On Rayleigh waves in incompressible orthotropic elastic solids

In this paper the secular equation for the Rayleigh wave speed in an incompressible orthotropic elastic solid is obtained in a form that does not admit spurious solutions. It is then shown that inequalities on the material constants that ensure positive definiteness of the strain-energy function guarantee existence and uniqueness of the Rayleigh wave speed. Finally, an explicit formula for the Rayleigh wave speed is obtained.     

Viscosity and Diffusion Effects at the Boundary Surface of Viscous Fluid and Thermoelastic Diffusive Solid Medium

This paper concentrates on the wave motion at the interface of viscous compressible fluid half-space and homogeneous isotropic, generalized thermoelastic diffusive half-space. The wave solutions in both the fluid and thermoelastic diffusive half-spaces have been investigated; and the complex dispersion equation of leaky Rayleigh wave motion have been derived. The phase velocity and attenuation coefficient of leaky Rayleigh waves have been computed from the complex dispersion equation by using the Muller's method. The amplitudes of displacements, temperature change and concentration have been obtained. The effects of viscosity and diffusion on phase velocity and attenuation coefficient of leaky Rayleigh waves motion for different theories of thermoelastic diffusion have been depicted graphically. The magnitude of heat and mass diffusion flux vectors for different theories of thermoelastic diffusion have also been computed and represented graphically.     

On a technique for deriving the explicit secular equation of Rayleigh waves in an orthotropic half-space coated by an orthotropic layer

The secular equation of Rayleigh propagating in an orthotropic half-space coated by an orthotropic layer has been obtained by Sotiropolous [Sotiropolous, D. A. (1999), The e®ect of anisotropy on guided elastic waves in a layered half-space, Mechanics of Materials 31, 215–233] and by Sotiropolous & Tougelidis [Sotiropolous, D. A. and Tougelidis, G. (1998), Guided elastic waves in orthotropic surface layer, Ultrasonics 36, 371–374]. However, it is not totally explicit and some misprints have occurred in this secular equation in both papers. This secular equation was derived by expanding directly a six-order determinant originated from the traction-free conditions at the top surface of the layer and the continuity of displacements and stresses through the interface between the layer and the half-space. Since the expansion of this six-order determinant was not shown in both two papers, it has been difficult to readers to recognize these misprints. This paper presents a technique that provides a totally explicit secular equation of the wave. The technique makes clear the way from the traction-free and continuity conditions to the secular equation and enables us to recognize the misprints appearing in the reported secular equation. The technique can be employed to obtain explicit secular equations of Rayleigh waves for many other cases. Moreover, the paper introduces a transfer matrix in explicit form for an orthotropic layer that is much simpler in form than the one obtained previously.     

Turbulent diffusion in a compressible medium

This paper examines the diffusion of impurity particles in a compressible turbulent medium and compares it to diffusion in an incompressible medium. The turbulent diffusion coefficients are calculated using exact formulas expressed in terms of the Green’s function describing impurity transport in an infinite homogeneous, isotropic, stationary turbulent medium. To obtain an approximate expression for the Green’s function, numerical solutions of the nonlinear DIA (direct interaction approximation) equation (which in this paper are obtained for the first time for the case of compressible turbulence) are employed. Two types of turbulence are examined, acoustic and a mixture of shock waves. These are described by different generalized spectra. Finally, it is shown that compressibility significantly enhances the diffusion coefficient in the case of acoustic turbulence and reduces it in the second case. Zh. éksp. Teor. Fiz. 114, 930–945 (September 1998)     

Nonlinear wave processes in media containing cracks partially filled with a viscous liquid

A nonlinear (in the cubic approximation) relaxation equation of state is derived for a rod containing cracks partially filled with an incompressible viscous liquid. The nonlinear effects of the self-action and interaction of low-and high-frequency longitudinal elastic waves propagating in such a rod are studied for the cases of identical and size-varied cracks. Linear and nonlinear acoustic parameters characterizing the self-action and interaction of elastic waves in a cracked rod are determined.     

Eigenfunction expansion method to characterize Rayleigh wave propagation in orthotropic medium with phase lags

ABSTRACT

The present paper deals with the propagation of Rayleigh surface waves in homogeneous, orthotropic thermoelastic half space in the context of three-phase lag model of thermoelasticity. A vector matrix differential equation is formed by employing normal mode analysis to the considered equations which is then solved by eigenfunction expansion method. The frequency equations for different cases are derived and the path of surface particles during Rayleigh wave propagation is found elliptical. Effect of phase lags on phase velocity, attenuation coefficient, and specific loss are presented graphically with respect to frequency as well as wave number.     

Transient radiation of acoustic waves at the atmosphere-earth interface

Using the contour integration method, we solved the problem on the transient radiation of acoustic waves by a mass-production source which moves uniformly in a gas in the direction normal to the interface of homogeneous gaseous and elastic half-spaces and vanishes at the moment of the solid surface contact. Asymptotic formulas are derived for the transient radiation field which are valid at the fronts of spherical and lateral waves. An exact analytical expression is obtained for the transient radiation field at observation points in the source trajectory. The total internal reflection effects in the transient radiation of sound are analyzed. Radiophysical Research Institute, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 40, No. 4, pp. 420–431, April, 1997.     

Incompressible and Compressible Limits of Coupled Systems of Nonlinear Schrödinger Equations

Recently, coupled systems of nonlinear Schrödinger equations have been used extensively to describe a double condensate, i.e. a binary mixture of Bose-Einstein condensates. In a double condensate, an interface and shock waves may occur due to large intraspecies and interspecies scattering lengths. To know the dynamics of an interface and assure the existence of shock waves in a double condensate, we study the incompressible and the compressible limits respectively of two coupled systems of nonlinear Schrödinger equations. The main idea of our arguments is to define a “H-functional” like a Lyapunov functional which can control the propagation of densities and linear momenta. Such an idea is different from the one using the standard Wigner transform to investigate the incompressible and the compressible limits of a single nonlinear Schrödinger equation. $\mathfrak{V}Recently, coupled systems of nonlinear Schr?dinger equations have been used extensively to describe a double condensate, i.e. a binary mixture of Bose-Einstein condensates. In a double condensate, an interface and shock waves may occur due to large intraspecies and interspecies scattering lengths. To know the dynamics of an interface and assure the existence of shock waves in a double condensate, we study the incompressible and the compressible limits respectively of two coupled systems of nonlinear Schr?dinger equations. The main idea of our arguments is to define a “H-functional” like a Lyapunov functional which can control the propagation of densities and linear momenta. Such an idea is different from the one using the standard Wigner transform to investigate the incompressible and the compressible limits of a single nonlinear Schr?dinger equation.      

Properties of surface waves in an elastic cylinder filled with a liquid

The properties of harmonic surface waves in an elastic cylinder filled with a liquid are studied. The case of elastic material for which the shear wave velocity is higher than the sound velocity in a liquid is considered. The wave motion is described based on the complete system of equations of the dynamic theory of elasticity and the equation of motion of an ideal compressible liquid. The asymptotic analysis of the dispersion equation in the region of large wave numbers and qualitative analysis of the dispersion spectrum showed that in such a waveguiding system there exist two surface waves, the Stoneley and the Rayleigh waves. The lowest normal wave forms the Stoneley wave on the internal surface of the cylinder. In this waveguide phase, velocities of all normal waves, except for the lowest one, have the velocity of sound in the liquid as their limit. Therefore, the Rayleigh wave on the external surface of the cylinder is formed by all normal waves in the range of frequencies and wave numbers in which phase velocities of normal waves of the composite waveguide and the lowest normal wave of the elastic hollow cylinder coincide.     

Guided elastic waves in a pre-stressed compressible interlayer

The propagation of guided elastic waves in a pre-stressed elastic compressible layer embedded in a different compressible material is examined. The waves propagate parallel to the planar layer interfaces as a superposed dynamic stress state on the statically pre-stressed layer and host material. The underlying stress condition in the two materials is characterized by equibiaxial in-plane deformations with common principal axes of strain, one of the axes being perpendicular to the layering. Both materials have arbitrary strain energy functions. The dispersion equation is derived in explicit form. Analysis of the dispersion equation reveals the propagation characteristics and their dependence on frequency, material parameters and stress parameters. Combinations of these parameters are also defined for which guided waves cannot propagate.     

Effects of compressibility on the Rayleigh-Taylor instability in Z-pinch implosions

1 IntroductionZ-pinch imploding plasma thin shells formed bygas puffs,cylindrical foils,and wire arrays can radiatehundreds of kilojoules of soft X-rays.They are highefficient and compact devices which have wideapplied prospects,specially,as intense X-ray…     

含孔隙分层半空间中瑞利波的传播特性研究

为了研究含孔隙介质分层半空间中瑞利波的传播规律,分析孔隙介质参数对瑞利波频散曲线的影响,本文进行了数值模拟研究。采用传递矩阵算法,计算了含孔隙分层半空间中一定频率范围内瑞利波所有模式的频散及激发强度曲线,并与均匀弹性固体分层半空间情况作了类比分析,在含孔隙覆盖层的两层模型和含低速孔隙夹层的三层模型下,详细研究了孔隙度、渗透率、层厚度等参数对瑞利波各模式的影响,发现孔隙度及层厚度的变化对频散曲线影响较大,而渗透率的变化对频散曲线影响较小。      

Acoustic scattering characteristics of a thick-walled orthotropic cylindrical shell at oblique incidence

The method of wave function expansion is adopted to study the scattering of a plane harmonic acoustic wave incident at an arbitrary angle upon an arbitrarily thick cylindrically orthotropic homogeneous cylindrical shell submerged in and filled with compressible ideal fluids. A laminate approximate model and the so-called state space formulation in conjunction with the classical transfer matrix (T-matrix) approach are employed to present an analytical solution based on the three-dimensional exact equations of anisotropic elasticity. The solution is used to correlate the perturbation in the material elastic constants of an air-filled and water-submerged aluminium cylindrical shell to the sensitivity of resonances associated with various modes of wave propagation appearing in the backscattered amplitude spectrum (i.e., axially guided, Lamb, Rayleigh and Whispering Gallery waves). The effects of shell wall thickness as well as inner fluid loading on the frequency response of the shell are also examined. A limiting case is considered and good agreement with the solution available in the literature is obtained.     

Ultrasonic plate waves in paper

A theoretical and experimental study of ultrasonic plate waves in machine-made paper is described. The paper is assumed to behave as a homogeneous orthotropic plate. The dispersion equation for orthotropic plate waves in principal directions is then developed analytically. It is shown that at low frequencies the orthotropic dispersion equation depends only on in-plane parameters. Using appropriate elastic constants, a computer is used to construct the dispersion curves and describe the internal motion for plate waves in paper. Finally, a portion of the dispersion curve is checked experimentally.     

On the fluid-dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation

The compressible and heat-conductive Navier-Stokes equation obtained as the second approximation of the formal Chapman-Enskog expansion is investigated on its relations to the original nonlinear Boltzmann equation and also to the incompressible Navier-Stokes equation. The solutions of the Boltzmann equation and the incompressible Navier-Stokes equation for small initial data are proved to be asymptotically equivalent (mod decay ratet ^–5/4) ast+ to that of the compressible Navier-Stokes equation for the corresponding initial data.     

分层半空间表面非线性瑞利波的激发*

针对非线性瑞利波在均匀分层半空间结构中的激发和传播规律进行研究。根据摄动理论和模态分解将分层半空间结构中瑞利波的二次谐波声场表示为二倍频瑞利波模式的线性组合,经由互易关系得到各模式的展开系数表达式。对不同分层半空间结构中瑞利波二次谐波的激发和传播特性进行讨论,结果表明相速度匹配的瑞利波模式其二次谐波分量随传播距离线性增长,非匹配模式的二次谐波分量则沿传播方向周期震荡传播。此外,文中定义非线性参数表征瑞利波模式产生的非线性程度,这有利于选择出具有明显非线性效应的匹配点,为后续检测工作提供理论依据,具有指导意义。     

Existence of Perturbed Equilibrium States in Some Two-Layer Systems with Density Inversion

In the system formed by a heavy elastic layer (lithosphere) and a half-space filled by an ideal incompressible fluid (asthenosphere), the possibility of the existence of equilibrium states with curved boundaries near an equilibrium of a system with rectilinear boundaries is investigated. Using an analysis of the characteristic equation, we obtain a relationship between the wave number of the desired static perturbation and the dimensionless parameters of the problem, namely, the dimensionless shear modulus, Poisson’s ratio, and the decompression. The assumption that the deformations are small imposes conditions on the ranges of modification of the quantities. For example, for a moderately compressible elastic material, the equilibrium which is called tectonic waves in geophysical applications is possible only in the long-wavelength range in the presence of the inversion of density and very strong decompression. The stability problem with respect to small dynamical perturbations of an (obtained) equilibrium with curved boundaries is stated. A wave dispersion relation connecting the complex frequency of oscillations with the wave number of perturbations and with the above dimensionless parameters of the system is derived.     

Nonlinear surface waves in soft, weakly compressible elastic media

Nonlinear surface waves in soft, weakly compressible elastic media are investigated theoretically, with a focus on propagation in tissue-like media. The model is obtained as a limiting case of the theory developed by Zabolotskaya [J. Acoust. Soc. Am. 91, 2569-2575 (1992)] for nonlinear surface waves in arbitrary isotropic elastic media, and it is consistent with the results obtained by Fu and Devenish [Q. J. Mech. Appl. Math. 49, 65-80 (1996)] for incompressible isotropic elastic media. In particular, the quadratic nonlinearity is found to be independent of the third-order elastic constants of the medium, and it is inversely proportional to the shear modulus. The Gol'dberg number characterizing the degree of waveform distortion due to quadratic nonlinearity is proportional to the square root of the shear modulus and inversely proportional to the shear viscosity. Simulations are presented for propagation in tissue-like media.     

On coupled waves in stratified hydromagnetic media — I

Linearized first-order coupled wave equations are derived which describe waves in horizontally stratified hydromagnetic media, with gravity allowed for. The cases considered are compressible media of finite scalar conductivity (for two special configurations of the magnetostatic fields), compressible media of infinite conductivity and cold incompressible media.     

A comparative study of the propagation of vectorial nonparaxial beams in the use of different approaches

Starting from the vectorial Rayleigh diffraction integral formula and using a simple expansion of the nucleus in the Rayleigh formula, an analytical propagation equation of vectorial nonparaxial Gaussian beams in free space is derived, which permits us to perform numerical calculations in comparison with the expression derived by Ciattoni et al. and with the direct numerical integration of the Rayleigh formula. It is found that as usual the use of expansion of vectorial Rayleigh diffraction integral is sufficient to provide satisfactory numerical results as compared with the direct integration of the Rayleigh formula. The above two analytical expressions are valid under certain conditions, however both are applicable in the far field.