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1 Journal of Aled Mathematcs and Physcs,, 13, 1, 1-17 htt://dx.do.org/1.436/jam Publshed Onlne October 13 (htt:// Numercal Smulaton of Acoustc-Gravty Waves Proagaton n a Heterogeneous Earth-Atmoshere Model wth Wnd n the Atmoshere B. G. Mhalenko, A. A. Mhalov, G. V. Reshetova The Insttute of Comutatonal Mathematcs and Mathematcal Geohyscs, Sberan Branch of the RAS, r. Akad. Lavrenteva, 6, Novosbrsk 639, Russa Emal: mh@sscc.ru Receved July 13 ABSTRACT A numercal-analytcal soluton for sesmc and acoustc-gravty waves roagaton s aled to a heterogeneous Earth-Atmoshere model. Sesmc wave roagaton n an elastc half-sace s descrbed by a system of frst order dynamc equatons of elastcty theory. Proagaton of acoustc-gravty waves n the atmoshere s descrbedd by the lnearzed Naver-Stokes equatons wth the wnd. The algorthm roosed s based on the ntegral Laguerre transform wth resect to tme, the fnte ntegral Fourer transform along the satal coordnate wth the fnte dfference soluton of the reducedd roblem. Keywords: Sesmc Waves; Acoustc-Gravty Waves; Naver-Stokes Equatons; Laguerre Transform; Fnte Dfference Method 1. Introducton In mathematcal smulaton of sesmc wave felds n an elastc medum, t s tycally assumed that the medumm borders on vacuum, and boundary condtons are sec- sesmc waves are assumed to be absolutely reflected, and the generaton of acoustc-gravty waves by elastc waves n the atmoshere and ther nteracton at the boundary are gnored. fed on a free surface. Secfcally, at the boundary, In the last decade, some theoretcal and exermental nvestgatons have shown that there s a strng correla- ton between the waves n the lthoshere and the at- moshere. Paer [1] descrbes the effect of acoustoses- Paers [,3] deal wth theoretcal nvestgatons of wave rocesses at the boundary between an elastc half-sace and an sothermal homogeneous atmoshere. In these aers, roertes of the surface Stoneley-Scholte and modfed Lamb waves are studed. In the resent aer we consder an effcent numercal algorthm to smulate and nvestgate the roagaton of mc nducton of an acoustcc wave roduced by a vbrator. sesmc and acoustc-gravty waves n a satally nho- mogeneous Atmoshere-Earth model. A ecularty of the algorthm s a combnaton of ntegral transforms wth a fnte-dfference method. A smlar aroach to solvng the roblem for a vert- cally nhomogeneous model n a cylndrcal system of coordnates wth no wnd n the atmoshere was cons- system dered n [4]. In the roblem statement, the ntal s wrtten down as a frst-order hyerbolc system n terms of the velocty vector and stress tensor n a 3D Cartesn system of coordnates. The medum arameters are assumed to be functons of only two coordnates, and the medum s assumed to be homogeneous n the thrd coordnate. Ths roblem statementt s called a.5 D one. The algorthm s based on the ntegral Laguerre transform wth resect to the temoral coordnate. Ths me- sectral method based on the Fourer transform, where, thod can be consdered to be an analog to a well-known nstead of the frequency, we have a arameter that s the degree of the Laguerre olynomals. The ntegral Laguerre transform wth resect to tme (n conto reduce trast to the Fourer transform) makes t ossble the ntal roblem to solvng a system of equatons n whch the arameter s resent only n the rght-hand sde of the equatons and has a recurrence relaton. Ths me- thod for solvng dynamc roblems of elastcty theory was frst consdered n aers [5,6] and then develoed for roblems of vscoelastcty [7,8] and orous meda [9]. The above-mentoned aers consder eculartes of ths method and the advantages of the ntegral Laand the guerre transform over the dfference methods Fourerr transform wth resect to tme. Coyrght 13 ScRes.

2 B. G. MIKHAILENKO ET AL. 13. Problem Statement The system of equatons for the roagaton of acoustc-gravty waves n an nhomogeneous non-onzed sothermal atmoshere n the Cartesan system of coordnates ( x, yz, ) wth the wnd drected along the horzontal axs x and vertcal stratfcaton along the axs z has the followng form: u x u x 1 P v x vx uz, (1) t x x z uy uy 1 P v x t x y, () uz uz 1 P g vx, (3) t x z P P P vx c vx uz uz, (4) t x t x z z u u x y u z vx t x x y z (5) uz F( x, y, z, t) z Here g s the acceleraton of gravty, () z s the reference atmoshere densty, c ( ) z s the sound seed, vx ( z ) s the wnd velocty along the axs x, u ( ux, uy, uz) s the velocty vector of dslacement of the ar artcles, P and are the ressure and the densty erturbatons, resectvely, due to a wave roagatng from a source of mass F( xyzt,,, ) ( r r ) ft ( ), where f () t s a gven tme sgnal n the source. Assume that the axs z s drected uwards. Zero subscrts for the medum hyscal arameters show ther values for the reference atmoshere. The atmosherc ressure P and the densty for the reference atmoshere n a homogeneous gravtatonal feld are: P g, ( z) 1ex( z/ H), z where H s the heght of the sothermal homogeneous atmoshere, and 1 s the densty of the atmoshere at the Earth s surface, that s, at z. The sesmc waves roagaton n an elastc medum s descrbed by the well-known system of frst order equatons of elastcty theory as the followng relaton between the dslacement velocty vector comonents and the stress vector comonents: u 1 F f() t, (6) t x uk u dvu. (7) t x xk k Here j s the Kronecker symbol, ( x1, x, x3) and ( x1, x, x3) are the elastc arameters of the medum, ( x1, x, x3) s the densty, u ( u1, u, u3) s the dslacement velocty vector, and j are the stress vector comonents. The equalty F( x, yz, ) F1ex Fey F3e z descrbes the dstrbuton of a source located n sace, and f () t s a gven tme sgnal n the source. The combned system of equatons for the roagaton of sesmc and acoustc-gravty waves n the Cartesan system of coordnates ( x, yz, ) ( x1, x, x3) can be wrtten down as u 1 u g v x F f() t K vx ez uz ex, t xk x1 x3 (8) uk u dvu K vx guz t x xk x1 (9) K vx dvu uz. (1) t x z Here j s the Kronecker symbol, ( x, z) s the densty, ( x, z) and ( x, z) are the elastc arameters of the medum, u ( u1, u, u3) s the dslacement velocty vector, and j are the stress tensor comonents; F( x, yz, ) F1ex Fey F3e z descrbes the dstrbuton of a source located n sace, and f () t s a gven tme sgnal n the source. The medum s assumed to be homogeneous along the axs y. System (1)-(5) for the atmoshere s obtaned from system (8)-(1) at Kатм 1, , P, c,. Set Kатм n system (9)-(1), and obtan the system of Equatons (6)-(7) for the roagaton sesmc waves n an elastc medum. In the roblem n queston, the atmoshere-elastc half-sace nterface s assumed to be the lane z x 3. In ths case, the condton of contact of the two meda at z s wrtten as u u, gu zz zz z z z z z t z t z. (11) xz z yz z The roblem s solved at the followng zero ntal data: u P, 1,,3. 1,,3. t j j t t t (1) All the functons of the wave feld comonents are assumed to be suffcently smooth so that the transforma-, Coyrght 13 ScRes.

3 14 B. G. MIKHAILENKO ET AL. tons resented below be vald. 3. Soluton algorthm At the frst ste, we use the fnte cosne-sne Fourer transform wth resect to the satal coordnate y, where the medum s assumed to be homogeneous. For each comonent of the system, we ntroduce the corresondng cosne or sne transform: a cos( kn y) W( x, znt,, ) W ( xyzt,,, ) d( y), sn( ky n ) n,1,,..., N ; (13) wth the corresondng nverson formula N 1 W( x, yzt,, ) W( x,, zt, ) W ( xnzt,,, )cos( ky n ) n 1 (14) or N W( x, yzt,, ) W ( xnzt,,, )sn( ky n ), (15) n 1 n where kn. a At rather a large dstance a, consder a wave feld u to the tme t T, where T s a mnmum roagaton tme of a ressure wave to the boundary r a. As a result of ths transformaton, we obtan N 1 ndeendent D unsteady roblems. At the second ste, we aly to the thus obtaned N 1 ndeendent roblems the ntegral Laguerre transform wth resect to tme W ( x, n, z) ( x, n, z, t)( ht) l W ( ht) d( ht),,1,,... wth the nverson formula ( x, nzt,, ) ( ht) ( xnzl,, ) ( ht) ( )! l ht l ht (16)! W W, (17) where ( ) are the orthogonal Laguerre functons. The Laguerre functons ( ) can be exressed n terms of the classcal standard Laguerre olynomals L ( ) ht (see aer [1]). Here we select an nteger arameter 1 to satsfy the ntal data and ntroduce the shft arameter h>. Then we have the followng reresentaton: ( ) ( ) ex( ) ( ) l ht ht ht L ht. We take the fnte cosne-sne Fourer transform wth resect to the coordnate x, smlar to the revous transform wth resect to the coordnate y wth the corresondng nverson formulas: or 1 W( xnz,, ) W(, nz, ) π (18) M W ( mnz,, )cos( kmx) π m1 M W( x, nz,, ) W ( mnz,,, )sn( k x), (19) m π m1 m where km. b It should be noted that the medum n ths drecton s nhomogeneous. The fnte dfference aroxmaton for the system of lnear algebrac equatons wth resect to z usng the staggered grd method was aled (see aer [11]) rovdng second order accuracy aroxmaton. Ths scheme s used for FD aroxmaton wthn the comutaton domans n the atmoshere and n the elastc half-sace, the fttng condtons at the nterface beng exactly satsfed. As a result of the above transformatons, we obtan N 1 systems of lnear algebrac equatons, where N s the number of harmoncs n the Fourer transform wth resect to the coordnate y. The sought for soluton vector W s reresented as follows: W( ) ( V ( ), V ( ),..., V ( )) T, 1 V [ ( m,..., M; z), xx( m,..., M; z), P( m,..., M; z ), (,..., ; )] T uz m M z. Then for every n -th harmonc ( n,..., N ) the system of lnear algebrac equatons can be wrtten down n the vector form: h ( A E) W( ) F ( 1). () Note that only the rght-hand sde of the obtaned system of algebrac equatons ncludes the arameter (the degree of the Laguerre olynomals) and has a recurrent deendence. The matrx A s thus ndeendent of. A sequence of wave feld comonents n the soluton vector V s chosen to mnmze the number of dagonals n the matrx A. The man dagonal of the matrx has the comonents of ths system multled by the arameter h (the Laguerre transform arameter). By changng the arameter h, the condton number of the matrx can be consderably mroved. Solvng the system of lnear algebrac equatons () determnes sectral values for all the wave feld comonents W ( mnz,,, ). Then, usng the nverson formulas for the Fourer transform (14), (15), (18), (19), and the Laguerre transform (17), we obtan a soluton to the ntal roblem (8)-(1). In the analytcal Fourer and Laguerre transforms, when K Coyrght 13 ScRes.

4 B. G. MIKHAILENKO ET AL. 15 determnng functons by ther sectra, nverson formulas n the form of nfnte sums are used. A necessary condton n the numercal mlementaton s to determne the number of terms of the summable seres to construct a soluton wth a gven accuracy. For nstance, the number of harmoncs n the nverson formulas of the Fourer transform (14), (15), (18), (19) deends on a mnmal satal wavelength n the medum and on the sze of the satal calculaton doman of the feld gven by the fnte lmts of the ntegral transform. In addton, the convergence rate of the summable seres deends on smoothness of functons of the wave feld. The number of the Laguerre harmoncs for determnng functons by formula (17) deends on a sgnal gven n the source f () t, the arameter h, and the tme nterval of the wave feld. Paers [5-8] consder n detal how one can determne the requred number of harmoncs and choose an otmal value of the arameter h. 4. Numercal Results Fgures 1-3 show the results of numercal calculatons of a wave feld as snashots at a fxed tme for the horzontal comonent of the dslacement velocty ux ( x, y, z ). Fgure 1 resents a snashot of the wave feld for ux ( x, y, z ) n the lane XZ at the tme t=15 sec. Ths model of the medum conssts of a homogeneous elastc layer and an atmosherc layer searated by a lane boundary. The hyscal characterstcs of the layers are as follows: the atmoshere: sound seed c 34 m sec 1. Densty versus coordnate z was calculated by the formula 3 ( z) 1ex( z/ H), where g cm 3, H 67 m; the elastc layer: ressure wave velocty c 3 m sec 1, shear wave velocty cs m sec 1, densty 1. g cm 3. A bounded doman, ( x, y, z ) = (15km, 15km, 1km), was used for the calculatons. A wave feld from a ont source (a ressure center) located n the elastc medum at a deth of ¼ of the length of a ressure wave wth coordnates ( x, y, z ) = (6 km, 7.5km,.75km) was smulated. The fgure shows the wave felds for the horzontal comonent u x of the dslacement velocty n the lane XZ at y y 7.5 km : wthout wnd (to), wth the wnd seed n the atmoshere of 5 m sec 1 (bottom). The elastc medum-atmoshere nterface s shown by the sold lne. Ths fgure demonstrates that n the elastc medum, n addton to the shercal P-ressure wave and the conc S-shear wave, there also roagates a non-ray shercal wave S *, and then there follows a surface Stoneley-Scholte wave. An acoustc-gravty wave refracted at the Earth-atmoshere boundary roagates n the atmoshere. At the boundary, ths wave generates the corresondng ressure and shear waves n Fgure 1. A snashot at t = 15 sec for the velocty comonent u x n the lane (XZ) wthout wnd (to), wth wnd (bottom) (wnd seed 5 m sec 1 ). the elastc medum. Fgures and 3 resent snashots of the wave feld when the sesmc waves velocty n the elastc medum s greater than the sound seed n the atmoshere. In ths model, the hyscal characterstcs of the elastc medum and the atmoshere are as follows: the atmoshere: sound seed c 34 m sec 1. Densty versus coordnate z was calculated by the formula 3 ( z) 1ex( z/ H), where g cm 3, H 67 m; the elastc layer: ressure wave velocty c 8 m sec 1, shear wave velocty cs 5 m sec 1, densty 1.5 g cm 3. A bounded doman ( x, y, z) ( km, 16 km, 14 km), was used for the calculatons. A wave feld from a ont source (the ressure center) located n the elastc medum at a deth of ¼ of the length of a ressure wave wth the coordnates ( x, y, z) (1km, 8km,.km) was Coyrght 13 ScRes.

5 16 B. G. MIKHAILENKO ET AL. Fgure 3. A snashot of a wave feld for the horzontal velocty comonent u x (x,y,z), at t = 1 sec wth wnd n the atmoshere (wnd seed 5 m sec 1 ). Fgure. A snashot at t = 1 sec for the velocty comonent u x n the lane (XZ) wthout wnd (to), wth wnd (bottom) (wnd seed 5 m sec 1 ). smulated. Fgure shows the wave felds for the horzontal comonent u x, of the dslacement velocty n the lane XZ at y y 8 km: wthout wnd (to), wth wnd seed n the atmoshere of 5 m sec 1 (bottom). The elastc medumatmoshere nterface s shown by the sold lne. Ths fgure shows that n the atmoshere, n addton to the concal P- ressure wave and the concal S-shear wave, there also roagates a non-ray shercal wave P *, and then there follows a surface Stoneley- Scholte wave. Fgure 3 resents snashots of a 3D wave feld at t 1 sec for the velocty comonent u x wth the wnd seed of 5 m sec 1 n the atmoshere. The numercal smulaton results have revealed some new eculartes of the wave roagaton wth wnd n the atmoshere. Secfcally, the nfluence of the wnd on the roagaton velocty of the surface Stoneley waves n an elastc medum has been demonstrated. The numercal results have also shown that the velocty of these waves ncreases downwnd and, hence, t decreases uwnd by a quantty equal to the wnd seed. The same nfluence of wnd s on a non-ray shercal exchange acoustc-gravty wave roagatng n the atmoshere from a source located n a sold medum. Another fact of the wnd nfluence that has been establshed s that the surface wave changes n the amltude along ts front. Ths manfests tself as an ncrease n the amltude n that art of the wave front that roagates downwnd and a decrease n the wave front roagatng uwnd but wth conservaton of the total wave energy. 5. Concluson The aroach roosed to the statement and soluton of the roblem makes t ossble to smulate the effects of the wave feld roagaton n a unfed mathematcal earth-atmoshere model and to study the exchange waves at ther boundary. The numercal smulaton of these rocesses makes t ossble to nvestgate the eculartes of the wnd effects on the roagaton of the acoustc-gravty atmosherc waves and surface Stoneley waves. 6. Acknowledgements Ths work was suorted by the Russan Foundaton for Basc Research (rojects no , and ). REFERENCES [1] A. S. Alekseev, B. M. Glnsky, S. I. Dryakhlov, et al., The Effect of Acoustc-Sesmc Inducton n Vbroses- Coyrght 13 ScRes.

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