ELASTIC WAVES PRODUCED BY LOCALIZED FORCES IN A SEMI-INFINITE BODY

Transcription

1 Romanian Repos in Physics, Vol. 6, No., P , ELASTIC WAVES PRODUCED BY LOCALIZED FORCES IN A SEMI-INFINITE BODY B.F. APOSTOL Depamen of Seismology, Insiue of Eah's Physics, Maguele-Buchaes, POBox MG-5, Romania afelix@heoy.nipne.o (Receied Noembe, 9) Absac. The popagaion of elasic waes in isoopic bodies is inesigaed by a new mehod, based on he elecomagneic Kichhoff poenials. This mehod is applied heein o elasic waes poduced in a semi-8 infinie (half-space) isoopic body by he acion of an exenal foce localized beneah, o on he body suface. The mehod leads o coupled inegal equaions fo he wae ampliudes, which ae soled fo he boh cases. The waes poduced by a foce localized beneah he suface ae saionay waes along he nomal o he suface. Fo a foce localized on he body suface wo ansese waes ae idenified, coesponding o he wo polaizaions (nomal and paallel o he popagaion plane). Anohe longiudinal wae appeas as an eigen-mode. The suface displacemen and he foce exeed on he suface ae compued in boh cases. All hese quaniies exhibi a chaaceisic decease wih he disance on he body suface and an oscillaoy behaiou. A bief discussion is included egading some possibiliies of exending he pesen mehod o eaing he effec of he inhomogeneiies on he waes popagaion. Key wods: semi-infinie elasic body; elasic waes; localized foces; Kichhoff poenials; nomal modes. PACS: 4..Bi; 46.5.Cc; 4..Ks; 6.. +d. INTRODUCTION The popagaion of elasic waes in bodies wih special, esiced geomeies was oiginally inesigaed by Rayleigh [] and Lamb [], and undewen aious deelopmens duing he ime []. I is someimes known as Lamb's poblem in seismology [4]. The poblem exhibis a ceain complexiy elaed o difficulies aising mainly fom he lack of an adequae eamen of he bounday condiions. Of paicula impoance is he deeminaion of he waes poduced in such elasic bodies by exenal foces, eihe localized on he body suface, o wihin he bulk, o exended oe ceain spaial olumes. Een moe ineesing, and moe difficul, is he poblem of eaing he effec of he inhomogeneiies, eihe localized o exended, on he wae popagaion infinie elasic bodies. Apa fom hei pacical impoance in engineeing, such poblems ae of gea eleance fo he effec of he seismic waes on he Eah's suface [5 9].

2 76 B.F. Aposol A new mehod is pesened hee fo sudying he wae popagaion in isoopic elasic bodies wih a finie (o paially finie) sucue, based on he Kichhoff poenials of he wae equaion wih souces, boowed fom elecomagneism. This mehod is employed heein o deemine he elasic waes poduced in a semi-infinie (half-space) body by exenal foces localized eihe beneah, o on he body suface. Fo he foce localized beneah he body suface he elasic waes ae saionay waes along he diecion pependicula o he body suface. Fo he foce localized on he suface we deemine wo ansese waes popagaing in he body and a longiudinal one which appeas as an eigenmode. The suface displacemen and he foce exeed on he suface ae compued in boh cases. All hese quaniies exhibi a chaaceisic decease and an oscillaoy behaio along he in-plane disance on he body suface. In boh cases, he pesen mehod leads o coupled inegal equaions fo he wae ampliudes, which ae soled. By means of he mehod pesened heein new esuls ae obained fo a poin-like foce localized unde he suface and one of Lamb's poblem (foce localized on he suface) is genealized. The genealizaion consiss in eaing a geneal disibuion of he foce acing on he body suface and a geneal oienaion of his foce. In addiion, in boh cases, he effecs of a localized pessue ae analyzed. Finally, a bief discussion is gien egading he exension of he pesen mehod o include he effec of he inhomogeneiies on he wae popagaion in elasic bodies wih finie geomeies. []. GENERAL THEORY The elasic waes in isoopic bodies ae goened by he equaion of moion ρ u= µ u+ ( λ + µ ) gad di u+ρf, () whee ρ is he densiy, u is he displacemen field, µ and λ ae he Lame coefficiens and f is an exenal foce pe uni mass (acceleaion). By a Fouie ansfom of he fom ( ) ( ) i KR, = d ω, ω e i u R u K () and a simila one fo he foce f, equaion () becomes K ( K ) ( )( ) whee we dopped ou he agumens be soled. Is soluion is gien by ρω +µ u= λ+µ Ku K +ρf, () ω K, ω fo simpliciy. Equaion () can easily

3 Elasic waes poduced by localized foces in a semi-infinie body 77 whee ( l )( Kf ) f u= K, ( ω K )( ω K ) ω l K (4) = µ, λ+ µ l = ρ ρ (5) ae he elociies of he ansese and, especiely, longiudinal waes. We can see fom equaion (4) ha fo a longiudinal foce f = f K/ K f = f K=K he displacemen field is longiudinal and has he eigenfequencies ω = l K, while fo a ansese foce, Kf =, he field is ansese and has he eigenfequencies ω= K. As i is well known, he Lame coefficiens can be expessed by he Young modulus E and he Poisson aio σ, Eσ E λ=, µ=, (6) ( + σ)( σ ) ( +σ) and, fo easons of sabiliy, E > and < σ < / (acually, fo usual bodies, < σ < /). In paicula, he aio l λ q = = + = µ σ saisfies he inequaliy q > / (acually q > ) []. In geneal, he soluion of he homogeneous equaion () ( fee waes ) mus be added o he paicula soluion gien by equaion (4) ( foced waes ). As i is well known, anohe, moe diec, mehod can be used fo soling equaion (), wihou esoing o Fouie ansfoms. The mehod consiss in wiing he displacemen u as a sum of wo funcions, u = u + u l, saisfying he condiions diu =, as fo ansese waes, and culu l = coesponding o longiudinal waes. Then, i is easy o see ha funcions u,l saisfy he wae equaions wih elociies,l, especiely. We pesen hee a hid mehod, which is used in he pesen pape, based on Kichhoff poenials. Indeed, making use of he noaions inoduced aboe we wie equaion () as f u u= q gad diu+, (8) whee we can ecognize he wae equaion wih souces q gad diu and f /. As i is well known, is (paicula) soluion is gien by he eaded (Kichhoff) poenial (7)

4 78 B.F. Aposol 4 ( R R R ) q gad di u, / u( R, ) = d + 4 π R R R + 4π (, / ) f R R R d R. R R Indeed, making use of he Fouie ansfom gien by equaion () and using also he well-known inegal ikr+ω i R / e 4π d R = () R ω K we ge easily he soluion gien by equaions () and (4). We apply heein his mehod of Kichhoff poenial, inspied fom he heoy of elecomagneism, [] o he elasic waes geneaed in a semi-infinie body by foces localized eihe beneah, o on he body suface. (9). SEMI-INFINITE BODY. SURFACE WAVES We conside a semi-infinie isoopic body exending oe he egion z <, wih a plane suface a z =. We may conside also a foce f acing on his body, eihe inside i o localized on is suface, and look fo soluions of equaion (), i.e. fo he waes popagaing in he body. Of couse, we assume ha u; f = fo z > (ouside he body). This is Lamb's poblem []. Usually, i is assumed ha foce f is localized, eihe on he body suface o beneah i, and he poblem is appoached by making use of a emakable popey of equaion (). Indeed, as i is well known, equaion () is anohe fom of he moe compac equaion wien as σij ρ u i = +ρfi, () x j whee σ =λu δ + µ u () is he sess enso, u ij ij ll ij ij u u i j = + x j x i is he sain enso and i, j, l, ec, denoe he coodinaes axes. We ake a small olume encicling a poin on he body suface and apply he Gauss heoem fo equaion (). I is easy o see ha, fo a foce disibued in he body olume, we ge ()

5 5 Elasic waes poduced by localized foces in a semi-infinie body 79 σ n =, (4) ij j fo any poin on he body suface, whee n is he uni eco nomal o he body suface. Equaion (4), which plays he ole of a bounday condiion, ells ha he body suface is fee (fom any foces). If hee exiss a foce localized on he body f δ z, hen i is easy o see ha he same pocedue leads o suface, say ( ) σ n + f =, (5) ij j i which ells again ha he oal foce on he suface is anishing. In addiion, equaion (5) ells ha he quaniy σ ijn j is he suface elasic foce (foce pe uni aea) acing on he suface. I is also easy o see ha he same pocedue can be applied, a leas in pinciple, o a foce localized ono a poin, a suface, egion, ec inside he body. In all hese cases we wie soluions fo he homogeneous equaion () and impose upon hem he coninuiy and he jump bounday condiion of he ype gien by equaion (5). This way, he geneal soluion of equaion () is obained. This was he mehod used by Lamb in eaing such poblems []. Apa fom less ineesing wo-dimensional poblems, Lamb eaed also paicula cases of a foce localized on he suface of a semi-infinie isoopic body and skeched an inegal epesenaion of he soluion fo a foce localized beneah he suface of such a body. I is woh noing also ha he fee-suface condiion gien by equaion (4) has been used by Rayleigh [] o idenify he damped fee suface waes ( e z, eal) popagaing on he suface of a semi-infinie body in he absence of any foce. A diffeen pocedue is adoped in his pape. A paicula soluion is obained by using Kichhoff poenial, as deemined by he exenal foce f ( foced waes ), and fee waes of he fom u l, ae added o i, i.e. soluions of he homogeneous equaion, in ode o saisfy he bounday condiions of he ype gien by equaions (4) and (5). We cay ou explici calculaions fo a poin-like foce localized beneah he body suface and fo a poin-like foce localized on he body suface. As we shall see, he damped fee suface waes ae no excied by localized exenal foces, hough hey may be excied by suface damped foces. 4. FORCE LOCALIZED BENEATH THE SURFACE We conside a foce (, ) = a ( ) δ( ) f R f R R (6)

6 8 B.F. Aposol 6 localized a deph d beneah he plane suface z = of a semi-infinie elasic body exending o he egion z <, such as R = (,, d). The chaaceisic lengh a is, much smalle han he elean disances, is inoduced on one hand fo easons of dimensionaliy and, on he ohe, fo haing a epesenaion of he spaial exension of he focus ono which he foce acs. In equaion (6) he posiion eco is gien by R = ( x, yz, ) = (, z) and denoes he ime. The popagaing spheical waes poduced by his poin-like foce in an infinie body ae well known (see fo insance Ref. [9]). We deie hee he waes poduced by such a souce in a semi-infinie body. We epesen he displacemen field as ( u ) ( z) u=, θ, (7) whee is he in-plane componen (paallel o he suface), u is he ansese componen (pependicula o he suface) and θ ( z) = fo z <, θ ( z) = fo z > is he sep funcion. We use Fouie ansfoms of he fom ( ) ( ) i k z = ω ω z i, ; d k, ; e, (8) k and a simila one fo u (, z; ), whee k is he in-plane waeeco. Usually, we leae aside he agumens k, ω, while peseing explicily he z-dependence of he funcions (k, ω; z) and u (k, ω; z). The diegence occuing in equaion (9) can hen be wien as u u ( z) u ( ) ( z) z (9) ω di = di + θ δ, whee we can see he occuence of specific suface conibuions associaed wih u () = u (z = ). We compue gad diu accoding o equaions (8) and (9) and inoduce i, ogehe wih he foce gien by equaion (6), in equaion (9). The ineening inegals educe o he known inegal [] i ωx/ c i i ( ) z d xj k x z e e, () z = whee J is he Bessel funcion of he fis kind and zeoh ode and ω = k. ()

7 7 Elasic waes poduced by localized foces in a semi-infinie body 8 In addiion, we inoduce he conenien noaions = k / k, = k / k and simila ones fo f,, whee k is a eco pependicula o k, kk =, and of he same magniude k. The foce em in equaion (9), which we denoe by F, can easily be ealuaed. Is Fouie ansfom is gien by a f F = sin z + d, () whee k =ω / >. Applying he pocedue descibed aboe we ge saighfowadly fom equaion (9) a f = F = sin z + d () and he se of coupled inegal equaions iqk i z z qk i z z = z ( z ) zu ( z ) + F d e d e, z qk i z z ik i z z = ( ) + ( ) + u dzu z e dzu z e F. z z In deiing hese equaions i is woh noing he non-ineibiliy of he deiaies and he inegals, accoding o he ideniy z z fo any funcion f (z), z > ; i is due o he disconinuiy in he deiaie of he i funcion e z z fo z = z. These equaions imply he elaionship (4) i z z i d ( ) e z z z f z = dz f ( z ) e if ( z) (5) u i i F = F. k z k z Inoducing u fom his equaion ino he fis equaion (4) and pefoming he inegaions by pas, we ge a single inegal equaion (6) iqω q q dzu z e e q F i z z iz ( + ) = ( ) + ( ) + ( ) iq i z z q iz qk i z z dzf( z) e F( ) e dzf( z) e. + + z Taking he second deiaie wih espec o z in his equaion we find (7)

8 8 B.F. Aposol 8 q F + = F + i k, z + q z (8) whee =ω / l k. Now, i is easy o ge he soluion fo. I is gien by a = fsin z + d + i fsgn ( z + d) cos ( z + d) ω (9) and, by equaions () and (6), ak u = fsin z + d + ikfsgn ( z + d) cos ( z + d). ω () We can see ha all hese soluions,, u ae saionay waes along he diecion pependicula o he suface, as geneaed by he saionay oscillaing foce gien by equaion (). In addiion, hey ae egula funcions fo, hough and u may incease indefiniely fo, (, ) u ( ) z + d ; his incease indicaes he ansiion o he damped egime. I is also woh noing he disconinuiy occuing a z = d. I is ineesing o noe ha he waeecos k and in he localized foce gien by equaion (6) ae independen and eal aiables. Ou of hem, he equaion of he elasic waes selecs only hose waeecos which saisfy he condiion ( k + ) ω =, and assigns hem o he allowed popagaing waes. This is he mahemaical mechanism hough which exended elasic waes ae geneaed by localized foces. Anohe obseaion is ha, he aboe waes being saionay, he polaizaion is meaningless fo hem, alhough hey ae associaed wih he elociy of he ansese elasic waes. On he ohe hand, we mus noice ha he soluion of he homogeneous equaion (8) is he fee longiudinal wae popagaing wih he waeeco, i.e. wih he elociy l of he longiudinal waes, and similaly, he soluion of he homogeneous wae equaion (8) is he fee ansese wae popagaing wih he waeeco and he elociy. 5. SURFACE DISPLACEMENT The displacemen of he suface z =, as caused by he foced waes obained aboe, can be compued by using he inese Fouie ansfoms of K = k,. As ( ) and u ( K ) gien by equaions (), (9) and (), whee ( ), K usually, we leae aside fo he momen he agumen ω in hese expessions. I is woh noing ha / k = ω is no an independen aiable. The Fouie i componens of he foce ae gien by ( ) = a e d f K f. We choose an in-plane

9 9 Elasic waes poduced by localized foces in a semi-infinie body 8 efeence fame wih one axis oiened along he in-plane adius (adial axis ) and anohe pependicula o he fome (angenial axis ). We denoe by α he angle beween he foce eco f and adius so ha he foce eco can be wien as ( f cos α, f sin α, f ), whee f denoes he in-plane (hoizonal) foce and f denoes he eical foce. We also denoe by ϕ he angle beween he in-plane waeeco k and adius, such ha k = k ( cos ϕ,sin ϕ) and k = k ( sin ϕ,cos ϕ ). Then, we can compue easily he foce pojecions f,, appeaing in equaions (), (9) and (). They ae gien by f = ( f cos ( α ϕ), f sin ( α ϕ ), f ). I is woh noing ha on changing k k, i.e. ϕ π+ϕ, he quaniies f, change sign, as hey should do; similaly, f, being he pojecion of he foce along he waeeco componen, mus change sign unde he eesal of he diecion of his componen,. Making use of he aboe noaions and of ( K) = k/ k + k / k we can obain immediaely he adial and angenial componens of he displacemen, ( K) and ( K), especiely. Howee, i is woh noing ha fo a eal displacemen he Fouie ansfoms mus saisfy he symmey elaionship ( K) = ( K ), and, similaly, u( K) = u( K ). Taking ino accoun he change of sign of he foce componens f,, unde his opeaion, we can see ha a faco sgn ( π ϕ ) mus be inoduced, in geneal, wheee elean, in ode o ge eal displacemens. The inegals wih espec o angle ϕ in he Fouie ansfoms imply he Bessel funcions J,. Some of hese inegals ae colleced in Appendix. The suface displacemen can be wien as whee ( ) ( ) a f a f I I I I 4 a f = cos α cos α, 4πω 4π 4πω a f a f = I sin α I I sin α, πω π a f a f ( ) = 6 4 cos α, u I I 4πω 4πω ω/ ω/ = = ω/ ω/ = 4 = ω/ k ω/ k 5 = 6 = ( ) ( ) I dk ksin d J k, I dk sin d J k, ( ) ( ) I dk sin d J k, I dkk cos d J k, ( ) ( ) I dk sin d J k, I dk sin d J k. () ()

10 84 B.F. Aposol We esimae hese inegals in he fas oscillaing limi ω/, ωd /. In his case, he main conibuion comes fom k and exends oe a ange k / fo d o k / d fo d. The leading conibuions fo d ae gien by a f a f a f cos α, sin α, cos α, () ω ω ω ( ) ( ) u ( ) whee oscillaing facos of he fom sin ωd /, cos ωd / ae lef aside. We can see he diecional chaace of he suface displacemen (hough angle α ) and he eical componen (u ) which is much smalle (by a faco ω / ) han he hoizonal componens. We shall see ha he diecional chaace as gien in equaion () fo he foced waes is amended by he conibuion of he fee waes. I is also woh noing ha he leading conibuion o he eical displacemen is caused by he in-plane foce f, and, in geneal, he eical componen of he foce bings a smalle conibuion. This is due o he saionay chaace of he waes along he eical diecion. Le us assume now a foce deied fom a localized pessue p. The foce componens ae hen gien by f = i pk/ ρ, f = and ( ) i f = i p/ ρ e d. In compuing he Fouie ansfoms of he suface displacemen we mus ake cae again of he geneal symmey elaions ( K) = ( K ) and u( K) = u( K ). We ge ω/ ( ) kk d( d) J ( k) a p 4πω ρ ω/ ( ) kk d ( ) a p πωρ ω/ ( ) kk ( d + d) J ( k) 4πω ρ a p d sin cos, d cos sin, d sin cos. (4) In he same limi ω/ ωd / he leading conibuions o he aboe displacemens ae gien by a p a p,,. 4 ω ρ ω ρ ( ) ( ) u ( ) (5) We can see ha he displacemens poduced by pessue fall off fase wih disance han he coesponding displacemens caused by a foce (equaion ()).

11 Elasic waes poduced by localized foces in a semi-infinie body FORCE EXERTED ON THE SURFACE We ae ineesed now in he foce exeed on he suface z = by he foced waes poduced beneah he suface. As shown aboe, he foce exeed by a displacemen field u pe uni aea of a suface wih uni nomal n is gien (in ou s noaions) by ρ f =σ n, whee σ =λu δ + µ u is he sess enso and i ij j ( / )( / / ) ij ll ij ij uij = ui xj + uj xi is he sain enso. Using he efeence fame defined by kk, and z we ge s a ( k ω ) = ( ) f, ω f cos d ikf sin d, f a f d ( k ) s, ω = cos, s a k ( k ω ) = ( ) f, kf cos d i fsin d. ω We noe ha he dilaaion anishes, + + u =, in accodance wih he fac ha hese soluions, gien by equaions (), (9) and (), ae consuced fom ansese waes. We compue he inese Fouie ansfoms of hese foces wih espec o he waeco k accoding o he pocedue descibed aboe fo he suface displacemen. The asympoic expessions ( ω/ ωd / ) ae gien by s a f s a f s a f ( ) ( ) ( ) (6) f cos α, f sin α, f cos α; (7) ω hey ae simila wih he suface displacemens gien by equaion (), excep fo an addiional faco ω. In he same manne we can compue he foce exeed on he suface by a localized pessue. 7. GENERAL SOLUTION The geneal soluion of he poblem is obained by adding o he paicula soluion gien by equaions (), (9) and () he soluions of he homogeneous equaions (8) and (8). These ae gien by a ansese wae and a longiudinal wae k i u,, e z = A A A (8)

12 86 B.F. Aposol i z u l = B,, B e, k (9) whee he consans A,, B ae deemined fom he condiion of a anishing oal suface foce, in accodance wih equaion (4). On equaions (8) and (9) one can check he ansesaliy condiion diu = and he condiion culu l = fo longiudinal waes. The fee waes soluions gien by equaions (8) and (9) ae wien in he efeenec fame defined by kk, and z. s We compue he suface foce ρ fi =λullδ i + µ ui, whee u = u + u l, and impose he condiion s i f s i + f =, (4) s whee f i, gien by equaions (6), coespond o he suface foce geneaed by he paicula soluion ( foced waes ). I is woh noing hee ha equaion (4) holds fo any poin on he suface z =, i.e. i is muliplied in fac by he faco e ik wih he same in-plane waeeco k. Since ω is he same fo boh he paicula soluion and he fee waes, and ω = ( + k ), ω = l ( + k ) in boh cases, i follows ha, ae he same, i.e. hey ae eal aiables, as coesponding o localized exenal foces. Consequenly, he fee waes ae popagaing waes. Damped fee suface waes (i.e. waes wih, puely imaginay) can be excied by damped exenal foces []. Condiion (4) leads o and he sysem of equaions A a f i = cosd (4) a ( k ) A + B= ( k f d + f d) sin i cos, ω ak ( ) = ( + ) k A k B f sin d ik f cos d, ω whose soluion can be wien as A ( k ) 4 k = + u, k ( k ) 4 k k B= u +, (4) (4)

13 Elasic waes poduced by localized foces in a semi-infinie body 87 = + 4 k. Incidenally, we noe hee ha = fo i and i gies he dispesion elaion ω ( k ) fo he Rayleigh suface waes. The suface displacemens bough abou by he fee waes ae gien by = A + B, = A and k u = A + B. We compue hei inese Fouie k ansfoms by he same pocedue as ha descibed in Secion 5. Unde he same condiions as hose employed in his Secion we ge he asympoic behaiou whee ( k ) u o o ( ) ( ) a f ω ( ) ( ) a f ω a f cos α, sin α, ω a f sin α, cos α, ω a f α ω l a f / cos, sinα ω o l (44) o fo he oal displacemen u = u + u. We can see ha he fee waes do no change he -dependence, bu inoduce an addiional diecional chaace. In addiion, he eical displacemen is affeced by facos depending on /. A l simila conclusion (excep fo he diecional chaace) holds fo a foce deied fom pessue. 8. FORCE LOCALIZED ON THE SURFACE We conside a semi-infinie isoopic elasic body exending oe he egion z > and assume a localized foce i k iω (, ) = a d ω (, ω) e δ( z) f R f k (45) k acing on he body plane suface z =, whee a is a chaaceisic lengh, R = (, z) and k is he in-plane waeeco. This is a genealizaion of one of Lamb's poblems []. We epesen he displacemen field as ( u ) ( z) u=, θ, (46) whee is he in-plane componen (paallel o he suface), u is he ansese componen (pependicula o he suface) and θ ( z) = fo z <, θ ( z) = fo z > is he sep funcion. The diegence occuing in equaion (9) can hen be wien as

14 88 B.F. Aposol 4 u diu= di + θ ( z) + u ( ) δ( z), z (47) whee we can see he occuence of specific suface conibuions associaed wih u = u z =. As befoe, we use a Fouie ansfom of he fom ( ) ( ) ( ) ( ) i k z = ω ω z i, ; d k, ; e, (48) k and a simila one fo u (, z; ) while peseing explicily he z-dependence of he funcions (, ω;z ) u ( k ω z). Usually, we leae aside he agumens k, ω, ω k and, ;. We compue gad diu accoding o equaions (47) and (48) and inoduce i, ogehe wih he foce gien by equaion (45), in equaion (9). As fo a foce localized beneah he suface, he ineening inegals educe o he known inegal [] i ωx/ c i i z ( ) d xj k x z e e, (49) z whee ω = k. (5) We use also he same conenien noaions = k / k, = k / k, and simila ones fo f,, whee k is a eco pependicula o k, kk =, and of he same magniude k. Then, equaion (9) educes o iaf iz = e (5) and o a se of wo coupled inegal equaions which ead and = iqk i z z qk i z z dz ( z ) e dzu ( z ) e = z + iaf iz + e qk i z z iq i z z d ( ) e d ( ) e u = z z z + z zu z + iaf iz + e. (5) (5)

15 5 Elasic waes poduced by localized foces in a semi-infinie body 89 We noe hee again ha in deiing hese equaions i is woh obseing he nonineibiliy of he deiaies and he inegals, accoding o he ideniy z z fo any funcion f (z), z > ; i is due o he disconinuiy in he deiaie of he i funcion e z z fo z = z. Fom equaions (5) and (5) we ge easily i z z i d ( ) e z z zf z = dzf ( z ) e if( z) (54) u = i ( ) i a f kf iz e. k z k Equaion (5) gies he ansese wae (fo eal) popagaing wih he elociy, accoding o equaion (5). Is polaizaion is nomal o he plane of popagaion (he plane deemined by he ecos k and ). This wae is usually known as he s-wae in he heoy of elecomagneism (fom he Geman wod senkech which means pependicula ). Fom equaion (5) we can see ha he s-wae becomes singula fo =. We pass now o he sysem of coupled equaions (5) and (5), and he elaionship gien by equaion (55). We inoduce u fom equaion (55) ino equaion (5) and ge ( q) z ( z ) iqω i z z + = d e + iaq i z i z z iaf iz + ( f kf) dz e e + q ( ) e. + z 4 This equaion can easily be soled by aking he second deiaie wih espec o z and using he non-ineibiliy equaion (54). We ge (55) (56) whee iaq + = e, z + iz ( ) ( f kf ) q (57) l ω = k. (58) Fo a longiudinal foce f k f = we obain fom equaion (57) longiudinal waes popagaing wih waeeco (fo eal) and wih he elociy l. Fo a geneal foce, equaion (57) has he paicula soluion

16 9 B.F. Aposol 6 and ia ( ) i z = f k f e (59) ω iak ( ) i z u = f k f e. (6) ω We can see ha and u gien aboe coespond o a ansese wae, k + u =, whose polaizaion lies in he plane of popagaion. This is called he p-wae, whee p sands fo paallel. We can see also ha is a egula funcion, while u may exhibi he same singulaiy as does fo =. 9. SURFACE DISPLACEMENT CAUSED BY A FORCE LOCALIZED ON THE SURFACE The displacemen of he suface z = can be compued by using he inese and u ( K ) gien by equaions (5), (59) and (6), K = k,. As usually, we leae aside fo he momen he agumen ω in Fouie ansfoms of, ( K) whee ( ) hese expessions. I is woh noing ha = ω / k is no an independen aiable. Fis, we conside a δ-ype foce localized on he suface, f( R) = ab f δ( ) δ( z), whee f is a consan eco and b is a chaaceisic localizaion lengh on he suface. Again, his is anohe genealizaion of one of Lamb's poblems []. The Fouie componens f( K) = ab f of his foce do no depend on K (bu hey may hae an ω-dependence). As befoe, we choose an in-plane efeence fame wih one axis oiened along he in-plane adius (adial axis ) and anohe pependicula o he fome (angenial axis ). We denoe by α he angle beween he foce eco f and adius. Then, he foce eco can be wien as f = ( f cos α, f sin α, f ), whee f denoes he in-plane (hoizonal) foce and f denoes he eical foce. Similaly, we denoe by ϕ he angle beween he in-plane waeeco k and adius, such ha k = k(cos ϕ, sin ϕ) and k = k ( sin ϕ,cos ϕ ). Then, he foce pojecions f,, eneing equaions (5), (59) and (6) can be wien as ( ) ( ) f = b f cos α ϕ, f = b f sin α ϕ, f = b f. (6) I is woh noing ha on changing k k, i.e. ϕ π+ϕ, he quaniies f, change sign, as hey should do; similaly, f, being he pojecion of he foce along

17 7 Elasic waes poduced by localized foces in a semi-infinie body 9 he waeeco componen, mus change sign unde he eesal of he diecion of his componen,. Making use of he aboe noaions and of ( K) = k/ k + k / k we can obain immediaely he adial and angenial componens of he displacemen, ( K ) and ( K), especiely. Howee, i is woh noing ha fo a eal displacemen he Fouie ansfoms mus saisfy he symmey elaionship ( K) = ( K ), and, similaly, u ( K) = u ( K ). Taking ino accoun he change of sign of he foce componens f,, unde his opeaion, we can see ha a faco sgn(π ϕ) mus be inoduced wheee elean. The Fouie componens of he displacemen can be wien as iab iab f cos cos sin sin sgn ω iab cos, kf ϕ ω ( k) = f ( α ϕ) ϕ ( α ϕ) ϕ ( π ϕ) iab iab f cos sin sin cos sgn ω (6) iab sin, kf ϕ ω iab iab k u ( k ) = kf cos ( α ϕ ) + f sgn ( π ϕ) ω ω ( k) = f ( α ϕ) ϕ+ ( α ϕ) ϕ ( π ϕ) (fo z = ). Now we can ake he inese Fouie ansfoms of hese quaniies. I is easy o see ha he inegals oe angle ϕ which conain facos sin ϕ and cos ϕ ae anishing. Fo he adial componen we ae lef wih ( ) ω/ k π k iab f ik cosϕ = sin ϕ d dϕsgn ( π ϕ) sin ϕcos ϕe ( π) ω (6) iab f ω/ π ik cosϕ dkk d cos e. ϕ ϕ ω ( π) The inegals in equaion (6) can be pefomed saighfowadly, by making use of he popeies of he Bessel funcions [, 4]. We ge ab ω ω = f sin α+ f J J. 4π ω ( ) ( ) In he limi ω / we ge (64)

18 9 B.F. Aposol 8 ab ω π sinα+ cos. ω 4 (65) ( ) ω ( f f ) ( ) / / / We can see ha he adial componen of he suface displacemen aains is maximum alue along a diecion pependicula o he diecion of he foce ( α=π /), as expeced fo a ansese wae geneaed by such a localized foce. I has a chaaceisic oscillaoy behaiou wih he in-plane disance and goes like / fo long disances. The empoal Fouie ansfom of he specum gien by equaion (65) fo f and f independen of ω (elaed o Fesnel inegals) exhibis a chaaceisic oscillaoy wae fon of he fom ( ) /, as expeced. Such qualiaie chaaceisics of he soluion o his poblem ae simila wih hose indicaed long ime ago by Lamb [] (See also Ref. []). Simila calculaions can be done fo he angenial componen ( ) ) and he can be obained fom equaions eical componen ( ) u. The esul fo ( ) (64) and (65) by puing fomally f = and eplacing sinα by cosα. The eical componen can be obained fom equaions (64) and (65) by eplacing sinα by and puing f =. Nex, we conside an in-plane localized pessue pb δ( ). The Fouie componens of he foce ae gien by ( ) f = i b p/ ρ k, f = f = and he Fouie componens of he displacemen ae ab p ab p ( k) = k( cos ϕ, sin ϕ) sgn ( π ϕ ), u( k ) = k. (66) ρω ρω and The inese Fouie ansfoms of hese displacemens gies ( ) ab pω ω ω ( ) = J + J, 6πρ ab pω ω ω u( ) = J J, 4πρ ω = whee J, ae Bessel funcions of he fis kind and second and, epeciely, hid / ode. The leading em ( ) in is anishing in he limi ω /, while u behaes like (67)

19 9 Elasic waes poduced by localized foces in a semi-infinie body 9 u ( ) / ab p ω ω π cos. ρ 4 ( ) ω / / (68) The eical componen of he suface displacemen has a wae fon of he fom ( ) /.. GENERAL SOLUTION FOR A FORCE LOCALIZED ON THE SURFACE The foce exeed on he suface by he foced waes is gien by s ρ fi =σ i =λ ullδ + µ u i i, whee he paicula soluion gien by equaions (5), (59) and (6) is used fo compuing he sain enso. Is componens ae gien by ( k ) ( ) a s = f ω f k f, s af f =, s a k = ( ) f f k f, ω whee f i ae gien by equaion (6). We can esimae his foce following he same pocedue descibed in he peceding Secion fo he displacemen. The esuls ae simila wih he coesponding displacemens. Fo insance, he asympoic expession fo he adial componen of such a foce is gien by s ab / ω π f ( ) ω/ ( / ) cos cos, ω f α (7) 4 which is simila wih equaion (65). s The conibuion f i of he fee waes, as gien by equaions (8) and (9), mus be added o his foce, in ode o saisfy he fee-suface bounday condiion gien by equaion (5). Fo he in-plane Fouie ansfoms his condiion eads s s i i i The soluions of his sysem of equaions ae (69) f + f + af =. (7)

20 94 B.F. Aposol ( ) ( k ) ( ) ( ) ( ) i af k ia i a k A = + kf f + f, ω k ω k A iaf =, i ak B= + kf + k f iafk, ω (7) = k + 4 k. The suface displacemen caused by he fee whee ( ) waes is gien by k = A + B, = A, u = A + B. (7) k In he asympoic limi ω / he main conibuion o he in-plane Fouie ansfoms of he quaniies gien by equaion (7) in he efeence fame defined by he adial axis, angenial axis and he eical axis z is bough by k. Wihin his appoximaion we find ω ω π cos 4 ab f ab f ( ) J / / ω ( ) fo he adial componen, a simila expesion fo he angenial componen and (74) ω ω π cos cos cos 4 ab f ab f ( ) α α / / l l ( ω) u J fo he eical componen of he displacemen. As we can see, he fee waes do no modify he asympoic -dependence of he displacemen caused by he foced waes (equaion (65)), bu inoduce an addiional angle dependence and ampliude facos elaed o he elociy of he longiudinal waes. A simila conclusion holds fo he displacemen caused by a poin-like pessue localized on he suface. (75). TEMPORAL DEPENDENCE The asympoic suface displacemens fo a foce localized beneah he suface conains fequency facos of he fom / ω, / ω ec. In addiion, fo d, hey hae also oscillaing facos of he fom sin ( ωd / ) ), cos ( ωd / )

21 Elasic waes poduced by localized foces in a semi-infinie body 95 (no included in equaions (44)). In he opposie case d hese oscillaing facos ae of he fom sin ( ω / ), cos ( ω / ), so we may ake a geneal behaiou of he ype sin ( ωr / ), cos ( ωr / ), whee R is a lengh elaed o he disance fom he souce o he poin on he suface. In addiion, he fee waes may bing also conibuions popagaing wih elociy l along he disance R, especially fo small alues of he in-plane adius. Consequenly, fo he ime dependence of he suface displacemen we hae o esimae, fo insance, inegals of he fom ω cos ωτ I = d ω, (76) ω whee τ R/, whee denoes a geneic elociy and ω is a ange of fequencies. I is easy o see ha fo small τ he inegal in equaion (76) is appoximaely gien by I ln ( ωτ). I ells ha he fon waes has an abup ise fo τ=, as expeced. Fo he suface displacemens caused by a foce localized on he suface he / chaaceisic fequency faco is ω and we hae o esimae inegals of he fom ω cos ωτ I = d ω, (77) ω By a change of aiable ωτ= z his inegal can be educed o a Fesnel inegal. The Fesnel inegal is gien by [] π + i iz de z =. (78) / We can see ha he wae fon goes like ( / ) / τ =.. CONCLUSIONS In conclusion, we may say ha we hae inoduced heein a new mehod of sudying he popagaion of he elasic waes in isoopic bodies, based on he Kichhoff poenials fo wae equaion wih souces, boowed fom he heoy of elecomagneism. The mehod implies coupled inegal equaions fo he waes ampliudes, which we soled. Making use of his mehod we hae deemined he waes poduced in an isoopic elasic semi-infinie body by an exenal foce

22 96 B.F. Aposol localized eihe on he body suface o beneah he suface a some disance d. In he lae case he waes ae saionay along he diecion pependicula o he body suface. We hae also compued he suface displacemen poduced by hese foces as well as he foce exeed on he suface as caused by a foce localized beneah. We hae esimaed hese quaniies in he fas oscillaing egime ( ω/ ωd /, whee ω denoes he fequency and is he elociy of he ansese waes). These quaniies exhibi a chaaceisic decease along he in-plane disance on he body suface and a chaaceisic oscillaoy behaiou. By making use of his mehod we hae genealized one of Lamb's poblem (foce localized on he suface of he body) and obained new esuls fo a poin-like foce localized beneah he body suface. Vaious ohe esuls can be obained by means of his mehod, fo aious ohe geomeies and foce disibuions. The pesen appoach can be exended o deemine he waes popagaing in elasic bodies wih special, finie geomeies, eihe as eigenmodes o caused by some exenal foces (boh localized o exended). Moe ineesing, we can exend he pesen appoach o include he effec of aious inhomogeneiies placed in elasic bodies, as caused by local aiaions in he body densiy o elasic consans. Indeed, suppose fo insance ha a small iegulaiy δρ occus in he densiy ρ in equaion (). The coesponding em δρ u can be ansfeed ino he hs of equaion (8) and can be eaed as a wae souce. I will bing an addiional conibuion o he poenial gien by equaion (9), which allows one o compue he changes bough by his inhomogeneiy boh in he eigenmodes and he elasic esponse of he body. Of paicula impoance is he case when his inhomogeneiy is placed on he body suface. Obiously, a simila eamen can be applied o inhomogeneiies occuing in he elasic coefficiens λ and µ, boh on he body suface o in he bulk. Some esuls in his diecion will be epoed in a fohcoming publicaion. Acknowledgmens. The auho is indebed o he membes of he Insiue fo Eah's Physics a Maguele-Buchaes fo encouaging suppo and o he membes of he Laboaoy of Theoeical Physics a Maguele-Buchaes fo many useful, enlighening discussions. APPENDIX A few inegals We gie hee a few inegals occuing in he calculaions descibed in he main ex:

23 Elasic waes poduced by localized foces in a semi-infinie body 97 π π icos z ϕ i J ( z) icos z ϕ π π izcosϕ izcosϕ i J ( z) J( z) π dϕcosϕ e = π, dϕsinϕ e =, π dϕcos ϕ e = π, dϕsinϕcosϕ e =, z π i cos π z ϕ π izcosϕ dϕsin ϕ e = J ( z), d sgn ( ) e, z ϕ π ϕ = π icos z ϕ icos z ϕ sin z dϕ sgn ( π ϕ) cos ϕ e =, dϕ sgn ( π ϕ) sin ϕ e = 4, z π π i cos izcosϕ z ϕ ( ) ( ) dϕsgn π ϕ cos ϕ e =, dϕsgn π ϕ sin ϕ e =, π icos z ϕ sin z dϕsgn ( π ϕ) sinϕcosϕ e = 4i. z z REFERENCES. Lod Rayleigh, On waes popagaed along he plane suface of an elasic solid, Poc. London Mah. Soc., 7, 4 (885); Scienific Papes, Vol., pp , Cambidge, London, 9.. H. Lamb, On he popagaion of emos oe he suface of an elasic solid, Phil. Tans. Roy. Soc. (London), A, 4 (94).. W. M. Ewing, W. S. Jadezky and F. Pess, Elasic Waes in Layeed Media (McGaw-Hill, New Yok, 857), and efeences heein. 4. M. A. Pelissie, H. Hoebe, N. an de Coeeing and I. F. Jones (eds.), Classics of Elasic Waes Theoy, Sociey of Exploaion Geophysiciss, Tulsa, OK, B. B. Bake and E. T. Copson, The Mahemaical heoy of Huygens' Pinciple, Claendon, Oxfod, J. D. Achenbach, Wae Popagaion in Elasic Solids, Noh Holland, K. E. Bullen, An Inoducion o he Theoy of Seismology, Cambidge Uniesiy Pess, Cambidge, K. Aki and P. G. Richads, Quaniaie Seismology, Theoy and Mehods, Feeman, San Fancisco, A. Ben-Menahem and J. D. Singh, Seismic Waes and Souces, Spinge, New Yok, 98.. L. Landau and E. M. Lifshiz, Theoy of Elasiciy, Couse of Theoeical Physics, Vol. 7, Elseie, Buewoh-Heinemann, Oxfod, 5.. M. Bon and E. Wolf, Pinciples of Opics, Pegamon, London, I. S. Gadsheyn and I. M. Ryzhik, Table of Inegals, Seies and Poducs, Academic Pess,, pp , 6.677;,.. B. F. Aposol, Resonance of he suface waes. The H/V aio, Rom. Reps. Phys., 6, 9 94 (8). 4. G. N. Wason, A Teaise on he Theoy of Bessel Funcions, Cambidge Uni. Pess, Cambidge, 9.