Elastic Surface Waves Research Articles

Some wave phenomena in elastic superconductors

This article formulates some wave phenomena in elastic superconductors, in which the effect of interaction of elastic waves with electromagnetic fields are taken into account. The effect of normal conduction current is also taken into consideration. It is shown that, due to the effect of dynamic motion, transverse magnetic waves may be induced by a transverse elastic wave in an elastic superconductor. Magnetic surface waves may also be induced by the Rayleigh elastic surface wave in the superconductor. The magnitude of the magnetic induction field associated with the induced magnetic wave can be significant at a certain excitation frequency. Furthermore, by considering the coupling effect between the elastic wave and the electromagnetic field, a study is given to investigate electromagnetoelastic surface waves in a semi-infinite elastic superconductor with the aid of the theoretical formulation given in the article. It is shown that the characteristics of the coupled wave in a superconductor can be quite different from that in a perfect conductor due to the presence of the Meissner effect in the superconductor.

Multiple Forward Scattering of Surface Waves: Comparison With an Exact Solution and Born Single-Scattering Methods

Summary We present a 2-D reformulation of surface wave scattering theory in terms of potentials, which allows an extension of the Born single-scattering approach to include multiple forward scattering. No additional numerical effort compared to single scattering is required for a computation of the wavefield over the whole heterogeneous region. Born single scattering for elastic surface waves and both multiple and single scattering for acoustic waves are also covered by the formulation. It is valid for fully anisotropic perturbations of the reference medium. We use the flexibility of our formulation to compare the different approximations with each other and, additionally, test all of them against an exact solution for the particular case of a cylindrical inclusion in a layered waveguide. Our numerical results, obtained for shear velocity contrasts of about 6 per cent, show that the method which includes multiple scattering is superior to the single-scattering methods if the scattering region extends over more than one wavelength. If coupling to higher modes is suppressed, the multiple-scattering method still yields nearly exact results for the vertical displacement. the influence of mode coupling and type conversion leads to only small errors in vertical displacement. Moreover, as we show for a cylinder with a diameter of two wavelengths, even an acoustic treatment of surface waves including multiple forward scattering may be more accurate than single scattering within an elastic treatment. For scatterer sizes below one wavelength the single-scattering approaches are accurate enough, while elastic and acoustic treatments of surface waves may differ considerably. The proposed multiple-scattering method is numerically very efficient, because the numerical effort mainly depends on the degrees of smoothness of the wavefield and the heterogeneity, and is not directly coupled to the wavelength.

Surface phonons and elastic surface waves

AbstractTheoretical investigations on the dynamics of the (001), (110) and (111) surfaces of some cubic metals (Ag, Cu, Ni) will be reviewed. Both, lattice dynamical and continuum theoretical results are obtained via a Green's function formalism. The main attitude of this paper is the comparison of our results with experiments and with results obtained via slab‐calculations. The calculation of elastic surface waves has been performed using a modified surface‐green‐function‐matching method. We have used two different approaches of calculation the bulk Green's function (a) using the spectral representation and (b) a method, what works on residues. The investigations are carried out using shortrange phenomenological potentials. The atomic force constants in the first surface layers are modified to describe surface phonon anomalies, observed by experiments. In the case of Ag (100) and Ag(110) we conclude that the detection of odd symmetry shear modes by Erskine et al. [1 a, b] was not very accurate.

Solitons in a Nonlinear Elastic Medium

The influence of nonlinearity on shear horizontal elastic (bulk and surface) waves in crystals is considered on the basis of a nonlinear equation derived from a microscopic scalar model. The continuum limit is compared with the equation obtained in the framework of nonlinear elasticity theory. Using the equation, it is demonstrated that nonlinear surface waves of the shear horizontal polarization are modulationally unstable, and, therefore, they have to produce surface solitary waves. The solitons are studied in the case of a nonlinear elastic plate when the effective asymptotic technique may be elaborated. In the main approximation of the asymptotic approach, the effective nonlinear Schr\odinger equation is derived. We predict that in the nonlinear elastic plate it is possible to observe bright or dark envelope solitons that propagate without distortion. The properties of the solitons depend on the carrier wave number and the thickness of the plate. We demonstrate also that there is a discrete set of the solitons produced by linear modes of the elastic plate.

Application of a microscope to Brillouin scattering spectroscopy

In order to make a microscopic observation of the surface elastic wave on inhomogeneous opaque materials, an apparatus for Brillouin scattering spectroscopy was constructed utilizing a commercial microscope. An optical system which serves both as an inlet of a laser beam and as an outlet of the scattered light was made and incorporated in the microscope. The apparatus was proved to be particularly useful to make microscopic-scale observations of surface elastic waves on metal films.

High-frequency elastodynamic boundary integral inversion using asymptotic phase transformation

Use of conventional finite element or boundary element computational methods for problems of time-harmonic elastic wave transmission, diffraction, and surface wave generation at a nonplanar fluid–solid interface become inefficient at high frequencies due to the fineness of discretization required for numerical convergence. A more efficient computational procedure has been developed that exploits the field phase as predicted by high-frequency asymptotic analysis (i.e., the geometrical theory of diffraction, or GTD). In this method, the boundary integral equation which governs elastic wave transmission is transformed by assuming a solution in the form of the GTD ansatz A(x)exp[iωp(x)], where p(x) is the field phase, ω is time harmonic frequency, and A(x) is the field amplitude. The phase p(x) is prescribed analytically through solution of the leading term eikonal equations of GTD. The transformed boundary integral equation is then solved numerically for the amplitudes A(x). The amplitudes A(x) will have a narrow spatial frequency spectral bandwidth, and hence can be efficiently calculated. This approach exploits the strengths of the GTD eikonal equation solutions while avoiding the analytical pitfalls encountered in the GTD transport equations.

Experimental study of sound scattering by elastic spheroids, and the excitation of their resonances

Experimental studies of ultrasonic scattering by submerged steel spheroids, both oblate and prolate, and of steel spheres have been carried out using incident short, sinusoidally modulated pulses. The time waveform and frequency responses of the scattering objects are obtained directly from experimental data via numerical processing. The sound pulses are incident in the axial direction and also in discrete oblique directions at every 5 deg from the axis. The sound paths of surface waves generated on the objects, causing a chain of echo pulses, are identified, echo arrival times are shown to agree with the surface wave interpretation, and resonanc spectra are extracted using the Numrich–de Billy method. For explaining the resonance spectrum, the phase-matching principle is employed. The target strength of the objects is extracted from experiment using various methods of interpretation (specular echoes, elastic-wave echoes, total echo energy, or entrance windows), and compared to the geometrical target strength. It is shown that specular echoes alone are not sufficient to explain the entire target strength, and the presence of elastic surface waves makes itself felt here.

Surface solitons on elastic surfaces: numerics

In previous work of the authors, the Whitam-Newell technique of study of small-amplitude, modulated, almost monochromatic signals was used to prove the existence of stable surface-wave solitary-like solutions of the envelope type on a structure made of a nonlinear elastic substrate and a superimposed elastic thin film. In the present work we present the results of numerical simulations performed either on the model one-dimensional nonlinear Schrödinger equation, or directly on the original two-dimensional elastic system using analytic solutions of the former as initial conditions in the latter. The numerical experiments confirm the solitary-wave behavior together with exponential decrease of amplitude with depth, as well as a very good solitonic behavior in the original mechanical system in the course of collision of the nonlinear surface waves. Surface elastic shock waves can also be exhibited when compensation between nonlinearity and dispersion is not precisely achieved.

The projection method for nonlinear surface acoustic waves

Linear elastic surface waves have complicated structure even in homogeneous halfspaces. Many applications involve waves on anisotropic, layered or piezoelectric materials. In analysing nonlinear effects, an economical procedure for deriving propagation equations is paramount. Since the asymptotic procedures naturally yield linear, inhomogeneous elliptic boundary value problems, compatibility conditions must be imposed to ensure solvability. These are provided by the “projection method”, which yields the propagation equations with minimal recourse to determining field perturbations introduced by nonlinearity. The method is applied to surface waves in dispersive and nondispersive systems.

Sound scattering by rough elongated elastic objects. I: Means of scattered field

By use of the recently published deformed cylinder formulation [T. K. Stanton, J. Acoust. Soc. Am. 86, 691–705 (1989)], the scattered field due to rough elongated dense elastic objects is derived. The (one-dimensional) roughness is characterized by axial variations of radius. Explicit expressions are derived describing both the mean and mean square of the stochastic scattered field for the rough straight finite length cylinder (broadside incidence) for both ka≪1 and ka≫1 (k is the acoustic wave number and a is the radius) while only the mean is calculated for the prolate spheroid, uniformly bent finite cylinder, and infinitely long cylinder for ka≫1 (again, all broadside incidence). The modal-series-based solution is used in the ka≪1 case as the modal solution simplifies to the sum of two terms (monopole and dipole-like terms). For ka≫1, a more convenient approximate ‘‘ray’’ solution is used in place of the modal series solution. The results show that (1) when ka≪1 the roughness-induced variations of the mean and mean-square scattered fields due to the rough straight finite cylinder depend on the roughness, but are independent of frequency—an effect that has no counterpart in the area of scattering by rough planar interfaces. (2) When ka≫1 the mean specular (geometrically reflected) and Rayleigh surface elastic waves of the scattered field of each object are attenuated due to the roughness and their variations are dependent upon the frequency. In addition, the (roughness-induced) attenuation of the Rayleigh wave depends on the number of times the wave has circumnavigated the object. The mean-square values for the straight finite cylinder are attenuated in a similar manner with the additional dependence upon the correlation distance of the surface.

The behaviour of elastic surface waves polarized in a plane of material symmetry II. Monoclinic media

A general analysis has been given in part I of symmetric elastic surface waves, characterized by the coincidence of the reference plane with a plane of material symmetry. There follows here a detailed account of the propagation of such waves in media with monoclinic, the minimum type of symmetry. The three definite integrals involved in the determination of the surface-wave function, and hence the speed of propagation, are evaluated and an explicit formula is obtained connecting a fourth integral, also required in the calculation of surface-wave properties, to the other three. Illustrative numerical results are presented, referring in all to 12 monoclinic crystals. The continuous transitions between subsonic and supersonic surface-wave propagation, encountered previously in cubic and transversely isotropic elastic media, occur in seven of the materials and thus emerge as a customary feature of symmetric surface waves. Computations of the associated displacement and traction fields and the paths of particles in the boundary of the transmitting body are also described.

The behaviour of elastic surface waves polarized in a plane of material symmetry. III. Orthorhombic and cubic media

The results obtained in part II for symmetric elastic surface waves in monoclinic media are specialized first to orthorhombic, then to cubic symmetry. A complete account is given of the configurations in which the S-tensors, and hence the surface-wave function and the secular equation, can be determined explicitly. The theoretical developments are complemented by extensive numerical computations, presented in the context of elastodynamic classifications of orthorhombic and cubic materials evolved by Musgrave and Chadwick & Smith respectively. The transitions between subsonic and supersonic surface-wave propagation encountered in part II recur in considerable abundance, but stand in a clear relationship to the classification only in the case of cubic media.

Secluded supersonic elastic surface waves

Supersonic surface waves are known to exist when the reference plane is a plane of material symmetry as in high-symmetry directions in anisotropic elastic materials of hexagonal symmetry. It is shown that supersonic waves propagating along an axial plane in general directions also exist. Conditions for existence and a few examples are presented. The secluded supersonic wave exists in a subspace defined by the condition for simple reflection. Investigation of this subspace reveals existence of exceptional waves, both simple and composite.

Elastic waves guided by an infinite plane crack

We study in this paper the propagation of elastic surface waves guided by an infinitely long crack. We investigate this problem by means of a boundary integral formulation. We state some new theoretical results (existence and other properties) and present a numerical method, as well as various numerical results, for the particular case of the plane crack.

The behaviour of elastic surface waves polarized in a plane of material symmetry. I. Addendum

This addition to a recent paper by Chadwick ( Proc. R. Soc. Lond . A 430, 213 (1990); hereafter referred to as part I) has been prompted mainly by the discovery of secluded supersonic surface waves propagating in configurations of transversely isotropic elastic media in which the reference plane is not a plane of material symmetry and coexisting with a subsonic surface wave. The occurrence of a supersonic surface wave travelling in a direction e 1 with speed v s implies that there are two homogeneous plane waves, with slowness vectors s i and s r such that s i . e 1 = S r . e 1 = v -1 s , which comprise the incident and reflected waves in a case of simple reflection at the traction-free boundary. Supersonic surface waves may therefore be found by searching within a suitably defined space of simple reflection, R . This is the approach which has led to the new results mentioned above and the principal conclusions of part I are re-examined here from the same point of view. It is found that, whereas the secluded supersonic surface waves in transversely isotropic media correspond to isolated points on a curvilinear projection of R which does not intersect the curve representing subsonic surface waves, the symmetric surface waves studied in part I define a curve which may lie partly inside and partly outside a projection of R in the form of a region, the interior points representing supersonic and the exterior points subsonic surface waves. This discussion is preceded by a simplification of the existence-uniqueness theorem proved in part I and followed by a reconsideration of the possibility that an inhomogeneous plane elastic wave can qualify as a surface wave. Such one-component surface waves do exist, but a symmetric surface wave necessarily contains two inhomogeneous plane waves.

Nonlinear surface elastic waves in a discrete model

The influence of nonlinearity on surface elastic waves is studied in the framework of a simple lattice model for cubic crystals including nonlinear interactions between the first, second, and third neighbors. It is shown that in the model nonlinear effects are of the same order as the dispersion ones and they may change the penetration length of surface waves. In particular, nonlinearity may support surface waves not existing in linear theory. Analytical results are obtained in two limit cases: deeply penetrating nonlinear surface waves and nonlinear waves strongly localized at the crystal surface. It is predicted that surface waves may be totally localized at the surface layer when the nonlinearity is of the order of a certain critical value.

Erratum: “Propagation of elastic surface waves along a cylindrical cavity and their excitation by a point force” [J. Acoust. Soc. Am. 72, 998–1004 (1982)

Abstract

Solitons in a nonlinear elastic medium.

The influence of nonlinearity on shear horizontal elastic (bulk and surface) waves in crystals is considered on the basis of a nonlinear equation derived from a microscopic scalar model. The continuum limit is compared with the equation obtained in the framework of nonlinear elasticity theory. Using the equation, it is demonstrated that nonlinear surface waves of the shear horizontal polarization are modulationally unstable, and, therefore, they have to produce surface solitary waves. The solitons are studied in the case of a nonlinear elastic plate when the effective asymptotic technique may be elaborated. In the main approximation of the asymptotic approach, the effective nonlinear Schr\odinger equation is derived. We predict that in the nonlinear elastic plate it is possible to observe bright or dark envelope solitons that propagate without distortion. The properties of the solitons depend on the carrier wave number and the thickness of the plate. We demonstrate also that there is a discrete set of the solitons produced by linear modes of the elastic plate.

On the nonlocal aspect of propagation of the elastic Goodier-Bishop surface waves

Plane wave reflections at the free surface of a half-infinite elastic space are examined assuming long-range action of cohesive forces, and a grazing incidence of the waves. Using KroenerEringen constitutive equations, nonlocal elastic moduli are determined, and the difference-differential equations of motion for the displacement potential functions established. Satisfaction of the homogeneous boundary conditions determines the ratios of the amplitudes of the incident and reflected waves. A more detailed analysis of the grazing incidence of P-waves in a nonlocal medium confirms the existence of the Goodier Bishop waves in the entire Brillouin frequency zone, at least for three different ranges of the particle interactions, and some experimental data available.

Propagation of elastic surface waves along a cylindrical cavity of arbitrary cross section

The propagation of elastic surface waves, guided by the free surface of an infinitely long cylinder of arbitrary cross section, is formulated as an eigenvalue problem for an unbounded self-adjoint operator. We prove the existence of a hierarchy of guided modes. Two of them propagate for any value of the wave number, whereas all of the others only exist beyond a cut-off wave number. For any fixed value of the wave number, only a finite number of modes propagate.