Free-Field (Elastic or Poroelastic) Half-Space Zero-Stress or Related Boundary Conditions

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1 Th 4 th World Confrnc on Earthquak Enginring Octobr 2-7, 2008, Bijing, China Fr-Fild Elastic or Porolastic) Half-Spac Zro-Strss or Rlatd Boundary Conditions Vincnt W L and Jianwn Liang 2 Profssor, Civil & Environmntal Enginring, Univ. of Southrn California, Los Angls, CA USA, vl@usc.du 2 Profssor, School of Civil Enginring, Tianjin Univrsity, Tianjin , China, liang@tju.du.cn ABSTRACT : Th boundary-valud problm for solving for wavs scattrd and diffractd from surfac and sub-surfac topographis hav attractd much attntion to arthquak and structural nginrs and strong-motion sismologists sinc th last cntury. It is of importanc in th dsign, construction and analysis of arthquak rsistant surfac and sub-surfac structurs in sismic activ aras that ar vulnrabl to nar fild or far fild strong-motion arthquaks. Th half-spac mdium can b lastic, or porolastic and fluid saturatd, th latr cas has attractd much nw rsarch in rcnt yars. Th prsnc of a surfac or sub-surfac topography, lik th cas of a surfac canyon, vally, canal or structural foundations, or an undrground cavity, tunnl or pip, will rsult in scattrd and diffractd wavs bing gnratd. Combind with th fr-fild input wavs, thy will togthr satisfy th appropriat strss and/or displacmnt boundary conditions at th surfac of th topography prsnt in th modl. For thos problms whr analytical solutions ar prfrrd in th studis, this oftn involvs a topography that is ithr: circular, lliptic, sphrical or parabolic in shap. This is bcaus in thos coordinat systms, th scattrd wavs ar xprssibl in trms of orthogonal wav functions, and th surfac of th topography oftn allows th orthogonal boundary conditions to b applid, so that th wav cofficints can b analytically dfind. Howvr, th prsnc of th half-spac boundary maks th problm much mor complicatd. Th scattrd wav functions ar no longr orthogonal on th flat half-spac surfac, and th zro-strss or rlatd boundary conditions ar no longr simpl nor straight forward to apply. This papr will xamin th availabl numrical and approximat mthods that hav bn attmptd or proposd, and th dirction all futur rsarch is taking us to solv this part of th problm. KEYWORDS: Diffraction, Fr-Fild, Half-Spac, Elastic, Porolastic. INTRODUCTION Larg topographis at half-spac surfacs, lik, canyons, vallys, canals and structural foundations, larg, undrground sub-surfacd structurs lik cavitis, pips, subways or tunnls, ar always prsnt in mtropolitan aras associatd with infrastructur dvlopmnts in cits. As a rsult of xcitation by incoming sismic wavs, thir prsnc will oftn gnrat additional wavs from diffraction and scattring, rsulting amplifications and d-amplification of th input wavs, which will affct th dformations, distributions and concntrations of strsss on narby ground surfacs and th structurs on thm. It is thus important to fully undrstand th thortical and nginring aspcts of th ffcts of such diffraction. This calls for a nd for accurat, analytical mthod to b dvlopd to study th in-plan rsponss of burid topographis to input wavs. In gnral, for any wav propagation boundary valud problms, it is ncssary to slct th right coordinat systms that will allow th vctor wav quations of th lastic wavs to b sparabl into scalar wav quations for th P- and SV- wav potntials. Howvr, bcaus of th complxity of th problm involvd, simpl analytic solutions to diffraction problms by ths undrground topographis, ar limitd in both quantitis and qualitis. For th cas of finit topographis in a finit rgion, lik that of

2 Th 4 th World Confrnc on Earthquak Enginring Octobr 2-7, 2008, Bijing, China machins, airplans or spac structurs lik satllits, ths would b wav boundary valud problms that can b solvd by a finit lmnt program, many vrsions of which ar widly availabl in industry. For th cass of problms involving infinit or smi-infinit mdium, lik th cas of all half-spac wav propagation problms, ths finit lmnt softwar will not b dirctly applicabl. A spcial larg finit lmnt may hav to b dfind to simulat th smi-infinit mdium. This will not always b satisfactory. This is not a nw problm. Solving for th scattrd wav potntials that nd to satisfy th zro-strss boundary conditions at th half-spac surfac is a classical boundary-valud problm that intrstd scintists, sismologists and nginrs, in particular arthquak nginrs, sinc th turn of th 20 th Cntury. Th zro-strss boundary conditions at th surfac of th half spac in th prsnc of surfac and sub-surfac topographis for in-plan cylindrical P- and SV- wavs hav always bn a challnging problms. Unlik th out-of-plan SH wavs, th imaging mthod cannot b applid to in-plan wavs. Th outgoing cylindrical P- and SV-wavs ar composd of Hankl functions of radial distanc coupld with th sin and cosin functions of angl. Togthr at th half-spac surfac th P- and SV- wavs functions ar not orthogonal ovr th smi-infinit radial distanc from 0 to infinity. For thm to simultanously satisfy th zro in-plan normal and shar strsss, sophisticatd analytic and numrical mthods will hav to b dvlopd. This papr will considr th various fw tchniqus that can b and hav bn usd to tackl this boundary-valud problm. Th fr-strss boundary conditions at th half-spac surfac to b solvd hr ar indd th most complicatd boundary conditions to b satisfid in th prsnt problm. Analytical mthods would rquir all th complicatd boundary conditions to b imposd and satisfid simultanously. Ovr th yars, numrical approximation to th gomtry is oftn mad. This is th cas whr th flat half-spac surfac is rplacd by a circular surfac of larg radius, th so-calld larg circl approximation to th half-spac surfac. Hr, a known xisting analytical formulation of th boundary-valud problm is prsntd, whr th Hankl wav functions ar xprssd in intgral forms, changing from cylindrical to rctangular coordinats, whn th zro-strss boundary conditions at th half-spac surfac can b applid in a mor straight forward sns. This is th so-calld Fourir Transform Mthod for solving th boundary-valud problm. It is somtims dsird to produc an altrnat, simplr approximat stimat of th solution to th boundary-valud problm without going at grat lngths to hav all th boundary conditions satisfid. It is thus intrsting to s what th solution would b lik if th half-spac boundary conditions ar not to b imposd and not to b satisfid, in othr words, if thy ar to b rlaxd. This is th so-calld mthod of Rlaxing th half-spac fr-strss boundary conditions Th prsnt mthod to b introducd may also srv as a gnral mthod for mor complicatd wav propagation problms, lik th diffraction problms involving surfac and sub-surfac topographis in a porolastic half-spac mdium. 2. THE MODEL: INCIDENT PLANE P WAVES As an xampl to illustrat th mthod, considr hr Figur ) th work of Lin t al 2008) of th diffraction of an undrground circular tunnl. Th following is a summary of th quations Th modl is a flat half-spac mdium with a circular rgion of a tunnl of outr radius a and innr radius b mad up of anothr lastic mdium. Th half-spac is supposd to b lastic, isotropic and homognous with mass dnsity ρ and Lamé lastic constants λ, μ. Th tunnl is supposd to b lastic with mass dnsity ρ and Lamé lastic constants λ, μ.

3 Th 4 th World Confrnc on Earthquak Enginring Octobr 2-7, 2008, Bijing, China Th xcitation consists of incidnt plan P-wavs ar assumd to hav an angl of incidnc γ α with rspct to th y-axis, frquncy ω and wav numbr k α = ω / c α, whr c α is th P-wav spd. Th prsnc of th half-spac flat ground surfac will rsult in both rflctd, plan P- & SV- wavs.. Thy ar givn by th r potntial function i r φ, φ and ψ rspctivly. Th total Fr-fild P- and SV-wavs potntials ar th sum of incidnt and rflctd plan wavs that satisfy th zro-strss flat ground surfac boundary conditions, ff ff rspctivly φ and ψ, ar givn by, with rspct to th r, θ ) coordinat systm Lin t al, 2008): y O x ψ r φ r ψ 2 s φ 2 s h r ψ s y φ s O θ b x φ i a φ t, φ t 2, t ψ t ψ 2 γ α Figur Th modl ff i r inθ ff r inθ = + = aj n n kr α ) = = bj n n kr ) n= n= ) φ φ φ ψ ψ whr a, b, n= 0,, 2, ar known cofficints. n n Th prsnc of th undrground tunnl will rsult in scattrd P- and SV- wavs in th half-spac. Thy ar rprsntd by Hankl functions of th first kind which rprsnt outgoing cylindrical wavs. Th P- and SV- scattrd potntials, masurd with rspct to th r, θ) coordinats, tak th form: φ s ) inθ s ) inθ = A, nhn kαr) ψ = B, nhn kr) n= n= 2) Th prsnc of th half-spac surfac will rsult in additional P- SV- wavs rflctd into th half-spac, of th form, with rspct to th r, θ ) coordinats, which ar finit vrywhr: φ s inθ s inθ 2 = A2, njn kαr) ψ2 = B2, njn kr) n= n= 3)

4 Th 4 th World Confrnc on Earthquak Enginring Octobr 2-7, 2008, Bijing, China Th Total Scattrd and Rflctd Wavs in th Half-Spac in cylindrical coordinats ar givn by, from quations ), 2) and 3): ff s s ) 2 n 2, n) n α, n n α n= φ = φ + φ + φ = a + A J k r) + A H k r) ff s s ) 2 n 2, n) n, n n n= ψ = ψ + ψ + ψ = b + B J k r) + B H k r) Th rgion within th tunnl is b r a, 0 θ 2π, with th outr circular surfac at r = a in contact with th half-spac mdium and th innr circular surfac r = b, th surfac of th innr cavity of th tunnl. Th prsnc of two circular surfacs will rsult in both outgoing and incoming cylindrical wavs, th total transmittd wavs insid th tunnl as: ) 2) n n α ) n n α )) φ = φ + φ = C H k r + C H k r t t t 2, 2, n= ) 2) n n ) n n )) ψ = ψ + ψ = D H kr + D H kr t t t 2, 2, n= inθ inθ inθ inθ 4) 5) 3. THE BOUNDARY CONDITIONS AND SOLUTIONS Th complt solution of this boundary-valud problm would consist of th boundary conditions: i) Zro-Strss at th Tunnl Innr Cavity Surfac r = b) ii) Strss & Displacmnt Continuity at th Tunnl Outr Surfac, and iii) Zro-Strss at th Half-Spac Surfac y = 0). Th first two sts of boundary conditions ar on circular surfacs in cylindrical coordinats, whil th last st is on th flat surfac in rctangular coordinats. For cylindrical wavs, th boundary conditions i) and ii) ar thus much mor trivial to apply. Th radrs ar rfrrd to Lin t al 2008) for thir dtaild quations. W will concntrat in this papr only on th discussion of th zro-strss boundary conditions. Almost 20 yars ago, an approximation to th flat surfac is proposd. L and Cao 989), Cao and L 990) usd a larg circular, almost flat surfac to approximat th half-spac surfac and prsntd numrical solutions to diffraction problms of surfac circular canyons of various dpths by incidnt plan P- and SV- wavs. Todorovska and L 990, 99a,b) solvd by th sam mthod for anti-plan SH wavs and incidnt Rayligh wavs on circular canyons. Latr, L and Karl 993a,b) xtndd th mthod to diffraction of P-, SV- wavs for diffraction by circular undrground cavitis. Also L and Wu 994a,b) usd th sam mthod of approximations for arbitrary-shapd two-dimnsional canyons. Finally, Davis t al 200) usd such an approximat mthod in thir cas studis of th failur of undrground pips. Rcntly, Liang t al 2000, 200a,b, 2002, 2003, 2004a,b,c) continud th analyss by th sam mthod from problms for circular-arc canyons and vallys, undrground pips to that of th porolastic half-spac. Currntly in ths last fw yars, th abov mthod was mt with svr criticism. It was flt that th larg circular approximation dos not rduc to th flat half-spac whn th radius, R, of th circular surfac approachs infinity, bcaus as R, th Bssl functions usd in th transformation would approach zro. It was flt that without a mthod to satisfactorily satisfy th fr-strss boundary conditions at th half-spac surfac, on would prfr rathr to rlax th fr-strss conditions than to us th larg circl approximations Todorovska and Yousf, 2006; Liang t al, 2006). With rspct to th {x, y} coordinats with origin at O at th half-spac surfac, th zro-strss boundary s s conditions at th half-spac surfac ar to b applid only to th scattrd P- and SV-wav potntials. φ + φ2

5 Th 4 th World Confrnc on Earthquak Enginring Octobr 2-7, 2008, Bijing, China s s and ψ + ψ 2. Thy ar rprsntd in cylindrical polar) coordinats, and th abov zro-strss conditions ar to b applid at th flat surfac y = 0 givn in rctangular coordinats. As statd at th introduction, applying such quations at th flat half-spac surfac to cylindrical wavs is not a trivial mattr, and prvious attmpt has bn mad to approximat th flat boundary surfac by a larg circular surfac. Th following approach is to lav th half-spac surfac flat and to follow th approach of Lamb s problm 904) to rprsnt ach mod or trm of th Hankl wav sris as an intgral in rctangular coordinats. In othr words, th Fourir transform of th th wav functions is takn. Tak th cas of th n mod of th P- and SV-wav scattrd potntials: n ) inθ i k ν n α ikx ν α y Hn kα r ) = ) ) dk iπν k α α 6) n k ) in i ν θ n ikx ν y Hn k r ) = ) ) dk iπν k whrν 2 2 α = k kα, and ν 2 2 = k k. Th cas of n = 0 in th abov intgral xprssions was first usd by Lamb 904) in his papr on th trmors. Hr in Equation 7), th Hankl wav functions in th {r, θ } cylindrical coordinats ar transformd to th {x, y } rctangular coordinats suitabl for th zro-strss boundary conditions at th flat half-spac surfac. Each intgrand in th abov intgrals is th Fourir Transform of th th n mod of th wav function. Using th {x, y} coordinat with origin at O on th half-spac surfac, th intgral taks th form, with y = y+ h, in th rgion 0 y h, or h y 0, apply now th transform to th whol st of trms for th P- and SV- potntials: s s ) inθ s ikx ν α y = r, ) = A, nhn kα r) = x, y) = [ a k)] dk n= s s ) inθ ikx y s ν =, ) =, n n ) =, ) = [ b )] n= φ φ θ φ ψ ψ r θ B H k r ψ x y k dk such that n k) a i ζα, n h)/ ν 0 A α, n = k) n= iπ 0 ζ, n h)/ ν 8) b B, n k να n ν h with, n h) ) α k ν n ν h ζ α =, and ζ, n h) ) =. kα k Th prsnc of both th undrground tunnl and th half-spac flat surfac abov will rsult in additional rflctd wavs gnratd Equation 3)). Thy will now b xprssd in intgral form, with rspct to th rctangular coordinat systm x, y) with origin O at th half-spac surfac, whr for y h in th half-spac, thy tak th form, with x = x, y = y+ h : s ikx+ ν s ikx y h α y ν h + ν ν α 2, ) = [ 2 )] ψ2, ) = [ 2 )] φ x y a k dk x y b k dk 9) In othr words, Equations 7) and 9) transform th scattrd cylindrical wavs in th half spac from wavs in cylindrical coordinats to that in rctangular coordinats, whr th fr-strss boundary conditions at th half-spac surfac, y = 0, ar straight forward, comparativly, to b applid. Such dtaild dscription of both th analytical procss and thir transformation back from rctangular to cylindrical coordinats, togthr with thir numrical implmntation, ar givn in Lin t al 2008) and will not b rpatd hr. 7) 4. RESULTS AND DISCUSSIONS

6 Th 4 th World Confrnc on Earthquak Enginring Octobr 2-7, 2008, Bijing, China A computr program in Visual Fortran to calculat th displacmnt amplituds and phrass on narby half-spac surfac abov th tunnl was dvlopd Lin t al, 2008). A Poisson Ratio of ν = 0.25 is also assumd unlss othrwis spcifid. Following all prvious work, it is convnint to xprss th frquncy of th incidnt, rflctd and scattrd wavs in trms of th dimnsionlss paramtr η dfind as follows ka ωa 2a 2a Trifunac,973): η = = = = 0) π πc c T λ Figur 2 shows, th horizontal and vrtical componnts of displacmnt amplituds, ux and u y, rspctivly along th surfac of th half-spac. Thy ar 3D plots of displacmnt amplituds plottd vrsus th dimnsionlss distanc x/a and dimnsionlss frquncy η = ωa π. c Figur 2 In th figur, th plan P-wavs hav an incidnt angl of γ = 30 o with rspct to th vrtical dirction. Th displacmnt amplituds along th surfacs ar plottd from x/ a= 4 to x/ a=+ 4, and for dimnsionlss frquncy η in th rang 0< η 6..Th circular tunnl of outr radius a is at a distanc of h blow th surfac with h/ a =.50. Th innr radius of th tunnl is b, with th ratio b/a = Th ratio of th mass dnsitis of th tunnl to that of th half-spac is ρ / ρ =.50, and th ratio of thir shar modulus is μ / 3.00 μ =, with th sam Poisson ratio of υ = υ = 0.25 for both th half spac and tunnl mdium. This maks th tunnl stiffr than th surrounding mdium. In th absnc of th undrground tunnl, for a uniform half-spac, th fr-fild ff ff amplitud of th ground displacmnt would b constant at ux =.2 and uy =.69 rspctivly in th x- and y- dirctions. L and Cao 989), Cao and L 990), in thir arlir work on wav diffractions around shallow circular canyons, obsrvd that th incidnt P- wavs at low frquncy, with wavlngth much longr than th topography in th displacmnt fild, do not sns nor s th prsnc of th obstacl as much as th highr frquncy with shortr wavlngth) incidnt P wavs. Th sam obsrvations wr mad by L and Karl 992, 993) for th sam undrground modl studid hr for incidnt P and SV wavs. Thr th undrground topography is a circular cavity. Th sam conclusion can b drawn from Figur 2 hr for both th horizontal and vrtical componnts, whr at low frquncis, 0< η <<, th displacmnt amplituds dviat slightly from th fr-fild amplituds plottd at η = 0 ), but as th frquncis incras, th oscillations of th

7 Th 4 th World Confrnc on Earthquak Enginring Octobr 2-7, 2008, Bijing, China amplituds incras, togthr with th amplification of th amplituds at ach frquncy gtting highr and highr. For incidnt angl of γ = 30 o, a displacmnt amplitud clos to 4 was obsrvd at som max u = 3.86 at η = 5.3, corrsponding to an amplification of almost 2. frquncy x ) Prvious works on th diffraction and scattring of anti-plan SH wavs by surfac and sub-surfac topographis all obsrvd a typical diffraction pattrn for th SH wavs, on that xhibit spcific charactristics both in front of and bhind th topography rsultd from th diffraction of th incoming wavs. Th lft and front sids of half-spac xhibit standing wavs pattrn, whil on th half-spac surfac to th right of th tunnl, with th tunnl acting as a barrir, a shadow zon is cratd, with wavs of dcrasing amplituds that ar also slightly smoothr, quit a contrast to th standing wav pattrn on th lft sid. Th prsnt and subsqunt figurs hr show that th sam wav pattrn can also b obsrvd for th in-plan P- and SV- wavs. Th only diffrnc may b that th in-plan pattrn is comparativly lss significant hr than of th SH wav cass. Th in-plan wavs also hav two componnts of motions, and ar furthr complicatd by th prsnc of wavs of both th longitudinal P-) and transvrs SV-) mod typs, so on of th two horizontal componnts can hav a mor significant pattrn. Figur 3 Figur 3 is a 2D plot of th displacmnt amplituds at th half-spac surfac abov th tunnl at 4 angls of incidnc for both th x- and y- componnts of motions. All plots ar at th sam dimnsionlss frquncy of η = ωa = Th sam ratios of shar π c modulus, μ / 3.00, μ = mass dnsitis, ρ /.50, ρ = and Poisson ratios, υ = υ = 0.25, will b usd as in prvious plots. Th sam ratios of dpth-to-radius h/ a =.50) and innr-to-outr radius b/a = 0.85) ar also usd. Th four angls of incidnc, from bottom to top ar rspctivly o o o o γ = 0 vritcal), 30, 60 and 85 almost horizontal) incidncs. Th lft column plots ar th horizontal x- componnt plots, whil th right column plots ar th vrtical y- componnt plots. Th graphs in ach plot hav th amplituds in th y-axis vrsus th dimnsionlss coordinat x/a in th horizontal axis, in th rang of [-4,4]. At ach plot, th horizontal dashd lin corrsponds to th fr-fild displacmnt amplitud at th half-spac surfac in th absnc of th undrground tunnl. Suprimposd on this ar th rsultant amplituds of th two typs of wavs, wavs calculatd with th half-spac strss-fr boundary conditions : imposd and 2: rlaxd fr-fild B.C. not applid). Thy ar rprsntd rspctivly by th wriggling solid lins and dashd lins and will b rfrrd from hr on as th Imposd wavs and Rlaxd wavs. Th two bottom plots for th x- and y- componnts ar forγ = 0 o, th cas of vrtical incidnc, which producs symmtric plots for both componnts of th Imposd solid) wavs and Rlaxd dashd) wavs, as xpctd physically. Both th Imposd and Rlaxd wavs hav vry

8 Th 4 th World Confrnc on Earthquak Enginring Octobr 2-7, 2008, Bijing, China similar wriggls, though th Rlaxd wavs do appar to hav highr maximum amplituds in th vrtical y-componnt of motion, though both maxima occur at th sam point of x/a on th half-spac surfac.. Th maximum amplitud at th dimnsionlss frquncy of η = 6.00, is around 3 for th imposd wavs and is almost 4 for th rlaxd wavs. A displacmnt amplitud of 4 hr, for a fr-fild amplitud of 2, thus corrsponds to an amplification of 2. Th nxt two sts of plots up abov, th cass of γ = 30 o and γ = 60 o, both show, for on of th componnts, th pattrn of standing, oscillatory wavs on th lft vrsus shadowy, smoothr wavs on th right. This is mor so for th y-componnt of th wavs at γ = 30 o, and th x-componnt of th wavs at γ = 60 o. As in th cas of vrtical incidnc, both th Imposd and Rlaxd wavs hav vry similar wriggls, and thir maximum amplituds ar much closr in location and amplituds. For γ = 60 o, th maximum amplitud of th Imposd wavs is ovr 4, whil that of th rlaxd wavs is blow but clos to 4. Th Imposd wavs thus hav an amplification of ovr 2. Th top plots ar for γ = 85 o, th cas of almost horizontal incidnc. It is not as intrsting as th othr thr cass of incidncs, as both th fr-fild and diffractd amplituds of th wavs ar comparativly lowr. 5. CONCLUSION In summary, this papr illustrats th succss of applying th flat half-spac zro strss boundary conditions onto th cylindrical wavs that ar scattrd and diffractd from surfac and sub-surfac circular topographis. It srv as a gnral mthod for mor complicatd wav propagation problms, lik th diffraction problms involving surfac and sub-surfac topographis in a porolastic half-spac mdium. REFERENCES H. Lamb 904) On th Propagation of Trmors ovr th Surfac of Elastic Solid, Phil. Trans. Royal Soc. A203, London, 904. V.R. Thiruvnkatahr & K. Viswananthan 965) Dynamic rspons of an Elastic Half Spac with Cylindrical Cavity to Tim Dpndnt Surfac Tractions ovr th Boundary of th Cavity, J. Math. Mch., , 965. M.D. Trifunac 973) A Not on Scattring of Plan SH Wavs by a Smi-Cylindrical Canyon, Int. J. of Earthquak Enginring and Struct. Dynamics,, , pp.4, 973. V. W. L 977) On Dformations Nar Circular Undrground Cavity Subjctd to Incidnt Plan SH Wavs, Symposium of Applications of Computr Mthods in Enginring, Aug 23-26, 95-96, Univ. of Southrn California., Los Angls, U.S.A.,, 977. V.W. L & M.D. Trifunac 979) Rspons of Tunnls to Incidnt SH Wavs, J. Eng. Mch., A.S.C.E.,05, , 7 Pags, 4 Rfs., Aug, 79. V. W. L & H. Cao 989) Diffraction of SV Wavs by Circular Cylindrical Canyons of Various Dpths, J. Eng. Mch., A.S.C.E., 59), , 22 Pags, 23 Rfs., Sp, 89 H. Cao & V. W. L 989) Scattring of Plan SH Wavs by Circular Cylindrical Canyons with Variabl Dpth-to-Width Ratio, Europan J. Earthquak Eng., III2), 29-37, 8 Pags, 0 Rfs., Dc, 89. H. Cao & V. W. L 990) Scattring and Diffraction of Plan P Wavs by Circular Cylindrical Canyons with Variabl Dpth-to-Width Ratio, Int. J. Soil Dynamics & Earthquak Eng., 93), 4-50, 0 Pags, 6 Rfs., May, 90. M.I. Todorovska & V. W. L 990) A Not on Rspons of Shallow Circular Vallys to Rayligh Wavs: Analytical Approach, Earthquak Eng. & Eng. Vibration, 0), 2-34, Mar, 90 M.I. Todorovska & V. W. L 99a) Surfac Motion of Shallow Circular Alluvial Vallys for Incidnt Plan SH Wavs -Analytical Solution, Int. J. Soil Dynamics & Earthquak Eng., 04), , 9 Pags, 30 Rfs., May, 9 M.I. Todorovska & V. W. L 99b) A Not on Scattring of Rayligh Wavs by Shallow Circular Canyons: Analytical Approach, ISET J. Earthquak Tchnology, 282), -6, 6 Pags, 5 Rfs., Jun, 9. V. W. L & J. Karl 992) Diffraction of SV Wavs by undrground, Cicular, Cylindrical Cavitis, Int. J. Soil Dynamics & Earthquak Eng., 8), 992), , 2 Pags, 8 Rfs., Jun, 92.

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