We analyze surface waves generated by a translating, oscillating surface disturbance atop a horizontal background flow of arbitrary depth dependence, with a focus on determining the Doppler resonance. For a critical value of the dimensionless frequency τ = ωV/g (ω: oscillation frequency, V: source velocity, g: gravitational acceleration) at which generated waves cannot escape. In the absence of shear the resonant value is famously 1/4; the presence of a shear current modifies this. We derive the theoretical and numerical tools for studying this problem, and present the first calculation of the Doppler resonance for a source atop a real, measured shear current to our knowledge. Studying graphical solutions to the (numerically obtained) dispersion relation allows derivation of criteria determining the number of far-field waves that exist in different sectors of propagation directions, from which the criteria for Doppler resonance follow. As example flows we study a typical wind-driven current, and a current measured in the Columbia River estuary. We show that modeling these currents as uniform or with a linear depth dependence based on surface measures may lead to large discrepancies, in particular for long and moderate wavelengths.

One important parameter in reconstructing and predicting the sea surface elevation from radar images is the surface current. The common method to derive the current is based on 3DFFT with which the (absolute) frequency is derived from a series of images and is fitted to the encounter dispersion relation that consist of the intrinsic exact dispersion relation for linear waves with an additional term that contains the current velocity to be found. The derived dispersion relation will be inaccurate because the images contain many inaccuracies from noise, shadowing, and other radar effects. This paper proposes an alternative method to determine the surface current. Following the method of the Dynamic Averaging and Evolution Scenario (DAES) as presented in [1], the idea is to choose the current velocity that minimizes the difference between an image at a previous time that has been evolved to the time of another image. In order to reduce inaccuracies, an averaging procedure over various images is applied. The method is tested on synthetic data to quantify the accuracy of the results. The robustness of the method will be investigated for several cases of different current parameters (speed and direction) for ensembles of seas with different peak frequency of characteristic sea states.

In linear Rankine panel method, the discrete linear dispersion relation is solved on a discrete free-surface to capture the free-surface waves generated due to wave-body interactions. Discretization introduces numerical damping and dispersion, which depend on the discretization order and the chosen methods for differentiation in time and space. The numerical properties of a linear Rankine panel method, based on a direct boundary integral formulation, for capturing two and three dimensional free-surface waves were studied. Different discretization orders and differentiation methods were considered, focusing on the linear distribution and finite difference schemes. The possible sources for numerical instabilities were addressed. A series of cases with and without forward speed was selected, and numerical investigations are presented. For the waves in three dimensions, the influence of the panels’ aspect ratio and the waves’ angle were considered. It has been shown that using the cancellation effects of different differentiation schemes the accuracy of the numerical method could be improved.

Understanding of the offshore wind wave status plays a guiding role in surrounding marine engineering constructions, marine traffic, sea farming, etc. Further study is beneficial to marine economy development, as well as to the academic value of wave theory. This paper primarily introduces the deduction of new wind wave growth relations. Firstly, a new relation formula between wave steepness and wave age was deduced by combining the 3/2 power law developed by Toba with the nonlinear dispersion relation deduced by Li, and by ignoring the effect of water depth. And when the higher-order term was ignored, the relation formula can be simplified as that based on linear dispersion. Secondly, based on the combination of this new relation formula with the significant wave energy balance equation, new wind wave growth relation formulae including the wave non-linear dispersion effect were deduced. When the deduced growth relation formulae were applied in offshore area of Jiangsu incorporating with Mitsuyasu’s empirical formula about the open sea fetch and wind speed, accurate open sea wave parameters of Jiangsu can be formulated by only considering one parameter, such as wind speed. Overall, as this methodology avoided the uncertainty about the fetch of open ocean and operation error during the calculation process, results gained from this report had higher accuracy than other published formulae, and results were validated by NCEP reanalyzing data of Jiangsu offshore area and other researches.

This article explores the feasibility of using a 3-dimensional Fast Fourier Transform (3D FFT) to obtain a frequency domain description of a spatio temporal measured short crested wave field. As 3D FFT is also the basic technique behind wave measurements by navigational X-band radars, the frequency components obtained by these radars could be used as initialization of a wave propagation model, enabling deterministic prediction of wave elevation on board of ships / offshore structures. Different methods are presented to use the dispersion relation to filter wave components obtained by the 3D FFT. The effect on the accuracy of data windowing and temporal measurement domain size are explored by simulations with linear synthetic wave data: It is investigated how well a synthetic wave field reconstructs after inverse transforming the filtered frequency components obtained by 3D FFT. A second paper [1] will consider the prediction outside the measurement domain by using the filtered 3D FFT components.

To evaluate wave forces and to estimate the motion of breakwater, a circular cylinder is investigated based on the linear wave theory in the present work. The cylinder possesses a porous sidewall, an impermeable bottom and a horizontal porous plate inside that is fixed in the cylinder to work as obstruct and make wave dissipation more effectively. To simplify the problem, the Darcy’s fine-pore model is applied to the boundary condition on the porous body surface. The boundary value problem is solved by means of the eigen-function expansion approach. The fluid domain is divided into three regions and different eigen-function series are used. The so-called dispersion relation for the region inside the cylinder is quite different from a conventional one due to the existence of the porous plate. It leads to eigen values of complex number. To obtain solutions for the radiation problems, particular solution should be constructed to take account of the normal velocity appearing on the porous boundary. The wave loads are evaluated by integrating the pressure difference on two sides of the wetted body surface. The theoretical works are in good consistence with the experimental results. The Haskind relations are examined for the porous body. It is found that the damping coefficient consists of two parts. In addition to the component of conventional wave-radiating damping, exists a second component caused by the porous effects.

In this paper, a recently derived (Zhou, 2008) fully nonlinear and higher-order dispersive Boussinesq-type model for wave generation and propagation is presented. This new model is an extension of the wave propagation model by Gobbi and Kirby (1999) and Gobbi et al. (2000) to include the time-varying seabed bathymetry. The resulting new version retains the 4th-order approximation of the dispersion relation and the velocity distribution in the vertical direction, and extends the application to both water wave propagation and wave generation by seabed disturbances such as submarine landslides. The model equations are solved numerically through a higher-order finite difference scheme. To examine the validity of the new model and the improvement due to the higher-order extensions, numerical simulations of two wave generation cases are carried out based on the new 4th order model and an existing lower order Boussinesq model. The results show that the higher order model provides the more accurate prediction for the generated waves, especially those in the trailing region of shorter wavelengths where the traditional lower order Boussinesq model becomes much less accurate.

We employ a fully nonlinear numerical model to generate and propagate long-crested gravity waves in a tank containing an incompressible invicid homogeneous fluid, initially at rest, with a horizontal free surface of finite extent and of finite depth. A non-orthogonal curvilinear coordinate system is constructed which follows the free surface and is “fitted” to the bottom topography of the tank and therefore tracks the entire fluid domain at all times. A waveform relaxation algorithm provides an efficient iterative method to solve the resulting discrete Laplace equation, and the full nonlinear kinematic and dynamic free surface boundary conditions are employed to propagate the solution. In addition, a monochromatic deterministic theoretical wave-maker, employing a Dirichlet type boundary condition, and a suitably tuned numerical beach are utilized in the numerical model. We first show that the model generates the appropriate dispersion relation for both the deep water and shallow water conditions. Subsequently, we calculate the energy and energy flux for small steepness waves and show that the model produces the expected linear results. We complete the paper by considering steeper waves where the linear results are suspect.