Elastic Surface Waves Research Articles

About embedded eigenvalues for a spectral problem arising in the study of elastic surface waves in a topographical waveguide

AbstractIn this paper, we are interested with the spectral study of an operator given by an elastic topographical waveguide, a deformed half‐space, of which the cross‐section is a local perturbation of a homogeneous half‐plane. We look for guided waves propagating more rapidly than Rayleigh waves (which mathematically would correspond to embedded eigenvalues) and prove that there are no guided waves propagating more rapidly than S‐waves. Thanks to the boundary of the deformed half‐plane and some reduced equations, these eventual eigenmodes must locally vanish. Adapting to our case a unique continuation principle for the elasticity system, we conclude that these eigenmodes vanish everywhere. Copyright © 2002 John Wiley & Sons, Ltd.

Linear and Nonlinear Elastic Surface Waves: From Seismic Waves to Materials Science

Linear and Nonlinear Elastic Surface Waves: From Seismic Waves to Materials Science

Scattering from a ribbed finite cylindrical shell

Scattering from a ribbed finite cylindrical shell is investigated with a theoretical formalism valid for long cylinders. An analytic expression is derived for the scattered pressure and it is shown that the scattered far field can be expressed as the sum of three components associated with specular reflection, scattering from helical waves, and Bloch–Floquet waves. Approximate solutions are proposed to reduce the computation load and resonance scattering is analyzed from the derived formulas leading to a relatively simple equation for the calculation of the Bloch–Floquet dispersion curves. Simple equations are proposed to determine the locations of the Bloch–Floquet waves for weak interactions of the ribs and the surface elastic waves. In the first Brillouin zone, it is found that the maximum level of Bloch–Floquet wave scattering depends only on the length of the cylinder as for the resonant modes of uniform cylindrical shells. Finally the relevancy of the method is examined by comparing numerical results to experimental data available in the published literature.

An expansion method for non-linear Rayleigh waves

We use ideas from analytic bifurcation theory to develop expansions for periodic small amplitude traveling surface elastic waves of permanent form in the half-plane (Rayleigh waves). We focus on the case of hyperelastic materials where the traveling wave problem has a variational structure, and solve numerically the equations describing the lowest order approximation to the traveling wave solutions. For the materials considered, there is evidence for solutions describing elastic displacements that have discontinuous derivative at the boundary of the domain.

Amplitude equations for non-linear Rayleigh waves

Abstract We investigate asymptotic equations describing small amplitude surface elastic waves in the half-plane (Rayleigh waves). For hyperelastic materials such model equations are Hamiltonian systems, and are seen to lead to the formation of singularities in the surface elastic displacement. At the time of singularity formation the Fourier spectra of the solutions exhibit power law decay, and the observed exponents suggest the existence of both differentiable and non-differentiable singular profiles.

Coexistence of Gravity Waves and Elastic Waves in Unconsolidated Sediments

Based on a linear viscoelastic model, we study the coexistence and transition of gravity waves and surface elastic waves in a half-space of unconsolidated sediment. The dispersion relation and surface response are calculated as a function of the period and rigidity of the system. The dispersion relation shows two branches. One branch corresponds to a Rayleigh wave for short-wave periods, and it becomes a leaky surface wave as the period increases. The second branch shows a linear dependence of the phase velocity with period as expected for gravity waves. As the period increases, this branch evolves into a leaky surface wave joining the first branch. The surface response function, which is related to the vertical displacement, is also calculated. Two distinct peaks are present in the surface response function. One corresponds to a classical Rayleigh wave, while the second is a gravity-driven mode whose amplitude is strongly dependent on the wave period and soil rigidity.

Elastic measurements of layered nanocomposite materials by Brillouin spectroscopy

Surface Brillouin spectroscopy makes it possible to measure surface elastic wave propagation parameters at frequencies up to 20 GHz or more. This enables us to measure the elastic properties of surface layers only a small fraction of a micrometre thick. The wavelength and incident angle of the light determine the wavenumber of surface elastic waves (SAW) that scatter the light inelastically, and their frequency can be found by measuring the change in wavelength of the scattered light. By analysing the elastic wave modes present in the surface, the elastic properties can be deduced. We have used this technique to measure the elastic properties of layered nanocomposite materials, which are widely used in the packaging industry. 12 μm polymer films (PET) were coated with glass oxide layers of thickness as little as 25 nm, to give transparent nanocomposite structures with excellent gas barrier properties. In order to understand and model the behaviour of these films under deformation, it is necessary to determine the elastic properties of the different layers. Evaluation of the elastic properties presents several challenges. First, the oxide layers are much thinner than the wavelengths of the surface phonons in surface Brillouin spectroscopy (and hence the depth probed), which usually lie in the range 250–500 nm. The anisotropic elastic properties of the PET substrate must therefore be measured accurately, and this can be done using bulk Brillouin spectroscopy. Second, a thin layer of metal (usually 10–20 nm) must be deposited on the glass surface so that the surface phonons scatter the light effectively. The elastic properties of the glass layer can then be deduced from surface Brillouin spectroscopy measurements, by simulating the surface wave modes of the metal/glass/polymer composite, and adjusting the parameters to give the best fit. In this way it is possible to observe how the properties of the glass vary as a function of thickness, and in turn to understand how to improve systematically the properties under deformation.

The Work of a Source Maintaining a Uniform Motion of a Load along an Elastically Supported Membrane

With all the progress that has been made in the highspeed surface electric transport, it has become necessary to take into account the wave processes taking place in a railroad track and in the ground under the track. In papers [1‐3] it is shown that a railroad track is mainly affected by surface waves whose velocity in a soft ground (peat) may be of the order of 200 km/h. Since the velocity of today’s high-speed trains may exceed 200 km/h, it is clear that the study of the surface elastic waves generated by a moving load is urgent. In this paper we investigate one aspect of this problem, namely, we analyze the work of a source providing the motion of a load with constant velocity. Since in the mechanics of elastic systems this problem is practically ignored, we consider one of the simplest models with a two-dimensional wave field represented by a membrane mounted on an elastic base. The load is modeled by a point object moving along the membrane uniformly and linearly. Based on the law of change of energy, we derived a relationship for the work of a source maintaining a uniform motion of the load. Neglecting the energy loss, we have shown that, for the load velocity V not exceeding the velocity of transverse waves c in the membrane, this work is equal to zero, and, in the opposite case ( V > c ), it is infinite. The latter result is a consequence of a jump discontinuity, which occurs in the membrane displacement at the boundary of the two-dimensional analog of the Mach cone formed in the membrane at V > c . The effect of the internal friction in the membrane (according to the Voigt rheological model [4]) on the load-generated field and the work of the source maintaining the uniform motion of the load had been studied. It was found that the internal friction eliminates all singularities of the field generated by the load. With an increase in friction, the source work grows at V c . We consider the oscillations of a membrane mounted on an elastic base and excited by a point object moving uniformly and linearly along the membrane. We assume that the friction in the contact is absent, and the vertical component of the object’s reaction to the membrane is constant and equal to P (Fig. 1). In this case, the equation for small vertical oscillations of the membrane has the form

Elastic wave propagation in a cylindrical bore situated in a micropolar elastic medium with stretch

Propagation of surface elastic waves in a cylindrical bore through a micropolar elastic medium with stretch is analysed in two cases. In the first case, the cylindrical bore is considered empty while in the second case, the bore is filled with homogeneous inviscid liquid. In both the problems, period equations are obtained in closed form. The problem of Banerji and Sengupta [2,3] has been reduced as a special case. Numerical calculations have been performed for a particular model and results obtained are presented graphically. It is noticed that the effect of micropolarity on dispersion curve is significant while the effect of micro-stretch on dispersion curve is not appreciable.

Dispersion relations of surface waves in semi-infinite periodic scattering arrays

Lord Rayleigh was the first in proposing the existence of surface waves in elastic systems at the end of the 19th century. Since then, people have observed elastic surface waves in very different contexts: from the huge wavelengthed seismic movements to the ultrasonic waves traveling along electronic devices. The aim of this work is to study the propagation of elastic surface waves through structures with periodically distributed inhomogeneities with cylinder symmetry. The theoretical study is based on variational tools developed by the authors. The periodicity led us to characterize the system by a band structure. In principle, those results can be used to design devices to forbid the transmission or to enhance and drive the propagation of elastic surface waves of selected frequencies. [Work supported by CICyT of Spain.]

Theory of acoustic magnetic resonance in the surface elastic waves absorption by metal multilayers

In view of the magnetoelastic interaction in a metal magnetic multilayer such as Fe/Cr the magnetoelastic contribution to the absorption of the surface elastic waves is calculated. This contribution is proportional to dynamic magnetic susceptibility χ and as a consequence its dependence on the magnetic field strength has a resonance character.

Anomalous velocity change of surface wave near the gelation point

We studied the properties of surface waves during the gelation process of tungstic acid and gelatin and silica gel by the time resolved surface wave measurements. The anomaly is clearly shown in the change of the wave number exponent x ( ω∼ k x , the values 1.5 and 1.0 indicate surface tension and elastic wave, respectively). x of the tungstic acid was about 1.4 in the sol state and rapidly decreased to 1.0 at the gelation point. x of gelatin was about 1.4 (same as tungstic acid) in the sol state and decreased gradually after the gelation point. On the other hand, x of silica gel remained near 1.5 after the gelation point.

Acoustic magnetic resonance in absorption and dispersion of surface elastic waves in multilayers

It is shown that magnetic resonances related to excitation of oscillations of the layer magnetization can appear in the spectrum and absorption of the Love acoustic surface waves propagating in a multilayered film with a Fe/Cr type magnetic ordering on a massive substrate. The magnetoelastic resonant contribution to the elastic shear modulus of the film is calculated and the relation between the resonant frequencies and the frequencies of electromagnetically excited uniform ferromagnetic resonance is established.

Direct Experimental Observation of the Crossover from Capillary to Elastic Surface Waves on Soft Gels

By using electrically excited surface waves we have studied the propagation of hydrodynamic modes on the surface of agarose gels in the frequency range ${10}^{1}--{10}^{3}\mathrm{Hz}$. Bulk rheological behavior has been also determined in the frequency range ${10}^{\ensuremath{-}2}--{10}^{2}\mathrm{Hz}$ by conventional rheometry. The propagation of two surface modes, capillary and elastic, is observed at low frequencies. Above a well defined crossover frequency, regular capillary behavior is observed. The experimental data are in excellent agreement with the theoretical predictions and the measured bulk viscoelastic behavior.

Magnetostatic analog of Love surface elastic waves

The possibility of the existence of a magnetostatic analog of Love surface elastic waves is predicted. They appear in situations where the conditions for the existence of magnetostatic volume waves hold in the upper layer of a ferromagnetic bilayer, but not in the lower layer.

Scattering of surface elastic waves at a vertical anisotropic layer

The propagation of surface elastic waves in an anisotropic inhomogeneous medium modelled by two vertically inhomogeneous anisotropic quarter-spaces and a thick vertically inhomogeneous anisotropic layer welded between the quarter-spaces is considered. The Green's function technique is applied to derive a set of equations for determination of scattering coefficients of the layer. Numerical results of Rayleigh-wave scattering by anisotropic versus isotropic layers are presented and discussed.

Intensity distribution of light emitted from a fiber tip mapped by short surface acoustic wave pulses

Laser generation of short surface acoustic wave (SAW) pulses with a fiber tip is demonstrated. In the experiments, a tapered optical fiber tip was used to guide the 532 nm radiation of a Nd : YAG laser with 180 ps pulse duration to the surface to be investigated. Due to the increasing spatial confinement of the laser radiation with decreasing tip–sample distance, SAW pulses with frequency components up to 1 GHz were excited in an aluminum film deposited on a quartz substrate. It is shown that thermoelastic excitation of elastic surface wave pulses provides a new tool for studying the spatial intensity distribution of the exciting laser light emanating from the aperture of a near-field optical device as a function of distance.

A sensor for liquids characterization based on elastic surface waves generated with a P(VF 2-VF 3) film in a non-piezoelectric media

Early works in elastic waves devices for sensors applications at liquid interfaces mainly utilize plate mode resonators, build with thin film technology. In general, these plates are fragile and difficult to handle during manufacture and use. Some of them are very sensitive to changes in ambient pressure, and airborne sound. We present a sensor which fabrication process was simplified using a piezo-polymer P(VF 2-VF 3) film stuck on a non-piezoelectric media. It presents great robustness and is adequate for liquid immersion. Equations for generation, propagation and detection of the a 0 mode of Lamb waves are presented. The measured values leads to the determination of propagation-path parameters and their changes due to liquid loading conditions.

RAYLEIGH SURFACE WAVES IN A PLATE

In the article the analysis of a monochromatic elastic surface wave in an unlimited plate of limited thickness whose opposite surfaces are planar and mutually parallel, made of an elastic isotropic material of a constant density is described. Equations are calculated for the wave number calculation — speed of surface wave propagation, and an equation for the calculation of the vector components deformation (component trajectory) of the elastic medium.

Coupling of land-based signals into the nearshore underwater acoustic field

Small, controlled land detonations on and near the beach recorded by land geophones and by offshore seismoacoustic sensors are used to characterize the coupling between land seismic and underwater acoustic fields. These data were collected in the Marine Physical Laboratory’s Adaptive Beach Monitoring program. The predominant arrival on land is an elastic surface wave. Although an ocean bottom interface wave can be seen clearly in seismoacoustic recordings just outside the surf zone, it attenuates to nearly background noise levels by the time it reaches a bottom hydrophone array 1.5 km offshore in 12-m water. Travel-time and frequency/wave-number analyses indicate that the hydrophone array’s received energy is composed mostly of dispersive body waves propagating as the lowest water-borne mode, with frequency-dependent phase and group velocities around 2.0 and 1.8 km/s, respectively. The frequency at which this lowest mode cuts off is the corner frequency of the high-pass filter representing the propagation from land to the underwater acoustic field. These results and numerical modeling, guided by insights obtained from classic work on acoustic propagation in a fluid wedge, are used to explain offshore underwater recordings of land vehicle activity. [Work supported by ONR, Code 32.]