Approximation by Multipoles of the Multiple Acoustic Scattering by Small Obstacles in Three Dimensions and Application to the Foldy Theory of Isotropic Scattering

The asymptotic analysis carried out in this paper for the problem of a multiple scattering in three dimensions of a time-harmonic wave by obstacles whose size is small as compared with the wavelength establishes that the effect of the small bodies can be approximated at any order of accuracy by the field radiated by point sources. Among other issues, this asymptotic expansion of the wave furnishes a mathematical justification with optimal error estimates of Foldy’s method that consists in approximating each small obstacle by a point isotropic scatterer. Finally, it is shown how this theory can be further improved by adequately locating the center of phase of the point scatterers and the taking into account of self-interactions. In this way, it is established that the usual Foldy model may lead to an approximation whose asymptotic behavior is the same than that obtained when the multiple scattering effects are completely neglected.     

Asymptotic dynamical equations for an ensemble of nonlinear dispersive waves

Self-consistent dynamical equations are derived for the propagation and interaction of an ensemble of short waves and a long wave propagating in a nonlinear dispersive medium. The method of multiple scales is applied to simple model systems to develop systematically an asymptotic perturbation analysis and to clarify the structure of the approximations that are involved. Some properties of these interaction equations are examined, taking into account their relationship to other existing equations for single or several waves. It is shown that the group velocity dispersion is of considerable importance to the dynamics of wave interactions.     

Non-linear Evolution of P-waves in Viscous–Elastic Granular Saturated Media

The Frenkel–Biot P-wave of the first type is a seismic longitudinal wave observed in rocks fully saturated with oil, water or high-pressure gas. The P-wave of the second type is observed in unsaturated soils and other porous media saturated with gas of low pressure. Their models include properties of the skeleton, that is, its elastic modules and its own viscosity. If the non-linear terms are accounted for, the asymptotic analysis, usual for weak non-linear waves, might be applied to get the wave spectrum evolution. The wetness of grains contacts in soils and such components of oil as tars or bitumen, which attached to the skeleton, can be described by generalized viscous–elastic stress–strain connections. The latter are nominated in such a way that creates the narrow frequency interval of wave of negative dissipation where the non-linear terms begin to play the main role besides the neutral stability for waves of zero wave number. The corresponding case, relevant to single continuum model, was analyzed in the literature. Here it is shown that the interpenetrating continua with interaction of the Darcy type provide the dissipation sink in the wave evolution equation. This generalization, (Tribelsky, M.I.: Phys. Rev. Lett. (2007, submitted)), can stabilize the asymptotic solution of the evolution equation, where the dispersion terms are omitted. The asymptotic solution of the equation is invariant to initial conditions and it means a transformation of initial wave spectra to unique one while wave is spreading in the viscous–elastic medium under consideration. This explains the phenomenon, observed in wave tests at marine beach, when any dynamics action (impact, explosion, and ultrasound action) created at some distance a wave of a single frequency (~25 Hz).     

A long wave low frequency motion model for a finitely sheared elastic layer

We consider two-dimensional long wave low frequency motion in a pre-stressed layer composed of neo-Hookean material. Specifically, the pre-stress is a simple shear deformation. Derivation of the dispersion relation associated with traction-free boundary conditions is briefly reviewed. Appropriate approximations are established for the two associated long wave modes. From these approximations it is clear that there may be either two, one or no real long wave limiting phase speeds. These approximations are also used to establish the relative asymptotic orders of the displacement components and pressure increment. Using these relative orders to motivate the introduction of appropriate a scales, an asymptotically consistent model long wave low frequency motion is established. It is shown that in the presence of shear there is neither bending nor extension, or analogues of their previously established pre-stressed counterparts. In fact, both the in-plane and normal displacement components have the same asymptotic orders and the derived governing equation is of vector form.     

Wave Dynamics of Saturated Porous Media and Evolutionary Equations

Nonlinear wave dynamics of an elastically deformed saturated porous media is investigated following the Biot approach. Mathematical models under research are the Biot model and its generalization by consideration of viscous stresses inside liquids. Using two-scales and linear WKB methods, the classical Biot system is transformed to a first-order wave equation. To construct the solution of the other system, an asymptotic modified two-scales method is developed. Initial system of equations is transformed to a nonlinear generalized Korteweg–de Vries–Burgers equation for quick elastic wave. Distinctions of wave propagation in the context of the Biot model and its generalization are shown.     

The flow structure of dilute gas-particle suspensions behind a shock wave moving along a flat surface

The asymptotic and numerical investigations of shock-induced boundary layers in gas-particle mixtures are presented.The Saffman lift force acting on a particle in a shear flow istaken into account.It is shown that particle migration across the boundary layer leads tointersections of particle trajectories.The corresponding modification of dusty gas model isproposed in this paper.The equations of two-phase sidewall boundary layer behind a shock wave moving at aconstant speed are obtained by using the method of matched asymptotic expansions.Themethod of the calculation of particle phase parameters in Lagrangian coordinates isdescribed in detail.Some numerical results for the case of small particle concentration aregiven.     

Longwave Approximation in Film Flow Theory

An asymptotic longwave model which takes dispersive terms into account is constructed for describing the motion of thin films with finite deviations from the middle surface. An exact periodic solution describing a nonlinear capillary wave is constructed within the framework of the model. Small deviations from the nonlinear capillary wave are described by a linear system with periodic coefficients. It is shown that for wave perturbation periods greater than a certain critical value the monodromy matrix of this system has eigenvalues whose absolute values are equal to unity. For perturbation periods less than the critical period the absolute value of one of the eigenvalues becomes greater than unity.     

Strong explosion in a combustible gas mixture

Assume that a planar, cylindrical, or spherical point explosion takes place in a combustible mixture of gases. As a result of the explosion a strong shock wave develops and triggers chemical reactions with the release of heat. The solution of the problem for the case in which the thickness of the heat release zone is neglected (the infinitely thin detonation wave model) was obtained in [1–3].It was emphasized in [4] that these solutions can be considered only as asymptotic solutions for time and distance scales which are large in comparison with the scales which are characteristic for the chemical reactions, and under the assumption that as the overdriven detonation wave which is formed in the explosion is weakened by the rarefaction waves it does not degenerate into an ordinary compression shock. Here the question remains open of the possibility of obtaining such asymptotic solutions with account for finite chemical-reaction rates.In conclusion the authors wish to thank E. Bishimov for carrying out most of the computations for this study.     

Traveling Wave Solutions for Delayed Reaction–Diffusion Systems and Applications to Diffusive Lotka–Volterra Competition Models with Distributed Delays

This paper is concerned with the traveling wave solutions of delayed reaction–diffusion systems. By using Schauder’s fixed point theorem, the existence of traveling wave solutions is reduced to the existence of generalized upper and lower solutions. Using the technique of contracting rectangles, the asymptotic behavior of traveling wave solutions for delayed diffusive systems is obtained. To illustrate our main results, the existence, nonexistence and asymptotic behavior of positive traveling wave solutions of diffusive Lotka–Volterra competition systems with distributed delays are established. The existence of nonmonotone traveling wave solutions of diffusive Lotka–Volterra competition systems is also discussed. In particular, it is proved that if there exists instantaneous self-limitation effect, then the large delays appearing in the intra-specific competitive terms may not affect the existence and asymptotic behavior of traveling wave solutions.     

Analysis of dispersion relations of a coupled thermoelasticity problem with regard to heat flux relaxation

Dispersion relations for a coupled thermoelasticity problem including a hyperbolic heat conduction equation are derived, and their asymptotic analysis is performed. Dependences of the wave number and characteristics of the vibration damping rate on frequency are obtained and compared with similar diagrams in the classical model.     

Asymptotic Research of Nonlinear Wave Processes in Saturated Porous Media

The problem of nonlinear wave dynamics of a fluid-saturated porous medium is investigated. The mathematical model proposed is based on the classical Frenkel--Biot--Nikolaevskiy theory concerning elastic wave propagation and includes mass, momentum, energy conservation laws, as well as rheological and thermodynamic relations. The model describes nonlinear, dispersive, and dissipative medium. To solve the system of differential equations, an asymptotic modified two-scales method is developed and a Cauchy problem for initial equations system is transformed to a Cauchy problem for nonlinear generalized Korteweg--de Vries--Burgers equation for modulated quick wave amplitudes and an inhomogeneous set of equations for slow background motion. Stationary solutions of the derived evolutionary equation that have been constructed numerically reflect different regimes of elastic wave attenuation: diffusive, oscillating, and soliton-like.     

Diffraction of solitary water waves around three-dimensional floating bodies

By using the matched asymptotic expansion method and the idea of edge layer, a mathematic model for describing the interaction between weakly nonlinear shallow-water waves and three-dimensional floating bodies is formed in the paper. As a numerical example, the diffraction of a solitary wave around a vertically floating circular cylinder has been investigated and the results are presented. The present method can further be extended to the study of wave diffraction around floating bodies of general shape. The project is supported by the National Natural Science Foundation of China.     

An explicit asymptotic model for the Bleustein–Gulyaev waveUn modèle explicite pour l'onde de Bleustein–Gulyaev

The antiplane motion of a transversely isotropic piezoelectric half-space is considered. An explicit asymptotic model is derived for the far field of the surface wave. It involves, in particular, a 1D hyperbolic equation for surface shear deformation propagating with the finite wave speed predicted for the first time by J.L. Bleustein and Yu.V. Gulyaev. Neumann and Dirichlet problems are formulated to restore interior mechanical and electric fields. The derivation utilizes asymptotic arguments combined with Lourier symbolic integration. Comparison with the exact solution is presented for surface impact loading. To cite this article: J. Kaplunov et al., C. R. Mecanique 332 (2004).     

On three-dimensional acoustic-gravity waves in model non-isothermal atmospheres

The propagation of acoustic-gravity waves is studied in a family of model non-isothermal atmospheres, including temperature profiles with any initial and asymptotic temperature and adjustable maximum temperature gradient. The equation for the vertical velocity of linear acoustic-gravity waves is solved exactly in terms of hypergeometric functions, the wave field being described to all orders in the scattering parameter kL, at all frequencies and distances, including wavelengths comparable to the scale of temperature change and atmospheric layers with a large temperature gradient. It is found that since the temperature is bounded, in the asymptotic regime waves grow in amplitude exponentially and phase increases linearly with altitude. The growth in amplitude is larger than exponential and the phase increases faster than linearly for atmospheres whose temperature increases with altitude, the effect being more marked for high frequency waves in regions of large temperature gradients. The accumulation of these effects leads to a wave field which is equivalent to the isothermal case at asymptotic temperature modified by a constant amplitude factor and phase shift which account for the history of propagation of the wave through the temperature gradients.     

Asymptotics of the Critical Regimes of Internal Gravity Wave Generation

The problem of constructing an asymptotic representation of the solution of the internal gravity wave field excited by a source moving at a velocity close to the maximum group velocity of the individual wave mode is considered. For the critical regimes of individual mode generation the asymptotic representation of the solution obtained is expressed in terms of a zero-order Macdonald function. The results of numerical calculations based on the exact and asymptotic formulas are given.     

An Asymptotic Model of Seismic Reflection from a Permeable Layer

Analysis of compression wave propagation in a poroelastic medium predicts a peak of reflection from a high-permeability layer in the low-frequency end of the spectrum. An explicit formula expresses the resonant frequency through the elastic moduli of the solid skeleton, the permeability of the reservoir rock, the fluid viscosity and compressibility, and the reservoir thickness. This result is obtained through a low-frequency asymptotic analysis of Biot’s model of poroelasticity. A review of the derivation of the main equations from the Hooke’s law, momentum and mass balance equations, and Darcy’s law suggests an alternative new physical interpretation of some coefficients of the classical poroelasticity. The velocity of wave propagation, the attenuation factor, and the wave number are expressed in the form of power series with respect to a small dimensionless parameter. The absolute value of this parameter is equal to the product of the kinematic reservoir fluid mobility and the wave frequency. Retaining only the leading terms of the series leads to explicit and relatively simple expressions for the reflection and transmission coefficients for a planar wave crossing an interface between two permeable media, as well as wave reflection from a thin highly permeable layer (a lens). Practical applications of the obtained asymptotic formulae are seismic modeling, inversion, and attribute analysis.     

Asymptotic solutions of higher approximations of fields of internal gravity waves in variable-depth stratified media

A problem of wave dynamics of internal gravity waves in a variable-depth stratified medium is considered. By using a modified method of geometrical optics (vertical modes—horizontal rays), wave modes of higher approximations of asymptotic solutions are constructed. It is demonstrated that the main contributions to the solution in real stratified media are made by the first terms of the corresponding asymptotic presentations.