Rayleigh wave propagation in curved waveguides

A JWKB asymptotic expansion describing inplane elastic waves is used to approximate a Rayleigh-like wave guided within a curved elastic waveguide whose curvature is small and changes slowly over a wavelength. The two lowest eigenmodes in a curved guide, taken together, constitute the Rayleigh-like wave. It is shown that this wave lies in the shadows of four, closely spaced, virtual caustics, two caustics per constituent eigenmode. If the curvature becomes too large one or more of the caustics ceases to be virtual and enters the guide after which a Rayleigh-like wave cannot propagate. The overall disturbance is shown to have an amplitude that is modulated because the wavenumbers of the constituent eigenmodes differ by a small amount. Moreover, the disturbance is shown to propagate with a wavenumber that, to leading order, has a linear dependence on the curvature causing the phase to be modulated, as well. Passing from a thin guide to a very thick one suppresses the amplitude modulation, making the phase modulation evident. Propagation into an environment of increasing curvature, for both thin and thick, shallowly curved guides is studied so that the modulations may be observed.     

On the Rayleigh Wave Speed in Orthotropic Elastic Solids

Recently, a formula for the Rayleigh wave speed in an isotropic elastic half-space has been given by Malischewsky and a detailed derivation given by the present authors. This study deals with the generalization of this formula to orthotropic elastic materials and Malischewsky’s formula is recovered as a special case. The formula is obtained using the theory of cubic equations and is expressed as a continuous function of three dimensionless material parameters.     

The matrix sign function for solving surface wave problems in homogeneous and laterally periodic elastic half-spaces

The matrix sign function is shown to provide a simple and direct method to derive some fundamental results in the theory of surface waves in anisotropic materials. It is used to establish a shortcut to the basic formulas of the Barnett–Lothe integral formalism and to obtain an explicit solution of the algebraic matrix Riccati equation for the surface impedance. The matrix sign function allows the Barnett–Lothe formalism to be readily generalized for the problem of finding the surface wave speed in a periodically inhomogeneous half-space with material properties that are independent of depth. No partial wave solutions need to be found; the surface wave dispersion equation is formulated instead in terms of blocks of the matrix sign function of i$i$ times the Stroh matrix.     

A Note on Perturbation Formulae for the Surface-Wave Speed Due To Perturbations in Material Properties

In a recent paper by Tanuma and Man, a two-term asymptotic formula was derived for the speed of surface waves propagating in an anisotropic elastic half-space whose elastic moduli differ only slightly from those for a (base) isotropic elastic material. This formula disagrees with that derived by Delsanto and Clark in an earlier paper using a different method. In this short note, we use a simple procedure to derive another two-term asymptotic formula for the surface-wave speed. Our formula takes the same compact form even if the base material is generally anisotropic. We show that when an error in the work of Delsanto and Clark is corrected, the three different methods do give equivalent results.      

Surface waves and surface stability for a pre-stretched, unconstrained, non-linearly elastic half-space

An unconstrained, non-linearly elastic, semi-infinite solid is maintained in a state of large static plane strain. A power-law relation between the pre-stretches is assumed and it is shown that this assumption is well motivated physically and is likely to describe the state of pre-stretch for a wide class of materials. A general class of strain-energy functions consistent with this assumption is derived. For this class of materials, the secular equation for incremental surface waves and the bifurcation condition for surface instability are shown to reduce to an equation involving only ordinary derivatives of the strain-energy equation. A compressible neo-Hookean material is considered as an example and it is found that finite compressibility has little quantitative effect on the speed of a surface wave and on the critical ratio of compression for surface instability.     

The attenuation of monochromatic surface waves due to the presence of an inextensible cover

The attenuation of surface gravity waves is an important process associated with air–sea and wave–current interactions. Here we investigate experimentally the attenuation of monochromatic surface gravity waves due to the presence of various surface covers. The surface covers are fixed in space such that they do not advect with the wave motion and are selected such that the bending modulus is negligible for the wave frequencies used in the experiment in order to minimize any flexural effects. Wave attenuation rates are found to be independent of wave steepness and the type of cover used over the tested parameter range. Results are consistent with the theoretical attenuation rate for an inextensible surface cover.     

Rayleigh waves of arbitrary profile in anisotropic media

The paper deals with surface wave propagation in an orthorhombic elastic half-plane. The general profile of the wave is considered, incorporating the anisotropy effects within the known representation in terms of a single plane harmonic function.     

A method to study interactions between narrow-banded random waves and multi-chamber perforated structures

A time-domain method, based on linear velocity potential theory, is presented to study the interaction between narrow-banded random waves and perforated structures. A simple relation is derived to estimate the jet length of flows through the perforated wall. The reflection coefficient of narrow banded random waves from perforated structures is calculated by assuming a Rayleigh distribution of the heights of incident random waves. For reflection of narrow-banded waves from a single-chamber perforated breakwater, a comparison of the predicted and measured reflection coefficients shows that the method presented in this paper can provide a prediction better than that of regular waves. Numerical results are also reported on the reflection of narrow-banded waves from multi-chamber perforated breakwaters.The project partially supported by the Hong Kong Research Grant Council (DAG03/04.EG39, DAG04/05.EG32).     

Transient development of axisymmetric surface waves

The initial value problem of axisymmetric wave motion generated by a cylindrical wave-maker immersed in an incompressible homogeneous fluid of constant finite depth is solved using the Laplace and Weber transforms. An asymptotic treatment of the problem for sufficiently long time and then for large distances is also carried out in detail to obtain a deeper understanding of the steady and transient motions.

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Sommario Viene risolto con l'uso della trasformata di Laplace e Weber il problema a valori iniziali di onde assialsimmetriche, generate da una sorgente d'onda cilindrica imnersa in un fluido incomprimibile, omogeneo, di profondità finita e costante. È, inoltre, sviluppata in dettaglio una trattazione asintotica del problema per tempi sufficientemente lunghi e grandi distanze, al fine di ottenere una migliore conoscenza sia del moto transitorio che di quello a regime.

     

Efficient computation of steady solitary gravity waves

An efficient numerical method to compute solitary wave solutions to the free surface Euler equations is reported. It is based on the conformal mapping technique combined with an efficient Fourier pseudo-spectral method. The resulting nonlinear equation is solved via the Petviashvili iterative scheme. The computational results are compared to some existing approaches, such as Tanaka’s method and Fenton’s high-order asymptotic expansion. Several important integral quantities are computed for a large range of amplitudes. The integral representation of the velocity and acceleration fields in the bulk of the fluid is also provided.     

The Patterns of Surface Capillary-gravity Short-crested Waves with Uniform Current Fields in Coastal Waters

A fully three-dimensional surface gravity-capillary short-crested wave system is studied as two progressive wave-trains of equal amplitude and frequency, which are collinear with uniform currents and doubly-periodic in the horizontal plane, are propagating at an angle to each other. The first- and second-order asymptotic analytical solutions of the short-crested wave system are obtained via a perturbation expansion in a small parameter associated with the wave steepness, therefore depicting a series of typical three-dimensional wave patterns involving currents, shallow and deep water, and surface capillary waves, and comparing them with each other.The project supported by the Foundation for the Author of National Excellent Doctoral Dissertation of China (200428), the National Natural Science Foundation of China (10272072 and 50424913), the Shanghai Natural Science Foundation (05ZR14048), and the Shanghai Leading Academic Discipline Project (Y0103). The English text was polished by Yunming Chen.     

Use of equivalent body forces for acoustic emission from a crack in a plate

A method to determine acoustic emission of surface waves from a crack near the free edge of a plate, is presented, in terms of the function f(t), which defines the time dependence of the crack opening process, the crack opening volume per unit thickness of the plate, and the elastic constants of the plate. The determination of the time-varying displacement is based on the use of equivalent body forces, which are shown to be two double forces. The acoustic emission of the crack, or the equivalent radiation from the double forces, has been obtained by a novel use of the elastodynamic reciprocity theorem. It is of interest that the normal surface-wave displacement at a position x₀ of the free edge comes out as depending on df/dt evaluated at x₀ for t > x₀/c_R, where c_R is the velocity of surface waves on the free edge.     

Surface waves in a deformed isotropic hyperelastic material subject to an isotropic internal constraint

An isotropic elastic half-space is prestrained so that two of the principal axes of strain lie in the bounding plane, which itself remains free of traction. The material is subject to an isotropic constraint of arbitrary nature. A surface wave is propagated sinusoidally along the bounding surface in the direction of a principal axis of strain and decays away from the surface. The exact secular equation is derived by a direct method for such a principal surface wave; it is cubic in a quantity whose square is linearly related to the squared wave speed. For the prestrained material, replacing the squared wave speed by zero gives an explicit bifurcation, or stability, criterion. Conditions on the existence and uniqueness of surface waves are given. The bifurcation criterion is derived for specific strain energies in the case of four isotropic constraints: those of incompressibility, Bell, constant area, and Ericksen. In each case investigated, the bifurcation criterion is found to be of a universal nature in that it depends only on the principal stretches, not on the material constants. Some results related to the surface stability of arterial wall mechanics are also presented.     

Explicit expression of polarization vector for surface waves,slip waves,Stoneley waves and interfacial slip waves in anisotropic elastic materials

We present explicit expression of the polarization vector for surface waves and slip waves in an anisotropic elastic half-space, and Stoneley waves and interfacial slip waves in two dissimilar anisotropic elastic half-spaces. An unexpected result is that, in the case of interfacial slip waves, the polarization vector for the material in the half-space _x2≥0$x_2\ge 0$ does not depend explicitly on the material property in the half-space _x2≤0$x_2\le 0$. It depends on the material property in the half-space _x2≤0$x_2\le 0$ implicitly through the interfacial slip wave speed υ$\upsilon$. The same is true for the polarization vector for the material in the half-space _x2≤0$x_2\le 0$.     

Surface and interfacial waves in anisotropic elastic quasicrystals

We present the Stroh formalism for two-dimensional subsonic steady-state motion of anisotropic quasicrystals. Using this new formalism and a series of identities and properties which follow, we investigate subsonic surface and interfacial waves in anisotropic quasicrystals. Our results suggest that there exist at most three subsonic surface wave speeds. This interesting observation is quite different from the unique surface wave speed known for anisotropic crystals. The degenerate case of decagonal quasicrystalline materials is discussed in detail.     

Dilational rheology of an air–water interface functionalized by biomolecules: the role of surface diffusion

As an attempt to develop a sensitive device for biomolecule detection, a micromechanical methodology based on the rheological change of an air–water interface is put forward (Berthier and Davoust, CEA/CNRS patent, PCT International Application WO 2003/080209 A3, 2003). Capillary waves induced from the vertical electrodynamic vibration of a brimful cylindrical tank filled with water stand as a good way to identify dilational elasticity and viscosity of an aging interface. Before, we were interested of the fact that complex wave number and the frequency of waves are obtained through an optical technique, which allows us to accurately recognize the whole interface geometry. These two parameters, a modeling based on a dispersion relation (Lucassen-Reynders and Lucassen, Adv Colloid Interface Sci, 2:347, 1969) and the surface mass transport equation, are jointly used to identify the surface diffusivity and the dilational rheology of the interface for a nonsoluble biochemical surfactant. Preliminary results obtained from a water surface functionalized by DNA, thanks to a lipidic monolayer, demonstrate the capabilities of the proposed methodology. The sensitivity of dilational rheology and the surface diffusivity on DNA adsorption on lipids is made evident.This paper was presented at the Annual Meeting of the European Society of Rheology, Grenoble, April 2005.     

Global wave-induced loads in abnormal waves: Comparison between experimental results and classification society rules

It is important to assess the consequences of ship encounters with abnormal waves due to the perceived dangers of such encounters. A starting point for this is the assessment of global loads, with a focus on examining how the design rules fare with respect to loads induced by abnormal wave encounters. This paper presents the results of an experimental investigation into the global wave induced loads experienced in a range of abnormal sea states. Results are obtained for a segmented, flexible backbone model of a typical naval frigate. Abnormal wave encounters result in a significant increase in the global wave-induced loads compared to the equivalent random sea, with slamming becoming considerably more severe. Through comparisons with the experimental measurements it is concluded that the design rules which allow for an extreme wave encounter provide a reasonable safety margin for the global loads in abnormal waves, although discrepancies occur towards the aft of the vessel. Further investigation of the amount and conditions in which the design rules may be exceeded by an abnormal wave encounter is required.