# 2017 • no. 3

# TABLE OF CONTENTS

## Mechanics

**Agayan K.L., Atoyan L.H., Kalinchuk V.V., Sahakyan S.L.**

Electro-magneto-elastic surface waves existence and propagation in a piezoelectric layered medium in the presence of an electric screen

**Babeshko V.A., Evdokimova O.V., Babeshko O.M., Gladskoy I.B., Gorshkova E.M., Zaretskaya M.V., Mukhin A.S.**

Discontinuous solutions of mixed problems and block elements

**Vatulyan A.O., Plotnikov D.K.**

On indentation of heterogeneous strip

**Evdokimov A.A.**

Allocation and movement of roots of the Lamb wave dispersion equation in complex plane

**Miakisheva O.A.**

The interaction of source-generated spherical waves with an elastic plate immersed in acoustic fluid

**Rubtsov S.E., Pavlova A.V.**

Cellular automata modeling of migration and gravitational sedimentation of impurity in a liquid flow

**Uglich P.S.**

On the residual stress definition in the round plate

## Physics

**Vekshin M.M., Nikitin V.A., Yakovenko N.A.**

Synthesis of optical waveguide with complicated cross-section shape and 3D waveguide structures in glass

**Kochergin V.S., Kochergin S.V.**

Variational algorithms for identifying power source of impulse point pollution

**Maniliuk Yu.V., Fomin V.V.**

Seiche oscillations in a partially enclosed basin

**Onishchuk S.A., Tumaev E.N.**

To the question of the formation of profiled silicon crystals at its growth by Stepanov’s method

## ABSTRACTS

*Subjects*: mechanics

**Agayan K.L.**^{1}, **Atoyan L.H.**^{1}, **Kalinchuk V.V.**^{2}, **Sahakyan S.L.**^{3}

**Electro-magneto-elastic surface waves existence and propagation in a piezoelectric layered medium in the presence of an electric screen**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2017, no. 3, pp. 5-14.

The problems of existence and behavior of electro-magneto-elastic surface waves in a structure consisting of a piezoelectric substrate, a piezoelectric layer and an adjoining dielectric medium on the top in the presence of an electric (or magnetic) screen are considered. The dispersion equation is derived and investigated.

*Keywords*: electro-magneto-elastic wave, layered piezoelectric structure, dynamic Maxwell’s equations.

^{1}Institute of Mechanics of National Academy of Sciences of the Republic of Armenia, Yerevan, 0019, Armenia

^{2}Southern Scientific Centre of Russian Academy of Science, Rostov-on-Don, 344006, Russia

^{3}Yerevan State University, Yerevan, 0025, Armenia

Corresponding author’s e-mail: kalin@ssc-ras.ru

- Mindlin R.D. A variation principle for the equations of piezoelectromagnetism in a compound medium. In:
*Complex variable analysis and its applications*. Moscow, 1978, pp. 397-400. - Yang J.S. Bleustein-Gulyaev waves in piezoelectromagnetic materials.
*Int. J. of Applied Electromagnetics and Mechanics*, 2000, iss. 12, pp. 235-240. - Li S. The electro-magneto-acoustic surface waves in a piezoelectric medium: the Bleustein-Gulyaev mode.
*J. Appl. Phys.*, 1996, vol. 80, iss. 9, pp. 5264-5269. - Bleustein J.L. A new surface wave in piezoelectric materials.
*Appl. Phys. Lett.*, 1968, iss. 13, pp. 412-414. - Gulyaev Y.V., Electroacoustic surface waves in solids.
*Sov. Phys. JEPT Lett.*, 1969, iss. 9, pp. 37-38. - Belubekyan M.V. Screened surface shear wave in piezoelectric half space of hexagonal symmetry. In:
*Proc. of 6*. Goris-Stepanakert, 2008, 21-26 Sept., pp. 125-130.^{th}Inter. Conf. “The problems of dynamics of interaction of deformable media” - Belyankova T.I., Kalinchuk V.V. Equations of dynamics of a prestressed magneto-electro-elastic medium. In:
*Proc. of SSC RAS Mechanics of solids*, 2016, no. 5, pp. 101-110. - Bagdasaryan, G.E., Danoyan, Z. N. Electro-magneto-elastic waves. Yerevan, 2006, 401 p.
- Kalinchuk V.V., Belyankova T.I., Levi M.O., Agayan, K.L. Some features of the dynamics of a weakly inhomogeneous magneto-elastic half space.
*Messenger of SSC RAS*, 2013, vol. 9, no. 4, pp. 13-17. - Campbell C.F., Weber R.J. Calculation of radiated electromagnetic power from bulk acoustic wave resonators. In:
*IEEE Int. Fr. Contr. Symp.*, 1993, pp. 451-458. - Mindlin R.D. Electromagnetic radiation from a vibrating quartz plate.
*Int. J. Solid. Struct.*, 1973, vol. 9, pp. 697-702. - Curtis R.G., Redwood M. Transverse surface waves on a piezoelectric material carrying a metal layer of finite thickness.
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*Linear piezoelectric plate vibrations*. Plenum, New York, 1969, 210 p. - Danoyan Z.N., Atoyan L.H., Sahakyan S.L. Electroelastic surface Love wave in a layered piezoelectric-dielectric structure in the presence of electric screen. In:
*Proc. of National Academy of Sciences of Armenia “Mechanics”*, 2013, vol. 66, iss. 2, pp. 25-33. (In Russian) - Belubekyan M.V., Ghazaryan K.B. To the question of existence of shear waves in a homogeneous elastic half space. In:
*Proceed. of National Academy of Sciences of Armenia “Mechanics”*, 2000, vol. 53, iss. 1, pp. 6-12. (In Russian) - Balakirev M.K., Gilinskii I.A.
*Waves in piezocrystals*, Novosibirsk, 1982, 239 p. - Avetisyan A.S., Karapetyan M.E. Screening of Gulyaev-Bleustein waves. In:
*Optimal control, stability and firmness of mechanical systems*. Yerevan, Armenia, Edition of YSU, 2002, pp. 97-101. (In Russian) - Yang J.S. Love waves in piezoelectromagnetic materials.
*Acta Mech.*, 2004, vol. 168, pp. 111-117. - Danoyan Z.N., Pliposian G.T. Surface electro-elastic Love waves in a
layered structure with a piezoelectric substrate and a dielectric layer.
*Int. J. of Solids and Structures*, 2007, vol. 44, pp. 5829-5847. - Yang J.S. Piezoelectromagnetic waves in a ceramic plate.
*IEEE Transact. On Ultrason., Ferroelectr., and Frequen. Control*, 2004, vol. 51, no. 8, pp. 1035-1039. - Lee P.C. A variational principle for the equations of
piezoelectromagnetism in elastic dielectric crystals.
*J. of Appl. Phys.*, 1991, vol. 69, pp. 7470-7473. - Royer D., Dieulesaint E.
*Elastic waves in solids*, Berlin-Heidelderg, 2000, 374 p.

*Subjects*: mechanics

**Babeshko V.A.**^{1}, **Evdokimova O.V.**^{1}, **Babeshko O.M.**^{2}, **Gladskoy I.B.**^{2}, **Gorshkova E.M.**^{2}, **Zaretskaya M.V.**^{2}, **Mukhin A.S.**^{2}

**Discontinuous solutions of mixed problems and block elements**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2017, no. 3, pp. 15-21.

A number of mixed boundary-value problems in the theory of elasticity are not traditional in the sense that unstable states of the system arise, leading to destruction. These include mixed problems with discontinuous boundary conditions, in which the behavior of contact stresses and displacements, which indicate the destruction of the medium, appear. In some cases, such boundary problems have unlimited energy. Examples of such mixed boundary problems are the contact problems for two rigid dies, which have approached the contact state but have not merged into one stamp, as well as two adjacent cracks, which disappear with a small distance between them. It is shown that such problems, arising in seismology, theory of strength, construction, have singular components, in some cases with unlimited energy, and can be solved by topological methods with pointwise convergence, in particular, by the block element method. Numerical methods based on the application of the energy integral to such problems are not applicable in connection with its divergence. In the case of cracks, taking into account the investigation of the properties of solutions of integral equations obtained earlier, it is proved that a set of cracks lying in the same plane whose vertices are removed by some distance will uncontrollably merge with logarithmic growth when the distance between the vertices reaches a certain minimum.

*Keywords*: block element, topology, integral and differential factorization methods, exterior forms, block structures, boundary problems, starting earthquakes.

^{1}Southern Scientific Center RAS, Rostov-on-Don, 344006, Russia

^{2}Kuban State University, Krasnodar, 350040, Russia

Corresponding author’s e-mail: babeshko41@mail.ru

- Vorovich I.I., Aleksandrov V.M., Babeshko V.A.
*Neklassicheskie smeshannye zadachi teorii uprugosti*[Nonclassical mixed problems in the theory of elasticity]. Moscow, Nauka Pub., 1974, 456 p. (In Russian) - Vorovich I.I., Babeshko V.A.
*Dinamicheskie smeshannye zadachi teorii uprugosti dlya neklassicheskikh oblastey*[Dynamic mixed problems of elasticity theory for nonclassical domains]. Moscow, Nauka Publ., 1979, 320 p. (In Russian) - Vorovich I.I., Babeshko V.A., Pryakhina O.D.
*Dinamika massivnykh tel i rezonansnye yavleniya v deformiruemykh sredakh*[Dynamics of massive bodies and resonant phenomena in deformable media]. Moscow, Nauka Pub., 1999, 246 p. (In Russian) - Kreyn M.G. Integral’nye uravneniya na polupryamoy s yadrom, zavisyashchim ot raznosti argumentov [Integral equations on the half-line with a kernel depending on the difference of the arguments].
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*Über eine Klasse singuläre Integralgleichungen*. S. B. Preuss. Acad. Wiss, 1932, pp. 696-706. - Nobl B.
*Metod Vinera-Khopfa*[The Wiener-Hopf method]. Moscow, Inostrannaya literatura Pub., 1962, 280 p. (In Russian) - Muskhelishvili N.I.
*Singulyarnye integral’nye uravneniya*[Singular integral equations]. Moscow, Nauka Pub., 1962, 600 p. (In Russian) - Gakhov F.D.
*Kraevye zadachi*[Boundary problems]. Moscow, Fizmatlit Pub., 1977, 640 p. (In Russian) - Babeshko V.A., Evdokimova O.V., Babeshko O.M. O blochnykh elementakh v prilozheniyakh [About block elements in applications].
*Fizicheskaya mezomekhanika*[Physical mesomechanics], 2012, vol. 15, no. 1, pp. 95-103. (In Russian) - Babeshko V.A., Evdokimova O.V., Babeshko O.M. K probleme fiziko-mekhanicheskogo predvestnika startovogo zemletryaseniya: mesto, vremya, intensivnost’ [To the problem of the physico-mechanical forerunner of the initial earthquake: place, time, intensity].
*Doklady Akademii nauk*[Rep. of the Academy of Sciences], 2016, vol. 466, no. 6, pp. 664-669. (In Russian) - Brychkov Yu.A., Prudnikov A.P.
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*Dinamika neodnorodnykh lineyno-uprugikh sred*[Dynamics of inhomogeneous linearly elastic media]. Moscow, Nauka Pub., 1989, 344 p. (In Russian)

*Subjects*: mechanics

**Vatulyan A.O.**^{1,2}, **Plotnikov D.K.**^{1}

**On indentation of heterogeneous strip**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2017, no. 3, pp. 22-29.

This paper presents a method for constructing approximate solutions to the problem of parabolic stamp indentation into heterogeneous elastic strip tightly engaged with the non-deformable base. The proposed method is based on the variational formulation of the problem and involves simplification of the potential energy functional through hypothesis of displacement fields introduction. Auxiliary problem of the action of a concentrated load on the boundary of an inhomogeneous strip is solved using the variational Lagrange principle. A system of two second order differential equations with variable coefficients with respect to the displacement vector components on the upper bound of the strip is obtained. Solution of the contact problem of parabolic stamp indentation into heterogeneous elastic strip in the case when the elasticity moduli depend only on the transverse coordinate is obtained. The characteristic relations such as “force - size of the contact area”, “indentation - size of the contact area” and the stress distribution in the contact zone for some different laws of heterogeneity are plotted.

*Keywords*: heterogeneous, elasticity, strip, variational method, indentation, contact problem.

^{1}Southern Federal University, Rostov-on-Don, 344006, Russia

^{2}National University of Science and technology “MISiS”, Moscow, 119049, Russia

Corresponding author’s e-mail: vatulyan@math.rsu.ru

- Epshtein S.A., Borodich F.M., Bull S.J. Nanoindentation in studying mechanical properties of heterogeneous materials. J. Min. Sci. 2016. Vol. 51, no. 3. P. 470-476.
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*FTPRPI*[J. Min. Sci.], 2016, no. 5, pp. 84-91. (In Russian) - Kossovich E.L., Dobryakova N.N., Minin M.G., Epsstein S.A., Aarkov K.V. Primenenie tehniki neprerivnogo nano- i mikroindentirovaniya dlya opredeleniya mehanicheskih svoistv mikrokomponentov uglei [Depth-sensing nano- and microindentation for characterization of coals microcomponents mechanical properties].
*Trudi XVIII Mejdunarodnoi konferencii ‘Sovremennie problemi mehaniki sploshnoi sredi’*[Works of 18th international conf. ‘Modern problems of continuum mechanics’]. Rostov-on-Don, 2016. vol. 2, pp. 30-33. (In Russian) - Bulichev S.I., Alehin V.P., Shorshorov M.H., Ternovskiy A.P., Shnirev G.D. Opredelenie modulya Unga po diagramme vdavlivaniya [Determination of Young’s modulus by indentation diagram]
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*Kontaktnie zadachi teorii uprugosti dlya neodnorodnih sred*[Contact problems of theory of elasticity for inhomogeneous media]. Moscow, FIZMATLIT, 2006. 240 pp. (In Russian) - Aizikovich S.M., Volkov S.S., Vasilev A.S. Osesimmetrichnaya kontaktnaya zadacha o vdavlivanii konicheskogo shtampa v poluprostranstvo s neodnorodnim po glubine pokritiem [The axisymmetric contact problem of the indentation of a conical punch into a half-space with a coating inhomogeneous in depth].
*Prikladnaya matematika i mehanika*[Journal of Applied Mathematics and Mechanics]. 2015, no 5, vol 79, pp. 710-716. (In Russian) - Vatulyan A.O., Plotnikov D.K. O nekotorih kontaktnih zadachah dlya neodnorodnih uprugih tel [On some contact problems for inhomogeneous elastic bodies].
*Trudi XVIII Mejdunarodnoi konferencii ‘Sovremennie problemi mehaniki sploshnoi sredi’*[Works of 18th international conf. ‘Modern problems of continuum mechanics’]. Rostov-on-Don, 2016. vol. 1, pp. 125-129. (In Russian) - Mihlin S.G
*Variacionnie metodi v matematicheskoi fizike*[Variational methods in mathematical physics]. Moscow, Science, 1970. 512 pp. (In Russian) - Grigoluk E.I., Tolkachev V.M.
*Kontaktnie zadachi teorii plastin i obolochek*[Contact problems of the theory of plates and shells]. Moscow, Mashinostroenie, 1980. 411 pp. (In Russian)

*Subjects*: mechanics

**Evdokimov A.A.**^{1}

**Allocation and movement of roots of the Lamb wave dispersion equation in complex plane**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2017, no. 3, pp. 30-37.

The classical Lamb problem is considered for steady-state harmonic waves in a free elastic layer under the applied load. An integral representation of the solution is reduced to an infinite series in terms of residues at the poles of Green’s matrix. These poles coincide with the roots of the Lamb’s dispersion equation and define eigenwaveforms that are normal modes of the layer. The pole arrangement on the continual dispersion curves in complex plane of wave number and their frequency dependency are studied. Two transformation mechanisms of the complex wave numbers into the real ones are observed. The first mechanism is regular. In this case, complex roots become real through the imaginary axis. Second one is irregular, when complex pole becomes real without passing imaginary axis and forms a backward wave. These Lamb waves have zero group velocity at the cutoff frequencies. Their appearances are accompanied by resonance phenomena.

*Keywords*: elastic layer, dispersion equation, Lamb wave.

^{1}Kuban State University, Krasnodar, 350040, Russia

Corresponding author’s e-mail: sfom@yandex.ru

- Lamb H. On Waves in an Elastic Plate.
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*Structural health monitoring with Piezoelectric Wafer Active Sensors*. 2nd ed. USA, Academic Press, 2014, 1024 p. (In Russian) - Viktorov I.A.
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*Dinamicheskie smeshannye zadachi teorii uprugosti dlya neklassicheskih oblastej*[Dynamic mixed problems of the theory of elasticity for nonclassical domains]. Moscow, Nauka Publish, 1979, 319 p. (In Russian) -
Glushkov E.V., Glushkova N.V. K opredeleniyu dinamicheskoy kontaktnoy zhestkosti uprugogo sloya [Definition of the dynamic contact stiffness of an elastic layer].
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*Opredeleniye i uchet singulyarnykh sostavlyayushchikh v zadachakh teorii uprugosti. Dis. d-ra fiz.-mat. nauk*[Definition and allowance for singular components in problems of the theory of elasticity. Dr. phys. and math. sci. diss.]. Krasnodar, 2000, 220 p. (In Russian) - Prada C., Clorennec D., Royer D. Wave energy transfer in elastic half-spaces with soft interlayers.
*J. Acoust. Soc. Am.*, 2008, vol. 124, no. 1, pp. 203-212.

*Subjects*: mechanics

**Miakisheva O.A.**^{1}

**The interaction of source-generated spherical waves with an elastic plate immersed in acoustic fluid**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2017, no. 3, pp. 38-45.

The interaction of acoustic waves excited by a point source with an elastic plate immersed in an acoustic fluid is considered. To estimate the range of applicability of various approaches, the mathematical models has been developed on the base of the simplified Kirchhoff–Love plate and the full set of 3D Navier-Lame equations for elastic laminate structures. Explicit integral representations for excited and scattered wave fields have been derived using the Fourier transform technique. In the far-field, the bulk and guided waves are described by an asymptotic representations obtained from those integrals using the stationary phase method and the residual technique. Among others, the derived solutions enable quantitative analysis of time-averaged wave energy fluxes in the considered source-structure system. The representations for the total source energy and its portions carried by reflected, transmitted and guided waves have also been obtained. The test comparisons with the numerical results of other authors are presented.

*Keywords*: elastic plate, integral and asymptotics representations, acoustic medium, point source, balance energy.

^{1}Kuban State University, Krasnodar, 350040, Russia

Corresponding author’s e-mail: miakisheva.olga@gmail.com

- Grandia W.A. NDE applications of air-coupled ultrasonic transducers.
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*Dinamicheskiye smeshannyye zadachi teorii dlya neklassichesikkh oblastey*[Dynamic mixed problems of the theory for nonclassical domains]. Moscow, Nauka Publ., 1978. 319 p. (In Russian) - Babeshko V.A. Glushkov E.V., Glushkova N.V. Metody postroyeniya matritsy Grina stratifitsirovannogo uprugogo poluprostranstva [Methods for constructing the Green matrix of a stratified elastic half-space].
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*Subjects*: mechanics

**Rubtsov S.E.**^{1}, **Pavlova A.V.**^{1}

**Cellular automata modeling of migration and gravitational sedimentation of impurity in a liquid flow**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2017, no. 3, pp. 46-52.

To date, the main direction of numerical methods for calculating turbulent flows is the solution of the averaged Navier-Stokes equations. Cellular automata (CA) models of gas dynamics make it possible to broaden the possibilities of studying diffusion and migration processes in the atmosphere and in the aquatic environment.

In the paper we present an approach to modeling the process of pollutant migration in a flow of liquid, special attention is given to simulation of the gravitational sedimentation mechanism of a heavy impurity. Thus, in terms of cellular automata, we constructed a model of migration of a single-component substance in a flow for a plane case. We also implemented a CA simulating the propagation of an impurity that settles under the effect of gravity.

The modeling of the flow of a substance using particles that characterize the presence of a mass unit at a given point in space involves their motion in the direction specified by the velocity vector, which changes when particles collide with obstacles or with each other. The fulfillment of the laws of energy, mass, and momentum conservation is provided by the formulated rules of particle movement and collision. When modeling the flow of a liquid with an impurity, each particle is endowed with a sign indicating whether it is “pure” (ie, an element of the main flow) or an impurity. If the mass of impurity particles is different from the mass of “pure” particles, greater, for example, it is necessary to take into account the gravitational interactions of the particles in the model. In this case, an additional phase appears in the elementary automata, implementing the effect of gravity, performed in an asynchronous mode and realizing the movement of the “heavy” impurity particles from upper to lower cells. This phase can be added to the main automaton in a certain number of cycles, which allows us to adjust the mass value of the impurity particles.

The results illustrating the evolution of the cellular automaton modeling the propagation of the “heavy” impurity entering into a flow of liquid over a certain time interval are presented. The transition from the physical description of the process to its CA model and back is accomplished by comparing physical characteristics with the average number of particles in a certain number of cells.

*Keywords*: cellular automaton model, flow, impurity, transport, gravitational sedimentation.

^{1}Kuban State University, Krasnodar, 350040, Russia

Corresponding author’s e-mail: kmm@fpm.kubsu.ru

- Marchuk G.I.
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*Dissipativnye struktury v reakcionno-diffuzionnyh sistemah. Jeksperiment i teorija*[Dissipative structures in reaction-diffusion systems. Experiment and Theory]. Izhevsk, IKI Pub., 2008, 300 p. (In Russian) - Bandman O. Mapping physical phenomena onto CA-models. In:
*AUTOMATA-2008. Theory and Applications of Cellular Automata*. London, Luniver Press, 2008, pp. 381-397. - Bandman O. Parallel Simulation of Asynchronous Cellular Automata Evolution.
*Proc. of 7*, 2016, Vol. 4173 of LNCS. Springer, pp. 41-47.^{th}Int. Conference on Cellular Automata, for Research and Industry (ACRI 2006) - Bandman O.L. Invarianty kletochno-avtomatnyh modelej reakcionno-diffuzionnyh processov [Invariants Cellular Automaton models of reaction-diffusion processes].
*Prikladnaja diskretnaja matematika*[Applied discrete mathematics], 2012, no. 3, pp 108-120. (In Russian) - Toffoli T. Cellular Automata as an Alternative to rather than approximation of Differential Equations in Modeling Physics.
*Physica D*, 1984, vol. 10, pp. 117-127. - Malinetskii G.G., Stepantsov M.E. Simulation of diffusion processes by means of cellular automata with Margolus neighborhood.
*Computational Mathematics and Mathematical Physics.*,1998, vol. 38, no. 6, pp. 973-975. - Rubcov S.E., Pavlova A.V., Shkurat I.I. O kletochno-avtomatnyh modeljah processa techenija zhidkosti pri nalichii prepjatstvij i primesi [About Cellular automata models of the process flow of the liquid in the presence of obstacles and impurities].
*Zashhita okruzhajushhej sredy v neftegazovom komplekse*[Environmental protection in the oil and gas sector], 2014, no. 7, pp. 39-44. (In Russian) - Pavlova A.V., Kalajdin V.V. Ob odnoj modeli rasprostranenija zagrjaznjajushhej primesi ot ploshhadnogo istochnika [On one propagation model contaminant from the area source].
*Zashhita okruzhajushhej sredy v neftegazovom komplekse*, 2012, no. 2, pp. 18-22. (In Russian) - Babeshko V.A., Zaretskaya M.V., Evdokimova O.V., Pavlova A.V., Babeshko O.M. Kruglyakova O.P., Kurilov P.I., Terenozhkin A.M., Gendina I.V. Ocenka vlijanija vulkanicheskih i prirodno-tehnologicheskih zagrjaznenij na jekosistemu Azovskogo morja [Assessing the impact of volcanic and natural-technological pollution on the ecosystem of the Sea of Azov].
*Zashhita okruzhajushhej sredy v neftegazovom komplekse*, 2010, no. 9, pp. 6-12. (In Russian)

*Subjects*: mechanics

**Uglich P.S.**^{1}

**On the residual stress definition in the round plate**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2017, no. 3, pp. 53-59.

The inverse coefficient problem of the initial stress definition in the oscillating round elastic plate is considered. The plate vibration is axisymmetric and it’s forced by the homogeneous oscillating load. The information about plate deflection on some fixed frequency serves as input data.

Two methods for the direct problem solving are introduced. The first method is based on the shooting method. The second is based on Ritz method and reduces the problem to the linear equations system. Results of the direct problem solution are presented for different distribution laws of the initial stresses. Boundary problems both for simply supported and for the clamped edge are considered. Results are compared with well-known analytical results for the plate without initial stresses. Results obtained using different methods show good coincidence. Eigen frequencies are also evaluated using two different methods.

The inverse problem is solved by the Galerkin method. Initial stress definition results are presented for different frequencies and boundary conditions.

*Keywords*: residual stress, inverse coefficient problems, elastic plates.

^{1}Southern Federal University, Rostov-on-Don, 344090, Russia

Corresponding author’s e-mail: puglich@inbox.ru

- Birger I.A.
*Ostatochnye napryazheniya*[Residual stresses]. Moscow, Mashgiz Pub., 1963. 232 p. (In Russian) - Nedin R., Vatulyan A. Concerning one approach to the reconstruction of heterogeneous residual stress in plate.
*Z. angew. Math. Mech.*, 2013. doi: 10.1002/zamm.201200195 - Timoshenko S., Woinowsky-Krieger S.
*Theory of plates and shells*. McGraw-Hill Book Company, 1959, 635 p. - Guz A.N., Mahort F.G., Guscha O.I.
*Vvedenie v akustouprugost’*[Introduction to acoustic firmness]. Kyiv, Naukova Dumka Pub., 1977, 580 p. (In Russian) - Filippov A.P.
*Kolebaniya deformiruemykh sistem*[Vibrations of deformable systems]. Moscow, Mashinostryenie Pub., 1970, 734 p. (In Russian) - Anikina T.A., Vatulyan A.O., Uglich P.S. Opredelenie kharakteristic neodnorodnoy plastiny [On the characterictics defining for inhomogeneous plate].
*Vychislitel’nye tekhnologii*[Computational Technologies], 2012, vol. 17, no. 6, pp. 26-36. (In Russian) - Vatulyan A.O.,
*Obratnye zadachi v mechanike deformiruyemogo tvyordogo tela*[Inverse problems in deformable solid mechanics]. Moscow, Fizmatlit Pub., 2007, 222 p. (In Russian)

*Subjects*: physics

**Vekshin M.M.**^{1}, **Nikitin V.A.**^{1}, **Yakovenko N.A.**^{1}

**Synthesis of optical waveguide with complicated cross-section shape and 3D waveguide structures in glass**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2017, no. 3, pp. 60-66.

Physics and mathematical modeling of multi-step processes of waveguide structures fabrication in glass K-8 with complex form of it cross-section, based on the combined, solution of two-dimensional nonlinear diffusion equation, electrostatic equation and wave equation has been made.

The equations were solved by finite-difference method. With the help of calculated values of silver ion concentration in glass the refractive index profile (proportional to concentration) was determined and, as a next step, optical characteristics of waveguide was calculated by analysis of it mode structure.

Model of partially buried waveguide has been made, which has asymmetric refractive index profile. The experimental sample of such waveguide has been made. The model describes the fabrication of waveguide channel in K-8 type glass, where the burial process is performed with the taper-shape mask covering the part of waveguide surface, which leads to different value of burial depth along waveguide longitudinal axis. It was shown that lateral shift of waveguide concentration profile, caused by direction and density of electrostatic flux at angle between waveguide axis and mask edge, having the value less than 1,5 degree, doesn’t affect the optical energy loss in the waveguide.

Also the modeling of technological processes of single-mode optical waveguides fabrication in two vertically placed toward each other layer in order to build 3D integrated-optic circuit. The simulation was made in order to select technological parameters with similar values, with the working wavelength 1.55 micrometer. The results of calculation showed that according to this purpose the concentration of melt for the second ion exchange must be half of it value for the first one; the gap width in mask for the first step - 2 micrometer, for the second exchange - 5 micrometer.

*Keywords*: integrated optics, ion exchange in glass, 3D waveguide optical
circuits.

^{1}Kuban State University, Krasnodar, 350040, Russia

Corresponding author’s e-mail: vek-shin@mail.ru

- Bogdanov S., Shalaginov M.Y., Boltasseva A., Shalaev V. M. Material
platforms for integrated quantum photonics.
*Opt. Mater. Express,*2017, no.~2, pp. 111-132. DOI: 10.1364/OME.7.000111 - Vekshin M.M., Culish O.A., Yakovenko N.A.
*Polarizatsionnye elementy i ustroistva integralnoi optiki*[Integrated-optic polarization elements and devices]. Krasnodar, KubSU, 2017, 240 p. - Ivanov V.N., Nikitin V.A., Nikitina E.P., Yakovenko N.A. Poluchenie
poloskovyh volnovodov s prognoziruemoj formoj secheniya metodom
ehlektrostimulirovannoj diffuzii [Fabrication of strip waveguides with
planned form of cross-section by field-assisted diffusion method].
*Zhurnal tekhnicheskoj fiziki*[Journal of Technical Physics], 1983, no.10, pp. 2088-2090. - Nikitin V.A., Yakovenko N.A.
*Elektrostimulirovannaya migratsia ionov v integralnoi optike*[Electric field-assisted ion migration in integrated optics]. Krasnodar: KubSU, 2013, 245 p. - Tervonen A., West B.R., Honkanen S. Ion-exchanged glass waveguide
technology: a review.
*Optical Engineering,*2011, no.~7, Paper 071107. DOI: 10.1117/1.3559213. - Rehouma F., Persegol D., Kevorkian A. Optical waveguides for evanescent
wave sensing.
*Applied Physics Letters*, 1994, vol. 65, pp. 1477-1479. DOI: 10.1063/1.113005 - West B.R., Madasamy P., Peyghambarian N., Honkanen S. Modeling of
ion-exchanged waveguide structures.
*Journal of Non-Crystalline Solids*, 2004. vol. 347, pp. 18-26. DOI: 10.1016/j.jnoncrysol.2004.09.013 - Jordana E., Ghibaudoa E., Boucharda A. \quad Design of a waveguide with optics axes tilted by 45
^{o}and realized by ion-exchange on glass.*Proc. of SPIE,*2016, vol. 9750, Paper 975009. DOI: 10.1117/12.2209260 - Zheng B., Hao Y.-L., Li Y.-B., Yang J.-Yi, Jiang X.-Q., Zhou Q.,
Wang M.-H. Manufacturing and characterization of buried optical waveguide
stack in glass substrate.
*Journal of Inorganic Materials*, 2012, no. 9, pp. 906-910. DOI: 10.3724/SP.J.1077.2012.11687. - Lüsse P., Stuwe P., Schüle J., Unger H.G. Analysis of vectorial mode fields in optical waveguides by a new finite difference method.
*J. Lightwave Technology,*1994, no. 3. pp. 487-494. DOI: 10.1109/50.285331

*Subjects*: physics

**Kochergin V.S.**^{1}, **Kochergin S.V.**^{1}

**Variational algorithms for identifying power source of impulse point pollution**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2017, no. 3, pp. 67-72.

The environmental situation in the Azov sea requires the establishment of reliable systems of environmental monitoring that can effectively evaluate its condition in areas exposed to technogenic impact. To obtain such information on the state of the object is the efficient application of modern computing technology. The decision of problems of different pollutions propagation in the sea is possible on the basis of methods of mathematical modeling and methods of solving inverse problems, when due to the assimilation of measurement data is required to identify certain parameters of the transport model. Currently, variational methods of assimilation and the method of adjoint equations are actively developing and are used to solve Oceanographic problems. Algorithms for data assimilation of measurements based, as a rule, on the minimization of a quadratic cost function, characterizing the deviation of model solutions from measurement data. Transport model of a passive impurity acts as constraints on variation of input parameters. In the present paper the variational methods of identifying power instantaneous point source pollution based on the solution of adjoint tasks and variational task. In addition to search for the desired value applied valuation method and the variational method of filtering of linear systems of equations. Numerical experiments were carried out using the hydrodynamic model of the Azov sea. The received fields of the currents used in the passive admixtures transport modeling. Numerical experiments have shown that using variational algorithms for identifying power instantaneous point source required a single iteration. Due to identification of the variables in the space and time parameters of the pollution source additional iterations are required. When implementing variational algorithms for identification in case of noisy measurement data is filtered. When implementing the valuation method and solving the corresponding system of equations necessary additional filtering. In General, the numerical experiments showed a reliable performance of the considered algorithms identify the power source of the contamination, in relation of transport model of passive tracer in the Azov sea.

*Keywords*: variational algorithm, identification of input parameters, passive admixture, transport model, Azov Sea, transport and diffusion of pollutions, assimilation of data measurements.

^{1}Marine Hydrophysical Institute, Sevastopol, 299011, Russia

Corresponding author’s e-mail: vskocher@gmail.com

- Malanotte-Rizzoli P., Holland W.R. Data constraints applied to models of the ocean general circulation. Part II: The Transient. Eddy-Resolving Case.
*Journal of Physical Oceanography*, 1988, vol. 18, iss. 8, pp. 1093-1107. - Yu. L., O’Brien J.J. Variational estimation of the wind stress drag coefficient and the oceanic eddy viscosity profile.
*J. Phys. Oceanogr.*, 1991, vol. 21, pp. 709-719. - Marchuk G.I. Osnovnyye i sopryazhennyye uravneniya dinamiki atmosfery i okeana [Basic and conjugate equations of dynamics of atmosphere and ocean].
*Meteorologiya i gidrologiya*[Meteorology and hydrology], 1974, no. 2, pp. 17-34. (In Russian) - Penenko V.V.
*Metody chislennogo modelirovaniya atmosfernykh protsessov*[Methods for numerical modeling of atmospheric processes]. Leningrad, Gidrometeoizdat Pub., 1981, 350 p. - Kochergin V.S., Kochergin S.V. Identifikaciya parametrov mgnovennogo tochechnogo istochnika zagryazneniya v Azovskom more na osnove metoda sopryazhennyh uravnenij [Identification of parameters of an instantaneous point source of pollution in the Azov sea on the basis of conjugate equations].
*Morskoj gidrofizicheskij zhurnal*[Marine hydrophysical journal], 2017, no. 1. P. 66-71. (In Russian) - Strahov V.N. Metod fil’tracii sistem linejnyh algebraicheskih uravnenij - osnova dlya resheniya linejnyh zadach gravimetrii i magnitometrii [Filtering method systems of linear algebraic equations is the basis for solving linear problems of gravimetry and magnetometry].
*Doklady AN SSSR*[Rep. of the USSR Academy of Sciences], 1991, vol. 320, no. 3, pp. 595-599. (In Russian) - Marchuk G.I.
*Matematicheskoye modelirovaniye v probleme okruzhayushchey sredy*[Mathematical modeling in the environmental problem]. Moscow, Nauka Pub., 1982, 320 p. (In Russian) - Eremeyev V.N., Kochergin V.P., Kochergin S.V., Sklyar S.N.
*Matematicheskoye modelirovaniye gidrodinamiki glubokovdnykh basseynov*[Mathematical modeling of hydrodynamics of deep-water basins]. Sevastopol, EKOSI-Gidrofizika Pub., 2002, 238 p. (In Russian) - Ivanov V.A., Fomin V.V.
*Matematicheskoye modelirovaniye dinamicheskikh protsessov v zone more - susha*[Mathematical modeling of dynamic processes in the zone of the sea - land]. Sevastopol: EKOSI-gidrofizika. 2008. 363 p. (In Russian) - Kochergin V.S., Kochergin S.V., Identifikaciya moshchnosti istochnika zagryazneniya v Kazantipskom zalive na osnove primeneniya variacionnogo algoritma [Identification of power source of pollution in the Kazantipsky Gulf through the application of a variational algorithm].
*Morskoj gidrofizicheskij zhurnal*[Marine hydrophysical journal], 2015, no. 2, pp. 79-88. (In Russian) - Kochergin V.S., Kochergin S.V. Ispol’zovanie variacionnyh principov i
resheniya sopryazhennoj zadachi pri identifikacii vhodnyh parametrov modeli
perenosa passivnoj primesi [The use of variational principles and the solution of the adjoint problem, identification of input parameters for models of transport of passive tracer]. In
*Ehkologicheskaya bezopasnost’ pribrezhnoj i shel’fovoj zon i kompleksnoe ispol’zovanie resursov shel’fa*[Ecological safety of coastal and shelf zones and complex use of shelf resources], iss. 22. Sevastopol’, MGI NANU, 2010, pp. 240-244. (In Russian)

*Subjects*: physics

**Maniliuk Yu.V.**^{1}, **Fomin V.V.**^{1}

**Seiche oscillations in a partially enclosed basin**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2017, no. 3, pp. 73-83.

The study of seiche oscillations in the open entrance basins is an important applied problem. Even relatively small level fluctuations caused by seiches can be accompanied by strong horisontal flows which impede navigation and are dangerous for moored ships. In this work within the framework of the linear shallow water theory the seiche oscillations are studied in a rectangular basin of constant depth. The analytical solution is obtained for the case of the nodal level line location at the entrance to the basin. The structures of the seiche oscillations is compared for closed and partially closed basins. Their similarities and differences are revealed. It is established that the seiche oscillation periods in a basin with an open entrance are always smaller than the periods of the corresponding modes in same dimensions and depth closed basin. It is shown that the transverse seiches have a two-dimensional structure in open entrance basin. Their wave flows velocities depend on the horizontal dimensions and depth of the basin. In a closed basin, the wave flows of transverse seiches are one-dimensional, their magnitude does not depend on the horizontal dimensions of the basin and is inversely proportional to the square root of the basin depth. In the open entrance basin nodal level lines of the longitudinal seiches are shifted from basin entrance as compared to the nodal lines in a closed basin. The lower mode of the seiche in the open entrance basin is the Helmholtz mode. The wave flow for the Helmholtz mode is always directed perpendicularly to the entrance to the basin and its maximum velocity does not depend on the width and length of the basin. It is directly proportional the initial deviation amplitude of the free surface and inversely proportional to the square root of the basin depth. The greatest velocities of the flows take place at the open boundary of the basin.The estimates of seiche periods and wave flow velocities are obtained for the Akhtarskiy frith (the Azov Sea).

*Keywords*: seiches, free waves, long waves, wave flows, open
entrance basin, partially enclosed basin, Helmholtz mode, analytical
solutions, the Azov Sea, Akhtarskiy frith.

^{1}Marine Hydrophysical Institute, Sevastopol, 299011, Russia

Corresponding author’s e-mail: uvmsev@yandex.ru

- Labzovskiy N.A.
*Neperiodicheskie kolebaniya urovnya moray*[Non-periodic sea level fluctuations]. Leningrad, Gidrometeorologicheskoe izdatelstvo Publ., 1971. 238 p. (In Russian) *Okeanograficheskaya entsiklopediya*[Oceanographic encyclopedia]. Leningrad, Gidrometeoizdat Publ., 1980. 304 p. (In Russian)- Rabinovich A.B. Seiches and harbor oscillations (Chapter 9). In Kim Y.C. (ed.)
*Handbook of Coastal and Ocean Engineering*. Singapoure, World Scientific Publ., 2009, pp. 193-236. - Chow, Ven Te (ed.)
*Advances in Hydrosciences*. New York and London, Academic Press, 1972, vol. 8. 359 p. - Manilyuk Yu.V., Cherkesov L.V. The influence of the gulf’s geometry on
seiche oscillations in an enclosed basin.
*Physical oceanography*, 1997, vol. 8, no. 4, pp. 217-227. - Zheleznyak M.I., Kantarzhi I.G., Sorokin M.V., Polyakov A.I. Rezonansnye harakteristiki akvatorij morskih portov [Resonance properties of seaport water areas].
*Inzhenerno-stroitel’nyj zhurnal*[Magazine of Civil Engineering], 2015, no. 5(57), pp. 3-19. DOI: 10.5862/MCE.57.1 (In Russian) - Kovalev D.P.
*Naturnye ehksperimenty i monitoring infragravitacionnyh voln dlya diagnostiki opasnyh morskih yavlenij v pribrezhnoj zone na primere akvatorij Sahalino-Kuril’skogo regiona*[Natural experiments and monitoring of infra-gravity waves for the diagnostics of dangerous marine phenomena in the coastal zone by the example of the water areas of the Sakhalin-Kuril region]. Diss. Dr. phys. and math. sci. YUzhno-Sahalinsk, 2015. 304 p. (In Russian) - Zyryanov V.N., Chebanova M.K. Gidrodinamicheskie effektyi pri vhozhdenii prilivnyih voln v estuarii [Hydrodynamic effects at the entry of tidal waves into estuaries].
*Vodnyie resursyi*[Water Resources], 2016, vol. 43, no. 4, pp. 379-386. DOI: 10.1134/S0097807816040187. (In Russian) - Cherkesov L.V., Ivanov V.A., Hartiev S.M.
*Vvedenie v gidrodinamiku voln*[Introduction to hydrodynamics and wave theory]. Sankt-Peterburg.: Gidrometeoizdat Publ., 1992. 264 p. (In Russian) - Rabinovich B.I, Levyant A.S. Chislennoe reshenie zadachi rascheta sejsh
na osnove RT-algoritma konformnogo otobrazheniya [Numerical solution of the
problem of seiche calculation based on RT-algorithm of conformal mapping].
*Prirodnye katastrofy i stihijnye bedstviya v Dal’nevostochnom regione*[Natural disasters and natural disasters in the Far East region], 1990, vol. 2, pp. 328-342. (In Russian) - Rabinovich A.B.
*Dlinnyie gravitatsionnyie volnyi v okeane*[Long ocean gravity waves: trapping, resonance, leaking]. S.-Peterburg, Gidrometeoizdat Publ., 1993. 325 p. (In Russian) - Vladimirov V.S.
*Uravneniya matematicheskoj fiziki*[Equations of mathematical physics.]. Moscow, Nauka Publ., 1981. 512 p. (In Russian) - Proudmen J.
*Dinamicheskaya okeanografiya*[Dynamical oceanography]. Moscow, Inostrannaya literatura Publ., 1957. 418 p. (In Russian) - Manilyuk Yu.V., Cherkesov L.V. Investigation of Free Liquid Oscillations in a Bounded Basin Representing an Approximate Model of the Sea of Azov.
*Physical oceanography*, 2016, vol. 2, pp. 14-23.

*Subjects*: physics

**Onishchuk S.A.**^{1}, **Tumaev E.N.**^{1}

**To the question of the formation of profiled silicon crystals at its growth by Stepanov’s method**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2017, no. 3, pp. 84-94.

In this article the processes of formation of single crystals during its growing by Stepanov’s method are studied. This growing method consists of drawing crystals through the crystal former (shaper). The presence of a form-building agent (i.e. non-free growth of crystals) leads to increase in concentration of defects in the crystal structure, which decreases the quality of solar cells. In this regard, the article explores the question: what forms of growth arise during the growing of silicon single crystals by Stepanov’s method. Typical defects that arise when growing single crystals of silicon in the form of ribbons and pipes are considered. Experimental investigations of defects in profiled crystals have shown that the defects are characterized by the formation of twining structures, i.e. two neighboring single crystals separated by an atomic plane. The influence of the seed orientation on twining formation process is studied. In particular, the peculiar angles between the growth directions and the twins formation are found. The formation of mosaic structure which competes with the twin structure and is displaced by the latter is observed. The appearance of spiral structures during the growth of single crystals of tubular forms is also observed.

To explain the features of formation we calculated (per one atom) the interaction energies of neighboring parallel atomic layers. The Lennard-Jones potential was chosen as the interaction potential. The results of the calculation showed that among the planes with small values of Miller’s indices the plane (112) possesses the smallest interaction energy. The family of planes equivalent to the (112) plane has a close orientation, and the angles between its directions are in good agreement with the angles in which direction the twins are formed. The presence of three close directions of twinning also explains the formation of spiraling structures when growing silicon single crystals with tubular cross section.

*Keywords*: solar cells, profiled silicon, Stepanov’s method, twinning, crystallographic planes, Miller indices.

^{1}Kuban State University, Krasnodar, 350040, Russia

Corresponding author’s e-mail: tumayev@phys.kubsu.ru

- Vasil’ev A.M., Landsman A.P.
*Poluprovodnikovye fotopreobrazovateli*[Semiconductor photovoltaics]. Moscow, Sovetskoe radio, 1971, 248 p. (In Russian) - Bogatov N.M., Zaks M.B. Perspektivy ispol’zovaniya polikristallicheskogo kremniya v fotoenergetike [Prospects for the use of polycrystalline silicon in photovoltaic]. In:
*Tezisy dokladov Vsesoyuznogo soveshchaniya “Perspektivy razvitiya i sozdaniya edinoy nauchno-tekhnicheskoy, proizvodstvennoy i ekspluatatsionnoy bazy Krasnodarskogo kraya po ispol’zovaniyu vozobnovlyaemykh istochnikov energii i problemy ikh ispol’zovaniya v narodnom khozyaystve strany”*[Theses of reports all-Union meeting “Prospects for the development and creation of a uniform scientific-technical, production and operational base in Krasnodar region on the use of renewable energy sources and the problems of their use in national economy”]. Gelendzhik, 1988, pp. 58-59. (In Russian) - Nikanorov S.P., Antonov P.I. Growth and properties of shaped crystals.
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*J. Cryst. Growth*, 1972, vol. 13/14, p. 84. - LaBelle H.E.
*USA Patent N 3915662*, 1975. - Schwuttke G.H. Low cost silicon for solar energy application.
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*Ustoychivyy rost kristallov*[Sustainable growth of crystals]. Mosow, Nauka Pub., 1988, 240 p. (In Russian) - Osip’yan Yu.A., Tatarchenko V.A. Poluchenie profilirovannykh kristallov kremniya, issledovanie ikh elektronnykh svoystv, defektnoy struktury, izgotovlenie solnechnykh elementov i opredelenie ikh parametrov [Receiving profiled crystals of silicon, study of their electronic properties, defect structure, fabrication of solar cells and determination of their parameters].
*Izvestiya AN SSSR. Seriya fizicheskaya*[Izvestiya an SSSR. A series of physical], 1983, vol. 47, no. 2, pp. 346-350. (In Russian) - Buzynin Yu.N., Orlov Yu.N., Buzynin A.N., Dement’ev Yu.S., Bletskan N.I., Sokolov E.B. Vyrashchivanie i strukturnye osobennosti monokristallicheskikh lent kremniya [Cultivation and structural characteristics of monocrystalline ribbons of silicon].
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*Izvestiya AN SSSR. Seriya fizicheskaya*[Izvestiya an SSSR. A series of physical], 1979, vol. 43, no. 9, pp. 1989-1991. (In Russian) - Shi Dzh., Geytos G., Abrosimov N.V., Brantov S.K., Erofeeva S.A., Tatarchenko V.A. Nekotorye strukturnye i elektrofizicheskie osobennosti kremnievykh lent, poluchennykh sposobom Stepanova [Some structural and electrical characteristics of silicon ribbons obtained by the method of Stepanov].
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