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This dissertation concerns capillary-gravity surface waves in a two-dimensional rectangular basin that is partially filled with water. To generate the surface waves, a harmonic forcing is applied to the vertical side walls of the basin. The dissertation consists of four parts which work with different assumptions on the frequencies of the forcing. The first part discusses the linearized model with Hocking's edge condition and gives an eigenvalue equation and an asymptotic expansion for the eigenvalues. Then, for the nonlinear problem, it is assumed that the frequency of the forcing is close to an eigenfrequency and the solution has an asymptotic expansion using a two time-scales approach. Under an edge condition, the first- and second-order approximations of the solution and a solvability condition from the third-order equations yield an ordinary differential equation for the amplitude of the solution. In part two, it is assumed that the frequency of the forcing applied to the boundary is close to the sum of two eigenfrequencies. In this case, the solvability conditions give a system of two differential equations for the complex valued amplitudes of the two eigenmodes. The system can be reduced to one real-valued differential equation. Its solutions yield the solutions of the original system and their properties. A condition for the existence of homoclinic orbits connecting the trivial equilibrium is obtained. These results are confirmed by numerical experiments. The third part is based on the results in the second part. Here, one of the eigenfrequencies is chosen to be much larger than the other one, and different orders of the amplitudes of the eigenmodes are assumed. The orders of the coefficients of the system found in the second part are obtained, and the resulting special case is discussed in detail. In particular, numerical examples of orbits that can be associated with homoclinic orbits connecting nontrivial equilibria are given. The behavior of solutions close to those orbits is demonstrated. In the fourth part, an additional frequency for the forcing terms given in parts two and three is introduced. In each situation, the modified systems are presented and discussed. / Ph. D.

Proposed designs for Very Large Floating Structures motivate us to understand water-wave propagation through arrays of hundreds, or possibly thousands, of floating structures. The water-wave problems we study are each formulated under the usual conditions of linear wave theory. We study the frequency-domain problem of water-wave propagation through a periodically arranged array of structures, which are solved using a variety of methods. In the first instance we solve the problem for a periodically arranged infinite array using the method of matched asymptotic expansions for both shallow and deep water; the structures are assumed to be small relative to the wavelength and the array periodicity, and may be fixed or float freely. We then solve the same infinite array problem using a numerical approach, namely the Rayleigh-Ritz method, for fixed cylinders in water of finite depth and deep water. No limiting assumptions on the size of the structures relative to other length scales need to be made using this method. Whilst we aren t afforded the luxury of explicit approximations to the solutions, we are able to compute diagrams that can be used to aid an investigation into negative refraction. Finally we solve the water-wave problem for a so-called strip array (that is, an array that extends to infinity in one horizontal direction, but is finite in the other), which allows us to consider the transmission and reflection properties of a water-wave incident on the structures. The problem is solved using the method of multiple scales, under the assumption that the evolution of waves in a horizontal direction occurs on a slower scale than the other time scales that are present, and the method of matched asymptotic expansions using the same assumptions as for the infinite array case.

Thesis (MScEng (Civil Engineering))--University of Stellenbosch, 2009. / Multi purpose reefs are a relatively new concept that incorporate functionalities of beach stabilization, breakwater/seawall protection, biological enhancement and recreational amenity. Economic benefits increase their attractiveness. There is, however, some degree of uncertainty in design guidelines as to the predictability of each of these aspects. With regards to recreational amenity enhancement, one such uncertainty exists in the ability to predict the reef configuration required to give a certain degree of surfability of a reef, and more specifically, to predict the shape of a plunging wave. An extensive survey of the relevant literature has been conducted to provide a background on multi purpose reefs and the uncertainties in predicting the success of multi purpose reefs in achieving their design objectives. A study of wave breaking has been done, along with an analysis of existing breaker height and breaker depth formulae. The effects of bottom friction, refraction, shoaling, winds currents and varying water level on wave breaking has been addressed. Surfability aspects were reviewed including a definition of breaking intensity which is defined by the wave profile in terms of vortex shape parameters, and other surfability parameters that influence the surfability of a reef. Background on numerical modelling methods has been given, along with a description and some trial runs of a new and promising method, Smooth Particle Hydrodynamics. Numerical models were run using the open source SPHysics package in order to assess the applicability of the package in measuring vortex shape parameters. The SPHysics package is, however, still in a stage of development, and is not yet suitable for reef studies with very long domains and with high numbers of particles (required for sufficient resolution in the plunging vortex). A theoretical examination was done on the relevant literature in order to gain an insight into the dynamics affecting the development of the plunging vortex shape. A case study of a natural surf reef was carried out in order to give qualitative estimation of the wave dynamics and reef structure required to give good quality surfing waves and high breaking intensity. The West- Cowell surfing reef factor was used as a tool in predicting wave focusing effects of a naturally occurring reef. Extensive two dimensional physical model laboratory studies were conducted in order to quantify the effects of the reef configuration and wave parameters on breaking intensity. Design guidelines were developed in order to assist in the prediction of breaking intensity for reefs constructed with surfing amenity enhancement as one of their design objectives. The results show that large underwater topographic features can significantly affect the shape and size of incoming waves. Refraction, focusing and shoaling can transform ordinary waves into waves deemed suitable for surfing. The West-Cowell surfing reef factor gives reasonable results outside its applicable range. The 2D physical model laboratory tests show significant variations in vortex shape parameters due to interactions between broken and unbroken waves in a wave train and also to the reflections developed in the flume. Results show that the predicted trends agree with the observations. The results also show that the junction between the seaward reef slope and the horizontal crest may have an effect on the wave shape in the form of a secondary crest between the primary crests. Design guidelines based on the results are presented, and show that breaker height formulae for smooth planar slopes show good agreement with the values of breaker heights measured in the physical model tests, and that existing breaker depth formulae show average agreement. The design guidelines could assist with more effective design of artificial reefs for surfing purposes.

<p>Low-frequency, long-range sound propagation over a seasurface has been calculated using a wide-angel Cranck-NicholsonParabolic Equation method. The model is developed toinvestigate noise from off-shore wind turbines. Thecalculations are made using normal meteorological conditions ofthe Baltic Sea. Special consideration has been made to a windphenomenon called low level jet with strong winds on rather lowaltitude.</p><p>The effects of water waves on sound propagation have beenincorporated in the ground boundary condition using a bossmodel. This way of including roughness in sound propagationmodels is valid for water wave heights that are small comparedto the wave length of the sound. Nevertheless, since only lowfrequency sound is considered, waves up to the mean wave heightof the Baltic Sea can be included in this manner.</p><p>The calculation model has been tested against benchmarkcases and agrees well with measurements. The calculations showthat channelling of sound occurs at downwind conditions andthat the sound propagation tends towards cylindrical spreading.The effects of the water waves are found to be fairlysmall.</p><p><b>Keywords:</b>wind turbine noise, off-shore wind power,long-range sound propagation, parabolic equation, scattering,water waves</p>

An understanding of extreme waves is important in the design and analysis of offshore structures, such as oil and gas platforms. With the increase of interest in the shipping of LNG, the design of import and export terminals in coastal water of slowly varying intermediate depth requires accurate analysis of steep wave shoaling. In this thesis, data from laboratory experiments involving random wave simulations on very gentle slopes have been analysed in terms of a model of large wave events, and the results interpreted by observation of the shape and magnitude of the large wave events. The auto-correlation function of the free surface elevation time histories, called NewWave, has been calculated from the wave spectrum and shown to fit very well up to the point where waves start to break (when compared to the ‘linear’ surface elevation time history). It has been shown that NewWave is an appropriate model for the shape of the ‘linear’ part of large waves provided kd > 0.5. A Stokes-like expansion for NewWave analysis has been demonstrated to match the average shape of the largest waves, accounting for the dominant vertical asymmetry. Furthermore, an appropriate local wave period derived from NewWave has been inserted into a Miche-based limiting criterion, using the linear dispersion equation, to obtain estimates for the limiting wave height. Overall, the analysis confirms the Miche-type criterion applies to limiting wave height for waves passing over very mild bed slopes. A derivation of general Green-Naghdi (GN) theory, which incorporates non-linear terms in its formulation, is also presented. This approach satisfies the boundary conditions exactly and approximates the field equations. The derived 2-dimensional vertical GN Level 1 model, capable of simulating steep waves on varying water depth, is validated against solitary waves and their interactions, and solitary waves on varying water depth and gives good qualitative agreement against the KdV equation. The developed and validated numerical model is used to simulate focussed wave groups on both constant depth and gentle slope. In general, the behaviour of waves simulated by the numerical model is very similar to that observed in the experimental data. There is evidence of vertical asymmetry as the water depth is reduced, owing to the non-linearity. Although the main physics is still controlled by linear dispersion, the higher order harmonics become increasingly important for shoaling waves. The numerical results also show a slope-induced wave set-up that keeps on increasing in amplitude as the wave group travels on the gentle slope.

This thesis presents Wronskian and Gram solutions to both the Korteweg-de Vries and Kadomtsev-Petviashvili equations, which are then scalable to arbitrarily large numbers of interacting solitons. Through variable transformation and use of the Hirota derivative, these nonlinear partial differential equations can be expressed in bilinear form. We present both Wronskian and Gram determinants which satisfy the equations. N=1,2,3 and higher order solutions are presented graphically; parameter tuning and the resultant behavioral differences are demonstrated and discussed. In addition, we compare these solutions to naturally occurring shallow water waves on beaches.

The forced water entry and exit of two-dimensional bodies through a free surface is computed for various 2D bodies (symmetric wedges, asymmetric wedges, truncated wedges and boxes). These bodies enter or exit water with constant velocity or constant acceleration. The calculations are based on the fully non-linear timestepping complex-variable method of Vinje and Brevig. The model was formulated as an initial boundary-value problem with boundary conditions specified on the boundaries (dynamic and kinematic free-surface boundary conditions) and initial conditions at time zero (initial velocity and position of the body and free-surface particles). The formulated problem was solved by means of a boundary-element method using collocation points on the boundary of the domain and solutions at each time were calculated using time stepping (Runge-Kutta and Hamming predictor corrector) methods. Numerical results for the deformed free-surface profile, the speed of the point at the intersection of the body and free surface, the pressure along the wetted region of the bodies and force experienced by the bodies, are given for the entry and exit. To verify the results, various tests such as convergence checks, self-similarity for entry (gravity-free solutions) and Froude number effect for constant velocity entry and exit (half-wedge angles 5 up to 55 degrees) are investigated. The numerical results are compared with Mackie's analytical theory for water entry and exit with constant velocities, and the analytical added mass force computed for water entry and exit of symmetric wedges and boxes with constant acceleration and velocity using conformal mapping. Finally, numerical results showing the effect of finite depth are investigated for entry and exit.

Thesis (Ph.D.(Civil Engineering))--University of the Witwatersrand, Faculty of Engineering, 1966.

In near-shore transforming seas, as waves approach the shoreline, wave shoaling and sometimes wave breaking take place due to the decreasing water depth. When a ship advances through the transforming seas, the ship body and waves interact with each other substantially and can lead to unknown motions of the ship hull. The physical process of how the wave transforms in the surf zone and how the vehicle actually behaves when it passes through the transforming seas is a complicated issue that triggers considerable research interest. The goal of my research is to characterize the dynamics of a high-speed surface ship model in transforming seas through a parametric numerical study of the shipwave interactions. In this study, the vehicle of interest is a surface effect ship (SES) and we aim to contribute to developing a methodology for simulating the transforming wave environment, including wave breaking, and its interactions with the SES. The thesis work uses a commercial software package ANSYS Fluent to generate numerical waves and model the interface between water and air using the volume of fluid (VoF) method. A ship motion solver and the dynamic mesh are used to enable the modeled ship to perform three degree-of-freedom (DoF) motion and the near-region of the ship hull to deform as well as re-mesh. Non-conformal meshes with hybrid compositions of different cell types and various grid sizes are used in the simulations for different purposes. Five user-defined functions (UDFs) are dynamically linked with the flow solver to incorporates ship/grid motions, wave damping and output of the numerical results. A series of steps were taken sequentially: 1) validation for ship motions including simulation of a static Wigley hull under steady flows to compare against previous experimental results by other researchers, and the comparison between the static SES model under steady flows and the moving SES model advancing in the calm water; 2) study of the ship with 3 DoF advancing in calm water of both constant depth and varying depth; 3) validation for numerical waves, including predictions of numerically progressive waves in both a regular tank and a tank with a sloped fringing reef to compare with theoretical and experimental results, respectively; 4) investigation of the transforming characteristics of the wave traveling over the sloped fringing reef, which mimics the near-shore wave environment and a study of the dynamics of the SES through transforming waves. We find that the flow solver used in this study reliably models the wave profiles along the ship hull. The comparison between a static SES in a current and a moving SES in calm water at the same Froude number shows that although the velocity fields around the vehicle are significantly different, the wave profiles inside and outside the rigid cushion of the vehicle are similar and the resistance force experienced by the vehicle in the two scenarios agree well over time. We conducted five numerical simulations of the vehicle traveling from shallow water to deep water across the transition zone for different Froude numbers. From the results, we find that as the Froude number increases, the wave resistance force on the vehicle becomes larger in both shallow water and deep water. In addition, the overall mean resistance force experienced by the vehicle over the whole trip increases with the Froude number. Statistical analysis of the wave motions suggests that the energy flux decreases dramatically in the onshore direction as the waves break. The more severe the wave-breaking process, the greater the decrease in energy flux. Both the increase of Froude number and the wave steepness apparently increase the resistance force on the vehicle in the shallow water. This thesis work captures the impact of the transforming characteristics of the waves and closely replicates the behavior of how waves interact with a ship in transforming seas through numerical modeling and simulation. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2017. / FAU Electronic Theses and Dissertations Collection