r o p a g a t i n g d o u b l ec o u p l e m o d e l i s a p p r o p r i a t e , w h e t h e r b o u n d a r i e s a r e p r e s e n t o r n o t .


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888 BUL L ET IN F THE SEISMOLOGICAL SOCIETY OF AMERICAACKNOWLEDGMENT

This work was supported by Grant No. AF-AFOSR-26-63 of the Air Force Office of ScientificResearch as part of the Advanced Research Projects Agency project VELA.REFERENCES

Balakina, L. M., Shirokova, H. I., and Vvedenskaya, A. V., Study of stresses and ruptures inearthquake foci with the help of dislocation theory, Publicat ions of the Dominion Observa-tory, Ottawa, 24, 321-327, 1960.Baker, B. B. and Copson, E. T., The Mathematical Theory of Huygens Principle, second editio~Oxford, Clarendon Press, 1950).Burridge, R., Lapwood, E. R., and Knopoff, L., Firs t motions from seismic sources near a freesurface, Bull. Seis. Soc. Amer., 54, 1875-1899.De Hoop, A. T., Representation theorems for the displacement in an elastic solid and their ap-plication to elastodynamic diffraction theory. D.Sc. thesis, Technische Hogeschool, Delft~1958.Knopoff, L., Diffraction of elastic waves, J. Acoust. Soc. Am., 28,217-229, 1956.Knopoff, L. and Gangi, A. F., Seismic reciprocity, Geophysics, 24, 681691, 1959.Knopoff, L. and Gi lbert, F. , Fir st motions from seismic sources, Bull. Seis. Soc. Am., 50: 117-134, 1960.Lamb, It. , On reciprocal theorems in dynamics, Loud. Math. Soc., 19, 144-151, 1889.Lamb, H., On the p ropagation of tremors over the surface of an elastic solid, Phil. Trans. Roy.Soc. London) A, 203, 1-42, 1904.Love, A. E. H., A treatise on the mathematical theory of elasticity, fourth edition New York:Dover Publications , 1944).Maruyama, T., On the force equivalents of dynamical elastic dislocations with reference to t heearthquake mechanism, Bull. Earthquake Res. Inst., Tokyo Univ., 41,467-486, 1963.Morse, P. M. and Feshbach, H., Methods of Theoretical Physics New York: McGraw Hill BookCo., 1953).Nabarro, F. R. N., The synthesis of elastic dislocation fields, Phil. Mag., 42, 1224-1231, 1951.Stauder, W., The focal mechanism of earthquakes, Advances in Geophysics, 9, 1-76, 1962.Whittaker, E. T ., A treatise on the analytical dynamics of particles and rigid bodies, fourth edi-tion New York: Dover Publications , 1944).

Manuscript received August 14, 1964.

Burridge and Knopoff 1964 BSSA BodyForceEquivalentsForSeismicDislocations

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