Time-harmonic Surface Wave Research Articles

Application of the reciprocity theorem to determine line-load-generated surface waves on an inhomogeneous transversely isotropic half-space

Free and forced time-harmonic surface waves are investigated for a half-space of transversely isotropic elastic material, where the axis normal to the free surface (the z-axis) is the axis of symmetry. In addition, the elastic constants and the mass density depend on the distance from the free surface. A surface wave is considered as being composed of a carrier wave that propagates over the free surface as it would over a membrane, while it carries along a system of depth-dependent motions. For free surface waves the analytical details, i.e., the calculation of the wave speed of the free surface waves, are worked out for the case that the elastic constants and the mass density vary exponentially with depth. For forced motion by a time-harmonic line load, the far-field surface wave motion is determined by an application of the reciprocity theorem for elastodynamics. Numerical results for displacements and stresses are graphically displayed. Fourier superposition has been used to determine the transient surface wave response to a pulse-type normal line-load.

Parameters determination for a linearly inhomogeneous half-space using characteristics of the time-harmonic surface waves

The paper presents a study of time-harmonic surface waves in a linearly inhomogeneous half-space. The study is based on the solution of that problem for an arbitrary (from 0 to 1/2) value of Poisson's ratio. Vertical vibrations due to a vertical harmonic force, which at large distances from the force represent Rayleigh-type waves, and transverse horizontal vibrations due to a horizontal force, which at large distances form waves of Love's type, are considered in detail. Material damping is taken into consideration. Inhomogeneity significantly affects relationships connecting wave characteristics and the frequency of vibration, and it is shown in the paper how this fact can be used for determining material properties (surface shear modulus, degree of inhomogeneity, damping ratio) with the help of experimental results concerning wave propagation over the surface of the half-space. It is shown that for forced waves the relationship between the wave phase angle and distance can significantly differ from a straight line, i.e. the wave number varies with distance. Therefore, it is desirable to relate experimental and theoretical results to such parts of wave propagation line, which correspond to same phase angle intervals. Copyright © 2000 John Wiley & Sons, Ltd.

Focusing of Rayleigh waves: simulation and experiments

A linear array of surface wave transducers has been developed to generate focused surface wave motion. A novel theoretical approach, whereby time-harmonic surface wave motion is represented by a carrier wave that satisfies a reduced wave equation on the surface of the body and supports the subsurface motion, is used to model the beam generated by a single element of the array. Comparison of theoretical and experimental results show that, for a single element, the opening angle of the beam is about 20 degrees and its cross-section can be represented by a Gaussian distribution of the normal displacements. For an eight-element array, the focused beam is subsequently obtained by superposition considerations. For the focused beam comparisons of theoretical and experimental results, in which the latter have been obtained by the use of a laser interferometer, show excellent agreement both for the normal displacements along a radial line and across the width of the beam. The array can be used for self-focusing of surface waves on a surface defect.

Explicit solutions for carrier waves supporting surface waves and plate waves

Time-harmonic surface waves in a half-space ( x 3 ≥ 0) and plate waves in a layer (− h ≤ x 3 ≤ h) of a homogeneous transversely isotropic linearly elastic solid, where the x 1 x 2-plane is the plane of transverse isotropy, have been reconsidered. The displacement components are expressed in terms of a carrier wave which defines the propagation in the x 1 x 2-plane, and which supports the sub-surface motion in the half-space and the thickness motion of the plate. The carrier wave can be any solution of the reduced wave equation for a membrane.

The effect of surface tension on the scattering of waves by a partially immersed vertical barrier in water of infinite depth

In this paper time-harmonic surface wave motion for progressive waves incident normally on and scattered by a partially immersed fixed vertical barrier in water of infinite depth is considered in the presence of surface tension. The problem for the velocity potential is solved, as others have been previously, by first supposing that the free-surface slopes at the barrier are prescribed and the formal solution in terms of these is obtained explicitly by complex-variable methods. To simplify the calculation the known solution corresponding to zero free-surface slopes at the barrier is subtracted out first and emphasis is placed on determining the residual potential. Finally, an appropriate dynamical edge condition is imposed on the formal solution to determine the required values of the edge-slope constants and hence fully solve the transmission problem. The problem was first examined some time ago using a complex-variable reduction procedure before the advent of this condition, although an explicit formal solution was not obtained, that earlier work forms a basis for the present investigation. It is noted in conclusion how the solution of the problem for waves generated by a partially immersed non-uniform heaving vertical plate may easily be obtained in a similar manner, since the formal solution required is just the residual potential determined in our main problem.

Reflection and Transmission of Obliquely Incident Surface Waves by an Edge of a Quarter Space: Theory and Experiment

The reflection and transmission of a plane time-harmonic surface wave which is obliquely incident on the edge of a quarter space is investigated theoretically, numerically, and experimentally. The theoretical formulation of the problem, which takes advantage of the translational invariance along the edge of the quarter space, is reduced to a system of singular integral equations along axes normal to the edge, for the defracted displacement components on the faces of the quarter space axes normal to the edge. The truncation of these equations leads to the definition of reflection and transmission coefficients, R and T. The equations are solved for R, T, and the diffracted displacements by the use of the boundary element method. A self-calibrated experimental technique is proposed which deploys four surface wave transducers, and which removes the effects of variable coupling between the transducers and the faces of the quarter space as the positions of the transducers are varied. The technique is particularly suited for the measurement of |R/T| as a function of the angle of incidence. Excellent agreement is observed between numerically and experimentally obtained values.

Wave‐source expansions for water of infinite depth in the presence of surface tension

In this paper two expansions are obtained by contour integration methods for the velocity potential describing two-dimensional time-harmonic surface waves due to a free-surface wave source on water of infinite depth in the presence of surface tension. First the series expansion at r = 0 is found and then the asymptotic expansion as Kr®¥, where K is the wave number for progressive waves and r the radial distance from the source. The corresponding expansions for the more important submerged wave source in terms of the radial distance from the image source in the free surface may then easily be deduced. The latter are required in a number of surface wave problems, particularly those of a short-wave asymptotic nature, and are also relevant in obtaining expansions for finite constant depth.

Numerical modelling of the diffraction of short time-harmonic linear waves

It is well known that the Green's identity formula relative to the two-dimensional Helmholtz equation can be taken as a basis for the numerical modelling of problems concerning the diffraction of time-harmonic surface waves, within the limits of the linear and inviscid theory for shallow- or constant-depth water. The early applications of this kind of mathematical approach have been given by DAUBRT and LEBR~TO~ (1), for the diffraction of objects placed in the open sea, and by HWA~G and TUCK (2), for the calculation of the wave-induced oscillations in a har- hour. In these applications, however, as well as in other applications, the whole problem was described by a single integral equation, written for a contour of either finite or infinite length. L~s~ (3) has given to the problem of the harbour oscillations another planning, charac- terized by two main features. First, the solution was obtained not from a single equa- tion, but from a set of two equations, the former related to the internal harbour basin, and the latter to the external sea. Secondly, the elevation field was obtained only after the evaluation of the normal-velocity field along the junction line between the two subdomains. This line of research has been further developed by L~ and RAICHL~N (4) for a similar problem, in which the internal domain was considered as the junction of more domains of simpler form, so that the whole problem was solved in terms of a set of more than two integral equations. MATTIOLI and TINTI (5) presented an analogous approach for the propagation of waves from the sea to a lagoon through a channel, partitioning the whole domain in a finite subdomain and two semi-infinite regions. Finally, HARTV.~ and EFRONY (e) extended the previous results to the case of a problem of wave diffrac- tion or radiation defined in the vertical plane. However, in all the applications cited until now, the subdivision of the original domain was not arbitrary. In fact in no case two junction lines intersect each other

Finite element matrices for seismic surface waves in three-dimensional structures

The finite-element technique of Lysmer and Drake has made possible the study of time-harmonic surface waves in irregular, two-dimensional structures. This paper provides the stiffness and mass matrices necessary for extending their technique to the study of time-harmonic surface waves in irregular, three-dimensional structures. The element stiffness matrix [ k ] and the element mass matrix [ m ] are obtained by symbolic integration for an isotropic, rectangular hexahedron; the elements of these 24 × 24 matrices are given in explicit form. In addition, the layer stiffness matrix [ K ] and the layer mass matrix [ M ] appropriate for horizontally layered, laterally homogeneous, three-dimensional models constructed from rectangular hexahedra are also given in explicit form. Modifications to both [ K ] and [ M ] are described which ensure surface-wave motion in the horizontally layered, laterally homogeneous, three-dimensional structure. Finally, an example is given which demonstrates the correctness of the modified forms of [ K ] and [ M ] and, by implication, the correctness of [ k ] and [ m ].

Guided Surface Waves on an Elastic Half Space

Three-dimensional wave propagation in an elastic half space is considered. The half space is traction free on half its boundary, while the remaining part of the boundary is free of shear traction and is constrained against normal displacement by a smooth, rigid barrier. A time-harmonic surface wave, traveling on the traction free part of the surface, is obliquely incident on the edge of the barrier. The amplitude and the phase of the resulting reflected surface wave are determined by means of Laplace transform methods and the Wiener-Hopf technique. Wave propagation in an elastic half space in contact with two rigid, smooth barriers is then considered. The barriers are arranged so that a strip on the surface of uniform width is traction free, which forms a wave guide for surface waves. Results of the surface wave reflection problem are then used to geometrically construct dispersion relations for the propagation of unattenuated guided surface waves in the guiding structure. The rate of decay of body wave disturbances, localized near the edges of the guide, is discussed.

The oblique reflection of a rayleigh wave from a crack tip

The reflection of a free-surface Rayleigh wave from the edge of a half-plane crack in an unbounded isotropic elastic solid is considered. The time-harmonic surface wave, propagating on one face of the crack, is obliquely incident on the edge so that the problem is three-dimensional. Application of a displacement representation theorem reduces solution of the problem to solution of an integral equation, which is solved by application of Laplace transform methods and the Wiener-Hopf technique. The transformed displacement has a simple pole corresponding to the reflected surface waves, and the residues are determined. The amplitudes and phases of the reflected surface waves on both faces of the crack have been calculated numerically, and are plotted as functions of the angle of incidence. Energies of the reflected waves are also shown in graphs.