Propagation of kelvin waves from a channel into a semibounded tank

The problem of propagation of Kelvin waves from a channel into a semibounded tank is considered. An exact solution of the problem is constructed using the Wiener — Hopf method. The solution is analyzed asymptotically and numerically. The Wiener — Hopf method was used in /1,2/ to solve the problem of diffraction of the Kelvin waves in tanks bounded by the infinite and semiinfinite parallel walls. Below a generalized method of matching /3/ is used to solve the problem of diffraction of Kelvin waves in the case when the walls confining the fluid meet at the right angles.     

Diffraction of a plane elastic wave by a wedge with a special form of boundary conditions

An exact solution of the antiplane problem of the diffraction of a plane elastic SH-wave with a step profile by a wedge is obtained. The stresses on the wedge sides are assumed to be proportional to a linear combination of the displacements, velocities and higher derivatives with respect to time of the displacements along the wedge axis. A solution of the problem is obtained using integral transformations with subsequent transformation using Cagniard's method. Solutions of the corresponding problems with boundary conditions of the Winkler and inertial types are considered. When a wave with a linear profile is incident on the wedge the stresses suffer a discontinuity of the second kind on the diffraction wave front; the same type of feature is observed in the problem with the inertial condition.     

Diffraction in a cylindrically orthotropic elastic solid containing a stress free crack

A cylindrically orthotropic elastic solid is excited by a point impulsive body force. The solid contains a semi-infinite stress free crack. The resulting anti-plane wave motion problem has been solved in the form of a finite series representing the incident and reflected pulses plus an integral representing the diffraction pulse. The series part of the solution has been previously treated. In the present investigation the diffraction integral is integrated when λ (which measures the anisotropy of the solid) is an odd integer number. The diffraction integral is also integrated when λ is half an odd integer, for the special case in which the source lies in the plane of the crack and parallel to the crack edge. The displacement jump across the circular diffraction wave front is given for unrestricted (positive) values of λ.     

Diffraction in a cylindrically orthotropic elastic solid containing a stress free crack

A cylindrically orthotropic elastic solid is excited by a point impulsive body force. The solid contains a semi-infinite stress free crack. The resulting anti-plane wave motion problem has been solved in the form of a finite series representing the incident and reflected pulses plus an integral representing the diffraction pulse. The series part of the solution has been previously treated. In the present investigation the diffraction integral is integrated when λ (which measures the anisotropy of the solid) is an odd integer number. The diffraction integral is also integrated when λ is half an odd integer, for the special case in which the source lies in the plane of the crack and parallel to the crack edge. The displacement jump across the circular diffraction wave front is given for unrestricted (positive) values of λ.     

High-frequency asymptotic method for investigating the problem of excitation of a thin dielectric coating of an ideally conducting cylinder (H-polarization)

An incomplete Galerkin projection method is constructed and proved as a generalization of the Sommerfeld method for the problem of plane diffraction of the field of a point source on an ideally conducting circular cylinder coated by a thin homogeneous dielectric layer with an arbitrary boundary, in the H-polarization case. An explicit asymptotic solution is obtained in the form of a generalized Watson series. Specific eesonance phenomena are investigated for a boundary layer with corner points. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 132–149.     

Fock asymptotics in the problem of diffraction by a moving smooth contour

In the paper the asymptotics are obtained of the solution of the diffraction problem in the Fock region for the incidence of a nonstationary modulated wave on a smooth moving contour. The formulas obtained generalize the solution of V. A. Fock to the stationary problem in a neighborhood of a point of tangency of an ray to the case of a nonstationary wave and a moving scatterer. A new characteristic of the problem is introduced the effective radius of curvature of the contour, depending on the geometric and kinematic properties of the scatterer. The boundaries are established of the applicability of the asymptotic formulas.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova AN SSSR, Vol. 195, pp. 69–81, 1991.In conclusion I want to thank the participants of the seminars of LGU and LOMI on the diffraction and propagation of waves for useful discussions.     

On an Integral Equation in the Problem of Diffraction of a Plane Wave on a Transparent Circular Cone

The problem of diffraction on a transparent convex cone is studied. A uniqueness theorem is proved for the case where the cone is illuminated by a compact source. For a circular cone, the solution is obtained in the form of Kontorovich-Lebedev integrals and Fourier series expansions. A singular integral equation is deduced for the Fourier coefficients, and its regularization is carried out. Bibliography: 13 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 101–123.     

On a version of the problem of choosing a vector subset

The NP-completeness is proved of the problem of choosing some subset of “similar” vectors. One of the variants of the a posteriori (off-line) noise-proof detection problem of an unknown repeating vector in a numeric sequence can be reduced to this problem in the case of additive noise. An approximation polynomial algorithm with a guaranteed performance bound is suggested for this problem in the case of a fixed space dimension.     

Radiation of short waves by a pair of plates backed by a semi-infinite set of ribbed stiffeners

The following acoustic diffraction problem is considered. The upper half-space is filled by an acoustic medium. Two semi-infinite thin plates are situated on the boundary of the upper half-space. One of the plates is backed by a semi-infinite periodical set of ribbed stiffeners. The source of an acoustic field is positioned on the other plate. The problem is reduced to an infinite system of linear algebraic equations. Such a system can be solved in a shortwave approximation. An expression for the acoustic field at a large distance from the junction of the plates is obtained. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 38–46. Translated by A. V. Badanin.     

Shortwave scattering by a diffraction echelette-grating

The two-dimensional problem of the scattering of a plane wave by a periodic perfectly conducting grating, an echelette with a right angle, is considered in the case of a high-frequency approximation (the wavelength is assumed to be small as compared with the period of the grating). The situation where the incident plane wave glides along one of the faces of a wedge is discussed. A ray-optical solution of the problem (a shortwave asymptotic result) is derived by the method of summing multiple diffracted fields, which is well known in the geometric theory of diffraction. The main result of this paper consists of obtaining simple formulas for the efficiency e_n of a diffraction order with maximal value of e_n, derived in the shortwave approximation. Numerical results are presented, and important optical properties obtained by asymptotic analysis are described. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 109–135. Translated by V. V. Zalipaev.     

Diffraction of a non-stationary linearly inhomogeneous acoustic wave which slides off a semi-infinite soft screen

An exact analytical solution of a new non-stationary scalar diffraction problem is obtained and analysed. A plane acoustic wave with a profile in the form of a delta function propagates along a semi-infinite soft screen. The wave amplitude varies linearly along the wave front. After reaching the end of the screen it “slides” off the screen, generating a diffraction field. A special modification of the Smirnov–Sobolev method is used to find this field. The solution is obtained in the form of an elementary function. It is shown that the sliding wave excites a travelling perturbation that is unlimited along the length of the screen. A similar phenomenon obviously also occurs when elastic waves slide from a cut (crack), which must be taken into account, in particular, in fracture theory.     

Solving the problem of contact equilibrium of a punch and an elastic body

The variational problem of contact equilibrium of a punch and an elastic body is considered. An equivalent formulation of the problem is given in variational inequality form. Existence and uniqueness of the solution is investigated in a particular case. A penalty method is proposed for approximate solution of the problem.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 56, pp. 97–103, 1985.     

Finite element method for the diffraction problem in a waveguide

The convergence of the finite element method (FEM) as applied to the diffraction problem in a waveguide in the case when there is no damping in the medium filling the waveguide is proved. A functional space that takes into account the partial radiation conditions is introduced to carry out the proof. A highly accurate approximation method for the partial radiation conditions is considered.     

Stability in diffraction tomography and a nonlinear “basic theorem”

The stability problem is studied for reconstruction of the refraction coefficient from boundary measurements of solutions of the Helmholtz equation at a fixed time-frequency. An answer is given in terms of Gabor means of the coefficient. A domain in the phase space is shown where the Gabor means can be stably reconstructed. As a corollary, a rigorous form is given to the basic theorem of diffraction tomography.     

Existence of an aggravated solution for a problem with delay

The aggravated mode (i.e., a mode in which a temporally global solution does not exist on certain spatial intervals, called heat localization regions) is considered for the case of instantaneous heat sources in the heat conduction problem with nonlinear thermal conductivity and a nonlinear heat source. The case with a delayed source is examined. An upper bound is established on the aggravation time. A theorem proves that the aggravation time in delayed problem is greater than the aggravation time in the "instantaneous" problem.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 67, pp. 26–30, 1989.     

The passage of a bending wave through an inhomogeneity in an absolutely rigid rib backing an elastic plate

The wave properties of a system consisting of an elastic plate and an absolutely rigid infinite rib with a defect on a segment are examined. An elastic inclusion and a gap are two kinds of defects under study. The Green's function method is applied to the diffraction problem and transforms it to singular integro-differential equations on an interval. For the case of short defects, the nonresonance and resonance asymptotics of the scattering pattern are obtained. These results show that the coefficient of penetration for a gap is much larger than that for an elastic inclusion if the frequency is nonresonant. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210. 1994, pp. 22–29. Translated by I. V. Andronov.     

Cyclic multiple-robot scheduling with time-window constraints using a critical path approach

An automated production system is considered in which several robots are used for transporting parts between workstations following a given route in a carousel mode. The problem is to maximize the throughput rate. Extending previous works treating scheduling problems for a single robot, we consider a more realistic case in which workstations are served by multiple robots. A graph model of the production process is developed, making it possible to apply PERT–CPM solution techniques. The problem is proved to be solvable in polynomial time.     

Diffraction of a plane wave by a narrow cone

The problem of diffraction of a plane scalar wave by a narrow cone is considered. The shape of the cone is arbitrary. The boundary condition is the Dirichlet or Neumann one. The wave scattered by the cone vertex arises as a result of the diffraction process. The subject of this paper is to calculate the wave amplitude. If the cone is narrow, it is possible to obtain simpler approximate formulas in comparison with Smyshlayev's one. The exactness of the approximate formulas is checked numerically. The etalon is a solution in explicit form in the axially symmetric case. The calculation shows that our formula is more exact in the case of the Dirichlet boundary condition than Felsen's formula. The approximate formula is a generalization of Felsen's one for circular cone to an arbitrary narrow cone in the case of the Neumann boundary condition. Bibliography: 6 titles. Dedicated to N. N. Uraltseva on her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 221, 1995, pp. 67–74. Translated by D. B. Dement'ev.     

Dynamical problems on longitudinal shear of a half-space with nonhomogeneities

Problems on diffraction of shear waves at cavities and rigid inclusions in a half-space with a fixed boundary and in one with a boundary which is free from forces are considered. The problem is reduced to a singular integral equation in the case of rigid inclusion in a half-space and to a Fredholm equation of the second kind in the case of a cavity.Translated from Dinamicheskie Sistemy, No. 9, pp. 47–54, 1990.     

On a certain nonstationary problem in a dihedral angle. II

An existence theorem is obtained for the initial-boundary value problem, considered in the previous paper, in the case of a positive parameter h in the boundary condition.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademiya Nauk SSSR, Vol. 188, pp. 178–185, 1991.