Seismoelectric effect in Lamb’s problem

abstract We consider a version of Lamb's Problem in which a vertical time-dependent point force acts on the surface of a uniform half-space. The resulting surface disturbance is computed as vertical and horizontal components of displacement, particle velocity, acceleration, and strain. The goal is to provide numerical solutions appropriate to a comparison with observed wave forms produced by impacts onto granite and onto soil. Solutions for step- and delta-function sources are not physically realistic but represent limiting cases. They show a clear P arrival (larger on horizontal than vertical components) and an obscure S arrival. The Rayleigh pulse includes a singularity at the theoretical arrival time. All of the energy buildup appears on the vertical components and all of the energy decay, on the horizontal components. The effects of Poisson's ratio upon vertical displacements for a step-function source are shown. For fixed shear velocity, an increase of Poisson's ratio produces a P pulse which is larger, faster, and more gradually emergent, an S pulse with more clear-cut beginning, and a much narrower Rayleigh pulse. For a source-time function given by cos2(πt/T), −T/2 ≦ T/2, a × 10 reduction in pulse width at fixed pulse height yields an increase in P and Rayleigh-wave amplitudes by factors of 1, 10, and 100 for displacement, velocity and strain, and acceleration, respectively. The observed wave forms appear somewhat oscillatory, with widths proportional to the source pulse width. The Rayleigh pulse appears as emergent positive on vertical components and as sharp negative on horizontal components. We show a theoretical seismic profile for granite, with source pulse width of 10 µsec and detectors at 10, 20, 30, 40, and 50 cm. Pulse amplitude decays as r−1 for P wave and r−12 for Rayleigh wave. Pulse width broadens slightly with distance but the wave form character remains essentially unchanged.

Some analytical aspects of viscoelastic Lamb’s problem for improving its numerical evaluation

Wave Motion ◽

10.1016/j.wavemoti.2012.08.010 ◽

2013 ◽

Vol 50 (2) ◽

pp. 226-232

Author(s):

Robert Arcos ◽

Jordi Romeu ◽

Arnau Clot ◽

Meritxell Genescà

Keyword(s):

Numerical Evaluation ◽

Lamb’S Problem

Fourth‐order finite‐difference P-SV seismograms

Geophysics ◽

10.1190/1.1442422 ◽

1988 ◽

Vol 53 (11) ◽

pp. 1425-1436 ◽

Cited By ~ 920

Author(s):

Alan R. Levander

Keyword(s):

Free Surface ◽

Finite Difference ◽

Fourth Order ◽

Equations Of Motion ◽

Dispersion Analysis ◽

Numerical Dispersion ◽

Staggered Grid ◽

Order System ◽

Surface Boundary ◽

Lamb’S Problem

I describe the properties of a fourth‐order accurate space, second‐order accurate time, two‐dimensional P-SV finite‐difference scheme based on the Madariaga‐Virieux staggered‐grid formulation. The numerical scheme is developed from the first‐order system of hyperbolic elastic equations of motion and constitutive laws expressed in particle velocities and stresses. The Madariaga‐Virieux staggered‐grid scheme has the desirable quality that it can correctly model any variation in material properties, including both large and small Poisson’s ratio materials, with minimal numerical dispersion and numerical anisotropy. Dispersion analysis indicates that the shortest wavelengths in the model need to be sampled at 5 gridpoints/wavelength. The scheme can be used to accurately simulate wave propagation in mixed acoustic‐elastic media, making it ideal for modeling marine problems. Explicitly calculating both velocities and stresses makes it relatively simple to initiate a source at the free‐surface or within a layer and to satisfy free‐surface boundary conditions. Benchmark comparisons of finite‐difference and analytical solutions to Lamb’s problem are almost identical, as are comparisons of finite‐difference and reflectivity solutions for elastic‐elastic and acoustic‐elastic layered models.

Comments on “Lamb's problem for fluid-saturated porous media”

Bulletin of the Seismological Society of America ◽

10.1785/bssa0820052263 ◽

1992 ◽

Vol 82 (5) ◽

pp. 2263-2273

Author(s):

M. D. Sharma

Keyword(s):

Porous Media ◽

Boundary Conditions ◽

Solid Phase ◽

Elastic Solid ◽

Porous Solid ◽

Surface Boundary ◽

Vertical Force ◽

Saturated Porous Media ◽

Concentrated Load ◽

Lamb’S Problem

Abstract Philippacopoulos (1988) discusses axisymmetric wave propagation in a fluid-saturated porous solid half-space. The disturbance is considered to be produced by the concentrated load P0exp(iωt) acting vertically at the surface. Boundary conditions chosen imply that a vertical force acting on the surface of fluid-saturated porous solid exerts no pressure on the interstitial liquid. These boundary conditions do not seem appropriate. In the present study, the boundary conditions have been changed in order to satisfy the concept of porosity. These are also in accordance with those discussed by Deresiewicz and Skalak (1963) for the special case of interface between liquid and liquid-saturated porous media. Analytic expressions have been derived for the displacements at the surface. The limiting case of a dry elastic solid is also deduced. Effects of intergranular energy losses due to solid phase and of dissipation due to flow of pore fluid are exhibited on the displacements at the surface. Contrary to Philippacopoulos (1988), the displacements in saturated poroelastic solids are found to be larger than those in a dry elastic solid with same Lamb's moduli.

Time-harmonic Lamb's problem for a system comprising a piezoelectric layer and piezoelectric half-plane

Journal of Sound and Vibration ◽

10.1016/j.jsv.2013.05.004 ◽

2013 ◽

Vol 332 (21) ◽

pp. 5375-5392 ◽

Cited By ~ 7

Author(s):

S.D. Akbarov ◽

N. Ilhan

Keyword(s):

Half Plane ◽

Piezoelectric Layer ◽

Lamb’S Problem ◽

Time Harmonic

Transient interior analytical solutions of Lamb’s problem

Mathematics and Mechanics of Solids ◽

10.1177/1081286519835266 ◽

2019 ◽

Vol 24 (11) ◽

pp. 3485-3513 ◽

Cited By ~ 5

Author(s):

Mohamad Emami ◽

Morteza Eskandari-Ghadi

Keyword(s):

Analytical Solutions ◽

Three Dimensional ◽

Time History ◽

Initial Boundary ◽

Initial Boundary Value Problem ◽

Integral Transforms ◽

Elastic Half Space ◽

Point Load ◽

Lamb’S Problem ◽

Interior Points

The classical three-dimensional Lamb’s problem is considered for an inclined surface point load of Heaviside time dependence. Attention is focused upon the acquisition of the transient elastodynamic analytical solutions for interior points through a unified method of analysis that is valid for arbitrary Lamé constants. The method of elastodynamic potentials is employed jointly with integral transforms to treat the corresponding initial boundary value problem. To derive the time-domain solutions, some integral equations are encountered, the solutions of which are found via a modified version of the Cagniard–Pekeris method. The final solutions are obtained as finite integrals that are amenable to numerical calculations. They are also expressed in the form of Green’s functions. The limit case of infinite time is investigated analytically to derive the closed-form expressions for the limits of the solutions as the temporal variable tends to infinity. As expected, the results are found to be equivalent to Boussinesq–Cerruti solutions in elastostatics. The elastodynamic solutions are also evaluated numerically to plot several time-history diagrams, depicting the transient motions of the interior points, especially of the points close to the boundary so as to illustrate the formation of forced Rayleigh waves at shallow depths within the elastic half-space.

Lamb’s problem: a brief history

Mathematics and Mechanics of Solids ◽

10.1177/1081286519883674 ◽

2019 ◽

Vol 25 (3) ◽

pp. 501-514

Author(s):

Mohamad Emami ◽

Morteza Eskandari-Ghadi

Keyword(s):

Special Interest ◽

Mathematical Theory ◽

Three Dimensional ◽

Elastic Solid ◽

Theory Of Elasticity ◽

Scientific Investigation ◽

Lamb’S Problem ◽

Solution Methods ◽

Dimensional Version ◽

History Of

In this review note, a historical scientific investigation is presented for Lamb’s problem in the mathematical theory of elasticity. This problem first appeared in 1904 in the pioneering paper of Professor Sir Horace Lamb (Lamb, H. On the propagation of tremors over the surface of an elastic solid. Philos Trans R Soc Lon 1904; 203: 1–42). Of special interest here are the analytical studies of the three-dimensional version of Lamb’s problem, which consists of a semi-infinite, homogeneous, isotropic elastic solid that is set in motion by the exertion of a dynamical point force applied suddenly on the surface of the domain. The objective of this paper is to offer a comprehensive introduction to Lamb’s problem for the reader, along with discussing its mathematical complexities. An account is given of the history of this ever-significant problem from its earlier stages to the more recent investigations via outlining and discussing different rigorous approaches and methods of solution that have been hitherto suggested. The limitations of different methods, if they exist, are also discussed. Eventually, various solution methods are compared considering their nature, advantages, and restrictions.

Lamb's problem for solids of general anisotropy

Wave Motion ◽

10.1016/s0165-2125(96)00016-9 ◽

1996 ◽

Vol 24 (3) ◽

pp. 227-242 ◽

Cited By ~ 48

Author(s):

C.-Y. Wang ◽

J.D. Achenbach

Keyword(s):

General Anisotropy ◽

Lamb’S Problem

Acoustic and elastic numerical wave simulations by recursive spatial derivative operators

Geophysics ◽

10.1190/1.3485217 ◽

2010 ◽

Vol 75 (6) ◽

pp. T167-T174 ◽

Cited By ~ 45

Author(s):

Dan Kosloff ◽

Reynam C. Pestana ◽

Hillel Tal-Ezer

Keyword(s):

Synthetic Data ◽

Analytic Solutions ◽

Numerical Dispersion ◽

Reverse Time ◽

Reverse Time Migration ◽

Data Set ◽

Lamb’S Problem ◽

Dynamic Elasticity ◽

Time Migration ◽

Recursive Operators

A new scheme for the calculation of spatial derivatives has been developed. The technique is based on recursive derivative operators that are generated by an [Formula: see text] fit in the spectral domain. The use of recursive operators enables us to extend acoustic and elastic wave simulations to shorter wavelengths. The method is applied to the numerical solution of the 2D acoustic wave equation and to the solution of the equations of 2D dynamic elasticity in an isotropic medium. An example of reverse-time migration of a synthetic data set shows that the numerical dispersion can be significantly reduced with respect to schemes that are based on finite differences. The method is tested for the solutions of the equations of dynamic elasticity by comparing numerical and analytic solutions to Lamb’s problem.

Solution of inverse Lamb's problem with an unknown source

Computational Mathematics and Modeling ◽

10.1007/bf01128355 ◽

1991 ◽

Vol 2 (1) ◽

pp. 44-47

Author(s):

A. V. Baev

Keyword(s):

Lamb’S Problem ◽

Unknown Source