Nonstationary love waves of the SH-type in an anisotropic elastic medium. A kinematic approach

In this paper, the propagation of Love waves in anisotropic elastic media is studied. These waves are a similar to the transverse surface SH waves in the isotropic case. Necessary conditions for the existence of Love waves of this polarization type near the surface Σ of an anisotropic elastic body are deduced. The algorithm developed here makes it possible to find the direction (s) of transverse surface wave propagation (at every point on the surface Σ). The algorithm employed is illustrated by some special anisotropic cases. The space-time method is used to construct the asymptotics of Love waves for those types of anisotropic media the eikonal equation of which is valid on the surface of an elastic body. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 262–276 Translated by Z. A. Yanson     

The effect of thermal relaxation on rayleight surface waves in a thermoelastic medium

We study the problem of propagation of Rayleigh surface waves in thermoelastic media on the basis of the generalized coupled theory of thermoelasticity, which takes account of the phenomenon of thermal relaxation for sharply nonsteady thermal loads, thereby eliminating the paradox of infinite velocity of propagation of heat. It thus makes it possible to take account of the finite velocity of propagation of heat in the study of the process of propagation of Rayleigh surface waves. Translated fromMatematichni Metodi i Fiziko-mekhanichni Polya, Vol. 40, No. 1, 1997, pp. 85–89.     

Rayleigh waves from a point source on a boundary free of tensions

The solutions of equations of elasticity theory that have a discontinuity only on a boundary free of tensions (Rayleigh waves) are considered. Initial data for the complex intensity of the surface Rayleigh waves are found in two simple media. The first elastic medium fills a half-space with Lamé parameters and density dependent on depth. The second medium is bounded by a curve determined by a natural equation. The parameters of the second medium depend on the arc length along the curve. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 132–149.     

Rayleigh wave scattering by a plane strip in a deep ocean

A theoretical formulation to study the problem of scattering of Rayleigh waves due to the presence of a rigid plane strip in a deep ocean is presented. A rigid plane strip (0 ≤z ≤ H, 0 ≤x ≤ l) is fixed in the surface of the ocean occupyingz ≥ 0. Fourier transformation and Wiener-Hopf technique are used to arrive at the solution. The scattered Rayleigh waves behave as cylindrical waves emerging out of the corner of the strip and its image in the free surface of the ocean. The scattered waves are obtained in terms of Bessel functions whose behaviour near and far from the strip is well-known. The numerical calculations for the scattered waves show that their amplitude increases rapidly for a small increase in the value of the wave number. Scattering of Rayleigh waves due to a thin plane vertical barrier and a thin barrier in the free surface of the ocean has been considered as the special cases.     

On the dispersion of nonstationary surface waves in anisotropic elastic media

The whispering gallery modes propagating along the surface of an anisotropic elastic body are investigated with the use of space-time caustic expansions and space-time ray series. Each surface mode modulated in amplitude and frequency, is interpreted as a wave packet, with its amplitude’s maximum moving at a group velocity. On the boundary surface, asymptotic expressions for the group velocity (as a function of time and coordinates) are derived, which are in agreement with analogous formulas for Rayleigh waves of SV type in the isotropic case. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 299–312.     

Propagation of Rayleigh waves of sv type in transversely isotropic elastic media

The asymptotics of high-frequency surface waves in elastic media is studied for a special case of anisotropy, namely, for transversely isotropic media (where the parameters of elasticity are invariant with respect to rotations about one of the coordinate axes). In the zeroth asymptotic approximation, the slow Rayleigh waves (of SV type) under study are polarized in the plane of the normal section of the surface. The principal term of the asymptotics (which has the form of a space-time (caustic) expansion) is found, and calculations related to the necessity of introducing two additional faster waves with complex eikonals are carried out. The conditions on the elasticity parameters of the medium that insure the origination of the surface waves in question are obtained. Due to the specific structure of the elasticity tensor under consideration, the boundary of the medium is necesarily plane. For appropriate values of elastic parameters, the resulting formulas coincide with the corresponding expressions in the isotropic case. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 278–293. Translated by Z. A. Yanson     

Creeping waves on a highly elongated body of revolution

Creeping waves play an important role in diffraction by a smooth convex body and give an asymptotic of the diffracted field in the shadow. Known results obtained by the boundary-layer method do not allow us to explain some of the properties of creeping waves on highly elongated bodies. In this paper, creeping waves on highly elongated bodies are studied in the case where the binormal curvature of the surface is asymptotically large. The asymptotics derived contains solutions of a differential equation of the Heun type. The analysis of the dispersion equation for the surface waves is carried out numerically. It is discovered that the magnetic creeping wave travels along the surface of a highly elongated body with much less attenuation than predicated by the usual theory. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 22–34. Translated by I. V. Andronov.     

Viscoelastic Rayleigh waves in a layered structure with a weak lateral inhomogeneity

Rayleigh waves in an almost layered viscoelastic medium are studied by using the “surface” ray method based on real rays. Viscoelasticity is described in terms of the Maxwell-Boltzmann-Volterra model and, for high frequencies, is treated as perturbed perfect elasticity. In addition to the leading term of the ray asymptotics, which corresponds to the balance of energy along rays, the correction term describing anomalous displacements of the Love type is discussed. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 7–13.     

Resonance waves in media with unwelded boundaries

It is shown that in layered media with weakened interfaces, resonance waves (Kr-waves) arise. Such waves have characteristic frequencies, and many other kinematic and dynamic features distinguish them from well-known Love and Rayleigh type waves. Comparison of theoretical results with experimental seismic data indicates that Kr-waves can be formed in real geological sections. We show theoretically and numerically that in a sedimentary basin of finite lateral extent, a horizontal Kr-wave can be trapped and resonates. This wave has a number of interesting properties: the wave spectrum has a resonance frequency, the wave group velocity is equal to an intermediate value between the shear and longitudinal velocities; the attenuation of the wave increases as the frequency varies from the resonance value. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 150–156.     

Dispersion properties of nonstationary SV Rayleigh waves near the surface of an inhomogeneous elastic body

The physical characteristics of the surface of nonstationary SV Rayleigh waves are studied and formulas are obtained for the phase and group velocities (as functions of time and the observation point on the surface) of the Rayleigh waves. These formulas are in accord with the familiar formulas for the phase and group velocities in the stationary case.Translated from Zapiski Nauchnykh Seminarov Leningradzkogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, AN SSSR, Vol. 173, pp. 172–179, 1988.     

Σ-Bounded algebraic systems and universal functions. I

We introduce the concept of a Σ-bounded algebraic system and prove that if a system is Σ- bounded with respect to a subset A then in a hereditarily finite admissible set over this system there exists a universal Σ-function for the family of functions definable by Σ-formulas with parameters in A. We obtain a necessary and sufficient condition for the existence of a universal Σ-function in a hereditarily finite admissible set over a Σ-bounded algebraic system. We prove that every linear order is a Σ-bounded system and in a hereditarily finite admissible set over it there exists a universal Σ-function.     

On nonstationary love waves near the surface of an anisotropic elastic body

Space-time (ST) ray expansions in the presence of a caustic are used for the construction of an asymptotics of surface waves similar to the SH Love waves. Certain conditions which ensure the existence of surface waves with SH polarization are deduced. Some special cases of anisotropic media are indiceted where these conditions are satisfied. Bibliography: 6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 203, 1992, pp. 166–172. Translated by Z. A. Yason.     

Nonlinear electrodynamic lensing of electromagnetic waves in a magnetic dipole field

We consider propagation of electromagnetic waves in magnetic dipole and gravitational fields proceeding in accordance with the nonlinear vacuum electrodynamics laws. We derive formulas describing the effect of nonlinear electrodynamic lensing of electromagnetic waves in the magnetic dipole field. We show that rotation of the magnetic dipole moment about an axis noncoincident with this moment leads to a nonlinear electrodynamic modulation of the electromagnetic radiation intensity by frequencies that are multiples of the dipole rotation frequency. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 1, pp. 85–94, January, 2007.     

Harmonic maps between annuli on Riemann surfaces

Let ρ _Σ = h(|z|²) be a metric in a Riemann surface Σ, where h is a positive real function. Let H _{r 1} = {w = f(z)} be the family of a univalent ρ _Σ harmonic mapping of the Euclidean annulus A(r ₁, 1):= {z: r ₁ < |z| < 1} onto a proper annulus A _Σ of the Riemann surface Σ, which is subject to some geometric restrictions. It is shown that if A _Σ is fixed, then sup{r ₁: ℋ _{r 1} ≠ ∅} < 1. This generalizes similar results from the Euclidean case. The cases of Riemann and of hyperbolic harmonic mappings are treated in detail. Using the fact that the Gauss map of a surface with constant mean curvature (CMC) is a Riemann harmonic mapping, an application to the CMC surfaces is given (see Corollary 3.2). In addition, some new examples of hyperbolic and Riemann radial harmonic diffeomorphisms are given, which have inspired some new J. C. C. Nitsche-type conjectures for the class of these mappings.     

A complex structure on the set of quasiconformally extendible non-overlapping mappings into a Riemann surface

Let Σ be a Riemann surface with n distinguished points p ₁,..., p _n . We prove that the set of n-tuples (φ₁,..., φ _n ) of univalent mappings φ _i from the unit disc $ \mathbb{D} $ \mathbb{D} into Σ mapping 0 to p _i , with non-overlapping images and quasiconformal extensions to a neighbourhood of $ \overline {\mathbb{D}} $ \overline {\mathbb{D}} , carries a natural complex Banach manifold structure. This complex structure is locally modeled on the n-fold product of a two complex-dimensional extension of the universal Teichmüller space. Our results are motivated by Teichmüller theory and two-dimensional conformal field theory.     

Sulle parabole iperosculatrici in un punto ad una superficie dello spazio affine

Riassunto Si studia il sistema dei piani per un punto diA di una superficie Σ. nello spazio affine ordinario, che segano Σ secondo curve aventi inA parabola iperosculatrice. Si determina poi il numero dei piani secanti Σ secondo curve che hanno inA, con la propria parabola osculatrice, contatto d’ordine superiore a 4.

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Resume On étudie le système des plans passant par un pointA d’une surface Σ. dans l’éspace affine ordinaire. qui coupent Σ selon des courbes ayant enA une parabole hyperosculatrice. On trouve ensuite le nombre des plans qui coupent Σ selon des courbes qui ont enA un contact d’ordre supérieur à 4 avec leur parabole osculatrice.

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Lavoro eseguito nell’ambito del Gruppo di ricerca N. 26 del Comitato nazionale per la matematica del C.N.R. (1963–64).     

Surface waves in electrostrictive materials under biasing fields

The theory of small incremental fields superposed on finite biasing fields in an electroelastic body is summarized first in this paper. The Stroh formalism for two-dimensional anisotropic elasticity and piezoelectricity is generalized to two-dimensional problems of an electroelastic body under biasing mechanical and electric fields. Application of this generalized Stroh formalism to study surface waves propagating in an electroelastic half-space under uniform initial fields indicates that there exist at most two surface waves even in the presence of biasing fields and these two surface waves correspond to the well-known Rayleigh and Bleustein waves in the mechanical nature. A simple procedure is established to distinguish these two types of waves. Application of the results to a model electrostrictive material is included for illustration. It is found that initial biasing fields can greatly influence propagation feature of surface waves.     

Scattering of waves by an expanding surface

In the present paper, the perturbance of a harmonic plane wave by an expanding surface originating at a fixed point is considered. This surface generates a wave with spherical front. An analytic expression of the wave field in the vicinity of the wave front is obtained both in the scalar and in the electromagnetic case. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 35–48. Translated N. S. Zabavnikova.     

Stability of the standing waves for a class of coupled nonlinear Klein-Gordon equations

This paper deals with the standing waves for a class of coupled nonlinear Klein-Gordon equations with space dimension N ≥ 3, 0 〈 p, q 〈 2/N-2 and p ＋ q 〈 4/N. By using the variational calculus and scaling argument, we establish the existence of standing waves with ground state, discuss the behavior of standing waves as a function of the frequency ω and give the sufficient conditions of the stability of the standing waves with the least energy for the equations under study.