Propagation of Torsional Surface Waves. in Heterogeneous Half-Space. with Irregular Free Surface

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1 Applied Mahemaical Sciences Vol. 7 no Popagaion of Tosional Sface Waves in Heeogeneos Half-Space wih Iegla Fee Sface M. M. Selim Depamen of Mahemaics Facly of Еdcaion Se Canal Univesiy Se Egyp selim3@yahoo.com Pesen addess: Depamen of Mahemaics Al-Aflaj Commniy College King Sad Univesiy P.O. Box 7 Al-Aflaj 9 Sadi Aabia Absac The effec of ieglaiy on he popagaion of osional sface waves in a heeogeneos elasic half-space has been sdied. The velociy eqaion has been deived. The velociies have been calclaed nmeically and ae shown gaphically. The sdy eveals ha he sface ieglaiy has a noable effec on he popagaion of osional sface waves. I is also obseved ha he velociy of popagaion of osional sface waves depends on he heeogeneiy pesen in he medim. Keywods: Tosional sface waves Ieglaiy Heeogeneos medim. Inodcion Waves popagaing along sfaces o inefaces ae vey impoan o seismologis eahqae engineeing in ndesanding of he cases of damage de o eahqaes. In fac sdy of sface waves in homogenos heeogeneos and layeed media has been of cenal inees o heoeical seismologiss nil ecenly. The basic lieae on he popagaion of elasic waves is he monogaph by Ewing e al.[6]. A lage nmbe of papes have been pblished in diffeen jonals afe he pblicaion of his boo. Veos [78] gives mch infomaion on he effec of heeogeneiy in he sdy of sface waves vibaions de o line-load. Alhogh mch infomaion is available on he popagaion of sface waves sch as Rayleigh waves Love waves

2 43 M. M. Selim and Sonely waves ec. he osional wave has no dawn mch aenion and vey lile lieae is available on popagaion of his wave. Lod Rayleigh [] in his emaable pape showed ha he isoopic homogenos elasic half-space does no allow a osional sface wave o popagae. Bhaachaya [] has been invesigaed he osional wave popagaion in a wo-layeed cicla cylinde wih impefec bond. The popagaion of osional wave in a finie pieoelecic cylindical shell has been discssed by Pal and Sama [4]. The popagaion of osional wave in an iniially sessed cylinde has been discssed by Dey and Da [3]. As exension of his wo Selim [6] has been invesigaed he osional wave popagaion in an iniially sessed dissipaive cylinde. The commendable wos by Dey e al. [4] and Dey e al. [5] in he sdy of sface waves in a nonhomogenos medim may be cied. Sdies of wave popagaion in elasic media wih iegla sfaces ae vey impoan leading o bee ndesanding of he behavio of sface wave popagaing on he acal Eah sface. I is heefoe ineesing o sdy he osional sface waves in medim wih iegla sface. The poblem of sface waves popagaion in an iegla media sing he pebaion echniqe indicaed by Eingen and Shbi [7] has been sccessflly employed by Mal [] Ka e al. [9] Chaopadhyay e al. [] and ohes. I seems ha no aemp has so fa been made o find o he effec of iegla sface on he popagaion of osional sface waves in a heeogeneos medim. A simila poblem fo slighly cved elasic half-space has been analyed by Handelman [8] and Pal e al. [3] fo he case of Raleigh wave popagaion. This pape has been famed o o show he effec of ieglaiy as well as heeogeneiy on he popagaion of osional sface waves in a heeogeneos medim. The velociy eqaion has been deived and he esls have been discssed. The sdy eveals ha he sface ieglaiy as well as heeogeneiy has a noable effec on he popagaion of osional sface waves. Fig.. Geomey of he poblem.

3 Popagaion of Tosional Sface Waves 43. Fomlaion of he poblem and is solion To sdy he osional sface waves a cylindical coodinae wih oigin on he sface and dieced nomally o he ineio of he half-space is consideed. Accoding o he poblem we assme he fee sface may be aen as ε f whee ε is small qaniy whose second and highe powes may be negleced Fig.. The heeogeneiy in he half-space is aen in he fom δ e fo igidiy ρ fo densiy ρ whee δ is consan has dimension of invese of lengh and ρ ae posiive consans and he eqaion of ieglaiy has been aen as whee χ h / a Fig.. ± χ fo a ε f fo > a The dynamical eqaions of moion fo he sysem ae [] ρ ρ ρ 3 whee ρ is he densiy and ae he coesponding sess componens in hei convenional sense ae he displacemen componens in adial cicmfeenial and axial diecions. In cylindical coodinaes he ess-sain elaions ae aen as λ Ω δ e 4 ij ij ij whee λ ae Lame's coefficiens and e ij i j j i and e ii Ω i j 3. 5 Dynamical eqaion of moion fo osional wave may be wien as:

4 43 M. M. Selim ρ 6 whee is he displacemen along diecion. Fo an elasic medim he sesses ae elaed o he displacemen componen by 7 Inseing elaions 7 in 6 one ges ρ 8 Assming hamonic wave solion i e ω he solion fo cicmfeenial displacemen in he medim becomes i U e ω V. 9 Using Eq.9 Eq.8 aes he fom V d dv d V d Eqaion called he paameic Bessel eqaion of fis ode and is geneal solion is BY AJ V whee J and Y denoe Bessel's fncions of he fis ode and of he fis and second ind especively. The second pa of Eq. is lef o fom he complee solion fo well-nown physical easons [5]. Then AJ V whee U C d du d U d U β 3 and ρ β and C ω is osional wave velociy in he medim.

5 Popagaion of Tosional Sface Waves 433 Fom Eq. in Eq.9 he displacemen componen in he heeogeneos half-space is given by AU J ei ω 4 whee U is he solion o d U du C U d d β 5 δ Using Eq. and appoximaion e δ Eq.5 ae he fom d U du C δ U 6 d d C δ whee C. ρ δ / Following he mehod of Dey e al. [4] we p U φ e in Eq.6 one ges d φ C γ φ d γ C δ. 7 δ whee γ. 4 C γ δ On ping α and ζ in Eq.7 we ge γ C δ d φ ζ α 4 φ ζ 8 d ζ Applying he sface wave condiion which is U a i.e. φ ζ a ζ. 9 The solion of Eq.8 may be aen as φ ζ A W α / ζ whee A is consan and W is Whiae fncion. Hence α / ζ γ δ δ / U AW α / e δ Fom Eq. in Eq.4 one ges

6 434 M. M. Selim γ δ δ / BW e J e i ω α / δ whee B AA is a consan. 3. Bonday Condiions In he absence of any exenal body foces in he bonday he sface of he medim is ep sess fee a ε f and also we assme ha he displacemen a is finie. The bonday condiions may be aen as d s d a ε f 3 Applying he bonday condiions 3 one ges d [ W γ δ / δ ] δ [ W γ δ / δ ] α / / d The bonday condiion 3 aes he final fom as α ε f 4 [ W γ δ εf / δ ] 4γ d α / δ dζ Wα / γ δ f / δ 5 Taing he asympoic expansion of he Whiae fncion and eaining p o linea em Eq.5 gives he velociy eqaion as C C 4δ 8γ δ Δ 3 δ Δ δ3 δ Δ δ 4γ γ 3 δ Δ 3 δ Δ 6 whee Δ ±χ. 4. Nmeical Resls In ode o demonsae he effec of ieglaiy on he popagaion of osional sface waves in he heeogeneos isoopic elasic half-space nmeical compaions of Eq.6 wee pefomed. The effec of ieglaiy and heeogeneiy on he phase velociy of osional sface wave is shown in Figs and 3. In hese figes he vaiaion of C / C agains / δ fo diffeen sies of ieglaiy χ. ±. and ±.4 is displayed. Fige gives he vaiaion of he phase velociy of osional sface wave agains /δ fo he posiive vales of χ. and. 4. Compaison wih he case of χ. I is obseved ha he phase velociy of osional sface waves deceases wih an incease in ieglaiy paamee χ fo he same vale of / δ. Fige 3 gives he same fo he negaive vales of χ. and. 4. The above nmeical esls fo he popagaion of osional sface waves in he heeogeneos half-space fo boh posiive and negaive vales of he ieglaiy

Since he acal Eah has iegla sface space so i is moe ealisic o conside he ieglaiy effec in he sdy of he popagaion of osional sface waves in heeogeneos Eah

7 Popagaion of Tosional Sface Waves 435 sface paamee χ eveal clealy ha as sface deceases. / δ inceases he velociy of osional 5. Conclsions I is conclded ha he sface ieglaiy has a noable effec on he popagaion of osional sface waves in heeogeneos medim wih iegla fee sface. Since he acal Eah has iegla sface space so i is moe ealisic o conside he ieglaiy effec in he sdy of he popagaion of osional sface waves in heeogeneos Eah medim. The sdy also shows ha he heeogeneiy pesen in he medim effec he popagaion significanly /δ Fig. Vaiaion of C / C vess / δ a χ... 4.

8 436 M. M. Selim /δ Fig. 3 Vaiaion of C / C vess / δ a χ Refeences [] Bhaachaya R. C On he osional wave popagaion in a wo-layeed cicla cylinde wih impefec bonding Poc. Indian nan. Sci. Acad. 4 A No [] Chaopadhyay A. and Pal A. Dispesion cves of SH waves cased by ieglaiy in he pesessed inenal sam Aca Geophys. Pol. XXXI No [3] Dey S. and Da A. Tosional wave popagaion in An iniially sessed cylinde Poc. Indian nan. Sci. Acad. 58 A No [4] Dey S. Gpa A. K. and Gpa S. Tosional sface waves in nonhomogenos and anisoopic medim J. Acos. Soc. Am [5] Dey S. Gpa S. Gpa A. K. and Pasad A. M. Popagaion of osional sface waves in a heeogeneos half-space nde a igid laye Aca Geophys. Pol. XLIX No [6] Ewing W.M Jadey W. S. and Pess F. Elasic waves in layeed media McGaw-Hill New Yo957. [7] Eingen A. C. and Shbi E. S. Elasodynamics Vol. II Academic Pess New Yo 975. [8] Handelman G. H. Pogess in applied mechanics Page Annivesay London 963. [9] Ka B. K. Pal A. K. and Kalyani V. K. Popagaion of love waves in an iegla dy sandy laye Aca Geophys. Pol. XXXIV No

9 Popagaion of Tosional Sface Waves 437 [] Lod Rayleigh The heoy of sond Dove New Yo945. [] Love A. E. H. The mahemaical heoy of elasiciy. Cambidge Univesiy Pess 97. [] Mal A. K. On he feqency eqaion fo Love waves de o abp hicening of csal laye Geofis. Pa. Appl [] Pal P. Ch. and Kma L. Popagaion of Rayleigh waves ove a hemoviscoelasic half space wih a slighly cved fee space Aca Geophys. Pol. XLVII No [3] Pal H. S. and Sama K. V. Tosional wave popagaion in a finie pieoelecic cylindical shell Poc. Indian nan. Sci. Acad. 43 A No [4] Redwood M. Mechanical Wave-gides Pegamon. Pess96. [5] Selim M. M. Tosional waves popagaion in an iniially sessed dissipaive cylinde Applied Mahemaical Science inpess 7. [6] Veos Ch. In plse vibaions of soil deposis wih vaiable shea modls I Sface waves In. J. Nme. Anal. Mehods Geomech 4 99a 9-. [7] Veos Ch. In plse vibaions of soil deposis wih vaiable shea modls I Sface waves In. J. Nme. Anal. Mehods Geomech 4 99b Received: Decembe 6 6

Chapter Finite Difference Method for Ordinary Differential Equations

Chapter Finite Difference Method for Ordinary Differential Equations Chape 8.7 Finie Diffeence Mehod fo Odinay Diffeenial Eqaions Afe eading his chape, yo shold be able o. Undesand wha he finie diffeence mehod is and how o se i o solve poblems. Wha is he finie diffeence