Isotropic Dissipative Medium Research Articles

Effect of Rotation, Magnetic Field and Initial Stresses on Propagation of Plane Waves in Transversely Isotropic Dissipative Half Space

The problem regarding the reflection of plane waves in a transversely isotropic dissipative medium is considered, in which we are studying about the reflection of incidence waves in initially stressed dissipative half space. After solving the governing equations, we find the two complex quasi-P (qP) and quasi-SV (qSV) waves. The occurrence of reflected waves is studied to calculate the reflection coefficient and the energy partition of incidence wave at the plane boundary of the dissipative medium. Numerical example is considered for the reflection coefficient and the partition of incident energy, in which we study about the effect of rotation, initial stresses and magnetic field.

On the existence conditions for a fast surface wave

Conditions are obtained for the existence of a fast-moving surface electromagnetic wave (with a speed close to the speed of light in the vacuum) on a flat interface between the vacuum and an isotropic dissipative medium with a permittivity . The interfaces considered include vacuum–seawater, vacuum–metal, vacuum–plasma, and vacuum–dielectric. Conditions for the existence of negligibly damped surface waves are considered for extremely high (vacuum–seawater, vacuum–metal) and very low (vacuum–plasma, vacuum–dielectric) values. It is shown that at least in these two limit cases, the phase wave velocity and the group wave velocity pass synchronously through the speed of light in the vacuum, which can be considered the reason why surface waves exist at the interface between vacuum and a collisionless plasma (with and ) and do not exist at the interface between the vacuum and a weakly absorbing dielectric (with and ). In the first limit case, it is shown that both the phase and group velocities pass at , implying that a surface wave exists at the vacuum-metal interface (with ), but that a surface wave (Zenneck's wave) cannot exist at the vacuum–seawater interface (with ).

Reflection of Plane Waves from a Free Surface of an Initially Stressed Transversely Isotropic Dissipative Medium

The governing equations of a transversely isotropic dissipative medium are solved analytically to obtain the speeds of plane waves. The appropriate solutions satisfy the required boundary conditions at the stress-free surface to obtain the expressions of the reflection coefficients of reflected quasi-P (qP) and quasi-SV (qSV) waves in closed form for the incidence of qP and qSV waves. A particular model is chosen for numerical computation of these reflection coefficients for a certain range of the angle of incidence. The numerical values of these reflection coefficients are shown graphically against the angle of incidence for different values of initial stress parameter. The impact of initial stress parameter on the reflection coefficients is observed significantly.

Attenuation vector in heterogeneous, weakly dissipative, anisotropic media

SUMMARY The attenuation vector plays an important role in investigating waves propagating in dissipative, isotropic or anisotropic media. It is defined as the imaginary part of the complex-valued traveltime gradient. The real part of the complex-valued traveltime gradient is referred to as the propagation vector. In this paper, simple expressions for the attenuation vector are derived for seismic body waves propagating in heterogeneous, weakly dissipative, anisotropic media. The ray-theory perturbation method is used, in which the dissipative medium is considered to be a small perturbation of a perfectly elastic medium. The general perturbation procedure proposed by Klimey (2002) is very suitable for this purpose, as it does not require ray tracing in dissipative media; the computation of rays in the reference perfectly elastic medium, followed by quadratures along them, is quite sufficient. It is shown that the waves propagating in a heterogeneous, weakly dissipative, anisotropic or isotropic medium are, in general, inhomogeneous. This means that their attenuation vector is not parallel to the propagation vector. This holds even for waves generated by a point-source in a homogeneous anisotropic weakly dissipative medium. An exception is a wave generated by a point-source in a homogeneous isotropic dissipative medium, where the generated wave is homogeneous. Thus, the commonly used concept of homogeneous waves can be applied neither to heterogeneous nor to anisotropic dissipative media. The situation is different for plane waves propagating in homogeneous dissipative isotropic or anisotropic media. In this case, the homogeneity or inhomogeneity of the plane wave may be chosen freely. Besides the attenuation vector, we also study the complex-valued ray-velocity vector. In heterogeneous media, the complex-valued ray-velocity vector is generally inhomogeneous, that is, its real and imaginary parts are not parallel. It is also shown that twice the scalar product of the attenuation vector with the ray-velocity vector in the reference medium yields 1/Q, where Q is the positionand direction-dependent quality factor. Quality factor Q does not depend on the inhomogeneity of the wave under consideration and offers a convenient measure of intrinsic material dissipation.

Quality factor Q in dissipative anisotropic media

In an isotropic dissipative medium, the attenuation properties of rocks are usually specified by quality factor [Formula: see text], a positive, dimensionless, real-valued, scalar quantity, independent of the direction of wave propagation. We propose a similar, scalar, but direction-dependent quality [Formula: see text]-factor (also called [Formula: see text]) for time-harmonic, homogeneous or inhomogeneous plane waves propagating in unbounded homogeneous dissipative anisotropic media. We define the [Formula: see text]-factor, as in isotropic viscoelastic media, as the ratio of the time-averaged complete stored energy and the dissipated energy, per unit volume. A solution of an algebraic equation of the sixth degree with complex-valued coefficients is necessary for the exact determination of [Formula: see text]. For weakly inhomogeneous plane waves propagating in arbitrarily anisotropic, weakly dissipative media, we simplify the exact expression for [Formula: see text] con-siderably using the perturbation method. The solution of the equation of the sixth degree is no longer required. We obtain a simple, explicit perturbation expression for the quality factor, denoted as [Formula: see text]. We prove that the direction-dependent [Formula: see text] is related to the attenuation coefficient [Formula: see text] measured along a profile in the direction of the energy-velocity vector (ray direction). The quality factor [Formula: see text] does not depend on the inhomogeneity of the plane wave under consideration and thus is a convenient measure of the intrinsic dissipative properties of rocks in the ray direction. In all other directions, the quality factor is influenced by the inhomogeneity of the wave under consideration. We illustrate the peculiarities in the behavior of [Formula: see text] and its accuracy on a model of anisotropic, weakly dissipative sedimentary rock. Examples show interesting inner loops in polar diagrams of [Formula: see text] in regions of S-wave triplications.

Radiation of a Hertzian dipole immersed in a dissipative medium

The general radiation formula for a Hertzian dipole immersed in an isotropic dissipative medium of infinite extent has been derived. As a boundary condition of the source, it is assumed that the dipole moment is a given quantity. When the conductivity of the medium is finite, the total radiating power is found to be infinite. Thus, in order to define a finite physically meaningful quantity, the dipole must be insulated. The total radiating power is then a function of the thickness of the insulator and the constants of the media. When the radius of the spherical insulator is large compared to a wavelength, the reflection coefficient of the wave traveling from the dielectric to the dissipative medium with the dipole as a source reduces to that of a plane wave as derived from Fresnel's equations. The similarity between this and the problem by Weyl (1919) is discussed.

Elastodynamic Response of a Unidirectional Composite Laminate to Concentrated Surface Loads: Part I

The quantitative features of the elastodynamic field produced in a unidirectional graphite/epoxy composite laminate by surface loads are studied. The material of the composite is modeled as a transversely isotropic dissipative medium with its symmetry axis parallel to the laminate surface. The surface displacement produced by a transient concentrated point load on the surface of a uniform half-space and a plate of finite thickness is calculated by means of a multiple transform technique followed by inversion through numerical integration. The two-dimensional (plane-strain) line load problems for the plate are also solved as special cases. In addition, an approximate solution based on the classical plate bending theory which includes transverse shear and rotary inertia is obtained and the range of validity of the approximate solution is determined through comparison with the exact results. The multiple transform technique is shown to be a viable alternative to finite element and other numerically intensive computational schemes for wave field calculations in composite laminates. Part I of the paper contains the formulation, and a discussion of the general features of the response, the numerical treatment and results are given in Part II.

Elastodynamic Response of a Unidirectional Composite Laminate to Concentrated Surface Loads: Part II

The quantitative features of the elastodynamic field produced in a unidirectional graphite/epoxy composite laminate by surface loads are studied. The material of the composite is modeled as a transversely isotropic dissipative medium with its symmetry axis parallel to the laminate surface. The surface displacement produced by a transient concentrated point load on the surface of a uniform half-space and a plate of finite thickness is calculated by means of a multiple transform technique followed by inversion through numerical integration. The two-dimensional (plane-strain) line load problems for the plate are also solved as special cases. In addition, an approximate solution based on the classical plate bending theory which includes transverse shear and rotary inertia is obtained and the range of validity of the approximate solution is determined through comparison with the exact results. The multiple transform technique is shown to be a viable alternative to finite element and other numerically intensive computational schemes for wavefield calculations in composite laminates. The mathematical formulation of the problem was described in Part I of the paper, numerical treatment and results are presented in this part.

A theoretical and experimental study of the insulated loop antenna in a dissipative medium

The performance of a bare antenna as a radiator in a dissipative medium can be significantly altered by placing a dielectric coating around the antenna. For certain antenna types and specific properties of the dissipative medium, the dielectric insulation has been shown to improve the performance of the antenna. In this paper the thin‐wire circular‐loop antenna centered in an insulating spherical cavity and immersed in an infinite homogeneous isotropic dissipative medium is analyzed. A Fourier series solution for the antenna current distribution is derived. The coefficients of the series are a combination of the coefficients for the loop in an infinite dielectric medium, which were determined previously by Wu [1962], and a second term, which is an infinite sum. The properties of the sum are examined, and expressions for the antenna input admittance, impedance, and electromagnetic field in the dissipative medium are obtained. Numerical results are presented for specific antenna sizes and dissipative media. For the special case of a loop with a uniform current distribution in an electrically small sphere the expression for the input impedance is shown to reduce to a solution previously given by Wait and Spies [1964]. The construction of an experimental model for the antenna is discussed, and antenna input impedances and current distributions measured with the model immersed in various salt‐water solutions are compared with theoretical calculations.

The current in bare circular loop antennas in a dissipative medium

The distribution of current in bare circular loop antennas has been evaluated from the theory of T. T. Wu. Computations have been made for loops in air and in an infinite homogeneous isotropic dissipative medium. Currents are shown graphically for loops with circumferences up to two wavelengths in media, with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\alpha/\beta</tex> ranging from zero to one, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k=\beta-j\alpha</tex> is the complex wave number.

The admittance of bare circular loop antennas in a dissipative medium

The normalized input admittance of thin bare circular loop antennas has been evaluated from the theory of T. T. Wu. Computations have been made for loops in air and in an infinite homogeneous isotropic dissipative medium. A comparison is also made with Storer's theory of the loop. Numerical results are given in the form of graphs for several wire sizes and for loops up to two and one-half wavelengths in circumference. The properties of the medium are represented by the ratio \alpha/\beta in the range from zero (perfect dielectric) to one (good conductor); \alpha and \beta are the imaginary and real parts of the complex propagation constant k=\beta-j\alpha = \omega\sqrt{\mu(\epsilon-j\sigma/\omega)} where \mu is the permeability, \epsilon the dielectric constant, and \sigma the conductivity of the medium.

The complete electromagnetic field of a half-wave dipole in a dissipative medium

The electromagnetic field of a dipole antenna immersed in a homogeneous isotropic dissipative medium is studied theoretically and experimentally. Expressions are obtained for the electric and magnetic fields both near and far in terms of currents and admittances that are more accurate than in conventional formulations. The elliptically polarized field near a half-wave dipole is evaluated theoretically and compared with experimental measurements over a wide range of values of attenuation.