Retrieving the elastodynamic Green's function of an arbitrary inhomogeneous medium by cross correlation

A correlation-type reciprocity theorem is used to show that the elastodynamic Green's function of any inhomogeneous medium (random or deterministic) can be retrieved from the cross correlation of two recordings of a wave field at different receiver locations at the free surface. Unlike in other derivations, which apply to diffuse wave fields in random media or irregular finite bodies, no assumptions are made about the diffusivity of the wave field. In a second version, it is assumed that the wave field is diffuse due to many uncorrelated sources inside the medium.     

A new approach of solving Green’s function for wave propagation in an inhomogeneous absorbing medium

A new approach is developed to solve the Green's function that satisfies the Hehmholtz equation with complex refractive index. Especially, the Green's function for the Helmholtz equation can be expressed in terms of a one-dimensional integral, which can convert a Helmholtz equation into a Schr?dinger equation with complex potential. And the Schr?dinger equation can be solved by Feynman path integral. The result is in excellent agreement with the previous work.     

Recovering the acoustic Green's function from ambient noise cross correlation in an inhomogeneous moving medium

We study long-range correlation of diffuse acoustic noise fields in an arbitrary inhomogeneous, moving fluid. The flow reversal theorem is used to show that the cross-correlation function of ambient noise provides an estimate of a combination of the Green's functions corresponding to sound propagation in opposite directions between the two receivers. Measurements of the noise cross correlation allow one to quantify flow-induced acoustic nonreciprocity and evaluate both spatially averaged flow velocity and sound speed between the two points.     

Field reconstruction in an anisotropic elastic medium

For steady-state vibrations of an anisotropic elastic body of finite dimensions, a method of the determination of the vibration energy flows in the body is proposed. The method is based on the measurements of the surface values of the stress and displacement vectors at a part of the boundary. The proposed algorithm of the wave field reconstruction is reduced to solving nonclassical boundary integral equations of the first kind with smooth kernels. The formulation of these equations does not require the determination of fundamental solutions, but represents a conditionally well-posed problem. The numerical realization of the proposed method is based on the Tikhonov regularization method and the idea of the boundary element method. Numerical experiments consisting in the reconstruction of the displacements and stresses at the boundary of a rectangular and a circular domains of austenitic steel are performed in the framework of a planar problem of the orthothropic elasticity theory.     

Solitary waves in an inhomogeneous cylindrical shell interacting with an elastic medium

A new generalized sixth-order nonintegrable equation is derived to model axisymmetric longitudinal wave propagation in an inhomogeneous cylindrical shell interacting with a nonlinear elastic medium. Exact soliton-like solutions to this equation are constructed with allowance for geometric and physical nonlinearities, both individually and in combination.     

Dyadic Green‘s function for an unbounded gyroelectric chiral medium in cylindrical coordinates

The dyadic Green‘s function for an unbounded gyroelectric chiral medium is formulated in the Fourier domain in cylindrical coordinates.After a triple Fourier integral of the dyadic Green‘s function has been obtained,the integral is reduced by performing the integration over the transverse Fourier variable.     

Propagation of the two-frequency coherence function in an inhomogeneous background random medium

The spatial and temporal structures of time-dependent signals can be appreciably affected by random changes of the parameters of the medium characteristic of almost all geophysical environments. The dispersive properties of random media cause distortions in the propagating signal, particularly in pulse broadening and time delay. When there is also spatial variation of the background refractive index, the observer can be accessed by a number of background rays. In order to compute the pulse characteristics along each separate ray, there is a need to know the behaviour of the two-frequency mutual coherence function. In this work, we formulate the equation of the two-frequency mutual coherence function along a curved background ray trajectory. To solve this equation, a recently developed reference-wave method is applied. This method is based on embedding the problem into a higher dimensional space and is accompanied by the introduction of additional coordinates. Choosing a proper transform of the extended coordinate system allows us to emphasize 'fast' and 'slow' varying coordinates which are consequently normalized to the scales specific to a given type of problem. Such scaling usually reveals the important expansion parameters defined as ratios of the characteristic scales and allows us to present the proper ordering of terms in the desired equation. The performance of the main order solution is demonstrated for the homogeneous background case when the transverse structure function of the medium can be approximated by a quadratic term.     

Green's function of radial inhomogeneous spheres excited by internal sources

Green's function in the interior of penetrable bodies with inhomogeneous compressibility by sources placed inside them is evaluated through a Schwinger-Lippmann volume integral equation. In the case of a radial inhomogeneous sphere, the radial part of the unknown Green's function can be expanded in a double Dini's series, which allows analytical evaluation of the involved cumbersome integrals. The simple case treated here can be extended to more difficult situations involving inhomogeneous density as well as to the corresponding electromagnetic or elastic problem. Finally, numerical results are given for various inhomogeneous compressibility distributions.     

The optical potential in the Green's function approach

Utilizing the so-calledλ ⁰⁰-approximation the effective single-particle potential (“mass operator”) is investigated with respect to its energy and momentum dependence. For the nucleon-nucleon interaction a localS-wave potential matching the experimental phase shifts is used. The dispersion behaviour of the real and imaginary parts of the generalized optical potential is studied. The potential on the energy-shell is in satisfactory agreement with the phenomenological fits.     

The role of surface tension in elastic wave scattering in an inhomogeneous poroelastic medium

It is well known that many porous media such as rocks have heterogeneities at nearly all scales. We applied Biot's poroelastic theory to study the propagation of elastic waves in isotropic porous matrix with spherical inclusions. It is assumed that the heterogeneity dimension exceeds significantly the pore size. Modified boundary conditions on poroelastic interface are used to take into account the surface tension effects. The effective wavenumber is calculated using the Waterman and Truell multiple scattering theory, which relates the effective wave number to the amplitude of the wave field scattered by a single inclusion. The calculations were performed for a medium containing fluid-filled cavities or porous inclusions contrasting in saturating fluid elastic properties. The results obtained show that when we consider elastic wave propagation in poroelastic medium containing soft inclusions, it is necessary to take into account the capillary pressure. The influence of the surface tension depends on the diffraction parameter and it is a maximum in the low frequency range.     

The role of surface tension in elastic wave scattering in an inhomogeneous poroelastic medium

It is well known that many porous media such as rocks have heterogeneities at nearly all scales. We applied Biot's poroelastic theory to study the propagation of elastic waves in isotropic porous matrix with spherical inclusions. It is assumed that the heterogeneity dimension exceeds significantly the pore size. Modified boundary conditions on poroelastic interface are used to take into account the surface tension effects. The effective wavenumber is calculated using the Waterman and Truell multiple scattering theory, which relates the effective wave number to the amplitude of the wave field scattered by a single inclusion. The calculations were performed for a medium containing fluid-filled cavities or porous inclusions contrasting in saturating fluid elastic properties. The results obtained show that when we consider elastic wave propagation in poroelastic medium containing soft inclusions, it is necessary to take into account the capillary pressure. The influence of the surface tension depends on the diffraction parameter and it is a maximum in the low frequency range.     

Green's functions for a volume source in an elastic half-space

Green's functions are derived for elastic waves generated by a volume source in a homogeneous isotropic half-space. The context is sources at shallow burial depths, for which surface (Rayleigh) and bulk waves, both longitudinal and transverse, can be generated with comparable magnitudes. Two approaches are followed. First, the Green's function is expanded with respect to eigenmodes that correspond to Rayleigh waves. While bulk waves are thus ignored, this approximation is valid on the surface far from the source, where the Rayleigh wave modes dominate. The second approach employs an angular spectrum that accounts for the bulk waves and yields a solution that may be separated into two terms. One is associated with bulk waves, the other with Rayleigh waves. The latter is proved to be identical to the Green's function obtained following the first approach. The Green's function obtained via angular spectrum decomposition is analyzed numerically in the time domain for different burial depths and distances to the receiver, and for parameters relevant to seismo-acoustic detection of land mines and other buried objects.     

An approximate Green's function for a locally excited fluid-loaded thin elastic plate

In the classic treatment of the line-driven, fluid-loaded, thin elastic plate, a branch cut integral typically needs to be evaluated. This branch cut arises due to a square root operator in the spectral form of the acoustic impedance. In a previous paper [J. Acoust. Soc. Am. 110, 3018 (2001)], DiPerna and Feit developed a methodology, complex layer analysis (CLA), to approximate this impedance. The resulting approximation was in the form of a rational function, although this was not explicitly stated. In this paper, a rational function approximation (RFA) to the acoustic impedance is derived. The advantage of the RFA as compared to the CLA approach is that a smaller number of terms are required. The accuracy of the RFA is examined both in the Fourier transform domain and the spatial domain. The RFA is then used to obtain a differential relationship between the pressure and velocity on the surface of the plate. Finally, using the RFA in conjunction with the equation of motion of the plate, an approximate expression for the Green's function for a line-driven plate is obtained in terms of a sum of propagating and evanescent waves. Comparisons of these results with the numerical inversion of the exact integral show reasonable agreement.     

Casimir stress in an inhomogeneous medium

The Casimir effect in an inhomogeneous dielectric is investigated using Lifshitz’s theory of electromagnetic vacuum energy. A permittivity function that depends continuously on one Cartesian coordinate is chosen, bounded on each side by homogeneous dielectrics. The result for the Casimir stress is infinite everywhere inside the inhomogeneous region, a divergence that does not occur for piece-wise homogeneous dielectrics with planar boundaries. A Casimir force per unit volume can be extracted from the infinite stress but it diverges on the boundaries between the inhomogeneous medium and the homogeneous dielectrics. An alternative regularization of the vacuum stress is considered that removes the contribution of the inhomogeneity over small distances, where macroscopic electromagnetism is invalid. The alternative regularization yields a finite Casimir stress inside the inhomogeneous region, but the stress and force per unit volume diverge on the boundaries with the homogeneous dielectrics. The case of inhomogeneous dielectrics with planar boundaries thus falls outside the current understanding of the Casimir effect.     

Reconstruction of the stiffness of an inhomogeneous elastic plate

The paper discusses the problem of reconstructing the inhomogeneous cylindrical, symmetric stiffness distribution of a round plate using information on the bias function for established oscillations, which is measured at a certain point. A solution is constructed to the direct problem using the Galerkin method and to the inverse problem of reconstructing the stiffness using an iterative approach based on the regularized linearization method. We present the results of calculation experiments on reconstructing different types of functions that show the efficiency of the proposed approach and make it possible to estimate changes in stiffness.     

Dislocations in an inhomogeneous anisotropic medium

The general expressions for the stress field of a dislocation situated near a welded, slipping and free boundary are derived employing the method of Fourier transformations. As an example the stress field of a screw dislocation near a simple type of welded boundary is explicitly given.