Wave Propagation in a Layer of Binary Mixture of Elastic Solids

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Barnard Pitts
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1 Jornl of Sold Mhns Vol. No. pp. 8- v ropgon n yr of nry Mr of Els Solds R. Kr M. nhl Dprn of Mhs Krshr Unvrsy Krshr-6 Ind Rvd prl ; pd 6 Jn SRC hs ppr onnrs on h propgon of vs n lyr of bnry r of ls solds sbd o srss fr bondrs. Slr ons for h lyr orrspondng o syr nd nsyr v ods r drvd n oplly spr rs. h plds of dspln oponns nd spf loss for boh syr nd nsyr ods r obnd. h ff of rs on phs vloy non offn spf loss nd pld ros for syr nd nsyr ods s dpd grphlly. prlr s of nrs s lso ddd fro h prsn nvsgon. IU r rnh. ll rghs rsrvd. Kyords: Mr; hs vloy; non offn; Spf loss; pld ros INRODUCION r s rl oposd of svrl dsn onsns. In h onn hors of rs h gr bsron h rl n b odld s onn s ndd by ssng h h onsns of r old b odld s sprposd onn so h h pon n h r s slnosly opd by rl pon of h onsn. h frs onn hory of rs s proposd by rsdll [] n rs of n nd hrodyn vrbls ssod h h onsn of h r. brf dsrpon s onnd n h rl rsdll nd opn []. srvy of onn hors h hv bn dvlopd o odl h hrohnl bhvor of rs onssng of vros onsns s prsnd n dl n rv rls by on [] n nd Crn [ ] dford nd Drhllr [6] nd n h boos of Sohyl [] nd Rgopl nd o [8]. In h hors for r of ls solds prsnd n on [] Grn nd Sl [] nd Sl [] h ndpndn onsn vrbls r h dspln grdn nd h rlv vloy nd h spl dsrpon s sd. h frs hory for r of ls solds bsd on h grngn dsrpon hs bn prsnd n dford nd Srn [ ]. In hs hory h ndpndn onsn vrbls r dspln grdn nd h rlv dspln. In h rn yrs n nrsng nrs hs bn dvlopd n h sdy of h lv proprs of hs hory Isn nd Qnnll []. I s orh nong h odl of nrpnrng sold onn s ppld n rsn nd Jhnr [] o drv hory of oposs hr h rlv dsplns of h ndvdl onsns s nfnsl. Isn [] drvd hory for bnry rs of ls solds n hh h ndpndn onsn vrbls r h dspln grdns dspln flds vol frons nd vol fron grdns. lso prsnd h lnr onsv ons n s of n sorop body h nr of syry nd sblshd nnss hor n h lnr dyn hory h no dfnnss sspon on h lsy nd no rsron on h nl srss. lso Isn [6] prsnd h bondry vl probls of h lnr hory for bnry rs of ls bods nd drvd fndnl solons n h lbr hory of hoognos nd sorop rs. Crl [] prsnd non-lnr hory for bnry rs nd dvlopd bs ons of h lnr hory. Corrspondng hor. l.: -6. E-l ddrss: rnsh_@rdffl.o IU r rnh. ll rghs rsrvd.

2 R. Kr nd M. nhl r Crl nd ssrll [8] sdd spl bhvor of dyn prosss n ls rs nd obnd prs drnon of h don of nfln nd h spl dy ss h dpndn dy r nsd h don of nfln. Isn [] drvd bs ons of non-lnr hory of h ondng vsols rs n grngn dsrpon nd sblshd bs ons of lnr hory. Qnnll [] onsdrd hory of vsols rs proposd by Isn [] nd drnd h dsspon ffs by h vsosy of r yp of onsn nd rlv vloy. Isn [] drvd onn hory of vsols opos hh s odld s r of ropolr ls sold nd ropolr Klvn-Vog rl. In h prsn ppr hv sdd h propgon of vs n lyr of r of o ls solds nd h nrl rsls r llsrd grphlly o sdy h bhvor of rvs for phs vloy non offn spf loss nd plds of dspln oponns for vros ods of v propgon. SIC EQUIONS onsdr h bs ons for h bnry r n h fror of lnrd hory nd ss h h onsns s nd s r h ls bods. Follong Qnnll [] h ons of lbr n h bsn of rnl body fors r ρ p s p ρ hr nd s r h oponns of srss nsor ρ nd ρ r h ss dnss nd r h oponns of dspln ssod h h onsns s nd s rspvly p r h oponns of nrnl body for. h onsv rlons for nrosyr hoognos nd sorop r r gvn by λ μ δ μ ζ α ν g δ ζ g γ ζ g s p rr rr ν δ ζ αg δ g γg ξ d rr ss hr g d nd α γ λ μ ν ζ ξ r prsrbd onsns nd δ s Kronr dl. Usng n nd h h hlp of obn h fld ons n rs of dspln flds s α ξ ρ Δ α r r Δ r r r r γ Δ γ r r Δ ξ ρ hr Δ s pln opror nd α μ ζ α λ μ α ν γ ζ γ ζ α ν ζ γ γ γ α ROEM FORMUION ND SOUION onsdr n nfn lyr of r of o ls solds hvng hnss. For o dnsonl probl ng orgn of h oordn sys y on h ddl srf of h lyr. h y pln s hosn o ond h h ddl srf nd h -s norl o long h hnss of h lyr. h srfs IU r rnh

3 v ropgon n yr of nry Mr of Els Solds IU r rnh r sbd o srss fr bondrs. pln s h pln of ndn so h nd ss h h solons r plly ndpndn of y.. y. hs h fld ons rd o ρ ξ α α ρ ξ α α 6 ρ ξ γ γ ρ ξ γ γ 8 For frhr onsdrons s onvnn o nrod h dnsonlss vrbls dfnd by μ μ s s ρ μ λ hr s onsn hvng dnsons of rlr frny. Usng h prssons rlng dspln oponns o h slr ponl fnons nd n dnsonlss for fr spprssng h dshs s h fld ons rd o } } } } 8 6 } } } hr /α α α /α / α ξ /α ρ 6 / / γ 8 / γ / ξ / ρ

4 R. Kr nd M. nhl IU r rnh h solon of 8- for h vs propgng long posv dron r } sn os sn os } sn os sn os } sn os sn os } sn os sn os hr nd r rspvly h v nbr nd phs vloy of v nd r rbrry onsns... λ / λ / λ N N M 6 8 } 6 8 M } } N } }.. Drvon of frny on h ppropr bondry ondons r: s s 6 Usng dnsonlss vrbls dfnd by n h prssons of srsss nd hn sng bondry ondons gvn by 6 h h hlp of nd for pln obn gh hoognos ons n gh nnons h ondon for h sn of non-rvl solon of hs ons gvs h frny on hr... n nd

5 v ropgon n yr of nry Mr of Els Solds M M M M M M α M α M ζ ζ M α γ γ M ζ γ ν α M λ α μ ν γ ζ M α ν M μ γ ζ γ γ γ ζ γ γ ; 8 M λ ζ SECIFIC OSS h spf loss s h ro of nrgy Δ dsspd n ng spn hrogh srss yl o h ls nrgy sord n h spn hn h srn s. h spf loss s h os dr hod of dfnng h nrnl fron for rl. Kolsy [] shos h spf loss Δ / s π s h bsol vl of h ro of h gnry pr of v nbr o h rl pr of v nbr.. Δ I π R MIUDES OF DISCEMENS h plds of dspln oponns of boh h onsns of r for syr nd s-syr ods of pln vs n b obnd s: sy os os sy sn sn sy sn sn sy os os sy os os sy sn sn sy sn sn os os sy IU r rnh

6 R. Kr nd M. nhl 6 RICUR CSE In h lng s f ngl h prsn of sond onsn of h r nd vnsh h prrs orrspondng o h oplng bn h o onsns of r hn ll rovr h rsls of lssl hory of lsy for sorop ls sold hh r slr o hos obnd n Grff [] by hngng dnsonlss ns no h physl ns. NUMERIC RESUS ND DISCUSSION In ordr o llsr horl rsls obnd n h prdng sons no prsn so nrl rsls ng λ. N ν. N μ.8 N α. N. ζ. N ξ. N ρ. Kg ρ.8 ll nrl opons r rrd o by ng.. Fg. dps h vrons of phs vloy h rsp o R.. rl pr of v nbr for syr nd nsyr ods of v propgon for r of o ls solds hrs Fg. rprsns h s son for sorop ls sold. Fg. dps h vrons of non offn h rsp o R for syr nd nsyr ods of propgon for r of o ls solds hrs Fg. rprsns h s son for sorop ls sold. Slrly Fgs. nd 6 rprsn h vrons of spf loss.r. v nbr for boh syr nd nsyr ods n s of r nd ls sold rspvly. Kg N Fg. Vron of phs vloy.r. v nbr n r for syr nd nsyr ods of propgon. Fg. Vron of phs vloy.r. v nbr n ls sold for syr nd nsyr ods of propgon. IU r rnh

7 v ropgon n yr of nry Mr of Els Solds Fg. Vron of non offn.r. v nbr n r for syr nd nsyr ods of propgon. Fg. Vron of non offn.r. v nbr n ls sold for syr nd nsyr ods of propgon. Fg. Vron of spf loss.r. v nbr n r for syr nd nsyr ods of propgon. In Fgs. -6 h sold lns rprsn vrons for frs syr od sll dshd lns rprsn vrons for sond syr od nd long dshd lns rprsn vrons for hrd syr od nd h orrspondng lns h nrl sybol rprsn vrons for h rspv nsyr ods. In Fg. nd 8 sold lns nd dshd lns rspvly rprsn vrons for dspln oponn nd hrs h orrspondng lns h nrl sybols rprsn vrons for nd rspvly. In Fg. sold ln nd IU r rnh

8 R. Kr nd M. nhl dshd ln rspvly rprsn vrons for dspln oponn nd for syr od hrs h orrspondng lns h nrl sybols rprsn vrons for nsyr od. Fg. 6 Vron of spf loss.r. v nbr n ls sold for syr nd nsyr ods of propgon. Fg. Vron of plds of dspln oponns n r for syr od. Fg. 8 Vron of plds of dspln oponns n r for nsyr od. IU r rnh

9 v ropgon n yr of nry Mr of Els Solds 6 Fg. Vron of plds of dspln oponns n ls sold for syr nd nsyr ods of v propgon. h phs vloy non offn nd spf loss of vros ods of propgon hv bn opd fro slr ons for vros vls of v nbr nd for dffrn bondry ondons. I s obsrvd fro Fgs. nd h h phs vloy of dffrn ods of v propgon srs fro lrg vls vnshng v nbr nd hn hbs srong dsprson nl h vloy flns o o h vl of Rylgh v vloy of rl hghr v nbr. h rson for h sypo pproh s h for shor vlnghs or hgh frns h rl pl bhvs nrsngly l h slb nd hn h oplng bn h ppr nd lor bondry srfs s rdd nd s rsl h proprs of syr nd s syr ods of v propgon bo or nd or slr. In h l for n nfn h slb h oon h ppr srf s no onfnd o h lor srf nd h dsplns bo lold nr h fr bondrs hs h b v dsprson rvs sypolly pproh hos for Rylgh vs. I s obsrvd fro Fgs. nd h for syr od h vls of non offn frs nrs nd hn drs for r of ls solds b h vls of non offn r hghr for h sond od s oprd o hos for frs od. lso for nsyr od h vls of non offn nrs h h nrsng nbr of od of propgon. For sorop ls sold h vls of non offn for h frs od of v propgon hb srong dsprson fro hghr vls for boh syr nd nsyr ods. h vls of non offn of sond od of propgon r hghr for nsyr od of propgon hn h rng < R < hrs h bhvor s rvrs byond hs rng. For h hrd od of v propgon h vls r hghr for syr od s oprd o hos for nsyr od. Fgs. nd 6 nd h h vls of spf loss drs h v nbr. Fg. nds h for h frs o ods h vls of spf loss r hghr for syr od s oprd o hos for nsyr od b h bhvor s rvrs for h hrd od. I s obsrvd fro Fg. h h bhvor nd rnd of vron of spf loss for frs nd hrd od s slr for boh syr nd nsyr ods of v propgon hrs for sond od h vls r hghr for nsyr od hn h rng < R < nd h bhvor s rvrs byond hs rng. Fgs. nd 8 rprsn h vrons of plds of dspln oponns of r.r. h hnss of pl. I s obsrvd h for syr od of propgon h pld of dspln oponns nrs.r. h hnss of pl hrs h plds of nd drs. For nsyr od h vron of pld for sho bhvor slr o h for syr od hrs h dspln oponn sho oppos bhvor. I s obsrvd fro Fg. h for ls sold h rnd of vron of plds of boh dspln oponns for nsyr od s slr h h pon h h vls of dspln oponn r hghr hn hos for b for syr od h vls of r grr hn hn h rng < <. hrs h bhvor s rvrsd byond hs rng. IU r rnh

10 R. Kr nd M. nhl 8 CONCUSION h hrrss of h dsprson rlons of pln vs propgng n lyr of bnry r of ls solds nd n sorop ls sold r prsnd. fr drvng slr on h bhvor of dsprson rvs of phs vloy nd non offn s sdd grphlly for vros ods of v propgon. h plds of dspln nd spf loss r lso opd fro h rlv prssons nd r shon grphlly for boh syr nd nsyr ods of v propgon. h nrlly opd rsls r fond o b n los grn h h horl rsls. CKNOEDGEMEN On of h hors Mss. Mnsh nhl s hnfl o Conl of Snf nd Indsrl Rsrh CSIR for fnnl sppor n rs of Jnor Rsrh Flloshp JRF. RFERENCES [] rsdll C. 6 Mhnl bss of dffson Jornl of Chl hyss : 6-. [] rsdll C. opn R. 6 ndbh dr hys /: Sprngr. [] on R.M. 6 hory of rs n: Conn hyss dd by.c. Erngn d rss N Yor. [] n R.J. Crn R.E. 6 Conn hors of rs: pplons Jornl of ppld Mhs : -. [] n R.J. Crn R.E. 6 Conn hors of rs: s hory nd hsor dvlopn Qrrly Jornl of Mhns nd ppld Mhs : -. [6] dford. Drhllr D.S. 8 hory of sbl nd srrd rs Inrnonl Jornl of Engnrng Sn : [] Sohyl I. 8 hrodyns of Irrvrsbl rosss n Fld Mrs bnr Vrlg bg. [8] Rgopl K.R. o. Mhns of Mrs orld Snf Sngpor. [] Grn.E. Sl.R. 66 Consv ons for nrng onn Inrnonl Jornl of Engnrng Sn : 8-. [] Sl.R. 6 pplons of hory of nrng onn Qrrly Jornl of Mhns nd ppld Mhs : -. [] dford. Srn M. l-onn hory for opos ls rls Mhn : 8-. [] dford. Srn M. ords dffsng onn hory of opos ls rls SME rnsons Jornl of ppld Mhns 8: 8-. [] Isn D. Qnnll R. Esn nd onnos dpndn rsls n h hory of nrng onn Jornl of Elsy 6: 8-8. [] rsn.f. Jhnr M. hory of opos odld s nrpnrng sold onn rhv for Ronl Mhns nd nlyss 6: -. [] Isn D. On h hory of rs of ls solds Jornl of Elsy : -68. [6] Isn D. 6 Esn hors n h hory of rs Jornl of Elsy : -6. [] Crl M. 8 On rs of nonspl ls solds Inrnonl Jornl of Engnrng Sn 6: [8] Crl M. ssrll F. On h spl bhvor n dyns of ls rs Eropn Jornl of Mhns /Solds : 6-. [] Isn D. On h hory of vsols rs Jornl of hrl Srsss : -8. [] Qnnll R. Esn nd ponnl dy n h lnr hory of vsols rs Eropn Jornl of Mhns /Solds : -. [] Isn D. hory of hrovsols oposs odlld s nrng ossr onn Jornl of hrl Srsss : 6-8. [] Kolsy. 6 Srss vs n Solds Clrndon rss Oford Dovr rss N Yor. [] Grff K.F. v Moon n Els Solds Clrndon rss Oford. IU r rnh

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