On the behavior near the caustic of a nonsteady wave field with singularity (A homogeneous generalized function) at the initial front

The Cauchy problem for the wave equation in the case of discontinuity on the initial front is investigated. The discontinuity is described by a homogeneous generalized function of degree λ. The transformation of the initial front while passing the space-time caustic is studied. The structure of the wave front and the space-time rays near the caustic is considered. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 122–133, 1990. Translated by N. Ya. Kirpichnikova.     

The space-time caustic of the wave field in an anisotropic medium

We examine the smooth caustic of space-time rays in an anisotropic medium, assuming first-order tangency of the rays to the caustic. An analytical expression of the wave field near the caustic is derived.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 148, pp. 5–12, 1985.I would like to acknowledge the scientific supervision of Prof. V. M. Babich.     

Mathematical simulation of acoustic wave refraction near a caustic

Issues related to the computation of wave fields in an acoustic medium near caustics are considered. A boundary condition on a caustic is established, and the Green’s function of a boundary value problem for the general case of a varying speed of sound is constructed. For this purpose, an auxiliary Goursat problem is considered and a system of its particular solutions is constructed using hyper-geometric functions. A Volterra integral equation for the Green’s function is obtained, and an algorithm for its expansion with respect to smoothness is described. A finite difference scheme approximating the solution of the differential problem with an unbounded coefficient is proposed. Numerical results are presented.     

An unloading wave in a medium with variable strain-wave velocities

A method is proposed for obtaining the equations of an unloading wave in the form of an algebraic polynomial in problems of the dynamics of an elastoplastic rod, in which strain waves propagate with variable velocities both when the stress increases and when it decreases.     

On the computation of a wave field in a multilayer medium

In this paper, theoretical seismograms of waves propagating in a medium comprising four layers that rest on a half-space are presented. The seismograms are computed by the method of summation of kinematic and dynamic wave analogs. On the basis of these computations, some conclusions concerning the behavior of the wave field at large distances are drawn. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 243–252. Translated by N. S. Smirnova.     

The discharge shock wave in a condensed ideally conducting medium with a magnetic field

Using a model equation of state containing a point of inflection, we solve the problem of the formation of a discharge shock wave in an ideally conducting medium with a magnetic field. Dissipative processes are neglected. Translated fromMatematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 1, 1998, pp. 51–53.     

An asymmetric wave field in a transversely isotropic elastic medium

A transversely isotropic homogeneous elastic medium excited by a point force perpendicular to the anisotropic axis is considered. The wave field in this medium is constructed and investigated. The front sets of the SV and SH waves are in contact with one another at a point. The front sets in the vicinity of this point are investigated additionally. If we consider the SH wave (or the SV wave) separately, then a false plane front set arises in this region. In considering the SH and SV waves in combination, this false front set disappears. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 163–174.     

Wave distribution in a variable medium with random inhomogeneities

We consider the behavior of a solution of the wave equation u_tt (t, x) – a² (t) u_xx (t, x)=f (t, x) with initial conditions u (0, x)=u₀, /t6t u (t, x) ¦_t=0 =u₁ (x), a andf being random functions; a(t) characteristizes the variable character of the medium;f(t, x) is the inhomogeneity, having the character of random walks.Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 75–78, 1988.     

Investigation of the wave field in an effective model of a layered elastic-fluid medium

For a medium that consists of alternating elastic and fluid layers, an effective model is constructed and investigated. This model is a special case of the Biot medium. The wave field is represented as Fourier and Mellin integrals. In the Mellin integral, the contour of integration is replaced by a stationary contour. In the expressions obtained, the order of integration is changed and the inner integral is calculated. The outer integral is equal to two residues. The corresponding poles are roots of two equations of fourth order. These roots lie on the right half-plane and may be complex or real. The representation obtained for the wave field is in agreement with the expressions derived by the Smirnov-Sobolev method. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 175–192.     

A point source ofSH wave in a medium with vertical boundary in the case where the velocity of wave propagation is a smoothly increasing function of depth

By the method of separation of variables, the solution of the equation that describes propagation of the SH waves in the case of a point source is constructed in the half-plane y≤0, The coefficients of this equation are piecewise constant functions. The asymptotics of the solution as w→+∞ is found. Bibliography: 6 titles. Translated fromapiski Nauchnykh Seminarov POMI, Vol. 218, 1994, pp. 44–55. This work was supported by the Russian Foundation of Fundamental Research (Grant 93-011-16148). Translated by T. N. Surkova.     

The wave field generated by a centre of rotation in an unbounded elastic medium with a semi-infinite conical crack

The dynamic stress intensity factor at the edge of a semi-infinite conical crack when the medium is loaded by a non-stationary centre of rotation is determined. A centre of rotation is understood to be a set of four forces of equal magnitude that act in the same plane and form pairs having the same direction of rotation.¹ If the magnitude of these forces is time dependent, i.e., their application is non-stationary, they form a non-stationary centre of rotation. The solution of the problem required the use of methods of integral transformations and discontinuous solutions, which reduced the problem to an integral differential equation in Laplace transform space. The combined use of the orthogonal polynomial method and time discretization to solve the equation enabled a formula for the stress intensity factor to be obtained.     

A filtration problem in a porous medium with variable permeability

Sunto Un problema di frontiera libera (moto di un fluido attraverso una diga in terra non omogenea) è ricondotto allo studio di una disequazione variazionale. Si danno condizioni sul coefficiente di permeabilità sufficienti ad assicurare che la parte bagnata della diga è un sottografico.

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Entrata in Redazione il 30 giugno 1976.

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This paper is partially supported by National Science Foundation Grant MPS 72-04959 A02, and by C.N.R. of Italy through L.A.N. of Pavia.     

Equations for a wave front,rays, and a phase function near a regular region of the caustic formed by space-time rays

Near a caustic, asymptotic formulas are obtained for the equations of a wave front and a phase function by means of a modification of the space-time ray method. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 203, 1992, pp. 92–100. Translated by N. Ya. Kirpichnikova.     

Behavior of a plasma with the collision rate proportional to the electron velocity in an external electric field

We solve the problem of the behavior of a gas plasma in a half-space analytically using the kinetic equation with the collision rate proportional to the modulus of the electron velocity. The plasma is in a variable external electric field. The specular reflection of electrons from the plasma boundary is used as a boundary condition. We use the solution to find the screened electric field. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 3, pp. 409–421, December, 2007.