Theory of Non-linear Waves in Multi-dimensions: with Special Reference to Surface Water Waves

The surface water waves are "modal" waves in which the "physical space" (t, x, y, z) is the product of a propagation space (t, x, y) and a cross space, the z-axis in the vertical direction. We have derived a new set of equations for the long waves in shallow water in the propagation space. When the ratio of the amplitude of the disturbance to the depth of the water is small, these equations reduce to the equations derived by Whitham (1967) by the variational principle. Then we have derived a single equation in (t, x, y)-space which is a generalization of the fourth order Boussinesq equation for one-dimensional waves. In the neighbourhood of a wave froat, this equation reduces to the multidimensional generalization of the KdV equation derived by Shen & Keller (1973). We have also included a systematic discussion of the orders of the various non-dimensional parameters. This is followed by a presentation of a general theory of approximating a system of quasi-linear equations following one of the modes. When we apply this general method to the surface water wave equations in the propagation space, we get the Shen-Keller equation.

A Note on Diffraction of Water Waves by a Nearly Vertical Barrier

A simplified perturbational analysis is employed, together with the application of Green's theorem, to determine the first-order corrections to the reflection and transmission coefficients in the problem of diffraction of surface water waves by a nearly vertical barrier in two basically important cases: (i) when the barrier is partially immersed and (ii) when the barrier is completely submerged. The present analysis produces the desired results fairly easily and relatively quickly as compared with the known integral equation approach to this class of diffraction problems.

Solution of a Boundary-Value Problem associated with Diffraction of Water Waves by a Nearly Vertical Barrier

An exact solution is derived for a boundary-value problem for Laplace's equation which is a generalization of the one occurring in the course of solution of the problem of diffraction of surface water waves by a nearly vertical submerged barrier. The method of solution involves the use of complex function theory, the Schwarz reflection principle, and reduction to a system of two uncoupled Riemann-Hilbert problems. Known results, representing the reflection and transmission coefficients of the water wave problem involving a nearly vertical barrier, are derived in terms of the shape function.

A New Approach to the Problem of Scattering of Water Waves by Vertical Barriers

Using a modified Green's function technique the two well-known basic problems of scattering of surface water waves by vertical barriers are reduced to the problem of solving a pair of uncoupled integral equations involving the “jump” and “sum” of the limiting values of the velocity potential on the two sides of the barriers in each case. These integral equations are then solved, in closed form, by the aid of an integral transform technique involving a general trigonometric kernel as applicable to the problems associated with a radiation condition.

Obliquely incident water waves against a vertical cliff

Utilizing the commutativity property of the Cartesian coordinate differential operators arising in the boundary conditions associated with the propagation of surface water waves against a vertical cliff, under the assumptions of linearized theory, the problem of obliquely incident surface waves is considered for solution. The case of normal incidence, handled by previous workers follow as a particular limiting case of the present problem, which exhibits a source/sink type behavior of the velocity potential at the shore-line. An independent method of attack is also presented to handle the case of normal incidence.

On the incoming water waves against a vertical cliff

The problems of obliquely incident surface water waves against a vertical cliff have been handled in both the cases of water of infinite as well as finite depth by straightforward uses of appropriate Havelock-type expansion theorems. The logarithmic singularity along the shore-line has been incorporated in a direct manner, by suitably representing the Dirac's delta function.

A note on surface water waves for finite depth in the presence of an ice-cover

A class of I boundary value problems involving propagation of two-dimensional surface water waves, associated with water of uniform finite depth, against a plane vertical wave maker is investigated under the assumption that the surface is covered by a thin sheet of ice. It is assumed that the ice-cover behaves like a thin isotropic elastic plate. Then the problems under consideration lead to those of solving the two-dimensional Laplace equation in a semi-infinite strip, under Neumann boundary conditions on the vertical boundary as well as on one of the horizontal boundaries, representing the bottom of the fluid region, and a condition involving upto fifth order derivatives of the unknown function on the top horizontal ice-covered boundary, along with the two appropriate edge-conditions, at the ice-covered corner, ensuring the uniqueness of the solutions. The mixed boundary value problems are solved completely, by exploiting the regularity property of the Fourier cosine transform.

On the solution of the problem of scattering of surface water waves by a sharp discontinuity in the surface boundary conditions

Closed-form analytical expressions are derived for the reflection and transmission coefficients for the problem of scattering of surface water waves by a sharp discontinuity in the surface-boundary-conditions, for the case of deep water. The method involves the use of the Havelock-type expansion of the velocity potential along with an analysis to solve a Carleman-type singular integral equation over a semi-infinite range. This method of solution is an alternative to the Wiener-Hopf technique used previously.

Exciting forces due to interaction of water waves with a submerged sphere in an ice-covered two-layer fluid of finite depth

Scattering of water waves by a sphere in a two-layer fluid, where the upper layer has an ice-cover modelled as an elastic plate of very small thickness, while the lower one has a rigid horizontal bottom surface, is investigated within the framework of linearized water wave theory. The effects of surface tension at the surface of separation is neglected. There exist two modes of time-harmonic waves - the one with lower wave number propagating along the ice-cover and the one with higher wave number along the interface. Method of multipole expansions is used to find the particular solution for the problem of wave scattering by a submerged sphere placed in either of the layers. The exciting forces for vertical and horizontal directions are derived and plotted against different values of the wave number for different submersion depths of the sphere and flexural rigidity of the ice-cover. When the flexural rigidity and the density of the ice-cover are taken to be zero, the numerical results for the exciting forces for the problem with free surface are recovered as particular cases. (C) 2011 Elsevier Ltd. All rights reserved.

Hydrodynamic interaction between two vertical cylinders in water waves

The hydrodynamic interaction between two vertical cylinders in water waves is investigated based on the linearized potential flow theory. One of the two cylinders is fixed at the bottom while the other is articulated at the bottom and oscillates with small amplitudes in the direction of the incident wave. Both the diffracted wave and the radiation wave are studied in the present paper. A simple analytical expression for the velocity potential on the surface of each cylinder is obtained by means of Graf's addition theorem. The wave-excited forces and moments on the cylinders, the added masses and the radiation damping coefficients of the oscillating cylinder are all expressed explicitly in series form. The coefficients of the series are determined by solving algebraic equations. Several numerical examples are given to illustrate the effects of various parameters, such as the separation distance, the relative size of the cylinders, and the incident angle, on the first-order and steady second-order forces, the added masses and radiation-damping coefficients as well as the response of the oscillating cylinder.

On the interaction of water waves and seabed by the porous medium model

The interaction of water waves and seabed is studied by using Yamamoto's model, which takes into account the deformation of soil skeletal frame, compressibility of pore fluid flow as well as the Coulumb friction. When analyzing the propagation of three kinds of stress waves in seabed, a simplified dispersion relation and a specific damping formula are derived. The problem of seabed stability is further treated analytically based on the Mohr-Coulomb theory. The theory is finally applied to the coastal problems in the Lian-Yun Harbour and compared with observations and measurements in soil-wave tank with satisfactory results.

Modelling of submarine landslides and generated water waves

The slide of unstable sedimentary bodies and their hydraulic effects are studied by numerical means. A two-dimensional fluid mechanics model based on Navier-Stokes equations has been developed considering the sediments and water as a mixture. Viscoplastic and diffusion laws for the sediments have been introduced into the model. The numerical model is validated with an analytical solution for a Bingham flow. Laboratory experiments consisting in the slide of gravel mass have been carried out. The results of these experiments have shown the importance of the sediment rheology and the diffusion. The model parameters are adjusted by trial and error to match the observed “sandflow”.

On the radiation and diffraction of linear water waves by an infinitely long rectangular structure submerged in oblique seas

The radiation and diffraction of linear water waves by an infinitely long rectangular structure submerged in oblique seas of finite depth is investigated. The analytical expressions for the radiated and diffracted potentials are derived as infinite series by use of the method of separation of variables. The unknown coefficients in the series are determined by the eigenfunction expansion matching method. The expressions for wave forces, hydrodynamic coefficients and reflection and transmission coefficients are given and verified by the boundary element method. Using the present analytical solution, the hydrodynamic influences of the angle of incidence, the submergence, the width and the thickness of the structure on the wave forces, hydrodynamic coefficients, and reflection and transmission coefficients are discussed in detail.

Scattering of linear water waves by a fixed and infinitely long rectangular structure parallel to a vertical wall in oblique seas

The scattering of linear water waves by an infinitely long rectangular structure parallel to a vertical wall in oblique seas is investigated. Analytical expressions for the diffracted potentials are derived using the method of separation of variables. The unknown coefficients in the expressions are determined through the application of the eigenfunction expansion matching method. The expressions for wave forces on the structure are given. The calculated results are compared with those obtained by the boundary element method. In addition, the influences of the wall, the angle of wave incidence, the width of the structure, and the distance between the structure and the wall on wave forces are discussed. The method presented here can be easily extended to the study of the diffraction of obliquely incident waves by multiple rectangular structures.

A unified criterion for initiation of sediment motion and inception of sheet flow under water waves

A unified criterion is developed for initiation of non-cohesive sediment motion and inception of sheet flow under water waves over a horizontal bed of sediment based on presently available experimental data. The unified threshold criterion is of the single form, U-o = 2 pi C[1 + 5(T-R/T)(2)](-1/4), where U-o is the onset velocity of sediment motion or sheet flow, T is wave period, and C and T-R are the coefficients. It is found that for a given sediment, U-o initially increases sharply with wave period, then gradually approaches the maximum onset velocity U-o = 2 pi C and becomes independent of T when T is larger. The unified criterion can also be extended to define sediment initial motion and sheet flow under irregular waves provided the significant wave orbital velocity and period of irregular waves are introduced in this unified criterion.