Cagniard's Technique Research Articles

On a Wavelet-Based Method for the Numerical Simulation of Wave Propagation

A wavelet-based method for the numerical simulation of acoustic and elastic wave propagation is developed. Using a displacement-velocity formulation and treating spatial derivatives with linear operators, the wave equations are rewritten as a system of equations whose evolution in time is controlled by first-order derivatives. The linear operators for spatial derivatives are implemented in wavelet bases using an operator projection technique with nonstandard forms of wavelet transform. Using a semigroup approach, the discretized solution in time can be represented in an explicit recursive form, based on Taylor expansion of exponential functions of operator matrices. The boundary conditions are implemented by augmenting the system of equations with equivalent force terms at the boundaries. The wavelet-based method is applied to the acoustic wave equation with rigid boundary conditions at both ends in 1-D domain and to the elastic wave equation with a traction-free boundary conditions at a free surface in 2-D spatial media. The method can be applied directly to media with plane surfaces, and surface topography can be included with the aid of distortion of the grid describing the properties of the medium. The numerical results are compared with analytic solutions based on the Cagniard technique and show high accuracy. The wavelet-based approach is also demonstrated for complex media including highly varying topography or stochastic heterogeneity with rapid variations in physical parameters. These examples indicate the value of the approach as an accurate and stable tool for the simulation of wave propagation in general complex media.

The traveling point load revisited

The nonaxisymmetrical boundary value problem of a point load of normal traction traveling over an elastic halfspace is revisited. The transient problem was first solved in the space-time domain in a paper by D.C. Gakenheimer and J. Miklowitz [Transient excitation of an elastic half space by a point load traveling on the surface, J. Appl. Mech. 36 (1969) 505–515] who used a modified Cagniard technique. As compared to this original work, the analysis in the present paper is more straightforward, thus allowing more insight into this important canonical problem. Some procedural matters in the analysis are set straight. Theoretical wavefronts are shown for each sonic load speed and, moreover, numerical results are presented for transonic and supersonic load motion.

Dynamic weight functions for a moving crack. I. Mode I loading

Dynamic weight functions are discussed, for arbitrary time-dependent loading of a plane semi-infinite crack extending at constant speed in an infinite isotropic elastic body. Then, the weight function appropriate to the case of general normal (or Mode I) loading is constructed explicitly, employing Fourier transforms to develop and solve a Wiener—Hopf problem. Transforms are inverted by a variant of Cagniard's technique. The weight function is then employed to develop a relationship, in the framework of first-order perturbation theory, between the Mode I stress intensity factor and a small but otherwise arbitrary time-varying deviation from straightness of the edge of the crack.

Generalized characteristic method of elastodynamics

A generalized characteristic method to solve two-dimensional time domain elastodynamics problems is developed. The (real) characteristic surfaces in x, y, t-space for the Navier's equations are continued to the complex domain, in which they are called generalized characteristics. The first order ordinary differential equations along the generalized characteristics which are equivalent to the Navier's equations are used to simplify an initial-boundary value problem of elastodynamics to a set of algebraic equations whose unknowns are the first order derivatives of the displacement components. The reflection coefficients of longitudinal and transverse waves, whose fronts can be of arbitrary convex shape, from a traction free boundary are also found to be algebraic expressions. The method makes the inversion of Laplace transform required by Cagniard's technique unnecessary and can be used directly to analyse the head wave and the regions of influence of the reflected waves. Closed form expressions for the time domain Green's function in the whole plane and the solution of Lamb's problem for a buried concentrated force are given as applications of the method. Formulae for the surface displacement take on a simpler form than those reported previously. Finally, some numerical results for the displacement of receivers on and below the surface are given. The displacements of reflected waves approach infinity when they are treated separately but superposing the displacements due to the reflected p- and s-waves shows that the transient Rayleigh pulse is finite.

Transient Interfacial Stress between Two Bonded Dissimilar Off-Angled Anisotropic Media to an Impulsive SH Source.

Transient interfacial stress between two bonded dissimilar anisotropic media to an impulsive line SH-source is considered. The principal axes of each anisotropic medium do not coincide with the line on the plane of the bonded interface. Applying the well-known Cagniard's technique, an exact closed form solution for the transient interfacial stress is derived. It is shown that the distribution of the interfacial stress is affected more by the off-angle in the source side medium than by that in the transmitted one.

The Transient Stress in an Elastic Half-space Subjected to a Semi-infinite Line Load varying as Unit Step Function on Its Surface

This paper presents an exact solution for the transient response in the interior of an elastic half space subjected to a normal or a tangential semi-infinite line load varying with time as a Heaviside unit step function on its surface. The solution is based on the stress function approach, the Laplace transform and the double Fourier transform method applied to a non-axisymmetric dynamical problem. The Laplace inverse transform which is performed by the three-dimensional Cagniard's technique gives an exact solution for the stress wave propagations. The results of numerical evaluation are shown graphically for the stress variation versus time in the interior of a half space.

Generation of acoustic waves by an impulsive point source in a fluid/solid configuration with a plane boundary

The space–time acoustic wave motion generated by an impulsive monopole point source in a fluid/solid configuration with a plane boundary is calculated with the aid of the modified Cagniard technique. The source is located in the fluid, and numerical results are presented for the reflected-wave acoustic pressure, especially in those regions of space where head-wave contributions occur. There is a marked difference in time response in the different regimes that exist for the wave speed in the fluid in relation to the different wave speeds (compressional, shear, Rayleigh) in the solid. These differences are of importance to the situation where the reflected wave in the fluid is used to determine experimentally the elastic properties of the solid.

On the exact evaluation of displacements within a semi-infinite solid due to a buried line source

Garvin’s variant of the Cagniard technique has been used to evaluate exactly the displacements at points within a homogeneous, isotropic, perfectly elastic, solid half space due to a buried explosive line source. The disturbance at any given time is confined to a region bounded by the free surface of the half space and theP wave front within it. The disturbance associated with the Rayleigh pulse affects a smaller region bounded by the free surface and the PP wave front.

Transient response of an elastic solid to a moving torsional load in a cylindrical bore an approximate solution

This paper is an attempt to apply Cagniard's technique to a transient problem of a dispersive medium. The transient response of an elastic solid to a moving torsional ring load in a cylindrical bore is considered. By making an approximation to the modified Bessel function, Cagniard's technique is applied. An approximate solution, which has explicit expressions for the wave front geometry and has a more suitable form for numerical computations, is obtained. The applicability of the solution is also discussed.

Generation of acoustic waves by an impulsive point source in a fluid/porous‐medium configuration with a plane boundary

The space‐time acoustic wave motion generated by an impulsive monopole point source in a fluid/porous‐medium configuration with a plane boundary is calculated with the aid of the modified Cagniard technique. To describe the macroscopic propagation of linear acoustic waves through the fluid‐saturated porous solid, four coupled partial differential equations are used that follow upon applying a volume averaging procedure to the two constituents, a perfectly elastic solid and an ideal fluid of the porous medium. The source is located in the fluid part of the configuration. Numerical results are presented for the reflected‐wave acoustic pressure, especially in those regions of space where head waves occur. There is a marked difference in time response in the different regimes that exist for the wave speed in the fluid in relation to the different wave speeds in the porous medium (fast compressional, slow compressional, shear). These differences are of importance to the situation where the reflected wave in the fluid is used to determine experimentally the acoustic properties of the porous medium.

Generation of acoustic waves by an impulsive line source in a fluid/solid configuration with a plane boundary

The space–time acoustic wave motion generated by a two-dimensional, impulsive, monopole line source in a fluid/solid configuration with a plane boundary is calculated with the aid of the modified Cagniard technique. The source is located in the fluid, and numerical results are presented for the reflected-wave acoustic pressure, especially in those regions of space where head wave contributions occur. There is a marked difference in time response in the different regimes that exist for the wave speed in the fluid in relation to the different wave speeds (compressional, shear, Rayleigh) in the solid. These differences are of importance to the situation where the reflected wave in the fluid is used to determine experimentally the elastic properties of the solid.

From Cagniard's method for solving seismic pulse problems to the method of the differential transform

Cagniard's method is well-known in mathematical geophysics. The integral transform method of Cagniard involves path deformations in the complex plane, residue calculus and, in some cases, branch cuts. The technique of the differential transform, which evolved from that of Cagniard, involves an integral-free transform and avoids the integral considerations associated with Cagniard's technique. These two techniques are examined here by considering their application to Garvin's problem, a classical pulse-propagation problem in geophysics. This examination demonstrates the superiority of the technique of the differential transform over that of Cagniard, thus confounding the belief that the intricate integral considerations involved in Cagniard's technique are unavoidable.

Transient Response of an Elastic Half Space to Moving Loads

In this paper the transient response of an elastic half space to a non-uniformly moving point load is considered and a solution procedure is developed for the problem of moving loads. By using the well-known Cagniard's technique, a general solution of displacement is obtained in the form of integral for the inner and the surface responses. Then, the title problem is reduced to a simple consideration on Arrival Time Function, the behaviour of which governs the range of integration. The solution obtained here should be applied to miscellaneous problems of a moving load, the speed of which is non-uniform and largely arbitrary.

Pulsed electromagnetic radiation from a line source in a two‐media configuration

The pulsed electromagnetic radiation from a two‐dimensional line source in a lossless two‐media configuration is investigated theoretically. The author's modification of Cagniard's technique is used to derive closed‐form expressions for the electromagnetic field components anywhere in the configuration. Numerical results are presented pertaining to the incident, the reflected, and the transmitted wave for different positions of the source and the observer, as well as for different contrasts of the two media. Also, the corresponding total‐field plots are shown. In an appendix, the modified Cagniard contours pertaining to a lossless N‐media configuration are discussed and some two‐media modified Cagniard contours are shown.

Transient disturbance produced in an elastic half-space by impulsive shearing traction distributed over a circular region

The displacement produced in a semi-infinite, homogeneous, isotropic elastic medium by the application of shearing traction over a circular portion of the half-space has been evaluated in exact form by Cagniard's Technique (Cagniard, 1962;Gakenheimer andMiklowitz, 1969).

Transient response of a layered elastic half space subjected to a reciprocating anti-plane shear load

The well-known Cagniard's technique seems very useful for analyzing the transient problem of an elastic body subjected to a moving load with uniform velocity, but, in the cases of non-uniform velocity it would be difficult to apply the technique directly to analyze the problem. For such cases, instead of the technique, the convolution theorem of the Laplace transform or the dynamic Betti-Rayleigh reciprocal theorem has been applied usefully by many authors.In this paper, an inversion scheme which enables us to apply the Cagniard's technique directly to those difficult problems is presented by solving the problem of a reciprocating anti-plane shear load applied to a layered elastic half space. In this treatment it will be found clearly that our technique is simpler and more systematic than any other techniques for solving the problem. Numerical results are shown in the form of curves and the wave-front singularities are also investigated.

Transient analysis of a three-dimensional plate by the ray grouping technique

The propagation of transient waves in an elastic plate excited by an arbitrary loading is investigated. Exact three-dimensional transform solutions are derived for a plate suddenly loaded on one of its bounding surfaces, with the inversion effected by Cagniard's technique. The solution is based on three canonical problems for finding the n-th reflected waves from only the information on the (n − 1)-th reflected waves. The technique automatically groups rays which arrive simultaneously at one point, thus simplifying the computations needed by the ray tracing technique. A nonaxially symmetric sample loading is selected for demonstrating the technique, and numerical results are presented. The Ray Grouping Technique can be extended to the layered medium case.

Response of an acoustic-elastic system to a transient source in the acoustic medium: Analytical modelling for the air-coupled wave

The present work considers an elastic solid half-space contiguous to an acoustic half-space. Since the acoustic half-space does not support shear, gases and liquids are included in the present study. By the analysis, one deduces the stresses in the solid produced by a spherical source of pressure embedded in the acoustic medium. The solution is obtained by using one-sided and two-sided Laplace transforms and by relying quite heavily on the properties of the Green's function for the acoustic medium. These properties make compatible the prescription of the source pressure on a spherical surface and the application of the boundary conditions at the plane acoustic-elastic interface. For an arbitrary pressure history input, the transform inversion is effected by Cagniard's technique to yield the exact solution.

On the displacements produced in a porous elastic half-space by an impulsive line load. (Non-dissipative case)

On the basis of Biot's dynamical theory of poroelasticity the disturbance produced by an impulsive line load in a porous elastic half-space is studied. Using the Laplace-Fourier transform we have solved for the displacement potentials in terms of which the displacements and stresses in the transformed space are expressible. The expressions for the solid displacements in the interior as well as on the surface of the half-space are obtained by Cagniard's technique. The displacements are expressed in terms of six algebraic terms — three of which are identified as the disturbance due to specific wave fronts and the others represent the head wave contributions. When specialized our results agree with those for the common elastic half-space.

The Response of a Fluid-Solid Interface to a Moving Disturbance

Abstract : The problem of a moving point force on a fluid-solid interface was considered by the authors in preceding papers. A related three-dimensional problem is studied in the present investigation. The moving point force is replaced here by a source which moves at a given distance through the fluid parallel to the interface. The steady-state solution to this problem for supersonic source velocities was obtained through the use of DeHoop's modification of Cagniard's technique, and expressions for the displacements at the interface are presented. (Author)