# Introduction n the paper are investigated resonances of prolate and oblate spheroidal bodies (entire and in the form of shells) by the three-dimensional and axissymmetrical irradiation. By the three-dimensional irradiation for the solution of the problem of the diffraction are used Debye's potentials. To resonances of elastic spheroidal bodies are devoted publications [1 -9]. Debye first proposed expanding the vector potential A # ?? in the scalar potentials U andV in his publication [10] devoted to studying the behavior of light waves near the local point or local line. Later, this approach was used in solving diffraction problems for cases of the electromagnetic wave diffraction of a sphere, a circular disk and a paraboloid of a revolution [11 -16], as well as for the diffraction of longitudinal and transverse waves by spheroidal bodies [7,17]. As applied to problems based on the dynamic elasticity theory, the introduction of Debye's potentials occurs as follows.The displacement vector u ? of an elastic isotropic medium obeys the Lame equation: where ? and µ are Lame constans, ? is the density of the isotropic medium and ? is the circular frequency of harmonic vibrations. According to the Author : Saint -Petersburg State Navy Technical University, Russia, 190008, Saint -Petersburg, Lotsmanskaya St., 3. e-mail: alexalex -2@yandex.ru Helmholtz theorem, the displacement vector u ? is expressed through scalar ? and vector ? ??? potentials: u grad curl = ? ? + ? ? ?? ?(2) Substituting Eg. (2) in Eg. (1), we obtain two Helmholtz equations, which include one scalar equation for ? and one vector equation for ? ??? : 2 0, h ?? + ? = (3) 2 2 0. k ?? + ? = ?? ? ?? ?(4) Here 1 / h c ? = is the wavenumber of the longitudinal elastic wave, 1 c is the velocity of this wave, 2 2 / k c ? = is the wavenumber of the transverse elastic wave and 2 c is the velocity of the transverse wave. In the three-dimensional case, variables involved in scalar equation ( 3) can be separated into 11 coordinate systems. As for Eq. ( 4), in the threedimensional problem, this equation yields three independent equations for each of components of the vector function ? ??? in Cartesian coordinate system alone. To overcome this difficulty, one can use Debye's potentials U and V , which obey the Helmholtz scalar equation 2 2 0; V k V ? + = 2 2 0. U k U ? + = (5) Vector potential ? ??? (according to Debye) is expanded in potentials V and U as 2 ( ) ( ), curlcurl RU ik curl RV ? = + ?? ? ?? ??(6) where R ?? is the radius vector of a point of the elastic body or the elastic medium. Let us demonstrate the efficiency of using Debye's potentials in solving the three-dimensional diffraction problem for the case of diffraction by an elastic spheroidal shell. The advantage of the representation (6) becomes evident, if we take into account that potentials V and U obey the Helmholtz J 2 2 2 2 2 2 2 ( / ) ( / ) 2 (/ )( / )( / ) ( / ) ( / ) R R B R R B R B ? ? ? ? ? ? ? ? ? = ? ? ? ? + ? ? ? ? ? ? ? + ? ? ? ? + 2 2 2 2 2 2 ( / )( / ) ( / )( / ) , R B R B k B ? ? ? ? ? ? ? ? + ? ? ? ? + (7) 2 2 1 2 2 2 0 [ ( 1 )] [( / )( / )( / ) ( / )( / )( / ) h R B R B ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ? + ? ? ? ? ? ? + ? ? ? ? ? ? ? + 2 2 2 2 ( / )( / )( / ) ( / )( / )( / ) ( / )( / ) R B R B B R ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? + ? ? ? ? ? + 2 1 2 ( / )( / )] (sin ) ( / ), B R ik V ? ? ? ? ? ? ? ? ? ? ? + ? ? (8) 2 2 1/2 1 2 2 0 2 [ ( 1 ) sin ] [ / )( / ) ( / )( / ) h R B R B ik ? ? ? ? ? ? ? ? ? ? ? ? = ? + ? ? ? ? ? + ? ? ? ? ? ? × [( / )( / ) ( / )( / )], V V ? ? ? ? ? ? ? ? ? ? + ? ? ? ?(9) where: 2 2 1/2 0 ( 1 ) ; 1 1;1 . B h U ? ? ? ? = ? + ? ? ? + ? ? +? Spheroidal components of the function ( , , ) ? ? ? ? ? ? ? ?? ? are expressed as follows [7]: 2 2 1/2 2 2 1/2 0 0 ( / ) ( 1 ) ( / )( 1 ) ( / ), R h h h h ? ? ? ? ? ? ? ? ? ? ? ? ? =? ? + + ? ? + ? ? (10) 2 2 1/2 2 2 1/2 0 0 ( / ) ( 1 ) ( / )( 1 ) ( / ), R h h h h ? ? ? ? ? ? ? ? ? ? ? ? ? =? ? + + ? ? + ? ?(11) , ? ? ? ? ?(12) where: 2 2 1/2 2 1/2 0 ( ) ( 1) ; h h ? ? ? ? = ? ? 2 2 1/2 2 1/2 ( )(1 ) . h ? ? ? ? =? ? Let us consider in the form of an isotropic elastic spheroidal shell (Fig. 1). All potentials, including the plane wave potential 0 , ? the scattered wave potential 1 , ? the scalar shell poten-tial 2 , ? Debye's potentials U and V and potential 3 ? of the gas filling the shell, can be ex-panded in spheroidal functions: (1) , , 0 1 0 1 , 1 0 2 ( , ) ( , ) ( , )co n m n m n m m n m n m i S C S C R C m ? ? ? ? ? ? ? ? = ? ? = ? ? (13) (3) , 1 , 1 , 1 0 2 ( , ) ( , )co m n m n m n m n m B S C R C m ? ? ? ? ? = ? ? = ? ? (14)(1) (2) , 2 , , , , 0 2 [( , ) ( , )] ( , )co m n m n m n l m n m n l l m n m C R C D R C S C m ? ? ? ? ? ? = ? ? = + ? ? (15) (1) , 3 , , 2 2 0 2 ( , ) ( , )co m n m n m n m n m E R C S C m ? ? ? ? ? = ? ? = ? ? (16) (1)(2) , , , , , 1 2 [( , ) ( , )] ( , )sin ; m n m n m n t m n m n t t m n m U F R C G R C S C m ? ? ? ? ? ? = ? = + ?? (17) (1) (2) , , , , , 0 2 [ ( , ) ( , )] ( , )co m n m n m n t m n m n t t m n m V H R C I R C S C m ? ? ? ? ? ? = ? = + ? ?(18) where: , 1 # ( , ) m n S C ? ? the angular spheroidal function; (1) , 1 ( , ), m n R C ? (2) , 1 ( , ) m n R C ? and(3) , 1 ( , ) m n R C ? ? radial spheroidal functions of first, second and third genders; 0 ; l C hh = 2 0 ; t C k h = 1 0 , C kh = k ? is the wavenumber of the sound wave in the liquid; 2 1 0 , C k h = 1 k ? is the wavenumber of the sound wave in the gas filling the shell; o h ? the half -focal distance; , , ,, , , , , , ,m n m n m n m n m n B C D E F , , , , , The corresponding expressions for boundary conditions have the form [7]: m n m n m n G H I ? are unknown expansion coefficients.0 1 1 1 0 1 2 ( ) ( / )( ) ( ) ( / ) ( ) [( / )( ) ( / )( )] ; h h h h h h ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ? ? + ? = ?? ? + ? ? ? ? ? ? ? (19) 1 1 1 1 1 2 ( ) ( / ) ( ) ( / ) ( ) [( / )( ) ( / )( )] ; h h h h h h ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ?? ? = ?? ? + ? ? ? ? ? ? ? (20) 0 2 2 1 1 0 0 1 2 ( ) 2 [( ) ( / ) ( ) ( / )] ; k h h h h u h u ? ? ? ? ? ? ? ? ? ? µ ? ? ? ? = ? ? + ? = ? ? + ? ? + ? ? (21) 1 2 2 1 1 1 1 3 2 2 [( ) ( / ) ( ) ( / )] ; k h h h h u h u ? ? ? ? ? ? ? ? ? ? µ ? ? ? ? = ? ? =? ? + ? ? + ? ? (22) 0 1 ; 0 ( / )( / )( / ) ( / )( / )( / ) ; h h u h h h u h ? ? ? ? ? ? ? ? ? ? ? ? ? ? = = = ? ? + ? ? (23)0 1 ; 0 ( / )( / )( / ) ( / )( / )( / ) , h h u h h h u h ? ? ? ? ? ? ? ? ? ? ? ? ? ? = = = ? ? + ? ?(24) where: 2 1/2 2 1/2 0 ( 1) (1 ) ; h h ? ? ? = ? ? 0 ? ? is the bulk compression coefficient of the liquid; 1 ? ? is the bulk compression coefficient of the gas filling the shell; 1 1 2 ( ) ( / ) ( ) [( / )( ) ( / )( )]; u h h h h h ? ? ? ? ? ? ? ? ? ? ? ? ? = ?? ? + ? ? ? ? ? ? ? 1 1 2 ( ) ( / ) ( ) [( / )( ) ( / )( )]; u h h h h h ? ? ? ? ? ? ? ? ? ? ? ? ? = ?? ? + ? ? ? ? ? ? ? 1 1 2 ( ) ( / ) ( ) [( / )( ) ( / )( )]. u h h h h h ? ? ? ? ? ? ? ? ? ? ? ? ? = ?? ? + ? ? ? ? ? ? ? The substitution of series ( 13) -(18) in boundary conditions ( 19) -( 24) yields an infinite system of equations for the determining of desired coefficients. Because of the ortogonality of trigonometric functions cos m? and sin m? , the infinite system of equations breaks into infinite subsystems with fixed numbers m Each of subsystems is solved by the truncation method. The number of retained terms of expansions ( 13) -( 18) is the greater the wave size for the given potential.The solution of the axissymmetrical problem of the diffraction at elastic spheroidal bodies was presented in [ # ? = The continuous elastic sphe-roid over the its conduct is very near to the ideal hard scatterer. This was seen by the compare-son of angular characteristics # ? = A relative section r ? of an elastic spheroid shows a r5esonance of a coincidence as this was and in a relative backscattering 0 ? (see Fig. 6), but a point of a maximum was by ). m A section of a radiation rad ? has an extremums in those points, what and a relative section of a scattering . r ? A comparison of curves 2 and 3 presented on a Fig. 10 with curve 1 of a Fig. 7 shows, what a relative backscattering section does not give sometimes of a full information about a resonant properties of elastic scattering. # IV. # Conclusions With the help of the numerical experiment are found low frequency resonances of elastic spheroidal bodies (entire and in the form of shells) both prolate and oblate by the three -dimensional and axissymmetrical irradiation. 2![scalar equation. It is convenient to represent components of ? ??? in the spherical coordinate system by expressing them through , U V and R ?? and then, using formulas of the vector analysis, to change to spheroidal components. The expressions for spherical I components of the vector function ( , , ) R ? ? ? ? ? ? ?? ? in terms of Debye's potentials have the form [7]:](image-2.png "2 Global") 1![Figure 1 : Elastic spheroidal shell in a plane harmonic wave field Expansion coefficients are determined from physical boundary conditions preset at two surfaces of the shell 0 (? and 1 , ? see Fig. 1) [7]:1. the continuity of the normal displacement component at both of the boundaries 0 ? and 1 ; ? 2. the identity between the normal stress in the elastic shell and the sound pressure in the liquid 0 ( )? or in](image-3.png "Figure 1 :") 2![Figure 2 : Moduluses of angular characteristics of scattering of spheroidal scatterers Same angular distributions, but by 1 3,1 C = (the elastic shell, 1 3, 0 C = ? for ideal sphe-roids) and 1 10, 0 C = according are presented at Fig. 3 and 4. Notations of curves at all three Fig. identical. The analysis of presented results shows, what by the angle of the irradiation 0 0 0 ? =](image-4.png "Figure 2 :") 32![Figure 3 : Moduluses of angular characteristics Figure 4 : Moduluses of angular chararacteristics of spheroidal scatterers of spheroidal scatterers](image-5.png "Figure 3 : 2 Global") 56272![Figure 5 : Relative backscattering cross sections of prolate spheroids](image-6.png "DFigure 5 :Figure 6 : 2 GlobalFigure 7 : 2 Global") ![? ? A full scattering cross section ?[7] is determined through a square of a modulus of a angular characteristic of a sound scattering ( , ) D ? ? :](image-7.png "D") ![is an area of a geometrical shade of a scatterer.With a help of an optical theorem a scattering cross section ? cab be found through a meaning of an imaginaty part of of an angular characteristic in a direction of a falling wave (a scattering "forward")At an analogy with the scattering cross section ? can introduce an idea of a section rad ? of an elastic or liquid body under an action of a point source[7]:](image-8.png "A") ![? ? is an angular characteristic of a sound radiation of a body under an action of a point source.At a basis of presented formulas was made an account of full ? and relative r ? scattering cross sections and a radiation cross section rad ? of spheroidal (prolate and oblate) bodies. On a Fig.9are presented relative sections of a scattering r ? of an ideal hard oblate spheroid (curve 1), of a steel oblate spheroid (curve 2) and of an ideal soft oblate spheroid (xcurve 3). In all three ca-ses a relation of a semiaxises](image-9.png "F") ![On a Fig. 10 are presented relative sections of a sections of a scattering r ? (curves 1 and 2) and a section of a radiation rad ? (curve 3) of prolate spheroidal bodies. A curve 1](image-10.png "") © 2013 Global Journals Inc. (US) ## Acknowledgments * Characteristics of sound scattering by elastic bodies of spheroidal form.// Tr. LKI., Ship ac AAKleshchev 1978 43 * Scattering of sound by elastic oblate spheroidal shell. // Acousical journal AAKleshchev 1975 21 938 * Resonance Response of Submerged Acousticaly Excited Thick and Thin Shells GCGaunard MFWerby // J.A.S.A., 77 1985 * Numerical study of material propertics of submerged objects using resonance response MFWerby GJTango // J.A.S.A. 1986 79 1260 * Correspondence between acoustical scattering from spherical and -on incident spheroidal shells MFWerby LHGreen // J.A.S.A. 1987 81 783 * Classification of resonances in the scattering from submerged spheroidal shells insonified at arbitrary angles of incidence MFWerby GCGaunaurd // J.A.S.A. 1987 82 1369 * Hydroacoustic Scatterers. First publication AAKleshchev 1992. 2012 St. Petersburg, Shipbuilding; St. Petersburg, Prima second publication. in Russia * Relative scattering cross sections Figure 10 : Relative scattering cross sections and of oblate spheroids the section of radiating of prolate spheroidal bodies 8. A. A. Kleshchev. With Reference to Low Frequency Resonances of Elastic Spheroidal Bodies J.Techn.Ac 9 1995 * About Low Frequency Resonances of Elastic Spheroidal Shells, Irradiating and Establishing by Harmonic and Pulse Signals AAKleshchev EIKuznetsova 2013 3 4 * Das Verhalten von Lichtwellen in der Nahe eines Brennpunktes oder Brennlinie PDebye Ann. Phys 30 755 1909 * Differencial and Integral Equations of Mechanics and Phisics PFrank RMises 1961 New York, Dover * Theorie der Beugung elektromagnetischer Wellen WFranc 1957 Springer Berlin-Gottingen-Heidelberg * Strenge Theorie der Beugung elektromagnetischer Wellen an der vollkommen leitenden Krescheibe. // Zs. Naturforsch JMeixner 1948 501 * Theory of Diffraction by a Paraboloid of Revolution VAFock Diffraction of Electromagnetic Waves by Some Bodies of Revolution. Moscow, Sov Radio 1957 in Russia * Electromagnetic Diffraction and Propagation Problems VAFock 1965 Oxford, Pergamon * AHoenl KMaue Westpfahl Theorie der Beugung Berlin, Springer 1961 * Diffraction of waves in elastic medium at elastic spheroid. // Tr. LKI TLGutman AAKleshchev 1974 91 31