Exact solutions of the linear problem of the steady-state waves created by a dipole in a flow of a stratified fluid

The problem of the steady-state waves which are formed when there is uniform flow of a non-viscous, incompressible, vertically stratified fluid round a dipole is considered in a linear formulation. Using the analytical properties of the solutions, two formulae are obtained for the vertical displacement field in the form of series of single integrals taken over the spectral curves. These formulae are simpler than those which have been previously proposed /1/ since the integrands do not contain special functions with logarithmic singularities and enable one to simplify the numerical analysis of the close domain of the wave field in which the asymptotic forms /2–4/ are applicable /5/.     

On the Propagation of Love Waves in Anisotropic Elastic Media

The Love waves concentrated near the surface of an anisotropic elastic body are studied. A uniform asymptotics of the wave field is constructed with the use of the nonstationary caustic expansion (Yu. A. Kravtsov's ansatz) in the form of a space-time ray series. Using three types of waves, which propagate along any direction in an elastic medium, as a vector basis, sufficient conditions for the existence of a nonzero asymptotic solution of the problem under study are obtained. The procedure for constructing asymptotic series is illustrated with the model of a transversely isotropic medium. Bibliography: 9 titles.     

On variations in the solutions of integro-differential equations of mixed problems of elasticity theory for domain variations

The behaviour of the solution of the boundary value problem for a pseudodifferential equation (PDE), Green's function of this problem, and also some of their local and global characteristics, during variation of the domain is investigated. Formulas are proposed that enable the solution of a broad class of PDE in a domain to be expressed in terms of the solution in the near domain. Local characteristics of the solution are expressed in terms of the local characteristics of the solution in the near domain. A double asymptotic form of Green's function for both arguments tending to the domain boundary occurs in the variation formula. The variation of this double asymptotic form as the domain varies is expressed in terms of this same asymptotic form. The system of variation formulas obtained is closed. It enables the PDE solution in the domain to be reduced to the solution of an ordinary differential equation in functional space. The local characteristics of the solution can also be found by this method without calculating the solution itself. If there is sufficient symmetry in the initial operator, then conservation laws in the Noether sense are obtained for its Green's function and its asymptotic form. The behaviour of the quantities under investigation is studied under inversion.

The investigation of variations of the solutions of problems for the variation of the domain occurs in the paper by Hadamard /1/, who studied the variation in conformal mapping and obtained a formula similar to (1.4). The formula for the variation of the solution of the boundary value problem for an elliptic differential equation is obtained in /2/. Variation formulas for the case of the operator of the problem about a crack and a circular domain are obtained in /3, 4/. The Irwin formula /5/ is obtained from formulas (1.4) and (1.21) by substitution.     

Nonlinear Cusped Caustics for Dispersive Waves

A multiple-scale perturbation analysis for slowly varying weakly nonlinear dispersive waves predicts that the wave number breaks or folds and becomes triple-valued. This theory has some difficulties, since the wave amplitude becomes infinite. Energy first focuses along a cusped caustic (an envelope of the rays or characteristics). The method of matched asymptotic expansions shows that a thin focusing region with relatively large wave amplitudes, valid near the cusped caustic, is described by the nonlinear Schrödinger equation (NSE). Solutions of the NSE are obtained from an asymptotic expansion of an equivalent linear singular integral equation related to a Riemann-Hilbert problem. In this way connection formulas before and after focusing are derived. We show that a slowly varying nearly monochromatic wave train evolves into a triple-phased slowly varying similarity solution of the NSE. Three weakly nonlinear waves are simultaneously superimposed after focusing, giving meaning to a triple-valued wave number. Nonlinear phase shifts are obtained which reduce to the linear phase shifts previously described by the asymptotic expansion of a Pearcey integral.     

Uniform asymptotic analysis for waves in an incompressible elastic rod II. Disturbances superimposed on a pre-stressed state

This paper studies the propagation of disturbances superimposedon a pre-stressed incompressible hyperelastic thin rod. Startingfrom the incremental equations given by Haughton and Ogden (1979,J. Mech. Phys. Solids 27, 179-212 and 489-512), we derive athree parameter-dependent one-dimensional rod equation as thegoverning equation. In particular, it is found that one parameterplays a crucial role. Depending on whether it is larger or smallerthan or equal to a critical value, the shear-wave velocity islarger or smaller than or equal to the bar-wave velocity. Inthe case that these two velocities are equal, there exist travelling-wavesolutions of arbitrary form. This implies that for this particularcase the initial disturbance would propagate along the rod withoutdistortion. To see the influence of the pre-stress in detail,we further consider an initial-value problem with an initialsingularity in the shear strain. The solutions are expressedin terms of integrals through the method of Fourier transform.We then conduct an asymptotic analysis for the solutions. Fora material point in a neighbourhood behind the shear-wave front,the phase function of these integrals has a stationary pointat infinity. Here, we use a technique of uniform asymtotic expansionto handle this case. An asymptotic expansion, correct up toorder O(t^-1), for the shear strain, which is uniformly validin a neighbourhood behind the shear-wave front, is derived.For material points in other spatial domains, the method ofstationary point is applicable, and asymptotic expansions (correctup to order O(t^-1)) are obtained. A novelty is that we are ableto deduce precise qualitative information about the waves inthe far field from our analytic results. Wave profiles for twoconcrete examples are also provided.     

Higher-Order Asymptotic Approximations for Surface Love Waves of SH Type in Tranversely Isotropic Elastic Media

A uniform asymptotics of the surface Love modes for a special case of anisotropy (tranverse isotropy) of an elastic media is obtained. In constructing the asymptotics of surface waves, the space-time (ST) ray method is employed. The wave field of each Love mode is represented as the sum of the ST caustic expansion involving the Airy functions with a real eikonal and two correction terms that are ST ray solutions, which in fact are inhomogeneous waves with complex eikonals. The eikonals and coefficients of the caustic and ray series are sought in the form of expansions in powers of two variables. The first variable is the distance from the surface, whereas the other characterizes the proximity of the caustic of a ray field to the boundary surface. Thanks to the specific structure of the elasticity tensor for a transversely isotropic medium, the boundary surface is necessarily a plane. A recursion process of computation of higher terms of the asymptotic expansion allows one to trace the conversion of the formulas obtained to the known ray solutions for isotropic elastic media. Relations between the elasticity parameters of a medium are obtained that ensure the existence of SH Love waves in a transversely isotropic medium and that are consistent with the conditions of the positiveness of the elastic energy of deformation. Bibliography: 6 titles.     

Radiation problem of normal stress loading of bore surface

The radiation of elastic waves at normal harmonic stress loading of a ring strip of the cylindrical bore-surface in the infinite elastic medium is considered. The wave field is represented by contour integrals. By using the saddle-point method stress and displacement of asymptotic expansions in the far field were obtained. It is shown that in the far field P-waves bring radial displacements and stresses, and perpendicular to these, displacements and stresses are caused by S-waves. In the paper orientation diagrams of radial and circular displacements and frequency dependencies of relative distribution of radiation power source on types of waves are presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic expansions of integrals of two Bessel functions via the generalized hypergeometric and Meijer functions

Asymptotic expansions of certain finite and infinite integrals involving products of two Bessel functions of the first kind are obtained by using the generalized hypergeometric and Meijer functions. The Bessel functions involved are of arbitrary (generally different) orders, but of the same argument containing a parameter which tends to infinity. These types of integrals arise in various contexts, including wave scattering and crystallography, and are of general mathematical interest being related to the Riemann—Liouville and Hankel integrals. The results complete the asymptotic expansions derived previously by two different methods — a straightforward approach and the Mellin-transform technique. These asymptotic expansions supply practical algorithms for computing the integrals. The leading terms explicitly provide valuable analytical insight into the high-frequency behavior of the solutions to the wave-scattering problems.     

The contact problem of elasticity theory for bodies with cracks

The plane contact problem of a stamp impressed into an elastic half-plane containing arbitrarily arranged rectilinear subsurface cracks is formulated and investigated by asymptotic methods. Partial or total overlapping of the crack edges is assumed. The problem reduces to a system of linear singular integrodifferential equations with side conditions in the form of equalities and inequalities. An analytic solution of the problem is obtained in the form of asymptotic power series /1/ in the relative dimension of the greatest crack. Dependences of the first terms of the asymptotic expansions of the desired functions on the mutual location of the cracks and the contact domains, the pressure and friction stress distributions, and the crack size and orientation are determined. Numerical results are presented.

Analysis of the influence of the stress-free boundary of the half-plane on the state of stress and strain of the elastic material near the cracks is presented in /2, 3/. The problem of a crack in an elastic plane whose edges overlap partially is also examined in /3/ by numerical methods.     

Asymptotic Solutions of a Fourth Order Differential Equation

In this paper, we derive uniform asymptotic expansions of solutions to the fourth order differential equation where x is a real variable and λ is a large positive parameter. The solutions of this differential equation can be expressed in the form of contour integrals, and uniform asymptotic expansions are derived by using the cubic transformation introduced by Chester, Friedman, and Ursell in 1957 and the integration-by-part technique suggested by Bleistein in 1966. There are two advantages to this approach: (i) the coefficients in the expansion are defined recursively, and (ii) the remainder is given explicitly. Moreover, by using a recent method of Olde Daalhuis and Temme, we extend the validity of the uniform asymptotic expansions to include all real values of x .     

Investigation of the high-frequency field of a plane electromagnetic wave inside a dielectric sphere

The field of a plane electromagnetic wave near a dielectric sphere is investigated. Contour integrals are separated out from the exact solution which at high frequencies represent waves multiply reflected from the inner surface of the sphere. The high-frequency asymptotic representation of the wave field in a neighborhood of the initial point of the caustic of the reflected and refracted rays is expressed in terms of standard special functions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 42, pp. 59–77, 1974.     

Time-dependent waves of rayleigh type near the surface of an inhomogeneous elastic body

The well studied, high-frequency Rayleigh waves polarized in a plane normal to a cross section of the surface of an inhomogeneous elastic body with phase speed close to the speed of transverse waves are generalized to the case of the time-dependent equations of elasticity. For the wave field uniform asymptotics are obtained in the form of space-time ray expansions of two types: with a real eikonal (for transverse waves diffracted at the surface) and with a complex eikonal (for a longitudinal wave damped away from the surface).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 156, pp. 168–183, 1986.     

A Systematic “Saddle Point Near a Pole” Asymptotic Method with Application to the Gauss Hypergeometric Function

In recent works [ 1 ] and [ 2 ], we have proposed more systematic versions of the Laplace’s and saddle point methods for asymptotic expansions of integrals. Those variants of the standard methods avoid the classical change of variables and give closed algebraic formulas for the coefficients of the expansions. In this work we apply the ideas introduced in [ 1 ] and [ 2 ] to the uniform method “saddle point near a pole.” We obtain a computationally more systematic version of that uniform asymptotic method for integrals having a saddle point near a pole that, in many interesting examples, gives a closed algebraic formula for the coefficients. The asymptotic sequence is given, in general, in terms of exponential integrals of fractional order (or incomplete gamma functions). In particular, when the order of the saddle point is two, the basic approximant is given in terms of the error function (as in the standard method). As an application, we obtain new asymptotic expansions of the Gauss Hypergeometric function ₂F₁(a, b, c; z) for large b and c with c > b + 1 .     

A generalized Cauchy problem for the linear differential equations of coupled physical - mechanical fields

A generalized Cauchy problem for a partial differential equation with constant coefficients, which is encountered in the study of physical processes in continuous media with widened physical - mathematical fields (see /1/) (generalized coupled thermoeleasticity /2/, coupled thermoeleasticity, porous media saturated with a viscous fluid /5/, mass and heat transfer /6/, linearized magnetoelasticity /7/, etc.) is considered. The characteristic properties of the solution of the problem, under certain constraints imposed on an equation by the stability condition, are studied. The presence of waves of higher and lower order is characteristic for the solution; in the course of time the lower-order waves are maintained and take a characteristic form. In the general case, the solution is represented in the form of integrals over the segments which link the singular points of Fourier - Laplace transforms with respect to time of the solution under consideration. The methods proposed enable an exact investigation to be made of the processes described by the equation for any time constants, and they also enable one to isolate the singularities at the fronts of propagating perturbations. As an application, the dynamic processes taking place in a thermoelastic subsapce (2) as a result of applying a mechanical and a thermal input at the boundary is studied. It is shown that in the case of unit perturbation of the boundary, the stress and temperature waves in the course of time assume a bell-shaped form and propagate with adiabatic velocity. A numerical analysis of the process which occurs due to sudden application of the force and of the thermal shock at the boundary is given.     

Improving the convergence of the modal decomposition of internal waves generated by a moving dipole

Existing results on the singularities in the neighbourhood of the plane of symmetry of the wave track of the fundamental solution of the linear equation of internal waves in Boussinesq form, obtained for exponential stratification are extended to cover arbitrary stratification. Use is made of well-known asymptotic expansions in powers of the mode number of the eigenvalues and eigenfunctions of the Sturm-Liouville problem. Apart from the principal singularity, available when using the method of “frozen coefficients”, the next singularity is singled out and it is demonstrated that this is an essential part of the approximate calculation of the wave pattern near the plane of symmetry of the track. 1999 Elsevier Science Ltd. All rights reserved.     

Effect of spherically symmetric mass flow from the surface of a particle on the force of interaction with a plane surface

A stationary velocity field of the flow of a gaseous medium generated by uniform radial injection from the surface of a spherical particle near a wall is considered in the Stokes' approximation. Bispherical coordinates are used to write the expression for the stream function. A formula is obtained for the force acting on the spherical particle when there is an arbitrary mass flow from its surface, generalizing earlier results /1, 2/. An expression for the force acting on the particle is obtained for the case of spherically symmetric injection from the surface of the particle, and asymptotic formulas at short and long distances from the wall are studied.

An analogous problem concerning the forces of interaction between two spherical particles of the same radius, when uniform injection of equal intensity takes place from their surfaces, is discussed. This is equivalent to the problem of the interaction of a spherical particle with a free surface. A general expression for the force of interaction, and its asymptotic forms for short and long distances, are obtained.     

Diffraction,Interference, and Depolarization of Elastic Waves. Caustic and Penumbra

A theoretical study of polarization-spectral anomalies of wave fields, or their deviations from the predictions based on simple plane-wave models, is presented. A simple unified method of numerical simulation of anomalies of nonstationary wave fields near a caustic and in the penumbra is described. It uses both the leading and correcting terms in asymptotic expansions. Examples of calculation of displacements and average polarization ellipses are given. Qualitative properties of wave fields are discussed. A review of earlier research on polarization anomalies of elastic waves is given. Bibliography: 38 titles.Dedicated to L. A. Molotkov on the occasion of his 70th birthday__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 297, 2003, pp. 136–153.     

Weakly supercritical dissipative structures on curved surfaces

Cellular, low amplitude structures appearing at cylindrical and spherical fronts of gaseous combustion and laser evaporation are described. In the case of a spherical front all these structures are found to be unstable. When the cylindrical front of gaseous combustion is expanded, we must expect the quasi one-dimensional structure homogeneous with respect to the ignorable coordinate to be replaced by a parquet-like pattern of rectangular cells, and later to reach a non-stationary regime. On the cylindrical front of laser evaporation the quasi one-dimensional structure of maximum amplitude is globally stable.

The best known hydrodynamic example of a kinetic problem connected with the formation of dissipative structures i.e. thermodynamically nonequilibrium stationary structures appearing as a result of the development of aperiodic instability in a spatially homogeneous state, are Benard cells /1,2/. New problems of this kind are connected with the instability of plane fronts of laser evaporation of condensed material, and of gaseous combustion /3–5/. The instability is aperiodic and appears at finite values of the wave number of the perturbation representing curvature of a plane front. The development of the instability leads to the formation of a stationary, periodically curved front /3/.

The purpose of this paper is to investigate such structures and their stability on cylindrical and spherical surfaces, and this corresponds to the problem of the propagation of a cylindrical or spherical flame through a gas, and of the laser evaporation of a spherical sample. Problems dealing with dissipative structures on curved surfaces are also of interest in biophysics, where a spherical surface models a cell membrane, while the cylindrical surface models the axon /6/.     

Focusing of shock waves in a highly viscous fluid

The method of combined asymptotic expansions is used to solve the problem of the focusing of a shock wave (in a weakly compressible medium of high viscosity. Asymptotic forms of the solution are constructed in a number of spatial zones. The focusing zone is described by its asymptotic form obtained by combining it with the solution corresponding to viscous geometrical acoustics. The reflection of a shock wave formed as a result of velocity jump near one of the foci of the ellipsoid of revolution is discussed as an example. Analytical relationships descrbing the focusing zone around the second focus are obtained. It is shown that at the focus itself the wave profile has an antisymmetric form, and the compression wave is followed by a rarefaction wave of the same form.