In this paper, we present an analytic solution to the shallow water wave equation for long waves propagating over a circular hump. As a useful tool in coastal engineering, the solution may be used to study the refraction of long waves around a circular hump. It may also be used as a validation tool for any numerical model developed for coastal wave refraction. As part of the verification process, we have compared our new analytical solution with a numerical solution obtained by using the finite difference method. The agreement between these two solutions is excellent. By using the analytic solution, we then examine the effect of the hump dimensions on wave refraction over a circular hump.

A nonlinear equation of the Korteweg–de Vries equation usually describes internal solitary waves in the coastal ocean that lead to an exact solitary wave solution. However, in any real application, there exists the Earth’s rotation. Thus, an additional term is required, and consequently, the Ostrovsky equation is developed. This additional term is believed to destroy the solitary wave solution and form a nonlinear envelope wave packet instead. In addition, an internal solitary wave is commonly disseminated over the variable topography in the ocean. Because of these effects, the Ostrovsky equation is retrieved by a variable-coefficient Ostrovsky equation. In this study, the combined effects of both background rotation and variable topography on a solitary wave in a two-layer fluid is studied since internal waves typically happen here. A numerical simulation for the variable-coefficient Ostrovsky equation with a variable topography is presented. Two basic examples of the depth profile are considered in detail and sustained by numerical results. The first one is the constant-slope bottom, and the second one is the specific bottom profile following the previous studies. These indicate that the combination of variable topography and rotation induces a secondary trailing wave packet.

Modified wake oscillator model for vortex-induced motion prediction of low aspect ratio structures Floating structures, such as Spar platform, are likely to experience significant motions while operation due to Vortex-induced Motion (VIM). The main features of VIM include aspect ratio and mass ratio of the structure. This paper focuses on further development of a previously published semi-empirical method called wake oscillator model (WOM) for VIM, which aims to derive a mathematical correlation in modelling all features of the phenomenon. Analytical solution is proposed to coupled WOM and structure oscillator model based on the Van der Pol equation by integrating mass ratio, m* = 4.36, and various low aspect ratios between 0.3 and 2.0. Additionally, a new approach was initiated to embed several empirical parameters linked to the aspect ratio in the existing WOM. It was found the aspect ratio parameter in the modified WOM significantly affected the performance of the model for estimation of VIM on floating structures.

Explorations by oil and gas companies have moved from shallow to deep water to satisfy consumers' changing needs. A circular-shaped cylinder structure placed in deep water is also known spar platform, is normally floated and subjected to severe conditions which cause unpleasant motions and fatigue damage to the structure. Wake oscillator model is considered a semi-empirical model that is most suitable for the evaluation of Vortexinduced Vibrations (VIV) structure features during the design stage. This work focuses on validation of wake oscillator model for VIV with previously semi-empirical method. Wake oscillator and structure oscillator models are coupled based on the Van der Pol Equation. The coupled models were evaluated and analytically validated. The results showed a qualitative and quantitative agreement with previous analytical, at the same maximum peak amplitude response, 0 = 0.22, with 1.2% percentage error . Meanwhile, experimental results from literature only showed a qualitative agreement but not in quantitative agreement. Analytical modelling may offer potential means for investigating VIV features and provide a fundamental analysis by using a cost saving but reliable method. Further works should be done to characterise the VIV and Vortex-induced Motions phenomenon for spar platform.