ΜΕΘΟ ΟΣ ΣΥΖΕΥΓΜΕΝΩΝ Ι ΙΟΜΟΡΦΩΝ ΓΙΑ ΤΗΝ ΕΠΙΛΥΣΗ ΠΡΟΒΛΗΜΑΤΩΝ ΙΑ ΟΣΗΣ ΗΧΗΤΙΚΩΝ ΚΥΜΑΤΩΝ ΣΕ ΘΑΛΑΣΣΙΟ ΙΑΣΤΡΩΜΑΤΩΜΕΝΟ ΠΕΡΙΒΑΛΛΟΝ.

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1 Ακουστική AcP4 ΜΕΘΟ ΟΣ ΣΥΖΕΥΓΜΕΝΩΝ Ι ΙΟΜΟΡΦΩΝ ΓΙΑ ΤΗΝ ΕΠΙΛΥΣΗ ΠΡΟΒΛΗΜΑΤΩΝ ΙΑ ΟΣΗΣ ΗΧΗΤΙΚΩΝ ΚΥΜΑΤΩΝ ΣΕ ΘΑΛΑΣΣΙΟ ΙΑΣΤΡΩΜΑΤΩΜΕΝΟ ΠΕΡΙΒΑΛΛΟΝ. ΣΥΓΚΡΙΣΗ ΑΠΟΤΕΛΕΣΜΑΤΩΝ ΚΑΙ ΑΞΙΟΛΟΓΗΣΗ ΜΕ ΜΕΘΟ Ο ΠΕΠΕΡΑΜΕΝΩΝ ΣΤΟΙΧΕΙΩΝ Γ. A. Αθανασούλης Κ. Α. Μπελιµπασάκης Σχολή Ναυπηγών Μηχ/γων Μηχ Σχολή Ναυπηγών Μηχ/γων Μηχ Εθνικό Μετσόβιο Πολυτεχνείο Εθνικό Μετσόβιο Πολυτεχνείο Πολυτεχνειούπολη Ζωγράφου Πολυτεχνειούπολη Ζωγράφου Αθήνα, 5773 Αθήνα, ΠΕΡΙΛΗΨΗ Θεωρούµε το πρόβληµα διάδοσης-σκέδασης ηχητικών κυµάτων σε θαλάσσιο διαστρωµατωµένο περιβάλλον που εκπέµπονται από αρµονική σηµειακή πηγή. Για την επίλυση του προβλήµατος εφαρµόζεται νέα µέθοδος συζευγµένων ιδιοµορφών, η οποία παράγεται από µεταβολική αρχή σε συνδυασµό µε κατάλληλη αναπαράσταση του πεδίου από σειρά τοπικών ιδιοµορφών, που έχει την ιδιότητα να ικανοποιεί µε ακρίβεια τις συνθήκες συναρµογής στις µη-οριζόντιες διεπιφάνειες και να συγκλίνει γρήγορα στην α- κριβή λύση. Παρουσιάζουµε αποτελέσµατα σε διάφορα παραδείγµατα υπολογισµών σε χαµηλές συχνότητες σε σύγκριση µε µέθοδο πεπερασµένων στοιχείων, από όπου διαφαίνεται η ακρίβεια και η αποτελεσµατικότητα της παρούσας µεθόδου. A COUPLED-MODE THEORY OR UDERWATER SOUD PROPAGATO A STRATED EVROMET. COMPARSO O RESULTS AD VALDATO VS. A TE ELEMET METHOD G. A. Athaassoulis Κ. Α. Belibassakis School of aval Arch. ad Marie Egg School of aval Arch. ad Marie Egg atioal Techical Uiversity of Athes atioal Techical Uiversity of Athes Zografos, Athes, 5773, Greece Zografos, Athes, 5773, Greece matha@cetral.tua.gr kbel@fluid.mech.tua.gr

2 Helleic stitute of Acoustics (HELA) Acoustics ABSTRACT We cosider uderwater acoustic wave propagatio ad scatterig i a axially symmetric cylidrical waveguide, cosistig of several fluid layers of variable thickess overlyig a impeetrable bottom. The problem is reformulated as a trasmissio problem by decomposig the domai ito three subdomais: the rageidepedet "ear" ad "far" parts, ad the rage-depedet (itermediate) part cotaiig the medium ad bottom irregularity. The pressure field i the rageidepedet subdomai, is expressed i terms of stadard ormal-mode series expasios. the itermediate subdomai a variatioal priciple is applied to the trasmissio problem, i cojuctio with a ehaced local-mode represetatio of the acoustic-pressure field, resultig i a cosistet coupled-mode system of equatios. This system cotais additioal equatios, associated with the additioal slopig-iterface modes, ad produces solutios cosistet with the iterface coditios ad the coservatio of eergy. umerical results are preseted i compariso with geeral EM solvers demostratig the efficiecy of the preset method.. troductio the preset work, a cosistet coupled-mode model, developed by the authors [], is used to solve the problem of uderwater acoustic wave propagatio ad scatterig i a multi-layered stratified acoustic eviromet, characterised by a peetrable bottom ad a umber of iterfaces of geeral shape, separatig layers with differet acoustic properties. The complete uderwater acoustic b.v.p. is reformulated as a trasmissio problem by decomposig the domai ito three subdomais: the rageidepedet "ear" ad "far" parts, ad the rage-depedet (itermediate) part cotaiig the bottom ad medium irregularity; see ig.. The pressure field i the two rageidepedet subdomais, is expressed i terms of stadard ormal-mode series represetatios. the itermediate subdomai, a variatioal priciple is applied to the trasmissio problem, i cojuctio with the ehaced local-mode represetatio of the acoustic pressure, resultig i a ew, cosistet, coupled-mode system of equatios. This system cotais oe additioal equatio, associated with each slopig-iterface mode, ad produces solutios cosistet with the slopig-iterface coditio ad the coservatio of eergy. umerical results are preseted for a two-layer sea eviromet, i the case of a steep upslope i shallow water, ad are compared with results obtaied by a geeral fiite elemet (EM) solver, Kampais ad Dougalis [], Dougalis et al [3], demostratig the applicability of the preset approach.. Differetial formulatio of the problem We cosider the rage-depedet, cylidrically symmetric marie eviromet show i ig.. or simplicity, we cosider two fluid layers, water of costat desity ρ ad sedimet of costat desity ρ > ρ, separated by the iterface J :z= h( r) ad overlyig a perfectly rigid horizotal boudary at z = H. We let c = c( r,z) be the speed of soud (discotiuous at the iterface) ad suppose that i the ear regio D ( r r ) ad i the far regio D ( r r ) the acoustic ad geometric parameters

3 Ελληνικό Ινστιτούτο Ακουστικής (ΕΛΙΝΑ) Ακουστική z r * D h h( r ) D h J D z = H r r r = r = igure. Domai decompositio ad otatio. The poit source is deoted by (*). < < ). The acoustic propagatio ad scatterig boudary-value problem i the domai < r, H z p = p r,z satisfyig are rage idepedet. (Thus, c ad h vary with r oly i D ( r r r ), is to determie a complex-valued fuctio ( ) δ ( r) p p + k ( r,z) p = δ ( z z ), p( r, ), ( r, H) π the iterface coditios r = =, (),(),(3) z + + ( ) = ( ) ( r, h(r) ) ( r, h(r ) ) p r, h(r) p r, h(r), ad the radiatio coditio, p p =, (4),(5) ρ ρ p( r,z ) ~ outgoig cylidrical waves, as r. (6) π f () we have itroduced the wave umber k = k(r,z) = ; i (5) c ormal derivative to the iterface z = h( r ). deotes the 3. Differetial formulatio of the problem The problem ()-(6) ca be reformulated as a trasmissio problem i the D with the aid of the followig geeral (ormal-mode) represe- bouded subdomai tatios of the acoustic field i D ad i ( ) p Z z Z z H k r 4ρ D, respectively, = ( ) ( ) ( ) ( ) ( ) = ( ) = ( ) ( ) + C Z z J k r, (7) = p = C Z z H k r, (8)

4 Helleic stitute of Acoustics (HELA) Acoustics ad by requirig the matchig of the field ad its ormal derivative at the commo vertical iterfaces r = r ad r r k ad { k } = =. formulas (7,8), the sets of umbers { } = { Z z } ad =, { Z ( )},.. z =,,.., ad the sets of fuctios of ( ),,.., are the eigevalues ad eigefuctios, respectively, of Sturm-Liouville problems, obtaied by sepa-,,.. ratio of variables i the subdomais D ad D. More details about the associated depth problem, ad its solutio i the case of two homogeeous layers: ρ z h ρ c r, z h c ρ H z h ρ c r, H < z < h = ( < < ) =, ( < < ) =, ( < < ) =, ( ) = c, ca be foud i [4]. The trasmissio problem admits a variatioal formulatio, expressed by the statioarity of a fuctioal of the form (see Ref. []), ( p,c { } { } ), C. (9) The variatioal priciple, δ =, ca the be used to obtai a alterative, semidiscrete (Katorovich) formulatio of the problem i terms of local modes. This family of local basis fuctios is obtaied by formulatig ad solvig local, vertical Sturm- Liouville problems i the iterval [ H, ]. The ehaced local-mode represetatio of the acoustic field p ( r,z ) i the variable-bathymetry/iterface domai D, developed i [], reads as follows where ( ) p r,z P r Z z;r P r Z z;r = ( ) ( ) ( ) ( ) ( ) = +, () Z z;r,, are obtaied as the eigefuctios of the followig local, vertical eigevalue problem (defied for each r < r < r ) : Z ( z;r) Z + ( k ( r,z) k (r)) Z( z;r ) =, H z, Z ( ;r ) =, ( H;r ) =, () z z i cojuctio with the matchig-iterface coditios + Z + Z Z( h( r ) ;r) = Z( h( r ) ;r), ( h( r ) ;r) = ( h( r ) ;r). () ρ z ρ z P r deote the amplitudes of the modes, ad the fuctios ( ) However, the local eigefuctios Z ( z;r ),, are icompatible with the slopig iterface coditio (5), wheever dh( r) dr. To remedy this icosistecy a additioal mode is itroduced i [], deoted by P ( r) Z ( z;r ) ad called the slopigiterface mode. The vertical structure of the slopig-iterface mode, Z ( z;r ), is a cotiuous fuctio satisfyig the followig coditios dz ( H ) Z ( r ) =, =, dz + Z + Z Z( h( r ) ;r) = Z( h( r ) ;r), ( h( r ) ;r) ( h( r ) ;r) =. ρ z ρ z the series expasio (), the first P r Z z;r < terms { ( ) ( )} =,,.. (3), correspodig to real horizotal eigevalues ( k > ), are the propagatig modes, ad the

5 Ελληνικό Ινστιτούτο Ακουστικής (ΕΛΙΝΑ) Ακουστική terms { P( r) Z( z;r )}, = +, +,.., correspodig to imagiary eigevalues ( k < ), are the evaescet modes. The slopig-iterface mode P ( r) Z ( z;r ) is ot eeded whe the iterface is flat. Each term i the expasio () satisfies the free surface coditio (), the boudary coditio (3) ad the iterface coditio (4), idividually. Thus, represetatio () reders all of them essetial coditios i relatio with the variatioal formulatio. Usig () i the variatioal priciple, we obtai the followig coupled-mode system of secod-order ordiary differetial equatios, with respect to the mode amplitudes (the Cosistet Coupled-Mode System): d P( r) dp( r) am ( r) + b m ( r) + cm ( r) P ( r ) =, m =,, 3,..., (4) = dr dr where all coefficiets are defied i terms of Z ( z;r ) i r < r < r. The system (4) cotais a additioal equatio, associated with the additioal slopig-iterface mode, ad produces solutios cosistet with the iterface coditios ad the coservatio of eergy. Eq. (4) is supplemeted by the followig ed coditios P r P r, P r = P r =, ( ) = ( ) = ( ) ( ) ( ) + ( ) = P ( r ) D P ( r ) P r A P r B, + = =, 3,..., (5) where the coefficiets A,B,D are defied i terms of the acoustic parameters at the edpoits ( r = r,r = r), ad ca be foud i Ref. []. 4. umerical results ad coclusios umerical results are preseted for the waveguide show i igs. ad 3, which models a smooth but steep upslope, i shallow water. The results cocer the calculated Trasmissio Loss (TL i db), as obtaied by the preset method (CCMM) ad by a geeral fiite elemet (EM) solver, based o a stadard Galerki/P discretizatio of the b.v.p., coupled with a exact, olocal absorbig boudary coditio at the exterior boudary of the waveguide, Kampais ad Dougalis [], Dougalis et al [3]. The source frequecy is 5Hz. the first case, preseted i ig., the pulsatig source is located at z = 5m (ear the free surface). The desity ad the soud speed of the seawater are take costat: ρ =, c = 5 m/s, ad the 3 gr/cm 3 desity ad soud speed of the sea bottom are: ρ =. 5gr/ cm, c = 7 m/s. this case, the umber of propagatig modes i D is =3. i ig.. the secod case, preseted i ig. 3, the source is located at z = 7m, i.e. very ear the bottom iterface, which i the ear regio lies i 75m depth. The desity ad soud speed of the seawater are also costat, with the same, as i the previous case, values. We ca observe from these figures that the agreemet betwee the two methods is excellet, i the whole domai, although the computatioal requiremets of the EM as compared to the preset method are oe order of magitude larger. O the other had, the EM, is iheretly more flexible to treat localized ihomogeeities. Thus, after further compariso ad validatio, both methods ca be used to complemet each other, i order to treat difficult situatios, such as acoustic scatterig problems from localized scatterers embedded i o-homogeeous waveguides.

6 Helleic stitute of Acoustics (HELA) Acoustics Refereces [] Athaassoulis G.A., Belibassakis K.A.,, A cosistet coupled-mode theory for uderwater soud propagatio i a geeral, stratified acoustic eviromet, i Proc. 6 th Europea Coferece o Uderwater Acoustics, ECUA, Gdask, Polad. [] Kampais.A., Dougalis V.A., 999, A fiite elemet code for the umerical solutio of the Helmholtz equatio i axially symmetric waveguides with iterfaces, J. Comp. Acoustics 7, 83-. [3] Dougalis V., Kampais., Mitsoudis D., A fiite elemet method for the approximatio of uderwater soud propagatio i geeral stratified eviromets, Proc. of Cof. Acoustics, HELA, Patras, Greece. [4] Boyles C.A., 984, Acoustic waveguides, Applicatios to Oceaic Sciece, Wiley, ew York. igure. (a) Compariso of EM ad CCEM i the case of a upslope eviromet. (b)trasmissio Loss (i db) at SD=RD=5m ad at RD=7m. igure 3. (a) Compariso of EM ad CCEM i the case of a upslope eviromet. (b)trasmissio Loss (i db) at RD=5m ad at SD=RD=7m.