Attenuating Resonance Modes in a Cylindrical Waveguide Placed in an Elastic Medium

Interference attenuating waves traveling in a cylindrical elastic waveguide, placed in an elastic medium, are considered. The group velocity of these waves is intermediate between that of the P wave and that of the S wave; the phase velocity equals that of the P wave. The frequency of the waves is almost constant and is determined by the requirement of constructive interference. The dispersion and attenuation of these waves are described. Bibliography: 3 titles.     

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.     

Surface Waves on a Cavity in a Semiinfinite Elastic Medium

Biot [5] examined the propagation of waves along the free surface of a cylindrical cavity in an elastic body of infinite extent and obtained a dispersion relation for the velocity of this wave in terms of the ratio of the wavelength to the cavity diameter. This paper contains solutions for waves in a semiinfinite elastic medium with a cylindrical cavity with axially symmetric harmonic loading of the plane surface. The solutions are expressed in terms of Lame potentials which are represented by combinations of integrals containing trigonometric kernels and kernels of Weber transforms. A solution is obtained for volume waves and Biot waves. The relative velocity and relative length of surface waves are studied as functions of the loading frequency.     

Plane wave approximation in linear elasticity

We consider the approximation of solutions of the time-harmonic linear elastic wave equation by linear combinations of plane waves. We prove algebraic orders of convergence both with respect to the dimension of the approximating space and to the diameter of the domain. The error is measured in Sobolev norms and the constants in the estimates explicitly depend on the problem wavenumber. The obtained estimates can be used in the h- and p-convergence analysis of wave-based finite element schemes.     

The Third-Harmonic Resonance for Capillary-Gravity Waves with O(2) Spatial Symmetry

It is well known that the addition of surface-tension effects to the classic Stokes model for water waves results in a countable infinity of values of the surface tension coefficient at which two traveling waves of differing wavelength travel at the same speed. In this paper the third-harmonic resonance (interaction of a one-crested wave with a three-crested wave) with O(2) spatial symmetry is considered. Nayfeh analyzed the third-harmonic resonance for traveling waves and found two classes of solutions. It is shown that there are in fact six classes of periodic solutions when the O(2) symmetry is acknowledged. The additional solutions are standing waves, mixed waves and secondary branches of “Z-waves.” The normal form and symmetry group for each of the solution classes are developed, and the coefficients in the normal form are formally computed using a perturbation method. The physical aspects of the most unusual class of waves (three-mode mixed waves) are illustrated by plotting the wave height as a function of x for discrete values of t.     

The classical and generalized love problems in a domain of low-frequency interference waves of signal type. I

Low-frequency interference waves propagating through a thin elastic layer, which is in rigid contact with an elastic half-space, are considered. A new method for computing the fields of low-frequency waves suggested by G. I. Petrashen allows us to compute the fields of these waves and to draw some conclusions concerning their behavior. In particular, we may conclude that the thin upper layer significantly distorts a signal sent deep into a medium. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 225, 1996, pp. 5–39. Translated by N. S. Zabavnikova.     

The Derivative Yajima–Oikawa System: Bright,Dark Soliton and Breather Solutions

In this paper, we study the derivative Yajima–Oikawa (YO) system which describes the interaction between long and short waves (SWs). It is shown that the derivative YO system is classified into three types which are similar to the ones of the derivative nonlinear Schrödinger equation. The general N ‐bright and N ‐dark soliton solutions in terms of Gram determinants are derived by the combination of the Hirota's bilinear method and the Kadomtsev–Petviashvili hierarchy reduction method. Particularly, it is found that for the dark soliton solution of the SW component, the magnitude of soliton can be larger than the nonzero background for some parameters, which is usually called anti‐dark soliton. The asymptotic analysis of two‐soliton solutions shows that for both kinds of soliton only elastic collision exists and each soliton results in phase shifts in the long and SWs. In addition, we derive two types of breather solutions from the different reduction, which contain the homoclinic orbit and Kuznetsov–Ma breather solutions as special cases. Moreover, we propose a new (2+1)‐dimensional derivative Yajima–Oikawa system and present its soliton and breather solutions.     

Waveforms in additional components of elastic bulk waves

Additional components in displacements of elastic wave fields are those which vanish in the case of propagation of homogeneous plane waves. For P-waves in a homogeneous isotropic solid, these are transverse components. Waveforms in additional components in simple models of non-time-harmonic elastic wave propagation with plane wavefronts are analyzed. It is demonstrated that the models based on homogeneous waves with a transverse structure and on inhomogeneous waves show a qualitative difference. Bibliography: 20 titles. Dedicated to V. M. Babich on the occasion of his 75th birthday Published in Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 90–98.     

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     

Energy decay rates of elastic waves in unbounded domain with potential type of damping

In this paper we first show that the total energy of solutions for a semilinear system of elastic waves in Rn with a potential type of damping decays in an algebraic rate to zero. We study the critical potential case and we assume that the initial data have a compact support. An application for the Euler-Poisson-Darboux type dissipation V(t,x) is obtained and in this case the compactness of the support on the initial data is not necessary. Finally, we shall discuss the energy concentration region for the linear system of elastic waves in an exterior domain.     

Nonlinear Waves in a Rod: Results for Incompressible Elastic Materials

A nonlinear intrinsic theory is used to describe the motions of a straight round elastic rod including the influence of radial shear and inertia. Consideration of steady wave motions reduces the two coupled partial differential equations to ordinary differential equations for which two integrals of the motion may be found. For incompressible elastic materials with the restriction of small strain gradients, but arbitrary finite strains, a large variety of exact solutions may be found by quadrature. These include large amplitude periodic waves (which may contain shocks), solitary waves, and in some cases waves that are transitional from one stress level to another. Such solutions may be found for uniform stress strain curves that are concave up or down or that contain inflections, and even for nonmontonic curves, which have been used to represent phase transitions.     

General exact solutions for linear and nonlinear waves in a Thirring model

In the present paper, we construct exact solutions to a system of partial differential equations iu_x + v + u | v | ² = 0, iv_t + u + v | u | ² = 0 related to the Thirring model. First, we introduce a transform of variables, which puts the governing equations into a more useful form. Because of symmetries inherent in the governing equations, we are able to successively obtain solutions for the phase of each nonlinear wave in terms of the amplitudes of both waves. The exact solutions can be described as belonging to two classes, namely, those that are essentially linear waves and those which are nonlinear waves. The linear wave solutions correspond to waves propagating with constant amplitude, whereas the nonlinear waves evolve in space and time with variable amplitudes. In the traveling wave case, these nonlinear waves can take the form of solitons, or solitary waves, given appropriate initial conditions. Once the general solution method is outlined, we focus on a number of more specific examples in order to show the variety of physical solutions possible. We find that radiation naturally emerges in the solution method: if we assume one of u or v with zero background, the second wave will naturally include both a solitary wave and radiation terms. The solution method is rather elegant and can be applied to related partial differential systems. Copyright © 2014 John Wiley & Sons, Ltd.     

Self-similar asymptotics of wave problems and the structures of non-classical discontinuities in non-linearly elastic media with dispersion and dissipation

Solutions of the non-linear hyperbolic equations describing quasi-transverse waves in composite elastic media are investigated within the framework of a previously proposed model, which takes into account small dissipative and dispersion processes. It is well known for this model that if a solution of the problem of the decay of an arbitrary discontinuity is constructed using Riemann waves and discontinuities having a structure, the solution turns out to be non-unique. In order to study the problem of non-uniqueness, solutions of non-self-similar problems are constructed numerically within the framework of the proposed model with initial data in the form of a “smooth” step. With time passing the solutions acquire a self-similar asymptotic form, corresponding to a certain solution of the problem of the decay of an arbitrary discontinuity. It is shown that, by changing the method of smoothing the step, one can construct any of the self-similar asymptotic forms, as was done previously in Ref. [Chugainova AP. The asymptotic behaviour of non-linear waves in elastic media with dispersion and dissipation. Teor Mat Fiz 2006;147(2):240–56] for media with terms of opposite sign, responsible for the non-linearity, although the set of admissible discontinuities and the structure of the solutions of the problems in these cases turn out to be different.     

Reflection and transmission of the surface <Emphasis Type="Italic">P</Emphasis>-waves on the line of jump in elastic parameters

Problems of diffraction of elastic surface waves of horizontal polarization (P-waves) on a line of jump in elastic parameters are considered. The corresponding coefficients of reflection, refraction, and transmission are obtained by means of the parabolic equation method. A comparison of the Rayleigh waves, SV-waves, and P-waves is carried out. Numerical values of the transformation matrix are found in the case where the whispering gallery wave transforms to a sum of whispering gallery waves and a homogeneous wave transforms to the first five modes of whispering gallery waves. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 342, 2007, pp. 106–137.     

Waves produced by sources distributed on an expanding disk

Explicit solutions of a nonhomogeneous wave equation are constructed. The solutions obtained describe waves generated by sources distributed on a disk that expands with velocities less than, equal to, or greater than the velocity of the wavefront. The structure and the directional transmission of waves are discussed. It is shown that the scalar solutions obtained can be applied to electromagnetic waves. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 38–47.     

Attenuation of normal waves in a cavity with a liquid in the presence of an elastic shell

The attenuation of normal waves is investigated in a cylindrical cavity of an unbounded viscoelastic medium, in which there is a thin elastic shell filled with a viscous compressible liquid. It is assumed that the absorption of waves in a liquid or solid is small. The effect of the shell thickness on the value of the absorption coefficient of the normal waves is studied.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 49–52, 1987.     

On the calculation of wave fields propagating in multilayer media

A program for evaluating elastic waves in multilayer media is composed on the basis of the algorithm previously stated. The waves under consideration undergo all possible acts of reflection and refraction on the interfaces of the media. Exchanges on these interfaces are also taken into account. Two examples illustrating the calculation of theoretical seismograms of elastic waves are presented. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 203, 1992, pp. 156–165. Translated by N. S. Zabavnikova     

New results for guided waves in heterogeneous elastic media

We prove the existence of guided waves propagating with a velocity strictly larger than the S (shear) wave velocity at infinity in the case of unbounded elastic media invariant under translation in one space direction and asymptotically homogeneous at infinity. These waves correspond to the existence of eigenvalues embedded in the essential spectrum of the self-adjoint elastic propagator.     

Turbulent solutions of a first order partial differential equation

A system of quasi-linear hyperbolic conservation laws which is hyperbolic but not strictly hyperbolic is studied. The system was derived as a model for the elastic string (B. Keyfitz and Kranzer, Arch. Rational Mech. Anal. 72 (1980), 210–241) and is assumed to be diagonizable. Interest is mainly in the large time behavior of the solution. Due to the nonlinearity of the system and the entropy condition, solutions converge to very simple elementary waves. Nonstrict hyperbolicity of the system may cause stronger nonlinear interactions between waves pertaining to different families; in particular, such interactions may regularize linear waves in the solution. The solutions are constructed using the random choice method.     

Reflected-refracted and head sh waves formed on a thin layer that is in welded contact with two elastic half-spaces

The reflection and refraction of SH waves by an elastic layer sandwiched between two elastic half-spaces are studied by using the numerical simulation on the basis of contour integration in the complex plane of the horizontal component of the slowness vector. The propagation of body, channel, head, and screened body waves are simulated in time and spectral domains. The wave fields associated with the propagation in the layer have strong attenuation, provided that the wave length is smaller than the value of the thickness of the layer. The stationary wave field of such waves is of resonance nature. Moreover, the maximum of the modulus of the spectral function is shifted to higher frequences as the epicentric distance increases. Thus, the attenuation of such waves depends on spectral characteristics of a source-receiver system. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 225, 1997, pp. 91–120. Translated by T. N. Surkova.