Wave Propagation In Isotropic Media Research Articles

Wave Propagation in Isotropic Media with Two Orthogonal Fracture Sets

Orthogonal intersecting fracture sets form fracture networks that affect the hydraulic and mechanical integrity of a rock mass. Interpretation of elastic waves propagated through orthogonal fracture networks is complicated by guided modes that propagate along and between fractures, by multiple internal reflections, as well as by scattering from fracture intersections. The existence of some or all of these potentially overlapping modes depends on local stress fields that can preferentially close or open either one or both sets of fractures. In this study, an acoustic wave front imaging system was used to examine the effect of bi-axial loading conditions on acoustic wave propagation in isotropic media containing two orthogonal fracture sets. From the experimental data, orthogonal intersecting fracture sets support guided waves that depend on fracture spacing and fracture-specific stiffnesses. In addition, fracture intersections have stronger effects on propagating wave fronts than merely the superposition of the effects of two independent fractures because of energy partitioning among transmitted/reflected waves, scattered waves and guided modes. Interpretation of the properties of fractures or fracture sets from seismic measurements must consider non-uniform fracture stiffnesses within and among fracture sets, as well as considering the striking effects of fracture intersections on wave propagation.

Numerical simulation of elastic wave propagation in isotropic media considering material and geometrical nonlinearities

In order to detect micro-structural damages accurately new methods are currently developed. A promising tool is the generation of higher harmonic wave modes caused by the nonlinear Lamb wave propagation in plate like structures. Due to the very small amplitudes a cumulative effect is used. To get a better overview of this inspection method numerical simulations are essential. Previous studies have developed the analytical description of this phenomenon which is based on the five-constant nonlinear elastic theory. The analytical solution has been approved by numerical simulations. In this work first the nonlinear cumulative wave propagation is simulated and analyzed considering micro-structural cracks in thin linear elastic isotropic plates. It is shown that there is a cumulative effect considering the S1–S2 mode pair. Furthermore the sensitivity of the relative acoustical nonlinearity parameter regarding those damages is validated. Furthermore, an influence of the crack size and orientation on the nonlinear wave propagation behavior is observed. In a second step the micro-structural cracks are replaced by a nonlinear material model. Instead of the five-constant nonlinear elastic theory hyperelastic material models that are implemented in commonly used FEM software are used to simulate the cumulative effect of the higher harmonic Lamb wave generation. The cumulative effect as well as the different nonlinear behavior of the S1–S2 and S2–S4 mode pairs are found by using these hyperelastic material models.It is shown that, both numerical simulations, which take into account micro-structural cracks on the one hand and nonlinear material on the other hand, lead to comparable results. Furthermore, in comparison to the five-constant nonlinear elastic theory the use of the well established hyperelastic material models like Neo–Hooke and Mooney–Rivlin are a suitable alternative to simulate the cumulative higher harmonic generation.

Optics of anisotropic nanostructures

The analytical formalism of Rokushima and Yamakita [J. Opt. Soc. Am. 73, 901–908 (1983)] treating the Fraunhofer diffraction in planar multilayered anisotropic gratings proved to be a useful introduction to new fundamental and practical situations encountered in laterally structured periodic (both isotropic and anisotropic) multilayer media. These are employed in the spectroscopic ellipsometry for modeling surface roughness and in-depth profiles, as well as in the design of various frequency-selective elements including photonic crystals. The subject forms the basis for the solution of inverse problems in scatterometry of periodic nanostructures including magnetic and magneto-optic recording media. It has no principal limitations as for the frequencies and period to radiation wavelength ratios and may include matter wave diffraction. The aim of the paper is to make this formalism easily accessible to a broader community of students and non-specialists. Many aspects of traditional electromagnetic optics are covered as special cases from a modern and more general point of view, e.g., plane wave propagation in isotropic media, reflection and refraction at interfaces, Fabry-Perot resonator, optics of thin films and multilayers, slab dielectric waveguides, crystal optics, acousto-, electro-, and magneto-optics, diffraction gratings, etc. The formalism is illustrated on a model simulating the diffraction on a ferromagnetic wire grating.

Including dispersion and attenuation directly in the time domain for wave propagation in isotropic media.

When sound propagates in a lossy fluid, causality dictates that in most cases the presence of attenuation is accompanied by dispersion. The ability to incorporate attenuation and its causal companion, dispersion, directly in the time domain has received little attention. Szabo [J. Acoust. Soc. Am. 96, 491-500 (1994)] showed that attenuation and dispersion in a linear medium can be accounted for in the linear wave equation by the inclusion of a causal convolutional propagation operator that includes both phenomena. Szabo's work was restricted to media with a power-law attenuation. Waters et al. [J. Acoust. Soc. Am. 108, 2114-2119 (2000)] showed that Szabo's approach could be used in a broader class of media. Direct application of Szabo's formalism is still lacking. To evaluate the concept of the causal convolutional propagation operator as introduced by Szabo, the operator is applied to pulse propagation in an isotropic lossy medium directly in the time domain. The generalized linear wave equation containing the operator is solved via a finite-difference-time-domain scheme. Two functional forms for the attenuation often encountered in acoustics are examined. It is shown that the presence of the operator correctly incorporates both, attenuation and dispersion.

Dispersion in anisotropic media modeled by three-dimensional TLM

The dispersion in anisotropic media modeled by three-dimensional TLM is investigated. Two nodes, the symmetrical condensed node with stubs and the symmetrical super-condensed node are considered. Simple closed-form expressions for the dispersion relations do not exist in general, therefore the investigations are restricted to wave propagation in isotropic media and to wave propagation along the mesh axes and the mesh diagonals. The dispersion analysis for the symmetrical super-condensed node yields a direct relationship between the relative permittivity and relative permeability and the parameters of the scattering matrix.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Wideband acoustic response of fluid-saturated porous rocks: Theory and preliminary results using waveguided samples

The complex dilatational (C*12) and shear (C*44) frequency-dependent elastic constants of nonporous and porous solids are measured in a frequency range 5kHz to 1 MHz. The measurements are performed in a continuous wave acoustic transmission bridge using cylindrical samples. Using samples with free external surfaces produces waveguided modes, enabling experiments to be conducted circumventing problems of diffraction and wavefront spreading, which are difficult to correct over broad frequency ranges. The theory of waveguided elastic wave propagation in isotropic media with complex elastic constants [generalization of the Pochammer–Cree solution, Pochammer, J. Math. (Crelle) 81, 324 (1876)] is presented. For solids with frequency-independent complex elastic constants (constant Q), solution of the resulting dispersion relations reveals the appearance of specific dissipation peaks associated with waveguide geometry. The dispersion relations are utilized in the inversion of torsional shear and extensional experimental results from nonporous and porous solids, thus obtaining their complex frequency-dependent effective elastic constants. A great variation in the elastic constants of Plexiglas with frequency is demonstrated. Evidence of the effects of pore scattering and the reduction of matrix moduli due to water adsorption is shown for porous glass. The effects of adsorption on the effective elastic moduli of porous solids are quantified utilizing the concept of surface elastic constants [M. E. Gurtin and I. A. Murdoch, Arch. Rat. Mech. Anal. 57, 291 (1975)]. The resulting equation reveals the importance of the surface to volume ratio in determining the effective elastic constants.