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J. W. C. Sherwood

Bio: J. W. C. Sherwood is an academic researcher from Imperial College London. The author has contributed to research in topics: Lamb waves & Plane wave. The author has an hindex of 2, co-authored 2 publications receiving 60 citations.

Papers
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Journal ArticleDOI
01 Feb 1958
TL;DR: In this article, a simple physical picture is given of plane waves possessing complex angles of propagation, which play an important role in the theory of elastic wave radiation, illustrated by studying continuous sinusoidal wave propagation parallel to the unstressed, plane boundary of a semi-infinite, homogeneous and isotropic solid medium, the Rayleigh and head waves being particular features of the investigation.
Abstract: A simple physical picture is given of plane waves possessing complex angles of propagation, which play an important role in the theory of elastic wave radiation. Their utility is illustrated by studying continuous sinusoidal wave propagation parallel to the unstressed, plane boundary of a semi-infinite, homogeneous and isotropic solid medium, the Rayleigh and head waves being particular features of the investigation. A novel study of the field due to an impulsive force acting at a line in the surface of a semi-infinite medium indicates a general method of solving important transient propagation problems encountered in seismology. The equivalent problem of an impulsive force acting at the edge and in the plane of a semi-infinite thin sheet has been simulated experimentally by detonating small explosive charges at the edge of an aluminium sheet. The displacements detected by a condenser microphone technique are in excellent agreement with the theoretical determinations.

51 citations

Journal ArticleDOI
TL;DR: In this article, a simple physical model for the displacement field radiated by an impulsive force acting at a line in the surface of a semi-infinite thin sheet is presented.
Abstract: Plane waves possessing complex angles of propagation play an important role in the theory of elastic wave radiation. A simple physical picture is given of these waves and their utility illustrated by employing them in the study of continuous sinusoidal wave propagation in the neighborhood of an unstressed, plane boundary in a semi‐infinite, homogeneous, and isotropic solid medium. The Rayleigh wave and the von Schmidt, or Head wave are particular features of the study. A simpler solution than Sauter's [Z. angew Math. Mech. (1950)] has been found for the displacement field radiated by an impulsive force acting at a line in the surface. By reciprocity this gives the surface displacement due to an internal line force. An equivalent problem is provided by an impulsive force acting at the edge and in the plane of a semi‐infinite thin sheet, provided that the bulk dilatation wave velocity is replaced by the thin sheet dilatation wave velocity. This has been simulated experimentally by detonating small explosive...

12 citations


Cited by
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Journal ArticleDOI
Alan Levander1
TL;DR: 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.
Abstract: I describe the properties of a fourth-order accurate space, second-order accurate time two-dimensional P-Sk’ finite-difference scheme based on the MadariagaVirieux 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.

1,429 citations

Journal ArticleDOI
TL;DR: In this paper, the effects of anisotropy on elastic wave propagation in anisotropic mediums are discussed and a review of the techniques used to solve them is presented.
Abstract: 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.

121 citations

Journal ArticleDOI
TL;DR: In this article, a review of the theoretical background to these methods and a comparison of the two popular algorithms, reflectivity and WKBJ seismograms, for a variety of earth models is presented.
Abstract: Synthetic seismograms, computed for realistic, horizontally stratified media, are now routinely used as an aid to seismic interpretation. This paper reviews the theoretical background to these methods and presents comparisons of the two popular algorithms, reflectivity and WKBJ seismograms, for a variety of earth models. The transformed wave equations are developed from the equations for a spherical, gravitating medium in a symmetric form suitable for body wave calculations. Four methods of solving these equations in general, inhomogeneous layers are described: the WKBJ and Langer asymptotic expansions and the WKBJ and Langer iterative solutions. Together with the earth-flattening transformation and the ray expansion, transformed solutions for body waves can then be obtained for realistic layered media. Four methods of inverting the frequency and wave number transformations are also described: the real and complex spectral and slowness methods. Although realistic seismic models are normally sufficiently complicated that numerical calculations are essential, before proceeding with numerical comparisons we have included a review of the canonical signals included in body wave seismograms. These analytic results for direct rays, partial and total reflections, turning rays on forward and reversed branches, head waves, interface waves, Airy caustics, and Fresnel and interface shadows are useful to anticipate and understand numerical problems and results. Finally, a comparison of Green's functions for crustal, mantle, and whole earth models, calculated using the WKBJ and reflectivity algorithms, is included.

107 citations

Journal ArticleDOI
TL;DR: In this paper, the possibilities and limitations of the theoretical investigation of elastic waves in horizontally layered media with the aid of the exact ray theory are studied. But this work is restricted to the propagation of sound waves from a line source in a layered-liquid medium.
Abstract: Summary The possibilities and limitations of the theoretical investigation of elastic waves in horizontally layered media with the aid of the exact ray theory are studied. In order to simplify the problem as far as possible, it is restricted to the propagation of sound waves from a line source in a layered-liquid medium. Three different types of the transition between two homogeneous liquids are investigated: a first-order discontinuity, a transition consisting of a prescribed number of homogeneous layers, and a transition layer with linear variation of sound velocity and density with depth. Theoretical seismograms are computed by exact formulas and by wave front approximations for the reflected waves in these models. From these results, the main conclusions about the applicability of the exact ray theory to studies of wave propagation in more complicated media are as follows: (1) The exact ray theory is applicable to the investigation of vertical and near-vertical reflections from inhomogeneous media without difficulties; and (2) continuously refracted waves in such media can be investigated by an approximation which is sufficient for many practical cases. The final aim of this study was to test whether the exact ray theory can be used to compute theoretical seismograms for seismological applications. Therefore, some approximations were developed which are useful for computations for realistic models of the Earth's crust and upper mantle. An Earth-flattening approximation is given which accounts for the influence of the Earth's curvature on the propagation of body waves, and the approximate computation of point source seismograms from line source seismograms is described. An example of theoretical seismograms is presented for an upper mantle model, published by Julian & Anderson (1968).

49 citations