Scattering of SH waves by a circular sectorial canyon

12 Pages·2013·7.88 MB·English

Geophysical Journal International
Geophys.J.Int.(2013)195,532–543 doi:10.1093/gji/ggt236
AdvanceAccesspublication2013July10
SH
Scattering of waves by a circular sectorial canyon
Kao-Hao Chang,1 Deng-How Tsaur2 and Jeen-Hwa Wang1
1InstituteofEarthSciences,AcademiaSinica,Nangang,Taipei11529,Taiwan,R.O.C.E-mail:[email protected]
2DepartmentofHarborandRiverEngineering,NationalTaiwanOceanUniversity,JhongjhengDistrict,Keelung20224,Taiwan,R.O.C.
Accepted2013June14.Received2013May9;inoriginalform2012November8
D
SUMMARY o
w
The scattering of plane SH waves incident on a circular sectorial canyon is considered. An nlo
a
accurateregion-matchingtechniqueisappliedtoderivearigorousseriessolution.Appropriate d
e
wGaevgeefnubnacuteiornpsoalyrenoemmpialolsy,eadretowdeelslcurtiibliezaendtitpolcaonrermecotltyiocnasp.tJuurdeicthioeussinbgausilsarfubnechtaiovniosu,irnivnosltvriensgs d from
fields near the canyon bottom. The enforcement of matching conditions on the auxiliary http
boundary leads to the determination of unknown coefficients. Plotted results demonstrate s
the influence of pertinent parameters on surface and subsurface motions. Both steady-state ://ac
a
and transient results are included. The solution technique proposed achieves a considerable d
e
m
reductioninthecomputationaleffort,facilitatingbenchmarkcomputations.Thederivedseries
ic
solution enriches the limited list of series solutions presently known for canyon problems .o
u
p
relatedtoSH-wavescattering. .c
o
m
Key words: Earthquake ground motions; Theoretical seismology; Wave scattering and /g
diffraction;Wavepropagation. ji/a
rtic
le
-a
b
s
tra
c
t/1
9
contributedtoaqualitativeunderstandingofseismicwavepropa- 5
ogy 1 INTRODUCTION gationandearthquake-inducedmotionsaroundcanyons. /1/5
ol Ruggedtopographicreliefhasbeenwellknownasoneofthelead- As to the more complicated terrain, sophisticated numerical 32
m /6
ingfactorssubstantiallyaffectingthespatialandtemporalvariations schemes are, no doubt, the powerful tools to deal with related 0
s 3
ei inseismicgroundmotions(e.g.Boore1973;Bouchon1973;Spu- issues. In recent decades, the tremendous progress in computer 93
S 4
dichetal.1996).Thenear-surfacecomplexityinlocalgeometric technologieshasboostedthegrowthofnumerousmeshdiscretiza- b
GJI discontinuities (e.g. canyons, ridges, slopes and cliffs) intensifies tion techniques. Some of the most popular methods utilized to- y g
u
thegreatdiversityofseismicwavefields.Thisisusuallycalledthe day are the finite-difference, finite-element, spectral-element and e
s
topographiceffectinthepertinentliterature(e.g.Gelietal.1988; boundary-element methods (see e.g. Tessmer et al. 1992; Geller t o
n
Kawase&Aki1990;Hartzelletal.1994;Scottetal.1997).Over &Takeuchi1998;Komatitsch&Vilotte1998;Satohetal.2001; 1
1
theyears,themeasurement,assessmentandpredictiononthepo- Ka¨ser&Dumbser2006;Khametal.2006;Ichimuraetal.2007;Ma A
tDenavtiiasl&toWpoegsrta1p9h7ic3;aCmeplelibfiic1a9ti8o7n;dHuoruinggheeatrathl.q2u0a1k0e)shhaavkeinbgee(ne.ogf. eKtraislt.e2k0e0t7a;lL.2e0e1e0t;aPl.el2t0ie0s8e;tLaol.m2b0a1r0d).eAtaltlh.o2u0g0h8;thLeeseeeatpaplr.o2ac0h0e9s; pril 2
0
continuedinteresttoresearchersinshallowgeophysics,seismology, areveryflexibleforarbitrarysurficialconfigurations,thereliability 19
civilengineering,etc. andaccuracyofterminalresultsfromthesecodesathandhaveto
Concavetopographieslikecanyonsareonecommontypeofnatu- bestrictlyverifiedwiththeexistingcanonicalsolutions.
rallandformsonEarth’ssurface.Sincetheearly1970s,variouscon- For2-DproblemsregardingtheSH-wavescatteringinthehalf-
ceptualmodelshavebeenbuilttoapproximatethecross-sectional plane,exactanalyticalsolutions,derivedbythemethodofsepara-
profilesofrealcanyons.SimplytaketheSH-wavecase,forexam- tionofvariables(MSV),whichissometimestermedasthemethod
ple,tworepresentativesintherangeoftheoreticalconcernsarethe of wavefunction expansions in engineering seismology, are lim-
semi-circularandsemi-ellipticcanyons(seeTrifunac1973;Wong& itedtosimplegeometricshapes.Todate,therehaveonlybeentwo
Trifunac1974).Withintherangeofnumericalconcerns,simplified casesavailable,thatis,thesemi-circularandsemi-ellipticcanyons
modelsforthetriangular,rectangular,trapezoid,cosine-shapedand (Trifunac1973;Wong&Trifunac1974).Thisisbecausewhenthe
Gaussian-shapedcases have beendevised (seee.g. Aki&Larner scatterer geometry coincides with any one of the separable coor-
1970; Sa´nchez-Sesma & Rosenblueth 1979; England et al. 1980; dinate systems, the MSV can be straightforwardly applied to the
Zhou&Chen2006).Sa´nchez-Sesmaetal.(2002)extensivelyre- governingwaveequationtoyieldcompleteexpressionsintermsof
viewedtheanalogousmodels.Indeed,thesesimplifiedmodelshave eigenfunctions(orwavefunctions).However,evenforsomesimple
532 (cid:2)C TheAuthors2013.PublishedbyOxfordUniversityPressonbehalfofTheRoyalAstronomicalSociety.
Circularsectorialcanyons 533
geometriccases,itisnearlyimpossibletoexactlysolvetheseprob- 2 THEORETICAL FORMULATIONS
lems via the MSV. That is to say, the possibility of determining
Consider a homogeneous, isotropic, linearly elastic, semi-infinite
a one-to-one relationship to each unknown coefficient has been
medium(withshearmodulusμandshearwavevelocityc)bounded
excluded. Examples for those whose surface geometries incom- s
by the horizontal ground surface, inlaid with an infinitely long,
pletely/completelyfittingtoacertainseparablecoordinatesystem
circular,sectorialcanyon(seeFig.1).Theradiusandcentralangle
are the truncated semicircular canyons, the partially filled semi-
of the canyon are a and β, respectively. An infinite train of unit-
circular/semi-elliptic alluvial valleys, the circular-arc mountains
amplitudeplaneSHwaves(withangularfrequencyω)isincident
andthesemi-elliptichills.Examplesforthosewhosecross-sections,
uponthiscanyonatanangleαtothey-axis.
notmatchinganyoneoftheseparablecoordinatesystems,arethe
As seen in Fig. 1, through introducing a circular-arc auxiliary
symmetricalV-shapedcanyons.Undersuchacircumstance,ifase-
boundary S , the half-plane is divided into two regions, an open
riessolutioncanbeconstructed,itmaybethebestcandidateforan a
region1andanenclosedregion2.Inthesetworegions,thesteady-
exactanalyticalsolution.Recently,abreakthroughintheapplication
state out-of-plane motions are required to satisfy the governing
oftheregion-matchingtechnique(RMT)tohandleaforementioned
Helmholtzequations,namely
tasks was made by Chang (2009), Tsaur & Chang (2009), Tsaur
(2011)andTsaur&Hsu(2013).ThroughthepromisingRMT,the ∇2u +k2u =0, j =1,2, (1) D
j j o
intrinsicdifficultyinutilizingtheMSVtoderiveachainofseries w
where∇2isthe2-DLaplacianandk =ω/c istheshearwavenum- n
solutionshasbeenconquered. ber.Thesubscripts j,where j =1and2,desnotethetotaldisplace- loa
With regard to the canyons of circular sectorial shapes, there d
ment fields in regions 1 and 2, respectively. The time-harmonic ed
sSeuecmhsantoidbeealliitzteledasthtaepnetioisnappapirdoxtoimrealteevlyanctlotosepitcostihnetchreoslist-esreacttuiroen. factor exp(iωt) is omitted in all the expressions throughout this from
paper.
ofPacoimaCanyon(seefig.2binWong&Jennings1975).Tothe h
authors’knowledge,theseriessolutiontothescatteringproblemof The zero-stress boundary conditions on the horizontal ground ttps
circularsectorialcanyonsisunavailablesofar.Thishasmotivated surfaceandthecanyonsurfaceare ://a
the authors to tackle this issue in a theoretical way. In this work, τ(1)= μ∂u1(r,θ) =0, for θ =±π and r >a, (2) cad
the RMT is applied to derive a series solution to the case under θz r ∂θ 2 em
SH-waveexcitation. ic.o
Fromamathematicalstandpoint,theproblemunderconsidera- ∂u (r,θ) π π u
tionconstitutesanexteriorellipticboundary-valueprobleminvolv- τr(z1)=μ 1∂r =0, for − 2 <θ <−2 +β and r =a, p.co
m
ing one salient corner (at the canyon bottom). From fundamental (3) /g
choenncdetphtastocferetlaaisntoqsutaatnitcistieasndsueclhasatosdsytrneasmsfiicesl,distmisaeyabsyetiollcboemhapvreed- τθ(2z)= μr ∂u2∂(θr,θ) =0, for θ =−π2 +β,π2 and r <a. ji/artic
inthevicinityofgeometricdiscontinuities(e.g.cornersandsharp le
(4) -a
edges).Thepresenceofthosediscontinuouspointsusuallymakes b
s
the conventional mesh-based methods computationally inefficient Thefree-fielddisplacementuF,existinginthehalf-planemedium tra
(e.g. an enormous amount of discretised meshes required), espe- withoutanysurface/subsurfaceirregularities,canbeexpressedasa ct/1
ciallywhenacquiringanacceptablesolutiontofieldsneargeomet- sumoftheincidentwavesandtheirspecularlyreflectedwavesfrom 95
ricsingularities.Whenoneattemptstogettheseriessolutionvia thehorizontalgroundsurface,thatis, /1/5
theRMT,theexclusionofsuchsignificantinformationmayleadto uF(r,θ)=exp[ikrcos(θ+α)]+exp[−ikrcos(θ−α)]. (5) 32
averylargesystemofequations.Althoughmostresultsobtained /60
by the RMT may have acceptable accuracy for engineering pur- EmployingtheJacobi–Angerexpansion(Abramowitz&Stegun 39
3
pose,furthereffortsareworthytobedonefromtheviewpointof 1972),eq.(5)canbere-expressedas 4 b
numericalcomputationforcondensingthesystemofequationsand (cid:2)∞ y g
improvingthesolutionaccuracy.Inviewofthis,anadditionaltreat- uF(r,θ)=2 εn(−1)nJ2n(kr)cos(2n)αcos(2n)θ ue
s
mentforthecornersingularityatthecanyonbottomisessential,and n=0 t o
iRtMisTinhteegrreaatfetderi.ntotheRMT.Suchintegrationiscalledtheaccurate −4i(cid:2)∞ (−1)nJ2n+1(kr)sin(2n+1)αsin(2n+1)θ, n 11 A
ThekeystepoftheaccurateRMTistocapturethedominantchar- n=0 p
acteristicsofstressfieldsnearthecanyonbottom.Thishighlights (6) ril 2
0
theimportanceofblendingstresssingularitiesintothesolutionpro- 1
9
cedure.Attheinitialstage,theRMTisfollowedandsubsequently,
thepresentproblemisrecastintermsofstressintegralequationsin-
steadofwavefunctions.Therefore,thesingularbehaviourofstress
fieldscanbeincorporatedwiththeRMTinastraightforwardway.
Meanwhile,theaccurateRMTmorestronglyenhancestheoverall
computationalefficiencyofsolutionsthantheRMT.
Themaincontributionsofthepresentworkarethreefold.First,
the proposed series solution is novel since it is likely to be non-
existentintheliterature.Second,thecanonicalgeometriesstudied
intheliteraturearebroadenedhereintocoverarangeofcircularsec-
torialshapes.Third,theanalyticalapproachadoptedhereprovides
an efficient validation tool for various numerical approximation
schemes. Figure1. Geometriclayoutoftheproblem.
534 K.-H.Chang,D.-H.TsaurandJ.-H.Wang
(cid:6) (cid:6)
nwh≥er1e)εanndisJtnh(e·)Ndeeunmotaensnthfeacnttohro(redqeuralBteoss1eliffunnc=tio0naonfdthtoefi2rsift −π2π ∂∂ur1 sin(2q+1)θdθ = −π2π+β P(θ)sin(2q+1)θdθ,
kind. Note that this expression inherently satisfies the stress-free 2 2
conditiononthehorizontalgroundsurface(eq.2). q =0,1,..., (13)
The total scattered field uS in open region 1 may be separated
intotwoparts,uS0 anduS2,thatis,uS =uS0+uS2.Thefirstpart
uS0 representsthescatteredfieldsexcludingtheeffectofregion2, (cid:6) π2 ∂u2 cosqν(cid:3)θ− π(cid:4)dθ =(cid:6) π2 P(θ)cosqν(cid:3)θ− π(cid:4)dθ,
anditalsocorrespondstothescatteredfieldsderivedbyTrifunac −π+β ∂r 2 −π+β 2
(1973)forasemi-circularcanyon,thatis, 2 2
q =0,1,.... (14)
(cid:2)∞ J(cid:6) (ka)
uS0(r,θ)=−2 ε (−1)n 2n H(2)(kr)cos(2n)αcos(2n)θ
n H(2)(cid:6)(ka) 2n Aftersomealgebra,itcanbeshownthat
n=0 2n ε
(cid:2)∞ J(cid:6) (ka) An = πnPˆnC1, (15)
+4i (−1)n 2n+1 H(2) (kr)sin(2n+1)α
H(2)(cid:6)(ka) 2n+1 D
n=0 2n+1 o
2 w
×sin(2n+1)θ, (7) Bn = πPˆnS, (16) nlo
a
d
awnhderperimHne(2s)(s·t)ainsdthfoerndtihffoerrednetriaHtiaonnkwelitfhunrecstipoenctotfoththeesaercgounmdeknitnodf, Cn = πε−nβPˆnC2, (17) ed from
respectivefunctions.
h
The second part uS2 is the contribution due to the presence of wherethetransformedfunctionsPˆC1,PˆC2andPˆSareasfollows: ttp
region 2. Through satisfying the governing Helmholtz equation (cid:6) π n n n s://a
(eq.1)andthezero-stressconditionsonthehorizontalgroundsur- PˆC1= 2 P(θ)cos(2n)θdθ, (18) ca
face(eq.2),andthentakingaccountoftheoutgoingscatteredwaves, n −π+β de
thescatteredwavefielduS2canbewrittenas 2 m
ic
uS2(r,θ)=(cid:2)n∞=0 AnHH22((n2n2))(cid:6)((kkra))cos(2n)θ PˆnC2=(cid:6)−π2π2+β P(θ)cosnν(cid:3)θ− π2(cid:4)dθ, (19) .oup.com
/g
+(cid:2)n∞=0 BnHH22((n2n2)+)+(cid:6)11((kkar))sin(2n+1)θ, (8) PˆnS =(cid:6)−π2π+β P(θ)sin(2n+1)θdθ. (20) ji/article
2 -a
wherethecomplexexpansioncoefficients AnandBnareunknown. To make progress and achieve numerical efficiency, the stress bs
Thedisplacementoftheresultantwavefieldu1intheopenregion function P(θ)maybeexpandedinaseriesofbasisfunctions,con- trac
1,whichisthecombinationoffreeandscatteredwavefields,canbe tainingpivotalinformationonthebehaviourofstressfieldsatthe t/1
expressedas auxiliaryboundary S .Hence, P(θ)maybeapproximatedasfol- 95
a /1
u1(r,θ)=uF(r,θ)+uS(r,θ) (9) lows: /53
(cid:2)∞ 2
Intheenclosedregion2,thedisplacementofwavefieldu2,satis- P(θ)= pmSm(θ), (21) /603
fyingtheHelmholtzequation(eq.1)andthetraction-freeconditions m=0 93
(eq.4),isgivenby where p arethesingularexpansioncoefficientstobedetermined, 4 b
m y
u2(r,θ)=(cid:2)n∞=0CnJJnn(cid:6)νν((kkar))cosnν(cid:3)θ− π2(cid:4), (10) sainndAguStlmaa(rθsb)aelaiheraenvttihoceuorsrinonefgruthwlaeirtfhboarasmnisrifn−ut1ne/cr3itoiaorsnatsh.neglreadoifal9d0i◦s,taPnc(eθ)rhfarosma guest o
n
in which ν =π/(π−β) and the complex expansion coefficients theendpointtendstozero(seeAppendixBfordetails).Pondering 1
1
C aretobedetermined.Moredetailsforthederivationofeq.(10) overthesingularnatureofthecornerpointatθ =−π/2+β,and A
n p
aresetoutinAppendixA. takingthetraction-freeconditionintoaccount,asetofappropriate ril 2
The radial stress must be continuous over the entire auxiliary basisfunctionssatisfyingthecorrectinversecubic-rootsingularity 0
1
boundaryS ,implying isgivenby 9
a
(cid:5)
∂u ∂u P(θ),− π +β <θ < π, C1/6[2θ−π/2(π−β)]
∂r1 = ∂r2 = 0, − π <2θ <−π +β,2 r =a, (11) Sm(θ)=ψm{(θ+π/2−2βm)[3π−2(β+θ)](π−β)/2}1/3, (22)
2 2
NoticethattheintroductionofthestressfunctionP(θ)mayallow where ψm =(2m)!(cid:12)(1/6)[π(cid:12)(2m+1/3)]−1, (cid:12)(·) denotes the
onetoincludethestresssingularity(nearthecanyonbottom)inthe gammafunctions,andC1/6[·]aretheGegenbauerpolynomialstyp-
2m
laterformulation. icallyhavingthefollowingproperty(see,e.g.Gradshteyn&Ryzhik
Multiplyingbothsidesofeq.(11)bythepropercosine/sinefunc- 2007,pp.798,eq.7.324):
tions,andintegratingoverthecorrespondingrangesgives (cid:6) 1 C1/6(σ)cos(hσ) π(−1)m(cid:12)(2m+1/3)
(cid:6) π2 ∂u1 cos(2q)θdθ =(cid:6) π2 P(θ)cos(2q)θdθ, q =0,1,..., 0 2(m1−σ2)1/3 dσ = (2m)!(cid:12)(1/6)(2h)1/6 J2m+1/6(h). (23)
−π ∂r −π+β Additionalremarkspertainingtotheselectionofsingularbasis
2 2
(12) functionsaregiveninAppendixC.
Circularsectorialcanyons 535
Aftersubstitutingeqs(21)and(22)into(18)–(20),onemayfind
thattheexecutionofFouriercosine/sinetransformstothesingular
basis functions S (θ) is required. Using the change of variables
m
andemployingeq.(23),threeintegraltransformsofS (θ)maybe
m
written,respectively,asfollows:
(cid:6)
π2 S (θ)cos(2n)θdθ =(−1)m+nJ2m+1/6[2n(π−β)], (24)
−π+β m [4n(π−β)]1/6
2
(cid:6) π2 S (θ)cosnν(cid:3)θ− π(cid:4)dθ =(−1)mJ2m+1/6(nπ), (25)
−π+β m 2 (2nπ)1/6
2
(cid:6)
π2 S (θ)sin(2n+1)θdθ =(−1)m+nJ2m+1/6[(2n+1)(π−β)].
−π+β m [(4n+2)(π−β)]1/6 D
2 o
(26) w
n
lo
Inthelightofeqs(24)–(26),thetransformedfunctionsshownin a
d
e
eqs(18)–(20)maybeexpressedasfollows: d
PˆnC1=(cid:2)m∞=0pm(−1)m+nJ2m[4+n1/(6π[2−n(βπ)]−1/6β)], n≥1, (27) from https
://a
PˆnC2=(cid:2)∞ pm(−1)mJ2(m2+n1π/6)(1n/6π), n≥1, (28) cadem
m=0 ic
.o
u
Freisguultrseo2f.TSruifrufnacaecm(1o9t7io3n).s(va)erηsu=sx0/.2a5a;t(bα)=η=601◦.f2o5r.comparisonwiththe PˆnS =(cid:2)∞ pm(−1)m+nJ2m[(+41n/6+[(22n)(+π1−)(βπ)]−1/6β)]. (29) p.com
m=0 /g
Tataαbl=e13.0C◦aalncdulηat=ed2re.sultsofthefirstcomponentsofwavefieldexpansioncoefficientsforβ=45◦ ji/article
-a
AccurateRMT bs
tra
M A0 B0 C0 ct/1
12 1.71507486+0.25135313i 2.11465312−1.24052281i 2.28676648+0.33513751i 95
20 1.71508502+0.25134337i 2.11464785−1.24054703i 2.28678003+0.33512449i /1/5
30 1.71508493+0.25134327i 2.11464762−1.24054697i 2.28677990+0.33512436i 3
2
42 1.71508493+0.25134325i 2.11464759−1.24054699i 2.28677991+0.33512434i /6
0
44 1.71508493+0.25134325i 2.11464759−1.24054699i 2.28677991+0.33512434i 3
9
3
RMT 4 b
y
50 1.71410826+0.25166114i 2.11460540−1.23928226i 2.28547768+0.33554819i g
u
100 1.71472627+0.25143148i 2.11458908−1.24009935i 2.28630169+0.33524197i e
s
500 1.71504652+0.25134894i 2.11463564−1.24050127i 2.28672869+0.33513192i t o
1000 1.71506992+0.25134519i 2.11464250−1.24052929i 2.28675990+0.33512692i n 1
1
A
Tataαbl=e23.0C◦aalncdulηat=ed1re2s.ultsofthefirstcomponentsofwavefieldexpansioncoefficientsforβ=45◦ pril 2
0
1
AccurateRMT 9
M A0 B0 C0
49 7.17640661+1.61364908i 6.83176172−3.95258994i 9.56854215+2.15153210i
60 7.17641493+1.61364728i 6.83176459−3.95259864i 9.56855324+2.15152970i
75 7.17641454+1.61364657i 6.83176375−3.95259869i 9.56855272+2.15152876i
95 7.17641426+1.61364619i 6.83176326−3.95259866i 9.56855235+2.15152825i
98 7.17641426+1.61364619i 6.83176326−3.95259866i 9.56855235+2.15152825i
RMT
50 7.18656337+1.62708650i 6.84934638−3.95397975i 9.58208449+2.16944867i
100 7.17872288+1.61751378i 6.83646096−3.95243003i 9.57163051+2.15668504i
500 7.17654539+1.61393512i 6.83209015−3.95254680i 9.56872718+2.15191350i
1000 7.17645845+1.61375043i 6.83187938−3.95257697i 9.56861126+2.15166723i
536 K.-H.Chang,D.-H.TsaurandJ.-H.Wang
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Figure3. Surfacemotionsversusx/afordifferentαwithβ=30◦atη=4.(a)α=0◦and30◦;(b)α=60◦and90◦;(c)α=−30◦and−60◦;(d)α=−90◦. .ou
p
.c
o
m
Notethatifn=0,eqs(27)and(28)havethefollowingforms: Multiplyingeq.(37)byasequenceoftestingfunctionsandinte- /g
gratingovertherange[−π/2+β,π/2]gives ji/a
PˆC1= 6 p , (30) (cid:6) (cid:6) rtic
0 21/3(cid:12)(1/6) 0 π π le
2 u S (θ)dθ = 2 u S (θ)dθ, q =0,1,.... (38) -a
1 q 2 q b
6 −π2+β −π2+β stra
Pˆ0C2= 21/3(cid:12)(1/6)p0. (31) Substituting eqs (9), (10) and (32)–(36) into (38), utilizing the ct/1
Wronskian relations (Abramowitz & Stegun 1972), applying eqs 95
Substitution of eqs (27)–(31) into (15)–(17) leads to the con- (24)–(26)andrearrangingyieldsasystemoflinearalgebraicequa- /1
nectionbetweentheexpansioncoefficientsofdisplacementfields tionsforsingularcoefficients p ,thatis, /53
m 2
(An,Bn andCn)andthoseofcornersingularity(pm),thatis, /6
(cid:2)∞ 0
3
A = 6 p , (32) pmUq,m =Vq, q =0,1,..., (39) 93
0 π 21/3(cid:12)(1/6) 0 m=0 4 b
y
wherealltheinvolvedfunctionsaregiveninAppendixD(seeeqs g
An = π2 (cid:2)m∞=0pm(−1)m+nJ2m[4+n1/(6π[2−n(βπ)]−1/6β)], n≥1, (33) tDeexc1phaannnidsqiuDoen2s).c.oAefffitecriternutnscpatmincgatnhebeinefivnailtueasteerdiebsyprthoepesrtlayn,dthaerdsimngautrliaxr uest on 1
1
Bn = π2 (cid:2)m∞=0pm(−1)m+nJ2m[(+41n/6+[(22n)(+π1−)(βπ)]−1/6β)] (34) amafiFtniooirtnenuinnmudmeicrbiecesarlmocfoamtnepdrumtthasetiioswnie,nitdgruihsntpicneagntisniangbdlteihc.eeIsinnqfienqai.rtee(3st9uru)m,nmcthaaetteidosunmatfo-- April 201
ter M−1 terms. Therefore, eq. (39) constitutes a system of M 9
6 equations with M unknowns. The number of truncation terms in
C = p , (35)
0 (π−β)21/3(cid:12)(1/6) 0 considerationdependsonlyontheaccuracyrequirement.Oncethe
singularcoefficients p arefound,thewavefieldexpansioncoeffi-
m
C = 2 (cid:2)∞ p (−1)mJ2m+1/6(nπ), n≥1. (36) ceiqesn(t3s2()i–.e(.36A)n., Bn andCn)canbeevaluatedstraightforwardlyvia
n π−β m (2nπ)1/6
m=0
Forthesakeofdeterminingthesingularexpansioncoefficients
p , the enforcement of displacement continuity condition across
m
theauxiliaryboundaryS isnecessary,thatis, 3 NUMERICAL RESULTS
a
π π AND DISCUSSIONS
u (r,θ)=u (r,θ), for − +β <θ < and r =a.
1 2 2 2 TogetthepropertruncationvalueM ineq.(39),anumberofcon-
(37) vergencetestsarecarriedoutfirst.Itisworthemphasizingthateach
Circularsectorialcanyons 537
resultsandthoseofTrifunac(1973).Obviously,resultsensurethat
there is no mistake in the formulism presented. In addition, one
canfindthatatarelativelyhighfrequency(seeFig.2b),theresults
ofthisstudyareveryclosetothoseofTrifunac(1973)whenthe
selectedvalueofβislargerthan170◦.However,atarelativelylow
frequency(seeFig.2a),inordertoarriveatagoodapproximation
totheresultofTrifunac(1973),β hastobelargerthan179◦.The
reasonisthatthecanyoncaneasilyhampertheincidentwaveswith
relativelyshortwavelengths,and,therefore,onlyasmallamountof
scatteredwavescanpenetrateintoregion2.
3.2 AccuracyandperformanceoftheaccurateRMT
Tables1and2demonstratethreesetsofcomputedvaluestothefirst
D
componentsofwavefieldexpansioncoefficientsinregions1and2at o
w
obliqueincidence(α=30◦).Thecentralangleofthecanyonisset n
tobeβ =45◦.Table1correspondstoalow-frequencycase(η=2), loa
d
whileTable2toahigh-frequencycase(η=12).Tomanifestthe ed
accuracyandefficiencyoftheaccurateRMT,calculatedresultsof fro
m
theseriessolutionderivedbytheRMT(seeAppendixE)arealso h
includedinTables1and2. ttp
s
plaAces sahcocwurnaciyn TatabMle=1,1th2e(aacc1u2ra×te1R2MmTatrgiuxarraenqtueiersed4)-,daencdim8a-l ://ac
a
decimalplaceaccuracyisattainableat M =42(a42×42matrix de
m
needed).Whenthewavefrequencybecomeshigher(seeTable2),4- ic
decimalplaceaccuracycanbeachievedatM =49,and8-decimal .o
u
placeaccuracyatM =95.Bycontrast,resultsobtainedbytheRMT p.c
convergeextremelyslowly.TheRMTgetsonly4-decimalplaceac- om
curacy at best for the low-frequency case, albeit a huge value of /g
M =1000(a1000×1000matrixsolved)adopted.AsseeninTa- ji/a
ble2,theRMTwithavasttruncationvaluemerelymaintains2-or rtic
le
3-decimal place accuracy at a very high frequency. Undoubtedly, -a
b
Tables1and2showthattheemploymentoftheaccurateRMTis stra
muchmoreefficientthanthatoftheRMT.Thereasontogetsuch c
goodresultsistheinclusionofnear-cornerstresssingularityinthe t/1
Figure4. Spectralvariationsinsurfacemotionsaroundthecanyonwith 95
β=60◦at(a)α=0◦and(b)α=45◦. solutionprocedure. /1/5
3
2
innersumwithindexnistruncatedto N or N −1terms(seeeqs 3.3 Surfaceandsubsurfacemotions /603
D3–D7andD13–D15inAppendixD),anditshouldbecalculated inthefrequencydomain 93
4
accuratelybynumericaltestingforitsconvergence, therebyleav- b
ingonlyoneparameter(i.e.summationindices M)toremovethe Fig.3displaysthesurfacedisplacementamplitudesversusthedi- y g
problemofrelativeconvergence.Numericalexperimentsshowthat mensionlesshorizontaldistanceforsevenincidentanglesatarel- ue
moretermsarerequiredasthewavefrequencyincreases. ativelyhighfrequency(η=4).FromFig.3,onecanfindthatthe st o
Inthefollowing,thedimensionlessfrequencyηisdefinedas relativelyhighlevelsofgroundshakingarelikelytooccurfrequently n 1
η= kπa = 2λa. (40) nodefeandrcietshpbeleauncpdepmseteronwletafatrmdcsoprtlhniteeurhd(oeUrsiLztCoenn)dtoasfltgtohreoinuccanrndeysaousnerf.aaTcsheet.hSmeuaarxfnaigmcleeummofovtiianolcunies- 1 April 2
0
ontheilluminatedsidebecomemoreoscillatorythanthoseonthe 1
9
shieldedside.Forhorizontalincidence(α=−90◦)inFig.3(d),the
peakamplitudecanbeuptoabout2.5timesthatofthefree-field
3.1 Validationforthelimitingcase
response.
When the size of region 2 becomes very small (i.e. β →180◦), For the sake of revealing the influence of dimensionless fre-
thesectorialcanyonapproximatestothesemi-circularone.Since quencies on surface motions, Fig. 4 gives the spectral variations
Trifunac (1973) derived an exact analytical solution to antiplane indisplacementamplitudesforverticalandobliqueincidence(i.e.
scatteringinducedbyasemi-circularcanyon,severalcasesshown α=0◦and45◦)withβ =60◦.AsseeninFig.4(a),theupperright
infigs6and8ofTrifunac(1973)aretakenasvalidationexamples. corner(URC)ofthecanyon(0<x/a<0.5)behavesasaconcen-
Computedresultspertinenttothecaseofη=0.25atobliqueinci- tratorcollectingthewaveenergy,especiallyforthehigh-frequency
dence(α=60◦)aregiveninFig.2(a),whilethoserelativetothe cases(6<η<12).InFig.4(b),similarfocusingphenomenoncan
caseofη=1.25inFig.2(b).Specificlocationsonthecanyonsur- alsobeobservedatlocationsclosetotheULCofthecanyon.The
facerangingbetweenx/a=−1and1aredisplayedbyaboldblack amplificationfactorcanreachthevaluesof2.3and2.0forvertical
line.Figs2(a)and(b)exhibitgoodconsistencybetweenthepresent andobliqueincidence,respectively.
538 K.-H.Chang,D.-H.TsaurandJ.-H.Wang
D
o
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ttp
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://a
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a
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e
m
Figure5. Subsurfacemotionsaroundthecanyonwithβ=60◦forη=8.(a)α=0◦;(b)α=90◦;(c)α=−60◦;(d)α=−90◦. ic.o
u
p
Figs5(a)–(d)exhibitthesubsurfacemotionsaroundthecanyon .co
atα=0◦,90◦,−60◦and−90◦,respectively,whenη=8.Thelocal m/g
maximumdisplacementamplitudesoccurontheright-handsideof ji/a
theflatgroundsurfaceforverticalincidence(α=0◦)inFig.5(a), rtic
and on the even surface of the canyon for oblique incidence in le
-a
Fig.5(c).Forthecaseofhorizontallyincidentwaves(seeFigs5b b
s
andd),seismicmotionsintheilluminatedregioncanbeamplified tra
c
atleasttwotimesmorethanthoseinthenearbyregion.Thismay t/1
beattributedtotheconstructiveinterferencebetweentheincident, 95
reflected,andscatteredwaves. /1/5
Overall, surface and subsurface motions in the vicinity of the 32
canyonaredependentnotonlyonthecentralangleofthecanyon /60
3
butalsoonthefrequencyandangleofincidenceofarrivingwaves. 9
3
4
b
y
g
u
3.4 Surfaceandsubsurfacemotionsinthetimedomain e
s
t o
Using the fast Fourier transform algorithm, one may pursue the n
1
time-domain responses from the preliminary results obtained in 1
the frequency domain. The incident signal is a symmetric Ricker Ap
wavelet(Ricker1945),whichisdefinedtobe ril 2
0
u(t)=(2π2f2t2−1)exp(−π2f2t2), (41) 19
c c
where f isthecharacteristicfrequencyandissettobe2Hzhere.
c
Computationsaremadeatanumberofdiscretefrequenciesranging
from0to8Hzwithanintervalof0.0625Hz.Thetimewindowis
chosenas16s.Thecanyonradiusaandshear-wavevelocityc are
s
settobe1kmand1kms−1,respectively.Thecentralangleofthe
canyonisβ =60◦.
Fig.6displaystwosyntheticseismogramsatverticalandoblique
incidence (α=0◦ and 30◦). At the top of Fig. 6(b), four signals
receivedafterthedirect-wavesignalsarelabelled,inturn,byarrows
B, L1, L2and L3,whileatthebottomofFig.6(b),threesignals
Figure 6. Synthetic seismograms for fc=2Hz at β=60◦. (a) α=0◦; aredesignatedbyarrowsB,R1andR2.FromFigs6(a)and(b),the
(b)α=30◦.
amplificationsofsurfacemotionscanbeuptoabout1.62and1.82
Circularsectorialcanyons 539
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ttp
s
://a
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ic
.o
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p
.c
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m
/g
ji/a
rtic
le
-a
b
s
tra
c
t/1
9
5
/1
/5
3
2
/6
0
3
9
3
4
b
y
g
Figure7. Snapshotsforsubsurfacemotionsaroundthecanyonwithβ=60◦atα=30◦.(a)t=3.0s;(b)t=3.5s;(c)t=4.0s;(d)t=4.5s;(e)t=5.0s; ue
(f)t=5.5s;(g)t=6.0s;(h)t=6.5s. st o
n
1
1
A
times those of the free-field response under vertical and oblique Ingeneral,thebottomandtwouppercornersofthecanyonbehave p
incidence,respectively. asnewwavesources.Akeyfeatureofthediffractionisthatapart ril 2
0
In order to show the discrimination between different signal ofthereflectedwavefrontfromthehorizontalgroundsurfaceiscut 1
9
sources,eightsnapshotsoftransientsubsurfacemotionsatα=30◦ offbythecanyonandthencontinuouslyregeneratesitself.
are shown in Fig. 7. Furthermore, all arrows marked in Fig. 6(b)
areappropriatelytaggedinFig.7.InFig.7(a),theincomingpulses
impingeonthecanyonbottom.InFig.7(b),ascatteredwavetrain
4 CONCLUSIONS
B (i.e. arrow B in Fig. 6b) radiates from the canyon bottom. In
Fig. 7(c), the incident pulses reach the URC of the canyon. The ArigorousseriessolutionwasderivedfortheSH-wavescattering
scatteredwavesarisingfromtheULCofthecanyonandthescat- by a circular sectorial canyon. A robust, accurate RMT proposed
teredwaves B combinetogetherontheleft-handsideofthehor- hereinhassuccessfullycombinedtheRMTwiththeforethoughtof
izontal ground surface. Fig. 7(d) shows two new scattered waves corner singularity. Selected basis functions explicitly include and
(seearrowsL1andR1)radiatedfromthetwouppercornersofthe can correctly model the known singularity in stress fields around
canyon.InFigs7(g)and(h),thescatteredwaves L2, L3and R2 thecanyonbottom.Asetofintegralequationsforstressfieldsatthe
generate.TheamplitudesofscatteredwavesL2andL3aresmaller auxiliaryinterfacehasbeenderivedandthensolvedbytheGalerkin
thanthoseofR2. method.Boththesteady-stateandtransientvariationsinsurfaceand
540 K.-H.Chang,D.-H.TsaurandJ.-H.Wang
subsurfacemotionshavebeenevaluatedandanalysed.Comparing Ka¨ser,M.&Dumbser,M.,2006.AnArbitraryhighorderdiscontinuous
withthecomputationalresultsusingtheRMT,theaccurateRMT galerkinmethodforelasticwavesonunstructuredmeshes—I.Thetwo-
indeedperformswellwithoutanyexcessivecomputationaleffort. dimensionalisotropiccasewithexternalsourceterms,Geophys.J.Int.,
Suchafast,convergenttechniquenotonlyprovidesareliablebench- 166,855–877.
Kawase,H.&Aki,K.,1990.TopographyeffectatthecriticalSVwaveinci-
markforassessingtheresultsofothernumericalapproaches,but
dence:possibleexplanationofdamagepatternbytheWhittier-Narrows,
alsohasapromisingpotentialtoattacksimilarproblemswithtrou-
California,earthquakeof1October1987,Bull.seism.Soc.Am.,80,1–22.
blesomegeometricsingularities.
Kham,M.,Semblat,J.F.,Bard,P.Y.&Dangla,P.,2006.Seismicsite-cityin-
teraction:maingoverningphenomenathroughsimplifiednumericalmod-
els,Bull.seism.Soc.Am.,96,1934–1951.
ACKNOWLEDGEMENTS
Komatitsch,D.&Vilotte,J.P.,1998.Thespectralelementmethod:anef-
The authors are deeply indebted to the Editor, Prof. Wolfgang ficient tool to simulate the seismic response of 2D and 3D geological
Friederichandareviewerfortheirencouragementsandinsightful structures,Bull.seism.Soc.Am.,88,368–392.
andvaluablesuggestionsandcommentsforimprovingthispaper. Kristek,J.,Moczo,P.&Galis,M.,2010.Stablediscontinuousstaggered
gridinthefinite-differencemodellingofseismicmotion,Geophys.J.Int.,
Besides,thefirstandthirdauthorssincerelyappreciatethefinancial
183,1401–1407.
supportgrantedbytheNationalScienceCouncil,Taiwan,R.O.C. D
Lee,S.J.,Chen,H.W.&Huang,B.S.,2008.Simulationsofstrongground o
(ProjectNos.NSC101–2811-M-001–167andNSC101–2116-M- w
motionand3DamplificationeffectintheTaipeibasinbyusingacompos- n
001–015,respectively). lo
itegridfinite-differencemethod,Bull.seism.Soc.Am.,98,1229–1242. a
d
Lee, S.J., Chan, Y.C., Komatitsch, D., Huang, B.S. & Tromp, J., 2009. e
d
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Circularsectorialcanyons 541
Tsaur, D.H., Chang, K.H. & Hsu, M.S., 2011. A series solution for the APPENDIX B: SINGULAR STRESS
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044502,doi:10.1115/1.4003745.
Inordertorevealtheexistenceofstresssingularityinthevicinity
Tsaur,D.H.&Hsu,M.S.,2013.SHwavesscatteringfromapartiallyfilled
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Wong,H.L.&Jennings,P.C.,1975.Effectsoftopographyonstrongground (or that in fig. 1 of Tsaur et al. 2010) and following eq. (13) of
motion,Bull.seism.Soc.Am.,65,1239–1257. Tsaur&Chang(2008)(oreq.10ofTsauretal.2010),theantiplane
Zargano,G.F.,Lerer,A.M.&Lyapin,A.M.,1979.WaveguidesofCompound displacementfieldswithintheregion2(i.e.cornerregion)canbe
Cross-sections,RostovUniversityPress,Rostov-na-Donu,formerUSSR
expressedas
(InRussian).
(cid:2)∞
Zhou,H.&Chen,X.,2006.AnewapproachtosimulatescatteringofSH
wavesbyanirregulartopography,Geophys.J.Int.,164,449–459. uˆ2(r1,θ1)= CˆnJ2nνˆ(kr1)cos(2n)νˆθ1
n=0
(cid:2)∞ D
APPENDIX A: DERIVATION OF EQ (10) + DˆnJ(2n+1)νˆ(kr1)sin(2n+1)νˆθ1, (B1) own
Fortheenclosedregion2,eq.(1)writteninpolarcoordinatestakes n=0 lo
a
thefollowingform where νˆ =π/(2βˆ) and r is the radial distance from the apex of de
(cid:7) (cid:8) 1 d
r1∂∂r r∂∂ur2 +r12∂∂2θu22 +k2u2=0. (A1) tshaetiswfieedsgtehetoHtehlmehooblstezrveeqduaptiooinnta.nNdottheethzeartoe-qst.re(Bss1c)oinndtriitniosnicsalolyn from h
U20s1in3g,stehcetiMonS9V.4()s,eweeet.agk.ePaivsaetpoa2ra0t1e0d,ssoelcuttiioonn1in6Cth;eAforfrkmen et al. bfeortAehnsstitiaodletdhsieorefscttrtheioessncsfiocrenaldensr,brteehgweiiorrinct.toemn,proenspenectstiivneltyh,earsadialandcircum- ttps://ac
a
u2(r,θ)= R(r)(cid:16)(θ), (A2) τˆ(2) =μ∂uˆ2(r ,θ ), (B2) dem
withtheseparationconstantasν˜2,wecandecomposeeq.(A1)into r1z ∂r1 1 1 ic.o
twolinearhomogeneousordinarydifferentialequationsinr andθ, μ∂uˆ up
respectively: (cid:7) (cid:8) τˆθ(12)z = r1 ∂θ12(r1,θ1). (B3) .com
R(cid:6)(cid:6)(r)+r−1R(cid:6)(r)+ k2− ν˜2 R(r)=0, (A3) Substitutingeq.(B1)into(B2)gives /gji/a
r2 kμ (cid:2)∞ rtic
τˆr(12)z(r1,θ1)= 2 CˆnJ2nνˆ−1(kr1)cos(2n)νˆθ1 le-a
(cid:16)(cid:6)(cid:6)(θ)+ν˜2(cid:16)(θ)=0. (A4) n=0 bs
difTfehreenrtaidaliaelqpuaarttio(enq..TAak3i)ngcainntboeaecacsoiulynttrtahnesffoarcmttehdatinthtoeBsoelsusteiol’ns −k2μ (cid:2)∞ CˆnJ2nνˆ+1(kr1)cos(2n)νˆθ1 tract/19
hastobefiniteattheoriginr =0,andconsequentlygettingridof n=0 5/1
thetermsinvolvingBesselfunctionsofthesecondkind,thegeneral kμ (cid:2)∞ /53
solutionofeq.(A3)canbeexpressedas + 2 DˆnJ(2n+1)νˆ−1(kr1)sin(2n+1)νˆθ1 2/60
n=0 3
iRn(wr)h=ichc1cJ1ν˜i(sktrh)e,arbitraryconstant. (A5) −k2μ (cid:2)∞ DˆnJ(2n+1)νˆ+1(kr1)sin(2n+1)νˆθ1. (B4) 934 by g
On the other hand, the angular part (eq. A4) has the following n=0 u
e
familyofsolutions: Accordingly,substitutingeq.(B1)into(B3)yields st o
n
wcc(cid:16)oh(n=θedr)iect=iocc2nco2aascn(toνdθ˜sπc=ν˜/3θ2πa)+r/.e2c,a3risnbisinterνra˜trθiyn,gceoqn.st(a4n)tisn.toApeqp.ly(iAn6g)tlheeadzsertoo-s((tAAre67ss)) τˆθ(12)z(r1,θ1)=+−μ2rrμνˆ1νˆ(cid:2)∞(cid:2)n∞=0(2nnCˆ+nJ12)nνDˆˆ(knJr(12)n+si1n)νˆ((2knr)1νˆ)θc1os(2n+1)νˆθ1. 11 April 2019
2 3sin(ν˜π/2) 1 n=0
(B5)
Employingeq.(A7),wemayrewriteeq.(A6)as
(cid:3) (cid:4)
π Fromthepower-seriesexpansionforBesselfunctionsofthefirst
(cid:16)(θ)=c˜3cosν˜ θ− 2 . (A8) kind(e.g.Jeffrey&Dai2008,pp.291,section17.2.1.1,eq.6),we
have
In order to determine ν˜, substituting eq. (A8) into the stress-free
conditionatθ =−π/2+β,eq.(4),yields (cid:3)ϕ(cid:4)ζ (cid:2)∞ (−1)m (cid:3)ϕ(cid:4)2m
nπ Jζ(ϕ)= 2 m! (cid:12)(ζ +m+1) 2 . (B6)
ν˜= , n=0,1,.... (A9) m=0
π−β
For−1<ζ <0,theleadingtermtendstoanunboundedvalue
Aftersettingν˜=nν,eqs(A5)and(A8)becomethebuildingblocks whentheargumentϕ approacheszero,indicatingtheexistenceof
withwhichweconstructthegeneralsolutionforregion2bylinear asingularityofthetypeϕζ.Fromeq.(B6),thestressfieldsaround
superposition,thatis,eq.(10). thecornerpointcanbeexpressedasapowerseriesinr .
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