The boundary integral equation method, also known as the method of singularities, is described. The Green function, consisting of Rankine and Kelvin parts, is introduced. Application of Green’s identity leads to an integral equation, which is solved numerically. Numerical aspects are covered, such as how to remedy the problem of irregular frequencies, or include coexisting current.

The hydrostatics in a fluid at rest are described. Such notions as center of buoyancy, metacentric radii, metacentric heights, hydrostatic stiffnesses are introduced.

Distinction is made between shallow water and deep water wave theories, depending on the value of the Ursell number. Potential flow theory is applied and the Stokes development is followed and first-order (linear), second-order, and third-order wave theories, in regular and irregular waves, are described. The concepts of phase and group velocities, mass transport, and energy flux are introduced. The application of stretching models to wave crest kinematics is described. At second-order distinction is made between bound (or locked) wave components (that accompany the first-order wave system) and free components (that travel independently). It is emphasized that, from third-order, such phenomena as mutual modifications of the phase velocities, or exchanges of energy, can take place between wave components. These interactions may lead to the occurrence of rogue waves, or to strong runups often observed at midships. The stream function wave model, which encompasses shallow and deep water cases, is presented. Finally the nonlinear Schr¨odinger equation that describes the time and space evolution of the wave envelope is applied to the prediction of the Benjaminis–Feir instability.

This chapter covers several types of flow instabilities of cylindrical bodies in current: vortex- induced vibrations (VIVs) and galloping, flutter, and wake-induced instabilities (WIO). VIVs mostly affect cylinders of circular cross section and they must be accounted for to assess the fatigue life of risers. The concept of reduced velocity is introduced and illustrative experimental values of VIV responses are given. Predictive methods are briefly described. Galloping instabilities appear at higher values of the reduced velocities for prismatic cylinders. Experimental results are given for a square cylinder and the quasi-static predictive method is outlined. Whereas, in galloping, only one degree of freedom is at hand, in the direction perpendicular to the free stream, in flutter an additional rotational motion comes into play. Finally Wake-Induced Instabilities are described, in the particular case of one circular cylinder in the lee of an upstream one.

The motion of a floating body in waves obeys a damped mass spring equation. The respective roles of mass, damping and stiffness need to be clearly understood. Harmonic and non-harmonic excitations are considered, together with different forms of damping: linear, quadratic, Coulomb.

The concepts of sea state and of short-term and long-term statistics are introduced. Wave by wave and spectral analyses are described; definitions are given of wave spectrum, significant wave height, mean wave period, and narrowness parameter. Theoretical distributions of wave heights (the Rayleigh law) are derived. The concept of return period is introduced. The other environmental parameters considered are the wind, the current, the internal waves, and the marine growth. The different definitions of mean wind velocity are explained. Typical wind profiles and wind spectra are presented.