Transient evolution of weakly nonlinear sloshing waves: an analytical and numerical comparison

The problem of water waves generated in a horizontally oscillating basin is considered, with specific emphasis on the transient evolution of the wave amplitude. A third-order amplitude evolution equation is solved analytically in terms of Jacobian elliptic functions. The solution explicitly determines the maximum amplitude and nonlinear beating period of the resonated wave. An observed bifurcation in the amplitude response is shown to correspond to the elliptic modulus approaching unity and the beating period of the interaction approaching infinity. The theoretical predictions compare favorably to fully nonlinear simulations of the sloshing process. Due to the omission of damping, the consideration of only a single mode, and the weakly nonlinear framework, the analytical solution applies only to finite-depth, non-breaking waves. The inviscid numerical simulations are similarly limited to finite depth.     

On the stability of lumps and wave collapse in water waves

In the classical water-wave problem, fully localized nonlinear waves of permanent form, commonly referred to as lumps, are possible only if both gravity and surface tension are present. While much attention has been paid to shallow-water lumps, which are generalizations of Korteweg-de Vries solitary waves, the present study is concerned with a distinct class of gravity-capillary lumps recently found on water of finite or infinite depth. In the near linear limit, these lumps resemble locally confined wave packets with envelope and wave crests moving at the same speed, and they can be approximated in terms of a particular steady solution (ground state) of an elliptic equation system of the Benney-Roskes-Davey-Stewartson (BRDS) type, which governs the coupled evolution of the envelope along with the induced mean flow. According to the BRDS equations, however, initial conditions above a certain threshold develop a singularity in finite time, known as wave collapse, due to nonlinear focusing; the ground state, in fact, being exactly at the threshold for collapse suggests that the newly discovered lumps are unstable. In an effort to understand the role of this singularity in the dynamics of lumps, here we consider the fifth-order Kadomtsev-Petviashvili equation, a model for weakly nonlinear gravity-capillary waves on water of finite depth when the Bond number is close to one-third, which also admits lumps of the wave packet type. It is found that an exchange of stability occurs at a certain finite wave steepness, lumps being unstable below but stable above this critical value. As a result, a small-amplitude lump, which is linearly unstable and according to the BRDS equations would be prone to wave collapse, depending on the perturbation, either decays into dispersive waves or evolves into an oscillatory state near a finite-amplitude stable lump.     

Modeling Acoustic Emission Signals Caused by Leakage in Pressurized Gas Pipe

Leakage in high pressure pipes creates stress waves which transmitted through the pipe wall. These waves can be recorded by using acoustic sensor or accelerometer installed on the pipe wall. Knowing how these waves vibrate pipe is very important in continuous leak source locating process. In this paper the pipe radial displacement caused by acoustic emission due to leakage is modeled analytically. The standard form of Donnell’s nonlinear cylindrical shell theory is used to derive the motion equation of the pipe for simply supported boundary condition. Using Galerkin method, the motion equation has been solved and a system of nonlinear equations with 7 degrees of freedom is obtained. A MATLAB code according to Runge-Kutta numerical method is generated to solve these equations and derive the pipe radial displacement. To check the theoretical results, acoustic emission testing with continuous leak source and linear array of two sensors positioned on two sides of the leakage source were carried out. The major noise of recorded signals was removed through the wavelet transform and filtering technique. For better analysis, fast Fourier transform (FFT) was taken from theoretical and de-noised experimental results. Comparing the results showed that the frequency which carried the most amount of energy is the same that expresses excellent agreement between the theoretical and experimental results validating the analytical model.     

Nonlinear dynamics of offshore structures under sea wave and earthquake forces

A study of dynamic response of offshore structures in random seas to inputs of earthquake ground motions is presented. Emphasis is placed on the evaluation of nonlinear hydrodynamic damping effects due to sea waves for the earthquake response. The structure is discretized using the finite element method. Sea waves are represented by Bretschneider’s power spectrum and the Morison equation defines the wave forcing function. Tajimi-Kanai’s power spectrum is used for the horizontal ground acceleration due to earthquakes. The governing equations of motion are obtained by the substructure method. Response analysis is carried out using the frequency-domain random-vibration approach. It is found that the hydrodynamic damping forces are higher in random seas than in still water and sea waves generally reduce the seismic response of offshore structures. Studies on the first passage probabilities of response indicate that small sea waves enhance the reliability of offshore structures against earthquakes forces.     

Nonlinear waves in an elastic tube with variable prestretch filled with a fluid of variable viscosity

In the present work, by employing the reductive perturbation method to the nonlinear equations of an incompressible, prestressed, homogeneous and isotropic thin elastic tube and to the exact equations of an incompressible Newtonian fluid of variable viscosity, we have studied weakly nonlinear waves in such a medium and obtained the variable coefficient Korteweg-deVries-Burgers (KdV-B) equation as the evolution equation. For this purpose, we treated the artery as an incompressible, homogeneous and isotropic elastic material subjected to variable stretches both in the axial and circumferential directions initially, and the blood as an incompressible Newtonian fluid whose viscosity changes with the radial coordinate. By seeking a travelling wave solution to this evolution equation, we observed that the wave front is not a plane anymore, it is rather a curved surface. This is the result of the variable radius of the tube. The numerical calculations indicate that the wave speed is variable in the axial coordinate and it decreases for increasing circumferential stretch (or radius). Such a result seems to be plausible from physical considerations, like Bernoulli’s law. We further observed that, the amplitude of the Burgers shock gets smaller and smaller with increasing time parameter along the tube axis. This is again due to the variable radius, but the effect of it is quite small.     

Nodal DG-FEM solution of high-order Boussinesq-type equations

A discontinuous Galerkin finite-element method (DG-FEM) solution to a set of high-order Boussinesq-type equations for modelling highly nonlinear and dispersive water waves in one horizontal dimension is presented. The continuous equations are discretized using nodal polynomial basis functions of arbitrary order in space on each element of an unstructured computational domain. A fourth-order explicit Runge-Kutta scheme is used to advance the solution in time. Methods for introducing artificial damping to control mild nonlinear instabilities are also discussed. The accuracy and convergence of the model with both h (grid size) and p (order) refinement are confirmed for the linearized equations, and calculations are provided for two nonlinear test cases in one horizontal dimension: harmonic generation over a submerged bar, and reflection of a steep solitary wave from a vertical wall. Test cases for two horizontal dimensions will be considered in future work.     

Imaging the acoustic nonlinear parameter with diffraction tomography

An approach to take diffraction into account in acoustic nonlinear parameter imaging is described. In this type of tomography, two collinear planar waves with different frequencies are used to insonify the object. The complex-amplitude of the generated secondary wave at the difference frequency is detected for producing the projections. The algorithmic formulation relies on the use of two simultaneous governing wave equations for the primary and difference-frequency waves. Three algorithms are presented respectively for reconstructing weakly, moderately, and strongly scattering objects. Illustrations of quantitative tomograms of the nonlinear parameter generated from simulated data show good correlations with theoretical results.     

Topics in nonlinear wave theory

The remarkable exact solutions, which have been found for the Kortewegde Vries and similar equations, are discussed using a perturbation approach. This leads quickly to the known solutions for multisoliton interactions, and it is hoped that it will prove useful in extending results to other cases. Then a numerical method for computing solutions to nonlinear wave equations, developed by Fornberg, is mixed with various theoretical ideas to explore a variety of problems. The problems include interacting waves, the evolution of initial steps and wells, and wavetrain instabilities.     

Evolution equation for nonlinear Scholte waves

The evolution equations for nonlinear Scholte waves (finite amplitude elastic waves propagating along liquid/solid interface), which account for the second order nonlinearity of a liquid, are derived for the first time. For mathematical simplicity the nonlinearity of the solid, which influence is expected to be weak in the case of weak localization of the Scholte wave, is not taken into consideration. The analysis of these equations demonstrates that the nonlinear processes contributing to the evolution of the Scholte wave can be divided into two groups. The first group includes nonlinear processes leading to wave spectrum broadening which are common to bulk pressure waves in liquids and gases. The second group includes the nonlinear processes which are active only in the frequency down-conversion (leading to wave spectrum conservation or narrowing), which are specific to the confined nature of the interface wave. It is demonstrated that the nonlinear parameters, which characterize the efficiency of various nonlinear processes in the interface wave, strongly depend on the relative properties of the contacting liquid and solid (or, in other words, on the deviation of the Scholte wave velocity from the velocities of sound in liquid and in solid). In particular, the sign of the nonlinear parameter responsible for the second harmonic generation can differ from the sign of the nonlinear acoustic parameter of the liquid. It is also verified that there are particular liquid/solid combinations where the nonlinear processes, which are inactive in the frequency up-conversion, dominate in the evolution of the Scholte wave. In this case distortionless propagation of the finite amplitude harmonic interface wave is possible. The proposed theory should find applications in nonlinear acoustics, geophysics, and nondestructive testing.     

Chaotic analyses of weakly damped parametrically excited cross waves with surface tension

Summary. The Wiggins-Holmes extension of the Generalized Melnikov Method (GMM) to higher dimensions and the extension of the Generalized Herglotz Algorithm (GHA) to non-autonomous systems are applied to weakly damped parametrically excited cross waves with surface tension in a long rectangular wave channel in order to demonstrate that cross waves are chaotic. The Luke Lagrangian density function for surface gravity waves with surface tension and dissipation is expressed in three generalized coordinates (or, equivalently, three degrees of freedom) that are the time-dependent components of three velocity potentials that represent three standing waves. The generalized momenta are computed from the Lagrangian, and the Hamiltonian is computed from a Legendre transform of the Lagrangian. This Hamiltonian contains both autonomous and non-autonomous components that must be suspended by applying an extension of the Herglotz algorithm for non-autonomous transformations in order to apply the Kolmogorov-Arnold-Moser (KAM) averaging operation and the GMM. Three canonical transformations are applied to (i) eliminate cross product terms by a rotation of axes; (ii) to transform to action-angle canonical variables and to eliminate two degrees of freedom; and (iii) to suspend the non-autonomous terms and to apply the Hamilton-Jacobi transformation. The system of nonlinear non-autonomous evolution equations determined from Hamiltons equations of motion of the second kind must be averaged in order to obtain an autonomous system that may be analyzed by the GMM. Hyperbolic saddle points that are connected by heteroclinic separatrices are computed from the unperturbed autonomous system. The non-dissipative perturbed Hamiltonian system with surface tension satisfies the KAM non-degeneracy requirements, and the Melnikov integral is calculated to demonstrate that the motion is chaotic. For the perturbed dissipative system with surface tension, the only hyperbolic fixed point that survives the averaged equations is a fixed point of weak chaos that is not connected by a homoclinic separatrix; consequently, the Melnikov integral is identically zero. The chaotic motion for the perturbed dissipative system with surface tension is demonstrated by numerical computation of positive Liapunov characteristic exponents.     

Strip theory for underwater vehicles in water of finite depth

This paper addresses the need to know the unsteady forces and moments on an underwater vehicle in finite-depth water, at small enough submergences for it to be influenced by sea waves. The forces are those due to the waves themselves, as well as the radiation forces due to unsteady vehicle motions. Knowledge of these forces and the mass distribution of the vehicle allow solution of the equations of motion at a single-frequency. Since the theory is linear, any incident wave field can be decomposed into the sum of many individual single-frequency sinusoidal waves. The motions due to each frequency component can then be added together to obtain the total predicted vehicle motions. The wave forces are due to the undisturbed sea wave plus those due to the diffracted wave necessary to satisfy boundary conditions on the vehicle. The long-used strip theory for ships, with the inviscid-flow approximation, is modified for finite depth and inclusion of lift forces on the vehicle fins. The two-dimensional solutions for the forces on each strip are found by a different method than is commonly used for strip theory. This form of the theory is easier to deal with and requires much less computing time than a fully three-dimensional approach. Experiments are conducted and their results are compared with the theory. Excellent agreement is found between the theoretical and experimental wave forces, including the diffracted wave. It is shown that inclusion of the forces on the fins not only improves the theoretical wave forces, but also brings the results of theory for the radiation forces and moments due to vehicle motions much closer to the experimental values that the theory without inclusion of fin lift forces.     

Hamiltonian formulation for solitary waves propagating on a variable background

Solitary waves propagating on a variable background are conventionally described by the variable-coefficient Korteweg-de Vries equation. However, the underlying physical system is often Hamiltonian, with a conserved energy functional. Recent studies for water waves and interfacial waves have shown that an alternative approach to deriving an appropriate evolution equation, which asymptotically approximates the Hamiltonian, leads to an alternative variable-coefficient Korteweg-de Vries equation, which conserves the underlying Hamiltonian structure more explicitly. This paper examines the relationship between these two evolution equations, which are asymptotically equivalent, by first discussing the conservation laws for each equation, and then constructing asymptotically a slowly-varying solitary wave.     

Forced Korteweg-de Vries equation in an elastic tube filled with an inviscid fluid

In the present work, treating the arteries as a prestressed thin walled elastic tube with a stenosis and the blood as an inviscid fluid, we have studied the propagation of weakly nonlinear waves in such a composite medium, in the long wave approximation, by use of the reductive perturbation method [C.S. Gardner, G.K. Morikawa, Similarity in the asymptotic behavior of collision-free hydromagnetic waves and water waves, Courant Institute Math. Sci. Report, NYO-9082 (1960) 1-30, T. Taniuti, C.C. Wei, Reductive perturbation method in non-linear wave propagation I, J. Phys. Soc. Jpn., 24 (1968) 941-946]. We obtained the forced Korteweg-de Vries (FKdV) equation with variable coefficients as the evolution equation. By use of the coordinate transformation, it is shown that this type of evolution equation admits a progressive wave solution with variable wave speed. As might be expected from physical consideration, the wave speed reaches its maximum value at the center of stenosis and gets smaller and smaller as we go away from the center of the stenosis. The variations of radial displacement and the fluid pressure with the distance parameter are also examined numerically. The results seem to be consistent with Bernoulli’s law for inviscid fluid.     

Wave propagation in fluid filled nonlinear viscoelastic tubes

Summary The present work considers one dimensional wave propagation in an infinitely long, straight and homogeneous nonlinear viscoelastic tube filled with an incompressible, inviscid fluid. In order to include the geometric dispersion in the analysis, the tube wall inertia effects are added to the pressure-area relation. Using the reductive perturbation technique, the propagation of weakly nonlinear waves in the long-wave approximation is examined. In the long-wave approximation, a general equation is obtained, and it is shown that by a proper scaling this equation reduces to the well-known nonlinear evolution equations. Intensifying the effect of nonlinearity in the perturbation process, the modified forms of these evolution equations are also obtained. In the absence of nonlinear viscoelastic effects all the equations reduce to those of the linear viscoelastic tube.     

一类具高阶非线性项的发展方程的准确周期解

本文导出了具高阶非线性项的Lienard方程的准确周期解并从理论上给予了证明,然后利用这些公式得到一大批具高阶非线性项的发展方程的各种Jacobi椭圆函数型的准确周期解,由此避免了一大类非线性发展方程求周期解时求解过程的重复。     

Evolution of weakly non-linear shallow water waves generated by a moving boundary

Summary In this study we describe the evolution of weakly non-linear shallow water waves in a rectangular channel of 16 m length which aregenerated by a moving boundary. We present a detailed comparison of computaticnal and observational waveheight-time series and thus verify the theoretical model as presented by Villeneuve and Savage [27].Three different types of wave generating devices were used: pistons with vertical and inclined frontal faces, submerged boxes and a rotating plate. Waveheight-time series are recorded at eight different positions along the channel by electrical resistivity gauges, and velocity profiles are determined at certain selective cross sections. Data of many wave experiments are presented in nondimensional form. This representation reveals that the initial wave forms depend upon a Froude number of the motion of the wave generator, the slope angle of the wedge-type pistons and on the dimensionless displaced volume. Evolving waveheight-time series that are recorded at the various gauges are compared with those obtained from computations by use of equations which generalize the Boussinesq equations to include time variations of the boundaries.The method of inverse scattering is applied both, experimental and numerical waveheight-time series are prescribed as initial data. Results are tested relative to two different observers one fixed in time the other fixed in space. Deviations are shown to be small in all cases.     

防波堤帷幕尺度对波浪力和消浪性能的影响研究

帷幕式防波堤下部透水,利于港内外的海水交换,防止水质恶化,维持原始水域生态平衡,因此模拟波浪与防波堤的相互作用具有重要的现实意义。该文基于高阶紧致插值CIP(Constrained Interpolation Profile)方法的二维波浪数值模型水槽,通过与文献结果进行比较,结果表明：该文数值模型能够有效地模拟孤立波在冲击帷幕式防波堤过程中的波浪破碎等强非线性现象。通过数值模拟,描述了帷幕式防波堤在不同的宽度和浸没深度情况下与孤立波之间的相互作用中流速和涡度的演变过程,并讨论了孤立波波高和帷幕式防波堤结构尺寸对反射、透射、耗散系数以及在帷幕式防波堤上冲击力的影响。研究表明：帷幕式防波堤的相对浸没深度越大时,消浪效果良好,同时帷幕式防波堤的相对宽度对消浪效果的影响较大。     

箱形船体波浪和波浪压力的非线性时域模型

该文将耦合计算模型应用于计算固定箱形船体上非线性波浪压力,研究了将该准三维模型应用于物体三维波浪压力的计算能力.为了提高计算强非线性波浪的能力,在方程中加入了高阶非线性项,并进行了箱体波浪压力的测量模型实验,以验证计算结果.计算模型将船体底面以下水域取为内域,其余水域为外域.内域用欧拉方程计算,外域采用Boussine...