Propagation of Scholte-Gogoladze surface waves along a curved fluid-solid interface

A high-frequency ray theory is presented for a type of small-amplitude waves (termed Scholte-Gogoladze waves), localized in a thin layer around an interface between elastic and fluid domains. The interface is assumed to be smooth, with typical radii of curvature much larger than the excitation wavelength. The technique employed in the paper is based on a boundary-layer version of the classical WKB-expansion (see V. M. Babich and N. Ya. Kirpichnikova, The Boundary-Layer Method in Diffraction Problems (Springer, Berlin 1979)). Bibliography: 20 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 324, 2005, pp. 229–246.     

Scattering of a spherical wave by a small ellipsoid

A point generated incident field impinges upon a small triaxialellipsoid which is arbitrarily oriented with respect to thepoint source. The point source field is so modified as to beable to recover the corresponding results for plane wave incidencewhen the source recedes to infinity. The main difficulty insolving analytically this low-frequency scattering problem concernsthe fitting of the spherical geometry, which characterizes theincident field, with the ellipsoidal geometry which is naturallyadapted to the scatterer. A series of techniques has been usedwhich lead finally to analytic solutions for the leading twolow-frequency terms of the near as well as the far field. Incontrast to the near-field approximations, which are expressedin terms of ellipsoidal eigenexpansions, the far field is furnishedby a finite number of terms. This is very interesting becausethe constants entering the expressions of the Lamé functionsof degree higher than three are not obtainable analyticallyand therefore, in the near field, not even the Rayleigh approximationcan be completely obtained. On the other hand, since only afew terms survive at the far field, the scattering amplitudeand the scattering cross-section are derived in closed form.It is shown that, in practice, if the source is located a distanceequal to five or six times the biggest semiaxis of the ellipsoidthe Rayleigh term of the approximation behaves almost as theincident field was a plane wave. The special cases of spheroids,needles, discs, spheres as well as plane wave incidence arerecovered. Finally, some theorems concerning monopole and dipolesurface potentials are included.     

Scattering on a resonator with a small opening

By the methods of the theory of extensions, one constructs an explicitly solvable resonator model with a small opening. One computes the high-frequency resonances, i.e., the eigenvalues of the dissipative operator, associated with the considered problem of resonance scattering in the framework of the Lax-Phillips theory.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 159–169, 1983.     

Scattering of antiplane shear wave by a propagating crack at the interface of two dissimilar elastic media

An analysis of the scattering of horizontally polarized shear wave by a semi-infinite crack running with uniform velocity along the interface of two dissimilar semi-infinite elastic media has been carried out. The mixed boundary value problem has been solved completely by the Wiener-Hopf technique. The effect of different values of the material parameter, the angle of incidence of incident wave and the crack propagation velocity on the stress intensity factor have been illustrated graphically.     

Two types of interface defects

The solution of a plane problem in the theory of elasticity for a two-component body with an interface, a finite part of which is either weakly distorted or is a weakly curved crack is constructed using the perturbation method. In the first case, it is assumed that the discontinuities in the forces and displacements at the interface are known, and, in the second case, the non-equilibrium nature of the load in the crack is taken into account. General quadrature formulae are derived for the complex potentials, which enable any approximation to be obtained in terms of elementary functions in many important practical cases. An algorithm is indicated for calculating each approximation. Families of defects are studied, the form of which is determined by power functions. The effect of the amplitude of the distortion and the shape of the interface crack on the Cherepanov–Rice integral as well as the shape of the distorted part of the interface on the stress concentration is investigated in the first approximation. An analysis of the applicability of the oscillating solution for a distorted interface crack is carried out. The results of the calculations are shown in the form of graphical relations.     

Scattering of the electromagnetic field on a finite impedance section of an interface

The scattering of a three-dimensional electromagnetic field on a local impedance section of a wavy surface is considered. The boundary-value problem for the system of Maxwell equations is reduced to solving a system of hypersingular integral equations. A numerical algorithm is developed and a program is designed for computing the electrodynamic characteristics of the given scattering problem. The accuracy of the simulation results is investigated. __________ Translated from Prikladnaya Matematika i Informatika, No. 25, pp. 56–69, 2007.     

Scattering of a SH-wave by an elastic fiber of nonclassical cross section with an interface crack

The problem of interaction of a plane time-harmonic SH-wave with an elastic fiber of quasi-square or quasi-triangular cross section, when an interface crack is present between an infinite elastic matrix and the fiber, is considered. The modified null-field method taking into account the asymptotic behavior of the solution at crack tips is exploited for obtaining numerical results. The effects of fiber shape, fiber/matrix material combination, debonding (crack size), and direction of wave incidence on the scattering amplitude in the far zone are analyzed. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 2, pp. 245–254, March–April, 2008.     

Null controllability in a fluid-solid structure model

We consider a system coupling the Stokes equations in a two-dimensional domain with a structure equation which is a system of ordinary differential equations corresponding to a finite dimensional approximation of equations modeling deformations of an elastic body or vibrations of a rigid body. For that system we establish a null controllability result for localized distributed controls acting only in the fluid equations and there is no control in the solid part. This controllability result follows from a Carleman inequality that we prove for the adjoint system.     

Concentration of electroelastic fields in a composite piezoceramic plate with defects crossing the interface

The characteristics of fracture and strength of a composite piezoceramic plate with defects in the form of cracks and holes situated in both of the plate components are investigated. The corresponding boundary-value problems of electroelasticity are reduced to systems of singular integral equations by constructing integral representations of the complex potentials. The results of numerical realization of the constructed algorithms are reported.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 6, pp. 777–786, November–December, 1998.     

An asymptotic analysis of the solution in the neighbourhood of the corner point of a crack along the kinked interface of two media

The asymptotic behaviour of an elastic field in the neighbourhood of the corner point of a crack at the interface of different materials is investigated within the framework of plane elasticity, taking into account the contact of its surfaces and the possibility of their mutual slippage with dry friction. The problem is solved by the method of complex Kolosov-Muskhelishvili potentials. The results obtained enable one to estimate the angular range of existence of contact zones and the singularity of the stresses close to the corner point of the crack. It is shown that the formation of contact zones, taking into account the friction forces accompanying slippage, depends essentially on the magnitude of the angle of the interface kinking the elasticity moduli of the materials and the friction coefficient. Numerical calculations are carried out and the stress and displacement distributions in the neighbourhood of the corner point are obtained.     

Bifurcation in the neighbourhood of a non-isolated singular point

The Lyapunov-Schmidt method for bifurcation problems has, until recently, been applied only to operator equations whose singular points are isolated in the solution set of the equation. For bifurcation at a multiple eigenvalue involving several parameters, however, singular points are often non-isolated. In this paper, the case of intersecting curves of singular points is considered. Under natural hypotheses on these curves, and assuming suitable transversality conditions on the first order nonlinearity of the operator, it is shown that the solution set of the equation may be completely determined locally in terms of the solutions of associated finite dimensional polynomial equations.     

Scattering by a Cylindrical Trap in the Critical Case

We study a two-dimensional analog of the Helmholtz resonator with walls of finite thickness in the critical case, for which there exists a frequency which is simultaneously the limit of poles generated both by the bounded component of the resonator and by a narrow communication channel. Under the assumption that the limit frequency is a simple frequency for the bounded component, by using the method of matched asymptotic expansions, we construct asymptotics for the two sequences of poles converging to this frequency. We obtain explicit formulas for the leading terms of the asymptotics of poles and for the solution of the scattering problem.     

The third neighbourhood of a point in a plane

Summary An unexceptionally birational model is constructed, in the form of a non singular algebraic threefold variety W, of the three dimensional aggregate of plane curve elements P₀P₁P₂P₃ of the third rank with origin at a fixed point P₀ of the plane. The geometric properties of W are then examined, and a base for surfaces and curves on W found. In particular the mapping on W of the singular (for instance cuspidal) curve elements is clarified. To Enrico Bompiani on his scientific Jubilee     

Scattering of the plane wave by a transparent wedge

In the paper it is proved that the problem of scattering of the plane wave by a transparent wedge has a unique solution, provided that the radiation condition should be meant in the following form: if one subtracts from the solution the incident wave and all reflected and refracted waves, then the remainder satisfies the radiation condition in integral form. The problem is scalar, the velocities of the wave inside and outside the wedge are not equal, the wave process is described by the classical Helmholtz equations, and the conjugation boundary condition is satisfied on the sides of the wedge. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 5–18.     

Scattering by a highly oscillating surface

The boundary function method [A. B. Vasil'eva, V. F. Butuzov, and L. V. Kalachev, The boundary function method for singular perturbation problems, SIAM Studies in Applied Mathematics, Philadelphia, 1995] is used to build an asymptotic expansion at any order of accuracy of a scalar time‐harmonic wave scattered by a perfectly reflecting doubly periodic surface with oscillations at small and large scales. Error bounds are rigorously established, in particular in an optimal way on the relevant part of the field. It is also shown how the maximum principle can be used to design a homogenized surface whose reflected wave yields a first‐order approximation of the actual one. The theoretical derivations are illustrated by some numerical experiments, which in particular show that using the homogenized surface outperforms the usual approach consisting in setting an effective boundary condition on a flat boundary. Copyright © 2014 John Wiley & Sons, Ltd.     

The lift on a small sphere in a linear shear flow near the interface of two immiscible fluids

Analytical expressions have been derived which predict, to lowest order, the inertial lift and the lateral migration velocity of a rigid sphere translating and rotating in a linear shear flow field near the flat interface of two immiscible fluids. This asymptotic analysis is primarily based on the assumption that the two Reynolds numbers defined by the gap width between the interface and the sphere, the shear rate and the translational slip velocity with which the spherical particle moves parallel to the interface are small. Furthermore, the radius of the sphere is assumed to be small compared to the gap width. To leading order in this creeping flow regime, the linear Stokes equations are obtained and a symmetry argument can be used to show that the Stokes solution does not predict any lift force. The transverse force experienced by the sphere and its migration velocity are due to the small but finite inertial terms in the Navier-Stokes equations, which can be studied by perturbation techniques. By applying a Green's function approach and matched asymptotic methods, which also incorporate the effects of the outer Oseen-like flow regime, the three components comprising the lift velocity have been calculated in closed form: the one induced by the shear rate only, the purely slip induced one and the one due to the interaction of the slip velocity with the shear flow field. The thus obtained expressions for the case of two immiscible fluids with arbitrary density and viscosity ratios extend the results that already exist in the literature for other flow configurations, such as an unbounded shear flow field [1] or a wall-bounded one, where the wall lies either within the leading order Stokes region [2] or in the outer Oseen region [3]. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)