Unbounded Homogeneous Medium Research Articles

The resolution of modal Doppler shifts in an acoustic waveguide

A method for estimating the source velocity from the acoustic signal field propagating in a dispersive oceanic waveguide is presented. The signal is observed at an omnidirectional receiver in the presence of additive white noise. The source transmits a continuous‐wave (cw) signal, and the source motion is assumed to be uniform (unaccelerated). The ocean is modeled as a waveguide that is horizontally stratified with an arbitrary sound‐speed profile in the vertical. The acoustic field generated by a moving point source in terms of normal modes predicts that each mode has a different Doppler shift, which contrasts with a single Doppler shift in a homogeneous unbounded medium. The modes are considered to remain phase locked with each other, i.e., coherent. The source velocity, including the number of modes involved, can be estimated through resolving Doppler shifts. The method is a time‐domain interpretation of a recently developed eigenstructure technique for multitarget direction finding with passive antenna arrays, in conjunction with smoothing preprocessing scheme to deal with coherent modes [Shan et al., IEEE Trans. Acoust. Speech Signal Process. ASSP‐33 (4) (1985)]. Simulation results that illustrate the performance of the method are presented.

Electromagnetic field of a rectangular patch of uniform and linear distributions of current

Closed-form expressions for a class of indefinite double integrals are presented. Using these formulas, exact analytical expressions for the magnetic vector potential and the electric and magnetic fields of a rectangular patch of uniform and linear distributions of electric current in an unbounded homogeneous medium are derived. The field expressions obtained are valid everywhere, in particular in the source region. Computed results verifying the correctness and accuracy of these expressions are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Radiation from an impulsive line source in an unbounded homogeneous anisotropic medium : van der Hijden, J H M T Geophys J R Astr SocV91, N2, Nov 1987, P355–372

Radiation from an impulsive line source in an unbounded homogeneous anisotropic medium : van der Hijden, J H M T Geophys J R Astr SocV91, N2, Nov 1987, P355–372

Radiation from an impulsive line source in an unbounded homogeneous anisotropic medium

Summary. The space-time elastic wave motion generated by an impulsive line source in a homogeneous anisotropic medium is calculated with the aid of the Cagniard-de Hoop method. Two types of sources are considered in detail, viz. a line source of expansion (model for an explosive source) and a line force (model for a mechanical vibrator). Numerical results are presented for the radiated particle velocity in the medium including those regions of space where shear-wave triplication occurs. There is a marked difference in the time response observed for the two types of sources and for the different positions of the receiver with respect to the source position. These waveform differences are important when the radiated wave is used to determine experimentally the elastic properties of the medium. As compared with the traditional Fourier-integral transform method to handle this problem, the computation time with the present method is considerably less.

Deep penetration of a hardronic internuclear cascade into matter at high energy and design of radiation shielding for strong-current accelerators

The author explain some physical laws of the propagation of a hadronic internuclear cascade in matter at high energy at large distances from the source and we derive some approximate asymptotic expressions for the distribution function of hadronic functions. We consider the propagation of an N-component internuclear cascade (N/sub 1/ species of neutral hadrons and N-N/sub 1/ charged hadrons) in an unbounded homogeneous medium with a flat monodirectional (along the z axis) and monoenergetic (energy E/sub 0/) source of particles of species j/sub 0/, located in the plane z = O. We assume that secondary articles produced in elementary acts of interaction and decay fly out in the direction of the motion of the primary particle, which is entirely natural in the high-energy range. The system of kinetic equations that describes the cascade propagation is given.

Integral transport theory for test particles in the presence of a time-dependent conservative force

An integral space-independent transport theory for test particles diffusing by collision against the field particles of an unbounded homogeneous host medium, and in the presence of a time-dependent external conservative force, is developed in the paper. Both the cases of a general and of a separable scattering-creation kernel are considered. Functional and constructive features of the theory are also discussed with particular regard to these case of/v-cross sections. Curves, representing the distribution function of the test particles considered and evaluated according to an appropriate specialization of the physical inputs, are shown.

On singular models of a thin inclusion in a homogeneous elastic medium

The equilibrium of an unbounded homogeneous and anisotropic elastic medium containing an inclusion, one of whose characteristic dimensions is much less than the other two is considered. It is assumed that the elastic moduli of the medium and the inclusion differ substantially. The principal terms of the expansion of the elastic fields in the neighbourhood of the thin inclusion are constructed in asymptotic series in small parameters of the problem: the ratio of the characteristic linear dimensions of the inclusion and the ratio of the characteristic values of the elastic moduli of the medium and the inclusion. The problem of constructing the principal terms of these expansions reduces to solving two-dimensional pseudo-differential equations derived by using the procedure of matching the external and internal asymptotic expansions /1, 2/. The results obtained enable two singular models of thin inclusions to be formulated for the cases when their elastic moduli are substantially greater and substantially less than the elastic modulus of the medium. It is shown that one of these models is equivalent to the model of a thin inclusion proposed in /3, 4/.

Strong motion prediction using mathematical modeling techniques

abstract The Parkfield, California, earthquake of 1966 produced the first recording of ground motion in the immediate vicinity of an earthquake fault. The simplicity of displacement pulse observed at station 2 aroused great interest among seismologists working in the area of earthquake source mechanism. Since then, numerous theoretical works on simulating strong motion have been carried out, starting with the use of a simple kinematic model assuming a uniform slip over a fault plane. The assumption of uniform slip had to be modified to explain the strong motion observed during the Parkfield earthquake of 1966, the Borrego Mountain earthquake of 1968, the San Fernando earthquake of 1971, the Imperial Valley earthquake of 1979, the El Asnam earthquake of 1980, and others. Through these studies, the simulation technique has been advanced to include a more realistic medium. In the beginning, most methods used Green9s function for an unbounded homogeneous medium. Then, the free surface effect, the effect of a sedimentary layer as well as laterally heterogeneous basin structure have been included in the simulation. The above simulation studies were, however, restricted to relatively low-frequency waves represented in a displacement or velocity seismogram. In order to predict an acceleration seismogram dominated by high-frequency waves, hybrid models have been proposed in which gross features of rupture propagation are specified deterministically, but the details of the process are described by a stochastic model specified by a small number of parameters. The application of these models to several California earthquakes revealed encouraging results that some of the key parameters of the model are stable among earthquakes. Mathematical modeling techniques will play an important role in the prediction of strong motion for a great earthquake in California as well as for major earthquakes outside California, based on the records of moderate earthquakes already acquired in California.

Sources of stress in two half-spaces

Abstract Solutions are given for two versions of the problem of displacements from a source of stresses (the domain of displacement or stress field peculiarities) in two isotropic linearly elastic half-spaces, on whose plane boundaries conditions relating the boundary values of the derivatives of the displacements of different orders are satisfied. The method of solution proposed permits expressing the displacement in the half-spaces in terms of indefinite integrals of the gradients of the displacements produced by the stress source in an unbounded homogeneous medium.

Time-dependent distortion of the distribution function via an operational formulation of the boltzmann equation

An operational formulation of the linearized Boltzmann equation is proposed for studying the effects induced by an external time-dependent conservative force on the distribution function of certain test particles, which diffuse by collisions against certain field particles of an unbounded homogeneous host medium. Firstly the theory of the method is expounded for both the cases of a general and a separable scattering-creation kernel. Applications are then presented for the two models of constant collision frequency and of constant mean free path, respectively. The relevant distortion factors for the distribution function are evaluated for two specializations of the external time-dependent force. Curves for the distribution function as functions of time and velocity are also given. The role played by the creation and removal laws in establishing the distortion effects is at last discussed.

Isotropic radiation from a steadily pulsating multidimensional distribution

A multidimensional dynamical configuration is considered, com prising an isotropic homogeneous unbounded medium steadily oscillating in response to a real constant frequency imparted by an embedded source. A Green function to the problem is first derived, and leads to the compounding of the general exact solution for an arbitrary spatial distribution of the source function. Specializations are made to yield an asymptotic solution, a spherically symmetric solution and an axisymmetric solution. All physical results satisfy an initially imposed radiation condition, whose consequence is subsequently accommodated during the construction of the Green function. Significances are attached and interpretations provided, namely, in terms of the associated isotropic waves. A cursory application is made to examine the oscillatory motion, when reduced to the isotropic state, of a rotating magnetoelastic system. An appendix is included to deal with a certain spherical integral encountered during the investigation into the axisymmetric case.

Transformation of earthquake displacement field for spherical earth

abstract An addition theorem for spherical waves is used to transform the displacement field due to an arbitrary shear dislocation in an unbounded homogeneous medium with origin at the focus to that in a coordinate system with origin at the center of the Earth. The final results are expressed in terms of the eigenvector solutions of the vector Naviér equation of dynamical elasticity. Some serious mistakes in recent papers on the subject have been pointed out.

The elastic field of an inclusion in an anisotropic medium

A new general method of solving problems of anisotropic elasticity, by successive approximations, is applied to find the elastic field of a basic physical problem. In this problem an arbitrary part of an unbounded homogeneous anisotropic medium is a misfitting inclusion that would undergo an arbitrary uniform strain if its surrounding material was absent. An inclusion of ellipsoidal shape is particularly considered; a convergent series of successive approximations is used to generate, and bound as restrictively as desired, the resulting uniform strain in this inclusion. Detailed bounds are given for a spherical inclusion in a medium of cubic anisotropy. Some applications are mentioned, especially the determination of the elastic field when the material in the ellipsoidal inclusion differs from that in its surroundings.

Advances in the theory of anisotropic elastic wave propagation

The qualitative effects of anisotropy on elastic waves propagating in a solid medium were well known to Lord Kelvin. Having been recognized, however, these effects were neglected as being of secondary importance in the dynamics of elastic mediums. This relegation of anisotropy to a secondary role in the dynamics of elastic mediums was undoubtedly justified, particularly in view of the relatively primitive state of experimental elasticity and seismology during Kelvin's time. It was not until after World War 2 that the effects of anisotropy again received serious attention. This was primarily because of the development of ultrasonic techniques for the measurement of dynamic elastic constants of pure crystals. In such experimental problems anisotropy no longer plays a secondary role. The study of how a disturbance, generated by a transducer on the surface of a crystal, spreads through the crystal led to the discovery, by Musgrave, that the wave surface, which forms the boundary of the spreading disturbance, could have cuspidal singularities. This had not been previously predicted, although it could have been predicted by Kelvin had he been more familiar with algebraic geometry. In another area of research (seismology), the post World War 2 years also saw a rise of interest in anisotropy, particularly in the effect of possible continental anisotropy on the propagation of Rayleigh waves. The increased experimental activity in crystal dynamics and the improvement of experimental seismology to the point where secondary effects became important resulted in a number of theoretical investigations into the propagation of plane, time harmonic waves in anisotropic mediums. By 1959 the state of the theoretical and experimental understanding of anisotropic elastic wave propagation had advanced to the point where rigorous wave theoretical calculations were in order. All the simple, solvable problems of isotropic dynamic elasticity, i.e. the initial value problem for an unbounded homogeneous medium, the mixed initial and boundary value problem for a surface line source on a half‐space, the normal mode problem for an elastic wave guide, etc., can be formulated and solved in detail in the anisotropic case. Furthermore, the solutions can be obtained by extensions of the usual transform methods used in isotropic problems. The results can be physically interpreted by means of classical differential geometry. This review summarizes those anisotropic problems treated since 1959 and the techniques developed to solve them.