Wave forces on a submerged horizontal plate–Part II: Solitary and cnoidal waves

This paper is the companion to Part I under the same title, and is mainly concerned with wave loads due to nonlinear waves of solitary and cnoidal type propagating over a submerged, horizontal and thin plate. Following the development of the nonlinear model (via the Level I Green–Naghdi theory) for the flow of an incompressible and inviscid fluid given in Part I, the wave-induced loads on the submerged, fixed (and rigid) plate are calculated, and results are compared with the available laboratory data, and with linear solutions of the problem. Dependence of the loads on wave conditions (wave height and wavelength) and plate characteristics (submergence depth and plate width) are studied for both the solitary and cnoidal wave cases.     

Re-polarization of elastic waves at a frictional contact interface—II. Incidence of a P or SV wave

This is Part II of a two-part paper which analyses the re-polarization of elastic waves at a frictional contact interface between two solids. The re-polarization of SH waves was solved in Part I by the use of the Fourier analysis. Here, in Part II, we consider the re-polarization of P or SV waves. It is assumed that the two solids are pressed together and, at the same time, loaded by anti-plane and in-plane shearing traction. If the incident wave is sufficiently strong, localized separation and slip may take place at the interface. As a result, the incident in-plane wave is re-polarized at the interface so that the anti-plane waves (SH waves) are induced. Using the method similar to that of Part I and considering the boundary conditions involving separation and slip, we manage to reduce the problem to a set of algebraic equations coupled with simple integral equations. An iterative method is developed based on the solution to the perfectly bonded interface. The locations and sizes of the separation and slip zones, the interface traction, the slip velocities, the global sliding velocities and the energy dissipation and partition are displayed for the case of two identical materials. It is found that the separation zones and the gaps are independent of the induced waves.     

Two-to-one resonant multi-modal dynamics of horizontal/inclined cables. Part I: Theoretical formulation and model validation

This paper is first of the two papers dealing with analytical investigation of resonant multi-modal dynamics due to 2:1 internal resonances in the finite-amplitude free vibrations of horizontal/inclined cables. Part I deals with theoretical formulation and validation of the general cable model. Approximate nonlinear partial differential equations of 3-D coupled motion of small sagged cables – which account for both spatio-temporal variation of nonlinear dynamic tension and system asymmetry due to inclined sagged configurations – are presented. A multi-dimensional Galerkin expansion of the solution of nonplanar/planar motion is performed, yielding a complete set of system quadratic/cubic coefficients. With the aim of parametrically studying the behavior of horizontal/inclined cables in Part II [25], a second-order asymptotic analysis under planar 2:1 resonance is accomplished by the method of multiple scales. On accounting for higher-order effects of quadratic/cubic nonlinearities, approximate closed-form solutions of nonlinear amplitudes, frequencies and dynamic configurations of resonant nonlinear normal modes reveal the dependence of cable response on resonant/nonresonant modal contributions. Depending on simplifying kinematic modeling and assigned system parameters, approximate horizontal/inclined cable models are thoroughly validated by numerically evaluating statics and non-planar/planar linear/non-linear dynamics against those of the exact model. Moreover, the modal coupling role and contribution of system longitudinal dynamics are discussed for horizontal cables, showing some meaningful effects due to kinematic condensation.     

On the effective behavior of nonlinear inelastic composites: II: A second-order procedure

A new method for determining the overall behavior of composite materials comprising nonlinear viscoelastic and elasto-viscoplastic constituents is presented. Part I of this work showed that upon use of an implicit time-discretization scheme, the evolution equations describing the constitutive behavior of the phases can be reduced to the minimization of an incremental energy function. This minimization problem is rigorously equivalent to a nonlinear thermoelastic problem with a transformation strain which is a nonuniform field (not even uniform within the phases). In part I of this paper the nonlinearity was handled using a variational (or secant) technique. In this second part of the study, a proper modification of the second-order procedure of Ponte Castañeda is proposed and leads to replacing, at each time-step, the actual nonlinear viscoelastic composite by a linear viscoelastic one. The linearized problem is even further simplified by using an “effective internal variable” in each individual phase. The resulting predictions are in good agreement with exact results and improve on the predictions of the secant model proposed in part I of this paper.     

Accuracy and efficiency of two numerical methods of solving the potential flow problem for highly nonlinear and dispersive water waves

The accuracy and efficiency of two methods of resolving the exact potential flow problem for nonlinear waves are compared using three different one horizontal dimension (1DH) test cases. The two model approaches use high‐order finite difference schemes in the horizontal dimension and differ in the resolution of the vertical dimension. The first model uses high‐order finite difference schemes also in the vertical, while the second model applies a spectral approach. The convergence, accuracy, and efficiency of the two models are demonstrated as a function of the temporal, horizontal, and vertical resolutions for the following: (1) the propagation of regular nonlinear waves in a periodic domain; (2) the motion of nonlinear standing waves in a domain with fully reflective boundaries; and (3) the propagation and shoaling of a train of waves on a slope. The spectral model approach converges more rapidly as a function of the vertical resolution. In addition, with equivalent vertical resolution, the spectral model approach shows enhanced accuracy and efficiency in the parameter range used for practical model applications. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

Chaos control of a nonlinear oscillator with shape memory alloy using an optimal linear control: Part I: Ideal energy source

In this paper, an optimal linear control is applied to control a chaotic oscillator with shape memory alloy (SMA). Asymptotic stability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function, which can clearly be seen to be the solution of the Hamilton–Jacobi–Bellman equation, thus guaranteeing both stability and optimality. This work is presented in two parts. Part I considers the so-called ideal problem. In the ideal problem, the excitation source is assumed to be an ideal harmonic excitation.     

Calculation and measurement of conical beams of three-dimensional periodic internal waves excited by a vertically oscillating piston

The structure of the wave field given by an exact solution of the linearized problem of radiation of three-dimensional periodic internal waves in a continuously stratified viscous fluid is analyzed numerically. The waves are generated by a piston, i.e., a disk lying on a fixed horizontal plane and oscillating in the vertical direction. The flow fields and the wave displacements are compared with the data of shadow visualization and measurements of the wave amplitudes made using a contact sensor. The calculated and observed wave patterns are in satisfactory agreement and the displacement distributions coincide correct to a fitting coefficient 0.7 < K < 1.1 characterizing the role of the nonlinear effects and other factors neglected in this model.     

Seismic responses for multi-storeyed buildings composed of nonorthogonal members under bi-directional horizontal earthouake ground motions

This paper presents a method for analyzing both linear and nonlinear seismic responses of multistoreyed buildings composed of nonorthogonal structural members under the action of bi-directional horizontal earthquake waves. Different kinds of restoring force model are used. An efficient computer programme for the computation of nonlinear seismic response of the structure was worked out. In order to illustrate the application of this method, an example of a ten-storeyed building of such sort is given.     

Lagrangian modelling of fluid sloshing in moving tanks

The paper considers the problem of sloshing of incompressible fluid in a moving 2-D rectangular tank under horizontal and vertical excitation. The problem is solved in Lagrangian variables by applying two approaches. First, a third-order asymptotic solution for resonant sloshing with a dominant mode is derived using a recursive technique. Then, fully nonlinear set of equations in the material coordinates is solved numerically by employing a finite difference method. Both methods are applied to a problem of high amplitude resonant Faraday waves and the obtained results are compared with experimental data known from the literature and a good agreement between the results of the two methods and the empirical data is demonstrated.     

Internal wave reflections and transmissions arising from a non-uniform mesh. Part III: The occurrence of evanescent waves in the Crank-Nicolson linear finite element scheme

The numerical scheme upon which this paper is based is the 1D Crank–Nicolson linear finite element scheme. In Part I of this series it was shown that for a certain range of incident wavelengths impinging on the interface of an expansion in nodal spacing, an evanescent (or spatially damped) wave results in the downstream region. Here in Part III an analysis is carried out to predict the wavelength and the spatial rate of damping for this wave. The results of the analysis are verified quantitatively with seven ‘hot-start’ numerical experiments and qualitatively with seven ‘cold-start’ experiments. Weare has shown that evanescent waves occur whenever the frequency of a disturbance at a boundary exceeds the maximum frequency given by the dispersion relation. In these circumstances the ‘extended dispersion’ relation can be used to determine the rate of spatial decay. In the context of a domain consisting of two regions with different nodal spacings, the use of the group velocity concept shows that evanescent waves have no energy flux associated with them when energy is conserved.     

A Dimension-Breaking Phenomenon for Water Waves with Weak Surface Tension

It is well known that the water-wave problem with weak surface tension has small-amplitude line solitary-wave solutions which to leading order are described by the nonlinear Schrödinger equation. The present paper contains an existence theory for three-dimensional periodically modulated solitary-wave solutions which have a solitary-wave profile in the direction of propagation and are periodic in the transverse direction; they emanate from the line solitary waves in a dimension-breaking bifurcation. In addition, it is shown that the line solitary waves are linearly unstable to long-wavelength transverse perturbations. The key to these results is a formulation of the water wave problem as an evolutionary system in which the transverse horizontal variable plays the role of time, a careful study of the purely imaginary spectrum of the operator obtained by linearising the evolutionary system at a line solitary wave, and an application of an infinite-dimensional version of the classical Lyapunov centre theorem.     

Interaction between a submerged evacuated cylindrical shell and a shock wave—Part II: Numerical aspects of the solution

The interaction between a submerged elastic circular cylindrical shell and an external shock wave is addressed. A linear, two-dimensional formulation of the problem is considered. A semi-analytical solution is obtained using a combination of the classical analytical approach based on the use of the Laplace transform and separation of variables, and finite difference methodology. The study consists of two parts. Part I focuses on the simulation and analysis of the acoustic fields induced during the interaction. Both the diffraction (absolutely rigid cylinder) and complete diffraction–radiation (elastic shell) are considered. Special attention is paid to the lower-magnitude shell-induced waves representing radiation by the elastic waves circumnavigating the shell. The focus of Part II is on the numerical analysis of the solution. The convergence of the series solution and finite-difference scheme is analysed. The computation of the response functions of the problem is discussed as well, as is the effect of the bending stiffness on the acoustic field. The membrane model of the shell is considered to analyse such effect, which, in combination with the models addressed in Part I, allows for the analysis of the evolution of the acoustic field around the structure as its elastic properties change from an absolutely rigid cylinder to a membrane. The results of the numerical simulations are compared to available experimental data, and a good agreement is observed.     

Interaction between a submerged evacuated cylindrical shell and a shock wave—Part I: Diffraction–radiation analysis

The interaction between a submerged elastic circular cylindrical shell and an external shock wave is addressed. A linear, two-dimensional formulation of the problem is considered. A semi-analytical solution is obtained using a combination of the classical analytical approach based on the use of the Laplace transform and separation of variables, and finite difference methodology. The study consists of two parts. Part I focuses on the simulation and analysis of the acoustic fields induced during the interaction. Both the diffraction (absolutely rigid cylinder) and complete diffraction–radiation (elastic shell) are considered. Special attention is paid to the lower-magnitude shell-induced waves representing radiation by the elastic waves circumnavigating the shell. The focus of Part II is on the numerical analysis of the solution. The convergence of the series solution and finite-difference scheme is analysed. The computation of the response functions of the problem is discussed as well, as is the effect of the bending stiffness on the acoustic field. The membrane model of the shell is considered to analyse such an effect, which, in combination with the models addressed in Part I, allows for the analysis of the evolution of the acoustic field around the structure as its elastic properties change from an absolutely rigid cylinder to a membrane. The results of the numerical simulations are compared to available experimental data, and a good agreement is observed.     

Finite element simulation of fully non‐linear interaction between vertical cylinders and steep waves. Part 1: methodology and numerical procedure

A methodology for computing three‐dimensional interaction between waves and fixed bodies is developed based on a fully non‐linear potential flow theory. The associated boundary value problem is solved using a finite element method (FEM). A recovery technique has been implemented to improve the FEM solution. The velocity is calculated by a numerical differentiation technique. The corresponding algebraic equations are solved by the conjugate gradient method with a symmetric successive overrelaxation (SSOR) preconditioner. The radiation condition at a truncated boundary is imposed based on the combination of a damping zone and the Sommerfeld condition. This paper (Part 1) focuses on the technical procedure, while Part 2 [Finite element simulation of fully non‐linear interaction between vertical cylinders and steep waves. Part 2. Numerical results and validation. International Journal for Numerical Methods in Fluids 2001] gives detailed numerical results, including validation, for the cases of steep waves interacting with one or two vertical cylinders. Copyright © 2001 John Wiley & Sons, Ltd.     

Quasi-periodic waves and quasi-solitary waves in stratified fluid of slowly varying depth

The nonlinear waves in a stratified fluid of slowly varying depth are investigated in this paper. The model considered here consists of a two-layer incompressible constantdensity inviscid fluid confined by a slightly uneven bottom and a horizontal rigid wall. The Korteweg-de Vries (KdV) equation with varying coefficients is derived with the aid of the reductive perturbation method. By using the method of multiple scales, the approximate solutions of this equation are obtained. It is found that the unevenness of bottom may lead to the generation of so-called quasi-periodic waves and quasisolitary waves, whose periods, propagation velocities and wave profiles vary slowly. The relations of the period of quasiperiodic waves and of the amplitude, propagation velocity of quasi-solitary waves varying with the depth of fluid are also presented. The models with two horizontal rigid walls or single-layer fluid can be regarded as particular cases of those in this paper.Project Supported by National Natural Science Foundation of China.