On Elastodynamic Diffraction of Waves From a Line-Load by a Crack

1984 ◽  
Vol 51 (2) ◽  
pp. 335-338 ◽  
Author(s):  
A. K. Gautesen
Keyword(s):  
Plane Wave ◽  
Simple Relation ◽  
Finite Width ◽  
Far Field ◽  
Angle Of Incidence ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Line Load ◽  
Fixed Angle ◽  
Plane Wave Incident

For the two-dimensional problem of elastodynamic diffraction of waves by a crack of finite width, we assume that the solution corresponding to incidence of a plane wave of either longitudinal or transverse motions under a fixed angle of incidence is known. We first show how to construct the solution corresponding to an in-plane line-load (the Green’s function) from this known solution. We then give a simple relation between the far field scattering patterns corresponding to a plane wave incident under any angle and the far field scattering patterns corresponding to the known solution. This relation is a generalization of the principle of reciprocity.

scholarly journals Διάδοση κυμάτων σε ανομοιογενή μέσα

2007 ◽  
Author(s):  
Κωνσταντίνος Αναγνωστόπουλος
Keyword(s):  
Field Equation ◽  
Plane Wave ◽  
Inverse Scattering ◽  
Scattering Problem ◽  
Field Operator ◽  
Sampling Point ◽  
Far Field ◽  
Two Dimensional ◽  
Time Harmonic

The scope of this doctoral thesis is, first, to develop an analytical, in principle, method for the solution of the two-dimensional scattering problem of time-harmonic elastic plane waves by a homogeneous orthotropic scatterer, second, to establish the complete theoretical framework, which is necessary for the application of the Linear Sampling Method (LSM) to the problem of reconstructing the support of twodimensional elastic anisotropic inclusions embedded in isotropic media and, third, to derive an extension of the Factorization Method (FM) to the inverse elastic scattering problem by penetrable isotropic bodies for time-harmonic plane wave incidence. Aconcise description of the contents of the thesis is outlined below. Chapter one contains a detailed bibliographical search, which is related to the analytical and numerical methods (with emphasis on the former) usually employed for the solution of the direct scattering problem by anisotropic elastic bodies as well as to those inverse scattering techniques, which are usually referred to as sampling and probe methods and, in particular, the LSM and the FM. Chapter two commences with a brief discussion of some fundamental results from the linearized theory of dynamic elasticity. The problem of a rigorous analysis of the elasticity equation governing the elastic behaviour of an orthotropic material in two dimensions is then addressed. This analysis, which is based on a suitable diagonalization applied to the underlying differential system and a plane wave expansion of the sought field, results in a Fourier series expansion for the displacement field describing the elastic deformations of the orthotropic medium and is complemented by the results of appendix A. A mathematical model for the solution of the associated transmission scattering problem, taking advantage of the aforementioned expansion, is also settled and analyzed. The details of its numerical treatment can be found in appendix B. Finally, numerical results for several inclusion geometries and a system thereof with material properties characterized by the cubic symmetryclass -a special case of the orthotropic class of symmetry- are presented. In chapter three, the LSM is extended to the case of a two-dimensional homogeneous anisotropic inclusion embedded in an isotropic background medium. The concepts of the elastic Herglotz function, the elastic far-field operator and the corresponding far-field equation, on which the formulation of the LSM heavily relies, are first introduced. Then, the proposed inverse scattering scheme is introduced and discussed in detail. By means of an appropriate operator decomposition of the far-field operator,the main theorem of the method, concerning the characterization of the behaviour of an approximate solution to the far-field equation at the boundary of the scatterer, is proved. In the end of the third chapter, the performance of the LSM is examined by applying it to a set of different geometric configurations of the elastic inclusion, filled with a cubic anisotropic material. An investigation of the effect of the various parameters entering the problem, such as the scatterer’s degree of anisotropy, the polarization of the elastic point source located at the sampling point and the noise level in the synthetic far-field data, on the reconstructed geometric profiles’ quality,is carried out. In the fourth chapter, the FM is elaborated for the shape reconstruction of a penetrable isotropic elastic body from the knowledge of the far-field pattern of the scattered fields for plane incident waves. The theoretical analysis is conducted in three dimensions and focuses on deriving a factorization of the far-field operator, which is the cornerstone for the applicability of the particular inversion scheme, and investigating thorougly the properties of the involved operators. This investigation gave birth to a number of interesting by-products and one of them, namely, a regularity estimate for the solution of a particular form of the corresponding interior transmission problem, is the subject matter of appendix C. By means of the proposed factorization, a series of theorems, which finally lead to an explicit characterization of the scattering obstacle, is then proved. In the end of the chapter, the performance of the investigated inverse scattering technique is demonstrated by applying it to specific two-dimensional elastic scatterer reconstruction problems involving different scatterer configurations and various choices for their constitutive parameters. The effect of using different levels of additive random noise in the forward synthetic data and combining results obtained for different polarizations of the elastic point source located at the sampling point, on the quality of the reconstructed profiles, is also examined. Finally, chapter five draws the conclusions that flow from the foregoing chapters and discusses the contribution of this doctoral thesis. A brief discussion about possible future studies is also included.

Perfect fluid forces in fish propulsion: the solution of the problem in an elliptic cylinder co-ordinate system

1961 ◽  
Vol 261 (1306) ◽  
pp. 316-328 ◽  
Keyword(s):  
Perfect Fluid ◽  
Elliptic Cylinder ◽  
Travelling Wave ◽  
Finite Width ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Fluid Forces ◽  
Low Aspect Ratio ◽  
Fish Propulsion ◽  
General Force

This paper is a discussion of perfect fluid forces involved in fish propulsion. First, the two-dimensional problem is solved in elliptic cylinder co-ordinates in which the surface, or strip μ = 0 is used to approximate a ‘fish’. A travelling wave with linearly increasing ampli­tude is imposed on the strip to represent the motion of the fish. The problem then is in­vestigated for a rigid strip of finite width oscillating about the forward end. Results of this calculation are used to correct the general force expression to the case of low aspect ratio. Experimental results are then discussed which verify the validity of the calculations.

A NOTE ON DIFFRACTION BY AN INFINITE SLIT

1960 ◽  
Vol 38 (1) ◽  
pp. 38-47 ◽  
Author(s):  
R. F. Millar
Keyword(s):  
Integral Equation ◽  
Normal Derivative ◽  
Cylindrical Wave ◽  
Narrow Slit ◽  
Angle Of Incidence ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Boundary Values ◽  
Transmission Coefficients ◽  
Differential Integral Equation

The two-dimensional problem of diffraction of a plane wave by a narrow slit is considered. The assumed boundary values on the screen are the vanishing of either the total wave function or its normal derivative. In the former case, a differential–integral equation is obtained for the unknown function in the slit; in the latter, a pure integral equation is found. Solutions to these equations are given in the form of series in powers of ε (where ε/π is the ratio of slit width to wavelength), the coefficients of which depend on log ε. Expressions are found for the transmission coefficients as functions of ε and the angle of incidence; these are compared with previous determinations of other authors.A brief outline is given for the treatment of diffraction of a cylindrical wave by the slit.

scholarly journals Diffraction of a plane wave by an almost circular cylinder

1961 ◽  
Vol 264 (1317) ◽  
pp. 246-268 ◽  
Keyword(s):  
Plane Wave ◽  
Cross Section ◽  
Circular Cross Section ◽  
The Body ◽  
Maximum Deviation ◽  
Dimensional Problem ◽  
Plane Wave Incident ◽  
Method Of Solution ◽  
Watson Transformation ◽  
The Mean

The two-dimensional problem of an E-polarized plane wave incident on a perfectly conducting cylinder of almost circular cross-section is treated , the maximum deviation of the perimeter of the cross-section from a strict circle being regarded mathematically as an infinitesimal quantity whose second and higher powers are neglected. In the body of the paper the method of solution uses infinite Fourier transform techniques, but an analysis involving a Watson transformation, more traditional in this type of problem , is given in appendix A. Attention is for the most part directed to the case in which the mean radius of the cylinder is large compared to the wavelength, and the form of the solution then appropriate is examined in some detail. In particular, initial terms of asymptotic expansions in inverse powers of the mean radius to wavelength ratio are obtained for the ‘specular’ and for the ‘creeping’ contributions to the far field. It is shown that the former contributionis in agreement with that derived by the Luneberg—Kline method, and the latter with the prescription proposed by Keller. Various Bessel function results are required, some of which are derived in appendices.

Reflection of spherical seismic waves in elastic layered media

Geophysics ◽  
1983 ◽  
Vol 48 (6) ◽  
pp. 655-664 ◽  
Author(s):  
Paul M. Krail ◽  
Henry Brysk
Keyword(s):  
Plane Wave ◽  
Seismic Waves ◽  
Plane Waves ◽  
Spherical Wave ◽  
Bright Spot ◽  
Angle Of Incidence ◽  
Complicated Manner ◽  
Plane Wave Incident ◽  
Wave Limit ◽  
The Impact

The solution of the elastic wave equation for a plane wave incident on a plane interface has been known since the turn of the century. For reflections from reasonably shallow beds, however, it is necessary to treat the incident wave as spherical rather than plane. The formalism for expressing spherical wavefronts as contour integrals over plane waves goes back to Sommerfeld (1909) and Weyl (1919). Brekhovskikh (1960) performed a steepest descent evaluation of the integrals to attain analytic results in the acoustic case. We have extended his approach to elastic waves to obtain spherical‐wave Zoeppritz coefficients. We illustrate the impact of the curvature correction parametrically (as the velocity and density contrasts and Poisson’s ratios are varied). In particular, we examine conditions appropriate to “bright spot” analysis; expectedly, the situation becomes less simple than in the plane‐wave limit. The curvature‐corrected Zoeppritz coefficients vary more strongly (and in a more complicated manner) with the angle of incidence than do the original ones. The determination of material properties (velocities and densities) from the reflection coefficients is feasible in principle, with exacting prestack processing and interpretation. For orientation, we outline the procedure for the simple case of a separated single source and detector pair over a multilayered horizontal earth.

scholarly journals Fourier Integral Transformation Method for Solving Two Dimensional Elasticity Problems in Plane Strain Using Love Stress Functions

2021 ◽  
Vol 8 (3) ◽  
pp. 333-346
Author(s):  
Charles C. Ike
Keyword(s):  
Plane Strain ◽  
Stress Function ◽  
Fourier Integral ◽  
Stress Fields ◽  
Shear Stresses ◽  
Finite Width ◽  
Two Dimensional ◽  
Line Load ◽  
Stress Components ◽  
Stress Functions

The Fourier integral method was used in this work to determine the stress fields in a two dimensional (2D) elastic soil mass of semi-infinite extent subject to line and strip loads of uniform intensity acting on the boundary. The two dimensional plane strain problem was formulated using stress-based method. The Fourier integral was used to transform the biharmonic stress compatibility equation to a fourth order linear ordinary differential equation (ODE) in terms of the stress function. The ODE was solved subject to the boundedness condition to obtain the bounded stress function. Cartesian stress components were obtained using the Love stress functions. Application of the stress boundary conditions for the case of line load of uniform intensity and the cases of uniformly distributed load on a strip of finite width gave the respective unknown constants of the Love stress functions; and hence the complete determination of the Cartesian stress components for the two cases considered. Inversion of the Fourier integral expressions obtained for the normal and shear stresses in the Fourier parameter gave respective expressions for the normal and shear stress fields for line and finite strip loads of finite width in the physical domain variables. The results obtained agreed with the results from previous studies which used displacement based methods.

scholarly journals Reflection of a Wave by a Cylindrical Mirror

1956 ◽  
Vol 9 (3) ◽  
pp. 145-150 ◽  
Author(s):  
Ll. G. Chambers
Keyword(s):  
Plane Wave ◽  
Scattered Field ◽  
Incident Field ◽  
Wave Incident ◽  
Two Dimensional ◽  
Cylindrical Mirror ◽  
Plane Wave Incident

The question of the reflection of a wave by a cylindrical mirror is of interest in a number of fields. It is a problem in which it is difficult to obtain an expression for the reflected or scattered field without recourse to physical assumptions which are sometimes somewhat dubious. An attempt was made by Sommerfeld to solve the problem of a plane wave incident upon such a perfectly conducting mirror by means of what he termed the “Non-Final Determination of Coefficients”. Unfortunately, a close examination of the problem renders it doubtful whether the method can be legitimately employed. It is possible, however, to solve the problem by expressing the scattered field in terms of the currents produced in the mirror, and finding the current generated in the mirror by an arbitrary incident field. The problem which we shall consider is the following two- dimensional one.

Modifications of the Godunov method for computing disturbance propagation in an elastic body

2016 ◽  
Vol 11 (1) ◽  
pp. 119-126 ◽  
Author(s):  
A.A. Aganin ◽  
N.A. Khismatullina
Keyword(s):  
Elastic Body ◽  
Godunov Method ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Order Accuracy ◽  
One Dimensional ◽  
Disturbance Propagation ◽  
Linear Waves ◽  
Longitudinal And Shear Waves ◽  
Contact Discontinuities

Numerical investigation of efficiency of UNO- and TVD-modifications of the Godunov method of the second order accuracy for computation of linear waves in an elastic body in comparison with the classical Godunov method is carried out. To this end, one-dimensional cylindrical Riemann problems are considered. It is shown that the both modifications are considerably more accurate in describing radially converging as well as diverging longitudinal and shear waves and contact discontinuities both in one- and two-dimensional problem statements. At that the UNO-modification is more preferable than the TVD-modification because exact implementation of the TVD property in the TVD-modification is reached at the expense of “cutting” solution extrema.

A STUDY OF TWO‐DIMENSIONAL HEAD WAVES IN FLUID AND SOLID SYSTEMS

Geophysics ◽  
1963 ◽  
Vol 28 (4) ◽  
pp. 563-581 ◽  
Author(s):  
John W. Dunkin
Keyword(s):  
Half Space ◽  
Vertical Displacement ◽  
Point Of View ◽  
Head Wave ◽  
Apparent Velocity ◽  
Two Dimensional ◽  
Multiple Reflections ◽  
Transient Wave ◽  
Line Load ◽  
Head Waves

The problem of transient wave propagation in a three‐layered, fluid or solid half‐plane is investigated with the point of view of determining the effect of refracting bed thickness on the character of the two‐dimensional head wave. The “ray‐theory” technique is used to obtain exact expressions for the vertical displacement at the surface caused by an impulsive line load. The impulsive solutions are convolved with a time function having the shape of one cycle of a sinusoid. The multiple reflections in the refracting bed are found to affect the head wave significantly. For thin refracting beds in the fluid half‐space the character of the head wave can be completely altered by the strong multiple reflections. In the solid half‐space the weaker multiple reflections affect both the rate of decay of the amplitude of the head wave with distance and the apparent velocity of the head wave by changing its shape. A comparison is made of the results for the solid half‐space with previously published results of model experiments.

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