A New Lagrangian Asymptotic Solution for Gravity–Capillary Waves in Water of Finite Depth

A third-order Lagrangian asymptotic solution is derived for gravity–capillary waves in water of finite depth. The explicit parametric solution gives the trajectory of a water particle and the wave kinematics for Lagrangian points above the mean water level, and in a water column. The water particle orbits and mass transport velocity as functions of the surface tension are obtained. Some remarkable trajectories may contain one or multiple sub-loops for steep waves and large surface tension. Overall, an increase in surface tension tends to increase the motions of surface particles including the relative horizontal distance travelled by a particle as well as the time-averaged drift velocity     

Non-linear interaction of water waves with submerged obstacles

The interaction of two-dimensional water waves with a fixed submerged cylinder is studied using a finite difference scheme with boundary-fitted co-ordinates. A mixed Eulerian–Lagrangian (MEL) formulation is used to satisfy the fully non-linear free surface conditions. The diffraction of small-amplitude water waves by a cylinder is examined for various wavelengths and amplitudes of the incident wave. Fourier analyses of the incident and diffracted waves are performed to determine their spectra. An example of a large-amplitude wave breaking over a cylinder is also studied. The non-linear numerical solutions are compared with those of experiments and linear theory where appropriate.     

Instability of capillary-gravity waves in water of arbitrary uniform depth

The Benjamin-Feir instability of periodic capillary-gravity waves on a liquid layer of arbitrary uniform depth is investigated. When surface tension is present, there is always instability for some wavenumber and liquid depth and bounds on the sideband frequencies for unbounded amplification are derived. The results are compared with the slow modulation theory using an averaged Lagrangian.     

Pressure induced by the interaction of water waves with nearly equal frequencies and nearly opposite directions

We present second-order expressions for the free-surface elevation, velocity potential and pressure resulting from the interaction of surface waves in water of arbitrary depth. When the surface waves have nearly equal frequencies and nearly opposite directions, a second-order pressure can be felt all the way to the sea bottom. There are at least two areas of applications: reflective structures and microseisms.Microseisms generated by water waves in the ocean are small vibrations of the ground resulting from pressure oscillations associated with the coupling of ocean surface gravity waves and the sea floor. They are recorded on land-based seismic stations throughout the world and they are divided into primary and secondary types, as a function of spectral content. Secondary microseisms are generated by the interaction of surface waves with nearly equal frequencies and nearly opposite directions. The efficiency of microseism generation thus depends in part on ocean wave frequency and direction. Based on the second-order expressions for the dynamic pressure, a simple theoretical analysis that quantifies the degree of nearness in amplitude, frequency, and incidence angle, which must be reached to observe the phenomenon, is presented.     

An unstructured-mesh-based finite element simulation of wave interactions with non-wall-sided bodies

An unstructured-mesh-based finite element method is employed to simulate two-dimensional nonlinear interactions between waves and non-wall-sided floating structures. The velocity potential theory is adopted and the potential is obtained at each time step through solving a matrix equation based on the Galerkin method. The boundary conditions on the free surface are satisfied in the Lagrangian form and the information is updated through the fourth-order Runge–Kutta method. Remeshing based on B-splines is applied regularly to avoid over-distorted elements, and smoothing based on a method using the energy of a curve defined through its nodes is applied to improve the stability of the results. Comparison is made with published results for transient wave motion in a tank to validate the present method. Extensive simulation is made for wedge-shaped bodies in vertical and horizontal motions, and comparison is made with the solution from second-order theory. Results are also provided for wedges in a tank, for wedges in large motion relative to water depth and for twin wedges.     

Burnett's equations from a (13+9N)-field theory

The Burnett equations are determined from a (13+9N)-field theory and the successive approximations up to the fifth-order of the Burnett's coefficients are given for gases whose particles interact according to a Lennard-Jones 6–12 potential and to an inverse power law potential.     

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.     

水陆两栖飞机波浪水面上降落耐波性数值分析

在规定的气象水文条件下,水陆两栖飞机起飞和降落的能力是决定其性能的重要因素,即耐波性能。采用ALE方法对流体域进行描述,运用基于微幅波理论的动边界数值造波方法模拟了不同波高和不同波长的动态海平面波浪,通过添加质量阻尼的消波方法抑制了固壁边界反射波对造波结果的影响,并采用罚函数耦合方法描述飞机与水体的耦合作用,研究了水陆两栖飞机在不同海情条件下波浪面上降落的纵摇运动、升沉运动以及底部压力等运动学和动力学特性,分析了水陆两栖飞机入水波浪的波长及波高对水陆两栖飞机耐波性能的影响,为飞机结构设计、水上降落操作规则制订及水陆两栖飞机耐波性物理水池试验提供参考。     

A Whitham–Boussinesq long-wave model for variable topography

We study the propagation of water waves in a channel of variable depth using the long-wave asymptotic regime. We use the Hamiltonian formulation of the problem in which the non-local Dirichlet–Neumann operator appears explicitly in the Hamiltonian, and propose a Hamiltonian model for bidirectional wave propagation in shallow water that involves pseudo-differential operators that simplify the variable-depth Dirichlet–Neumann operator. The model generalizes the Boussinesq system, as it includes the exact dispersion relation in the case of constant depth. Analogous models were proposed by Whitham for unidirectional wave propagation. We first present results for the normal modes and eigenfrequencies of the linearized problem. We see that variable depth introduces effects such as a steepening of the normal modes with the increase in depth variation, and a modulation of the normal mode amplitude. Numerical integration also suggests that the constant depth nonlocal Boussinesq model can capture qualitative features of the evolution obtained with higher order approximations of the Dirichlet–Neumann operator. In the case of variable depth we observe that wave-crests have variable speeds that depend on the depth. We also study the evolutions of Stokes waves initial conditions and observe certain oscillations in width of the crest and also some interesting textures and details in the evolution of wave-crests during the passage over obstacles.     

变深度浅水域中非定常船波

以Green—Naghdi(G—N)方程为基础，采用波动方程／有限元法计算船舶经过变深度浅水域时非定常波浪特性．把运动船舶对水面的扰动作为移动压强直接加在Green-Naghdi方程里，以描述运动船体和水面的相互作用．以Series60 CB＝0．6船为算例，给出自由面坡高，波浪阻力在船舶经过一个水下凸包时变化规律，并与浅水方程的结果进行了比较．计算结果表明，当船舶经过凸包时，波浪阻力先增加，后减少，并逐渐趋于正常．同时发现，当船速小于临界速度时(Fr＝√gh＜1．0)，G—N方程给出的船后尾波比浅水方程的结果明显，波浪阻力也比浅水方程的结果有所提高，频率散射必须考虑．当船速大于临界速度时(Fr＝√gh＞1．0)，G—N方程的计算结果与浅水方程差别不大，频率散射的影响可以忽略．     

Composite steady waves in a gravity fluid stream of finite depth

The problem of plane steady gravitational waves of finite amplitude, caused by a periodically distributed pressure over the surface of an ideal incompressible gravity fluid stream of finite depth, is considered. It is assumed that these waves do not vanish as the pressure becomes constant, but become free waves, which exist at constant pressure and special values of the stream velocity. As in [1], where a stream of finite depth is considered, such waves will be designated composite as contrasted with forced waves which vanish together with the variable part of the pressure. A general method is given for computing the composite wave characteristics. The first three approximations are computed to the end. An approximate equation for the wave profile is found.     

Flexural gravity wave motion over poroelastic bed

Using Biot’s consolidation theory, effect of poroelastic bed on flexural gravity wave motion is analyzed in both the cases of single-layer and two-layer fluids. The model for the flexural gravity waves is developed using linear water wave theory and small amplitude structural response in finite water depth. The effects of permeability and shear modulus of poroelastic bed and time period on flexural gravity wave motion are studied by analyzing the dispersion relation, phase speed, plate deflection, interface elevation and pressure distribution along water depth. Various results for surface gravity waves are analyzed as special cases. The study reveals that bed permeability retards the hydrodynamic pressure distribution along the water depth significantly compared to shear modulus whilst, floating plate deflection decreases significantly with change in shear modulus compared to permeability of the poroelastic bed. The present study can be generalized to analyze various wave–structure interaction problems over poroelastic bed.     

Steady Periodic Hydroelastic Waves

This is a study of steady periodic travelling waves on the surface of an infinitely deep irrotational ocean when the top streamline is in contact with a light frictionless membrane that strongly resists stretching and bending and the pressure in the air above is constant. It is shown that this is a free-boundary problem for the domain of a harmonic function (the stream function) which is zero on the boundary and at which its normal derivative is determined by the boundary geometry. With the wavelength fixed at 2π, we find travelling waves with arbitrarily large speeds for a significant class of membranes. Our approach is based on Zakharov’s Hamiltonian theory of water waves to which elastic effects at the surface have been added. However we avoid the Hamiltonian machinery by first defining a Lagrangian in terms of kinetic and potential energies using physical variables. A conformal transformation then yields an equivalent Lagrangian in which the unknown function is the wave height. Once critical points of that Lagrangian have been shown to correspond to the physical problem, the existence of hydroelastic waves for a class of membranes is established by maximization. Hardy spaces on the unit disc and the Hilbert transform on the unit circle play a role in the analysis.     

Multiple scattering of elastic waves in metal-matrix composite materials with high volume concentration of particles

A new approach is proposed to investigate the propagation of compressional (P) and shear (SV) waves in metal-matrix composite materials with high volume concentration of particles. The theory of quasicrystalline approximation and Waterman's T matrix formalism are employed to treat the multiple scattering resulting from the particles in composites. The addition theorem for spherical Bessel functions is used to accomplish the translation between different coordinate systems. The analytical expression of the Percus–Yevick correlation function is also given. Closed form solutions for the effective propagation constants and the dynamic effective elastic modulus of materials are obtained in the low frequency limit. At higher frequencies, only numerical results of them are presented. Numerical examples show that the phase velocities of P and SV waves in the composite materials with low volume concentration in the low frequency are in good agreement with the results in previous literatures. The effects of the incident wave number, the volume fraction of particles and the material properties of the particles and matrix on the dynamic effective elastic modulus are also examined.     

One-soliton shaping and inelastic collision between double solitons in the fifth-order variable-coefficient Sawada–Kotera equation

Nonlinear Dynamics - The main concern of the present article is to study the fifth-order variable-coefficient Sawada–Kotera (VcSK) equation which describes the motion of long waves in shallow...     

Behavior of perturbations of solitary and periodic waves on the surface of a heavy liquid

In [1] a system of equations was obtained for the case of a potential motion of an ideal incompressible homogeneous fluid; the system described the propagation of a train of waves in a medium with slowly varying properties, the motion in the train being characterized by a wave vector and a frequency. A solitary wave is a particular case of a wave train in which the length of the waves in the train is large. In [2, 3] a quasilinear system of partial differential equations was obtained which described two-dimensional and three-dimensional motion of a solitary wave in a layer of liquid of variable depth. It follows from this system that if the unperturbed state of the liquid is the quiescent state, then some integral quantity (the average wave energy [2–4]), referred to an element of the front, is preserved during the course of the motion. This fact is also valid for a train of waves, and can be demonstrated to be so upon applying the formalism of [1] to a Lagrangian similar to that used in [2]. In the present paper we obtain, for the case of a layer of liquid of constant depth, a solution in the form of simple waves for a system, equivalent to the system obtained in [3], describing the motion of a solitary wave and also the motion of a train of waves. We show that it is possible to have tilting of simple waves, leading in the case considered here to the formation of corner points on the wave front. We consider several examples of initial perturbations, and we obtain their asymptotics as t→∞. We make our presentation for the solitary wave case; however, in view of our statement above, the results automatically carry over to the case of a train of waves.     

QALE‐FEM for modelling 3D overturning waves

A further development of the QALE‐FEM (quasi‐arbitrary Lagrangian–Eulerian finite element method) based on a fully nonlinear potential theory is presented in this paper. This development enables the QALE‐FEM to deal with three‐dimensional (3D) overturning waves over complex seabeds, which have not been considered since the method was devised by the authors of this paper in their previous works (J. Comput. Phys. 2006; 212 :52–72; J. Numer. Meth. Engng 2009; 78 :713–756). In order to tackle challenges associated with 3D overturning waves, two new numerical techniques are suggested. They are the techniques for moving the mesh and for calculating the fluid velocity near overturning jets, respectively. The developed method is validated by comparing its numerical results with experimental data and results from other numerical methods available in the literature. Good agreement is achieved. The computational efficiency of this method is also investigated for this kind of wave, which shows that the QALE‐FEM can be many times faster than other methods based on the same theory. Furthermore, 3D overturning waves propagating over a non‐symmetrical seabed or multiple reefs are simulated using the method. Some of these results have not been found elsewhere to the best of our knowledge. Copyright © 2009 John Wiley & Sons, Ltd.     

A higher‐order Boussinesq‐type model with moving bottom boundary: applications to submarine landslide tsunami waves

A two‐dimensional depth‐integrated numerical model is developed using a fourth‐order Boussinesq approximation for an arbitrary time‐variable bottom boundary and is applied for submarine‐landslide‐generated waves. The mathematical formulation of model is an extension of (4,4) Padé approximant for moving bottom boundary. The mathematical formulations are derived based on a higher‐order perturbation analysis using the expanded form of velocity components. A sixth‐order multi‐step finite difference method is applied for spatial discretization and a sixth‐order Runge–Kutta method is applied for temporal discretization of the higher‐order depth‐integrated governing equations and boundary conditions. The present model is validated using available three‐dimensional experimental data and a good agreement is obtained. Moreover, the present higher‐order model is compared with fully potential three‐dimensional models as well as Boussinesq‐type multi‐layer models in several cases and the differences are discussed. The high accuracy of the present numerical model in considering the nonlinearity effects and frequency dispersion of waves is proven particularly for waves generated in intermediate and deeper water area. Copyright © 2006 John Wiley & Sons, Ltd.