# Extreme Wave Loading on a Vertical Circular Cylinder
## 1 Introduction  
Wave-induced loads on platform columns and offshore wind turbine columns need to be properly considered to ensure a reliable design of these structures. The former are support structures of, e.g., tension-leg platforms (TLPs), gravity-based structures (GBSs), and semisubmersible rigs, which are widely used for offshore oil and gas exploration. The latter become increasingly important due to the rapid expansion in the global offshore wind sector. The relevant interaction of surface water waves and cylindrical structures has been the subject of intense interest for decades. Yet, the physics and statistics of such interaction are complex, which remain to be fully resolved for strongly nonlinear loads as experienced in severe conditions. Typically, the physics of wave loading can be split into a quasi-static (non-breaking) and a dynamically slamming (breaking) component [1]. These have different magnitudes and timescales as they are dominated by different physical phenomena. For non-breaking waves, the relevant phenomena that cannot be well explained by the linear wave theory include the so-called'springing'- [2] and 'ringing'-type [3] responses of a vertical cylinder. These are typically large transient deformations of astructure that have natural frequencies substantially higher than the dominant frequencies of surrounding waves. Extensive research (e.g., [2; 3]) suggested that these are associated with higher-order forces occurring at frequencies that are close to integer multiples of the fundamental/dominant wave frequency. More specially, second-order forces are found to be responsible for exciting the'springing'-type responses and third-order forces for the 'ringing'-type responses. In addition, a secondary load cycle, i.e., an oscillation after the main peak at about one-quarter of the wave period, was observed [4; 5; 6]. Numerical investigations by [7] indicate that the high-frequency forces (up to the sixth order) are also responsible for the occurrence of this secondary load cycle. Thus, an accurate description of higher-order nonlinear characteristics of an extreme non-breaking wave loading on a cylindrical structure is of importance. Currently, diffraction analyses up to second order conducted in the frequency domain have been widely applied; see, e.g., [8; 9]. For using the perturbation analysis to capture the third-order wave forces, the work of [3] is noted. While for wave forces above the third order, fully nonlinear numerical models solving the Laplace equation (e.g., [10; 11]) or the Navier-Stokes equations (e.g., [12; 13]) in the time domain are developed. The huge computational resources required in these time-domain approaches limit their applications to the offshore industry as engineering models. The forcing model developed by [14] is thus extremely useful. They found that the nonlinear part of non-breaking wave loading on a bottom-mounted cylinder can be described by a Stokes-type expansion. Thus, the higher-order wave forces (up to fifth order) can be estimated from the linear component alone. The linear forces can be obtained quickly by using the linear diffraction analysis mentioned above. In contrast to non-breaking wave loading, breaking waves are notoriously difficult to model numerically and physically due to the violence of the hydrodynamics and scaling issues. Various models have been put forward for calculating the time history of the loading, see for instance, a Wagner-type model in [1] and CFD (computational fluid dynamics)-type models in [15; 16]. However, for dynamically sensitive structures, the time-integrated impulse from the force on a short time scale (e.g., breaking wave loading) rather than the exact loading time history is generally considered to be sufficient at the early stage of design [17]. Estimating and modeling the underlying physics of the impulse are more robust and tractable. Pressure-impulse theories solving the Laplace equation were derived for wave impact on a two-dimensional and three-dimensional vertical wall by [18; 19], respectively.