Computation of interface wave motions by reciprocity considerations

The current theoretical study deals with computation of Stoneley waves along a solid–solid interface and Scholte waves (also called Scholte-Gogoladze) along a solid–liquid interface by reciprocity considerations. Closed-form solutions of the wave motions generated by a time-harmonic line load applied in two bonded elastic half-spaces of different material properties are derived in a simple manner. In order to perform direct applications of reciprocity theorems, we introduce in this article new expressions for the displacements of free interface waves. Reciprocity relations between an actual state, interface wave motion generated by a time-harmonic line load, and a virtual state, an appropriately chosen free wave traveling along the interface, are derived. Scattered amplitudes of Stoneley waves and Scholte waves due to the load are thus computed. To show application of the obtained results, scattering of Stoneley wave by a delamination at the interface is then studied.     

Application of the reciprocity theorem to scattering of surface waves by a cavity

Scattering of surface waves by a cylindrical cavity at the surface of a homogenous, isotropic, linearly elastic half-space is analyzed in this paper. In the usual manner, the scattered field is shown to be equivalent to the radiation from a distribution of tractions, obtained from the incident wave on the surface of the cavity. For the approximation used in this paper, these tractions are shifted to tractions applied to the projection of the cavity on the surface of the half-space. The radiation of surface waves from a normal and a tangential line load, recently determined by the use of the reciprocity theorem, is employed to obtain the field scattered by the cavity from the superposition of displacements due to the distributed surface tractions. The vertical displacement at some distance from the cavity is compared with the solution of the scattering problem obtained by the boundary element method (BEM) for various depths and widths of the cavity. Comparisons between the analytical and BEM results are graphically displayed. The limitations of the approximate approach are discussed based on the comparisons with the BEM results.     

Reciprocity and power-flow theorems for the scattering of plane elastic waves in a half space

In this paper we present six field-reciprocity relations, and an additional six power-reciprocity relations for the far-zone scattered field if time-harmonic plane elastic waves are incident upon an obstacle in a half space. The incident waves are successively selected as to be either a P-wave, a SV-wave or a Rayleigh surface wave, each propagating in a prescribed direction. In the derivations we employ an explicit integral representation for the far-zone scattered field amplitude. The latter is obtained by expanding the half-space Green's displacement tensor in the far-zone region. Then, starting with the general Betti-Rayleigh theorem, the reciprocity identities are systematically inferred by inspection. We also present energy-conservation relations due to a single incident P-, SV- or Rayleigh wave.     

楔形空间中任意形状孔洞对平面SH波的散射：IBEM求解

采用间接边界元法(IBEM),求解了楔形空间中任意形状孔洞对平面SH波的散射问题。基于单层位势理论,首先在孔洞表面及其附近楔形空间表面上施加虚拟均布荷载,以构造散射波场。进而由自由表面零应力条件,建立方程求解得到虚拟荷载密度。总波场由楔形空间自由波场和散射波场叠加得到。研究表明：IBEM方法能够精确高效求解楔形空间中弹性波散射问题。楔形空间孔洞对波的散射特征依赖于波入射角、无量纲入射波频率、楔形夹角、孔洞位置及其形状;孔洞周围波的相干效应十分显著,空间表面位移幅值及孔边动应力集中因子比半空间情况放大一倍有余,分别达到8.5和10.0;该研究为楔形空间中更为复杂的P、SV波散射问题求解奠定了基础。     

On the energy radiated by Rayleigh waves

A relation is derived between the energy radiated by straight-crested Rayleigh waves and the amplitude of the vertical component of the displacement at the surface. This relation involves a constant of proportionality which depends in a complicated manner on Poisson's ratio, but a simple, quadratic, approximation is derived which is accurate to within 1%. The corresponding relation between the radiated energy and the amplitude of surface displacement for cylindrical surface waves is also derived and is used to show the influence of Poisson's ratio on the partition of energy between the body waves and the surface wave from two point sources.     

Surface acoustic wave measurements of small fatigue cracks initiated from a surface cavity

A model for the low frequency scattering of a surface acoustic wave by a surface cylindrical cavity with two corner cracks is presented. It is applied to determine the depth of the small fatigue cracks initiated from a pit-type surface flaw. The general scattering formalism based on the elastodynamic reciprocity principle is employed. The effect of the cylindrical cavity on the surface wave reflection from cracks is considered using an approximate stress intensity factor for the corner cracks. In situ surface acoustic wave measurements have been performed during fatigue tests for an Al 2024-T3 sample. The surface wave signal was acquired continuously at different cyclic load levels. The model is verified by comparing calculated reflection signals and spectra with those from experiments. The depths of fully and partially open cracks are determined from the predicted and the measured surface wave reflections. The surface wave reflection is observed to be sensitive to crack closure.     

Scattering and excitation of SH-surface waves by a protrusion at the mass-loaded boundary of an elastic half-space

A rigorous theory of the scattering and excitation of SH-surface waves by a protrusion at the mass-loaded boundary of an elastic half-space is presented. The boundary value problem (which is of the third kind) is solved by employing two suitably chosen Green functions. One of them is represented as a Fourier type of integral, the other is taken to be the Bessel function of the second kind and order zero. The procedure leads to a system of three, coupled, integral equations. This system is solved numerically. In case of an incident bulk wave, the amplitude of the launched surface wave is computed; in case of an incident surface wave, its transmission and reflection factor are computed. For both cases, an expression for the far-field radiation pattern of the scattered bulk wave is derived. A reciprocity relation is shown to exist between the amplitude of the launched surface wave and the far-field bulk wave radiation pattern. Numerical results are presented for a triangularly-prismatic protrusion; they are compared with the results pertaining to a corresponding indentation in the mass-loaded boundary, that have been obtained in a previous paper.     

Fully nonlinear modeling of radiated waves generated by floating flared structures

The nonlinear radiated waves generated by a structure in forced motion, are simulated numerically based on the potential theory. A fully nonlinear numerical model is developed by using a higher-order boundary element method （HOBEM）. In this model, the instantaneous body position and the transient free surface are updated at each time step. A Lagrangian technique is employed as the time marching scheme on the free surface. The mesh regridding and interpolation methods are adopted to deal with the possible numerical instability. Several auxiliary functions are proposed to calculate the wave loads indirectly, instead of directly predicting the temporal derivative of the velocity potential. Numerical experiments are carried out to simulate the heave motions of a submerged sphere in infinite water depth, the heave and pitch motions of a truncated flared cylinder in finite depth. The results are verified against the published numerical results to ensure the effectiveness of the proposed model. Moreover, a series of higher harmonic waves and force components are obtained by the Fourier transformation to investigate the nonlinear effect of oscillation frequency. The difference among fully nonlinear, body-nonlinear and linear results is analyzed. It is found that the nonlinearity due to free surface and body surface has significant influences on the numerical results of the radiated waves and forces.     

Doppler effects of an oscillating line source in shear flow with a free surface

The linearised water-wave radiation problem for the oscillating 2D submerged source in an inviscid shear flow with a free surface is investigated analytically. There is a nonzero surface velocity. The depth is infinite and the vorticity is uniform. The amplitudes radiated from the source are calculated analytically. Due to Doppler effects, there may be up to four different emitted waves, and there is resonance with zero group velocity and infinite amplitude.     

Existence and uniqueness of a solution to a problem of steady composite waves on the surface of a liquid flowing over a wavy bottom

Composite waves on the surface of the stationary flow of a heavy ideal incompressible liquid are steady forced waves of finite amplitude which do not disappear when the pressure on the free surface becomes constant but rather are transformed into free nonlinear waves [1]. It will be shown that such waves correspond to the case of nonlinear resonance, and mathematically to the bifurcation of the solution of the fundamental integral equation describing these waves. In [2], a study is made of the problem of composite waves in a flow of finite depth generated by a variable pressure with periodic distribution along the surface of the flow. In [3], such waves are considered for a flow with a wavy bottom. In this case, composite waves are defined as steady forced waves of finite amplitude that, when the pressure becomes constant and the bottom is straightened, do not disappear but are transformed into free nonlinear waves over a flat horizontal bottom. However, an existence and uniqueness theorem was not proved for this case. The aim of the present paper is to fill this gap and investigate the conditions under which such waves can arise.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 88–98, July–August, 1980.     

Exact Solitary Water Waves with Vorticity

The solitary water wave problem is to find steady free surface waves which approach a constant level of depth in the far field. The main result is the existence of a family of exact solitary waves of small amplitude for an arbitrary vorticity. Each solution has a supercritical parameter value and decays exponentially at infinity. The proof is based on a generalized implicit function theorem of the Nash–Moser type. The first approximation to the surface profile is given by the “KdV” equation. With a supercritical value of the surface tension coefficient, a family of small amplitude solitary waves of depression with subcritical parameter values is constructed for an arbitrary vorticity.     

Non-Darcy natural convection from a vertical wavy surface in a porous medium

We examine the combined effect of spatially stationary surface waves and the presence of fluid inertia on the free convection induced by a vertical heated surface embedded in a fluid-saturated porous medium. We consider the boundary-layer regime where the Darcy-Rayleigh number, Ra, is very large, and assume that the surface waves have O(1) amplitude and wavelength. The resulting boundary-layer equations are found to be nonsimilar only when the surface is nonuniform and inertia effects are present; self-similarity results when either or both effects are absent. Detailed results for the local and global rates of heat transfer are presented for a range of values of the inertia parameter and the surface wave amplitude.     

Modeling nonlinear Rayleigh wave fields generated by angle beam wedge transducers—A theoretical study

Nonlinear Rayleigh wave fields generated by an angle beam wedge transducer are modeled in this study. The calculated area sound sources underneath the wedge are used to model the fundamental Rayleigh sound fields on the specimen surface, which are more accurate than the previously used line sources with uniform or Gaussian amplitude distributions. A general two-dimensional nonlinear Rayleigh wave equation without parabolic approximation is introduced and the solutions are obtained using the quasilinear theory. The second harmonic Rayleigh wave due to material nonlinearity is given in an integral expression with these fundamental Rayleigh waves radiated by the wedge transmitter acting as a forcing function. Multi-Gaussian beam (MGB) models are employed to simplify these integral solutions and to extract the diffraction and attenuation correction terms explicitly. The effect of nonlinearity of generating sources on the second harmonic Rayleigh wave fields is taken into consideration; simulation results show that it will affect the magnitude and diffraction correction of the second harmonic waves in the region close to the Rayleigh wave sound sources. This research provides a theoretical improvement to alleviate the experimental restriction on analyzing the effects of diffraction, attenuation and source nonlinearity when using angle beam wedge transducers as transmitters.     

Transient radiation by a submerged fluid-filled cylindrical shell

The radiation by a submerged fluid-filled cylindrical shell in response to a transient external pressure pulse is considered, and a semi-analytical model based on the Reissner–Mindlin shell theory is employed to simulate the interaction numerically. Two types of radiated waves that have been previously seen in experimental images for a submerged evacuated cylindrical shell are observed in both the external and internal fluids, the symmetric Lamb waves S₀ and the antisymmetric Lamb (or pseudo-Rayleigh) waves A₀. The third type of radiated waves is also observed that has not been explicitly imaged either experimentally or numerically for a submerged evacuated cylindrical shell, and it is demonstrated that these waves are the Scholte–Stoneley waves A. The effect that the complex structure of the radiated field has on the wave phenomena in the internal fluid is analyzed for shells of several different thicknesses, and the results of this analysis are summarized in the form of diagrams suitable for the use at the pre-design stage.     

Comments on the relation between surface wave theory and the theory of reflection

The sextic Stroh formalism, previously extensively used in the analysis of subsonic phenomena, has been used for the analysis of reflection phenomena and leaky surface waves in the first transsonic range of velocities. In particular the behaviour of the reflection problem at the limiting velocity is studied. It is shown that when the condition of free surface can be satisfied without the inhomogeneous partial wave, a situation which would appear to be the natural limiting case of a surface wave of infinite penetration, the body wave alone satisfies the condition of free surface. This result illuminates the Barnett-Lothe existence theorem for subsonic surface waves. The close connection between the reflection problem and the leaky surface wave problem becomes very apparent in the formalism used. It is shown that for a point on a branch of leaky waves where the solution is undamped, the conditions for simple reflection, i.e. reflection only involving the two body waves, are also present. In the vicinity of such a point reflection is accompanied by resonance excitation of leaky waves. The paper concludes with some explicit calculations for transversely isotropic solids.     

A lattice Boltzmann method for viscous free surface waves in two dimensions

We propose a new method based on the combination of the lattice Boltzmann equation (LBE) and the kinematic boundary condition (KBC) method to simulate viscous free surface wave in two dimensions. In our method, the flow field is modeled by LBE, whereas the free surface is explicitly tracked by the local height function, which is calculated by the KBC method. The free surface boundary condition (FSBC) for LBE is revised from previous researches. Interpolation‐supplemented lattice Boltzmann (ISLB) method is introduced, which enables our approach to be applied on arbitrary, nonuniform mesh grids. Five cases are simulated respectively to validate the LBE–KBC method: the stationary flow and the solitary waves simulated by the revised‐FSBC are more accurate than the one obtained by the former‐FSBC; numerical results of standing waves show that our method is compatible to the existing two‐dimensional finite‐volume scheme; cases of small amplitude Stokes wave and waves traveling over a submerged bar show good agreement on wave celerity, wavelength, wave amplitude and wave period between numerical results and corresponding analytical solutions and/or experiment data.Copyright © 2012 John Wiley & Sons, Ltd.     

Nonlinear vibrations of a vapor film in the presence of intense heat fluxes

Waves propagating along the interface between a thin vapor film and a liquid layer in the presence of a heat flux are investigated. The boundary conditions on the vapor-liquid phase surface take into account the temperature dependence of the pressure and the possibilities of formation of the metastable state of the superheated liquid and mass flow. Variations in the saturation pressure as functions of the temperature and mass flux lead to generation of weakly damped periodic waves of low amplitude whose velocity can be much higher than the velocity of the gravity waves. The waves ensure stability of the vapor film beneath the liquid layer in the gravity field. The finite-amplitude waves on the surface of the vapor film differ from the Stokes surface waves on the free surface of isothermal fluid. Instability regimes related with superheating of the liquid ant its explosive boiling when the amplitude of an initially small wave increases to infinity in a finite time can develop in a certain working-parameter regime.     

Blast-induced waves in soil around a tunnel

The paper is concerned with the numerical modelling of waves in saturated granular soil around a cylindrical tunnel with a tubbing-type lining. The waves are induced by a short-term large-amplitude load on the tunnel wall. The soil is assumed to be fully saturated or contain a small amount (a few volume percent) of free gas in the pore water. The behaviour of the solid skeleton is described by a constitutive relation of the hypoplasticity theory. The presence of gas makes the soil stiffness stronger pressure dependent and leads to the formation of a shock front in the soil in the immediate vicinity of the tunnel. Free gas in the pore water is shown to be an adverse factor from the viewpoint of both the residual state of the soil and the deformation of the lining. A permanent reduction of the effective stresses in the soil caused by the wave is stronger for the soil with gas. For a sufficiently large loading amplitude, the wave may lead to the vanishing of the effective stresses and liquefaction of the soil. It is also shown that the presence of gas increases the displacement and the velocity of the tubbings.     

Head-sea low-frequency diffraction of wave groups by a slender body

A progressive wavetrain with a narrow frequency band is known to be accompanied by long set-down waves travelling with the groups. In the scattering of the wavetrain from a structure free long waves can be radiated. An asymptotic theory is presented for the head-sea low-frequency diffraction of set-down waves of a wavetrain by a slender body. To match the outer with the inner solutions in a simplified manner, new transversal multiple-scales are chosen and the modulation process is described. The present theory allows a calculation of the low-frequency wave force without solving any two-dimensional boundary value problem.