Diffraction of Pulse Sound Signals on Elastic. Spheroidal Shell, Put in Plane Waveguide

Transcription

1 Adv. Stdies Teor. Pys., Vol. 7, 3, no. 5, HIKARI Ltd, ttp://dx.doi.org/.988/astp Diffraction of Plse Sond Signals on Elastic Speroidal Sell, Pt in Plane Wavegide A. A. Klescev Saint Petersbrg State Navy Tecnical University Rssia, 98, Saint Petersbrg, Lotsmanskaya st., 3 alexalex-@yandex.r Copyrigt 3 A. A. Klescev. Tis is an open access article distribted nder te Creative Commons Attribtion License, wic permits nrestricted se, distribtion, and reprodction in any medim, provided te original work is properly cited. Abstract. Wit te elp of te Forier transform and of te metod of te imaginary sorces and imaginary scatterers is solved te problem of te scattering of te plse sond signal by te elastic speroidal sell, pt in te plane wavegide. Keywords: diffraction, wavegide, plse, elastic speroidal sell, sorce, scatterer, imaginary.. Introdction At te basis of te metod of te imaginary sorces and imaginary scatterers is solved te problem of scattering of te plse signals of te elastic speroidal sell, accommodated in te plane wavegide wit te ideal bondary conditions. Te implse signals pt te energy, terefore tey are propagating wit te grop velocity, wic lie in te principles of te metod of te imaginary sorces and imaginary scatterers.. Te metod of te imaginary sorces and imaginary scatterers for te elastic speroidal sell, pt in te plane wavegide Te scattering of sond by te bodies, placed in te wavegide, are investigated in te papers [,, 8, 9, -6]. In te paper [] were calclated te

2 698 A. A. Klescev spectral caracteristics of te idial speroid, placed in te sond cannel, by te plse irradiation; in te papers [] and [8] wit te elp of te metod of te imaginary sorces and scatterers are fond te vertical distribtions of te scattered sond field of te ideal soft speroid, placed in te plane wavegide, at te irradiation is by te armonic signal. In te article [6] wit te elp of te Forier transform and caracteristics of te stationary continos) sond signal are calclated te plses, scattered by te ideal prolate speroid. Let's pt te elastic speroidal sell into te liqid layer wit te tickness H and te constant sond velocity. At te pper bondary of te wavegide is flfilled Diricle condition, at te lower bondary Neiman condition. Te axis of te rotation of te prolate speroidal sell will be orientated parallel to te bondaries of te wavegide and perpendiclar to te plane of te Figre. Te dimesions of te scatterer, distance from it to te bondaries and te tickness of te wavegide H are spposed to be sc tat we can do witot taking into consideration te scattering of te second order of te waves reflected from te bondaries of wavegide are not taken into accont in te frter process of te diffraction. Te centre of te scatterer is fixed at te distance z from te bottom, at te orizontal distance R from it and on te dept H z Fig. ) is placed te pointsorce Q of te implse sond signal. Using te metod of te imaginary sorces and scatterers [. 8], are fond te scattered plse signal in te point Q. Te sond plse signals were te two appearance: wit te armonic and freqencymodlated filling. Te spectrm S ) πν of te sond plse of te sorce Ψ i t) wit te armonic filling as te appearance [3] : iν n ν S πν ) ) sin π n ), ) π ν ν ) ν were: ν te freqency of te filling of te implse; n te nmber of te oscillation periods of te armonic signal in te plse; ν te circlar freqency. Te spectrm S ) πν is connected wit Ψ t) by te retrn Forier transform: i ) π ) Re πν )exp πν ) πν ) Ψ i t S i d ) Te spectrm of te reflected signal Ss πν ) is by te prodct of te spectrm S ) πν at te corresponding meanings of te anglar caracteristic of te scattering of te speroidal sell D,, ν ) and te anglar coordinates of te point of te observation). Let s consider te scatterer in te form of te elastic speroidal sell Fig.. ). All te po-tentials, inclding te plane wave potential, te scattered wave potential, te scalar sell potential, te Debye potentials U и V and te

3 Diffraction of plse sond signals 699 potential 3 of te gas, filling te sell, can be expanded in te speroidal wave fnctions [ 8 ]: Fig.. Te mtal disposition of te plse point-soirces and scatterers in te plane wavegide.

4 7 A. A. Klescev m m,n m,n m, n ; m n m n i ε S C, ) S C, ) R ) C, ) cos m 3 m,n m,n m, n ; m n m B S C, ) R ) C, ) cos m m,n l m,n m, n l m,n m, n l ; m n m ) ) S C, cos m [ C R C, ) D R ) C, )] 3 m,n m, n m,n ; m n m E R ) C, ) S C, ) cos m 3) U m n m S ) ) C, sin m [ F R C, ) G R ) C, )] m,n t m,n m, n t m,n m, n t ; V m n m S ) ) C, cos m [ H R C, ) I R ) C, )] m,n t m,n m, n t m,n m, n t ; were: C k ; C k, k is te wavenmber of te sond wave in te gas, filling t te sell; coefficients. B m, n, m, n C, D m, n, E m, n, F m, n, G m, n, H m, n, I m, n - are nknown expansion Fig.. Te elastic speroidal sell. Te expansion coefficients are determined from te pysical bondary conditions preset at te two srfaces of te sell and, see Fig. ) [ 8 ]:

5 Diffraction of plse sond signals 7 ) te continity of te normal displacement component at bot of te bondaries and ; ) te identity between te normal stress in te elastic sell and te sond pressre in te liqid ) or in te gas ); 3) te absence of te tangential stress at bot of te sell bondaries and. Te corresponding expressions for te bondary conditions ave te form [ 8 ] ) ) ) ) ) ) ) ) ) [ ] ; 3 ) ) [ ) ) )] ; k k µ λ λ [ ) ) ) )] ; k k 3 µ λ λ 4) ) ) ) ) ) ) ; O ) ) ) ) ) ). O Here λ and µ - are te Lame constants of te sell material; λ - is te blk compression coefficient of te liqid; λ - is te blk compression coefficient of te gas, filling te sell [ 8 ]: ) / ) ) [ / ) ) / ) )]; ) ) ) ) ) ) ) [ ] ; ) ) ) ) ) ) ) [ ]. Te sbstittion of te series 3) in te bondary conditions 4) yields te infinite

6 7 A. A. Klescev system of te eqations for te determining te desired coefficients. Becase of te ortogonality of te trigonometric fnctions cos mφ and sin mφ, te infinite system of te eqations breaks into infinite sbsystems wit fixed nmbers m. Eac of te sbsystems is solved by te trncations metod. Te nmber of te retained terms of te expansions 3) is te greater, te greater te wave size for te given potential Te anglar caracteristic of elastic speroidal sell D, ) ad calclated by te formla [ 8 ]: n, ) / ) ) m, n m, n, ) cos, m n m D ik i B S C m 5) were: C k te wave dimension, k te wave nmber in liqid, te alf focal distance; S m, n C, ) te normalized anglar speroidal fnction; ), ) 3) Rm, n C and R, ) m, n C te radial speroidal fnctions of te first and tird forms correspondingly; ε m m ), m ); te radial coordinate of te scatterer; te anglar coordinate of te sorce. Figre sows te relative backscattering cross sections σ for te different speroidal bo-dies ideal and elastic ones) and for te two angles of te irradiation θ and θ 9 ). Te Crve corresponds to te steel prolate speroidal sell, irradiated along te axis of te revol-tion Z θ ). Crve corresponds to te ideal soft speroid wit te same irradiation angle. Crve 3 represents te relative backscattering of te steel speroid, irradiated at te angle θ 9 tree dimensional problem). Crve 4 corresponds to te rigid prolate speroid:, 5, θ 9 ; crve 5 is for te soft speroid, 5, θ 9. All te crves, sown in Fig. 3 represent fnctions of te wave size of te scatterers C k.

7 Diffraction of plse sond signals 73 bodies. Fig. 3. Te relative backscattering cross sections of prolate speroidal Acknowledgements Tis worc spported as part of researc nder State Contract no P4 of April,, witin te Federal Target Program Scientific and scientific pedagogical personnel of innovative Rssia for te years Conclsions In te paper is sown te effectiveness of te metod of te imaginary sorces and imaginary scatterers for te plse seqence, got from speroidal body and based at te se of te grop velocity of te sond.

8 74 A. A. Klescev References. A. Bostrom, In col. Artic. / Ed. by Varadan V. K., Varadan V. V., Acostic, Electromagnetic and Elastic Wave Scattering Focs on te Matrix Approac, Pergamon press, New-York, 98. G. A. Grinblat, A. A. Klescev, Te scattering and te emission of te sond by te bodies, placed in te wavegide, J. Tecn. Acost., 994), A. A. Karcevic, Spektrm and Analysis, GITTL, Moska, A. A. Klescev, Debye and Debye Type Potentials in Diffractions, Radiation and Elastic Waves Propagation Problems, Acost. Pys., 58, ), A. A. Klescev, Diffraction, Radiation and Propagation of Elastic Waves, Profprint, S.-Petersbrg, A. A. Klescev, E. I. Kznetsova, Diffraction of Implse Sond Signals on Speroidal Body, Pt in Plane Wavegide, I.J.T.M.P.,, ), A. A. Klescev, E. I. Kznetsova, Interaction of Acostic Scatterers, Acost. Pys., 57, ), A. A. Klescev, Hydroacostic Scatterers, Prima, St.- Petersbrg,. 9. A. A. Klescev, I. I. Klkin, Te spectral caracteristics of te scattering of te sond by te body, placed in te sond cannel, Sov. Pys. Acost.,, 974), A. A. Klescev, L. S. Seiba, Te scattering of te sond wave by te ideal prolate speroids, Sov. Pys. Acost., 6, 97), A. A. Klescev, Scattering of Low Freqency Plsed Sond Signals from Elastic Cylindrical Sells, Acost. Pys., 57, ), A. A. Klescev, Te diffraction of te sond beam at te elastic speroidal sell, placed in te plane wavegide and interacting wit te bondaries, Proc. Symp. Interact of Acost. Waves wit Elast. Bodies, Tallinn, TGU 989), 3 6.

9 Diffraction of plse sond signals Y. A. Kravtsov, V. M. Kzkin, V. G. Petnikov, Te approximate approac to te problem of te diffraction of te waves in te wavegide wit te smootly altering parameters, News of ig. edc. inst. Radiopysics, 6, 983), Y. A. Kravtsov, V. M. Kzkin, V. G. Petnikov, Te diffraction of te waves at te reglar scatterers in te mltimode wavegides, Sov. Pys. Acost., 3, 984), S. O. Kvyatkovskiy, Te convergency of te metod of te T matrix and Rayleig ypo-tesis, News of ig. edc. inst. Radiopysics, 3, 987), S. O. Kvyatkovskiy, Te diffraction of te sond waves at te scatterer in te wavegide, Sov. Pys. Acost., 34, 988), Received: May 4, 3