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A P P E N D I X A: O N T H E O R T H O G O N A L I T Y O F S U R F A C E W A V E E I G E N F U N C T I O N S IN C Y L I N D R I C A L C O O R D I N A T E S
A . l I N T R O D U C T I O N
In developing an expression for th e surface wave G reen ’s function in two d im en sions using rep resen tatio n th eorem s, Ile rre ra (1964) d e m o n stra te d th a t R ayleigh (an d Love) wave eigenfunctions for a laterally hom ogeneous, p lan e-stratified h alf space are o rth o g o n al w ith respect to a p a rtic u la r d ep th integral. He n o ted th a t th e co n trib u tio n of a given surface wave m ode to th e G re e n ’s function was d e te r m ined by an a p p ro p ria te norm alizatio n of this integral. M cG arr and A lsop (1967) em ploying a deriv ation originally used by Takeuchi etal. (1962) to co m p u te g roup velocities, showed th a t for plane waves th e choice of n o rm alizatio n was eq uivalent to im posing u n it energy tra n s p o rt across a plane of u n it w idth an d infinite d e p th p e rp e n d ic u la r to th e direction of p ro p ag atio n . H errera’s arg u m en t was ex ten d ed to plane waves in th ree dim ensions by M alischew sky (1970) who th ereb y d e m o n stra te d R ayleigh, Love and m u tu al o rth o g o n ality betw een Love and Rayleigh waves. M ore recently, Aki and R ichards (1980) invoked H errera’s original proof to derive surface wave te rm s for th e tw o-dim ensional G re e n ’s function, and recognised th a t w ith th e a p p ro p ria te choice of s tre ss/d isp la c e m e n t-re la te d q u an tities th e hom ogenous wave e q u a tio n in b o th two and th ree dim ensional problem s can be reduced to th e sam e sy stem of first-order differential eq u atio n s. T hey suggested th a t th e dev elo p m en t of surface wave o rth o g o n ality proceeding directly from stress and d isp lacem en t ex pressed in a cylindrical co o rd in ate sy stem is not possible. T h e purp o se of this n o te
is to d e m o n stra te th a t this developm ent, alth o u g h m ore com plicated alg eb raically th a n for th e tw o-dim ensional case, does indeed exist and th a t it gives rise to b o th a fu n ctio n al o rth o g o n ality relation betw een Love and Rayleigh eigenfunctions and a slightly m ore specific expression of Rayleigh wave eigenfunction o rth o g o n ality . A lthough not d e m o n strated here, these relations can be used to derive th e surface wave te rm s to th e three-dim ensional G re e n ’s function in th e sam e fashion as H ererra (1964). In th e final section we estab lish th e physical significance of eigenfunction
n o rm alizatio n in a cylindrical c o o rd in a te reference fram e.
A . 2 B E T T I ’S I D E N T I T Y A N D L O V E -R A Y L E I G H O R T H O G O N A L I T Y
We will begin as H errera (1964) w ith B e tti’s id e n tity and ex am in e first th e o rth o g o n ality betw een Love an d Rayleigh eigenfunctions an d then proceed to or th o g o n a lity betw een Rayleigh m odes as th e b o th derivatio n s follow a sim ilar a r gum en t b u t th a t in th e form er case is som ew h at less involved. C onsider any two displacem en t fields, u an d v w ith harm on ic tim e depen d ence e~lLjt satisfy in g th e hom ogeneous elastic wave eq uation th ro u g h o u t a laterally hom ogeneous, s tr a ti fied half-space V. B e tti’s id e n tity asserts th a t th e two fields and th e ir asso ciated tra c tio n s (t(u), t(v )) over S satisfy
J
d S [t(u) • v - t(v ) • u] = 0, ( A l )w here S is a closed surface w ithin V. We n o te th a t ( A l ) is essentially an expression of energy conservation reflecting th e fact th a t no body forces exist w ith in S. Now let u an d v rep resen t two ind ividual (sta n d in g wave) Fourier-B essel co m p on en ts of Rayleigh an d Love waves, respectively, b o th of which satisfy th e p rescribed conditions in a cylindrical co o rd in a te system an d which we define as