A distinguishing factor of marine dynamics is the presence of the air-water interface. In order to determine the dynamic fluid forces acting on floating bodies - the wave exciting forces and the radiation forces (i.e. added mass and damping) - in addition to the hydrostatic forces, a lower order model of water waves based on the velocity potential and a linearized form of Bernoulli’s equation is given. The air-water interface is defined by two boundary conditions: kinematic and dynamic boundary conditions. Examining limits of the free surface boundary conditions allows a limiting process in the estimation of fluid added mass without having to solve a free surface boundary value problem. A low order model of plane progressive waves is simply a harmonic function in the lateral plane multiplied by an exponentially decaying function in the vertical coordinate. Application of the linear free surface conditions yields the important dispersion relation - a relation between the temporal wave frequency and the spatial wave frequency.

The presentation is necessarily brief and references for a more comprehensive development are listed.

Hydroelastic problems involve dynamically coupled, structurally elastic, hydrodynamic systems. The fluid can have many effects such as added inertia, additive hydrostatic stiffness, increased system damping, or external excitation (e.g. wave impact, variable current forces, etc.). This chapter illustrates some of the aspects of hydroelastic problems by deriving fundamental relationships and discussing a specific example - ship springing. Springing vibration is differentiated from whipping vibration by the source of excitation. Springing is excited by synchronous matching of the natural frequency with the incident wave encounter frequency while whipping is transient vibrations due to impact/slam loads. A well-developed energy method - the Rayleigh-Ritz method - is applied in the determination of fluid-structure resonance. For general marine vibrations, energy methods may be used when free surface effects are small or negligible. Fluid inertia effects are calculated using strip theory and Lewis form coefficients. Limitations of strip theory are discussed. A spherical globe mounted on a flexible pole submersed in water is given as an example of a hydroelastic system.

The analysis in this chapter of marine platform motions is directly applicable to any floating system such as ships, offshore platforms, floating wind turbines, or wave energy devices. The basic underlying model is the classic linear spring-mass-damper system. The mass will be augmented by the added mass of the fluid; the damping will be the result of the dissipation of energy by waves; the linear spring will be due to hydrostatic effects plus any external stiffness such as mooring lines; and the exciting forces are due to incident waves. Depending on the body shape and mass distribution, the equations of motion can be dynamically/statically coupled. Wave excitation is comprised of Froude-Krylov and diffraction components. Solutions to the equations of motion in the frequency domain are expressed as RAO’s. The RAO is a linear operator representing the dynamic response of a system (e.g. displacement, acceleration, bending moment, etc.) per unit input, typically the incident wave amplitude. Once the rigid body dynamics are expressed as RAO’s, other quantities or dynamics of interest may be determined, e.g. relative motion, dynamic bending and shear.

A reduced order model for marine vehicle dynamics is the simple linear spring-mass-damper system. However, the various terms in the equation of motion differ in detail from their mechanical counterparts. The usual balance between mechanical inertial, damping, and stiffness loads with external forcing is maintained, but now includes additional effects reflecting the presence of the fluid. Individual coefficient matrices correspond to the mass of the platform plus the mass of the water being accelerated; the linear damping coefficient of the system due to viscous effects and the generation of radiating waves due to platform motion; a linear restoring force/moment coefficient due to hydrostatic pressure and/or mooring lines; and an external exciting force/moment due to incident waves, wind, tow lines, etc. Ideal fluid theory is introduced to model the hydrodynamic forces implicit in the marine system’s equations of motion. The purpose is not to give a detailed derivation of basic hydrodynamics, but rather to describe the assumptions necessary to apply the useful ideal, potential theory and understanding when the theory will be successful and, equally important, when it will not.

This chapter discusses lift, added-mass, and history forces for an isolated particle. Shear-induced and spin-induced lift is considered, along with angular particle torque. The lift for fixed torque and equilibrium spin are discussed. In addition, added-mass and history force are considered for solid particles and bubbles.

In this chapter the wave body interaction is extended to the second order in the wave amplitude. This leads to wave loads, and responses, taking place at the sum and difference frequencies of the components in the incoming wave system. The time averaged wave load (the drift force) is first considered in regular waves. The different formulations of the drift force (near-field, far-field, middle-field, Lagally) are described. The sensitivity of the drift force to a coincident current is emphasized and the extension of the diffraction radiation theory to a superimposed current is presented, together with the so-called Aranha’s formula that offers a practical approximation to the wave drift damping effect. Second-order diffraction theory is detailed in regular waves, with practical application to a bottom-mounted vertical cylinder. The concept of Quadratic Transfer Function (QTF) is first introduced in the case of bichromatic seas and then applied to the low frequency loading in irregular waves, where the validity of frequently used approximations (e.g. Newman approximation) is discussed. A section is then devoted to the Rainey equations which can be viewed as a second-order extension of the inertia term in the Morison equation. Finally practical calculation of the slow-drift and springing responses is considered.

In this chapter linearized potential flow theory is applied to the prediction of wave loads upon marine structures, and of their wave response. The linearized diffraction radiation theory is presented, leading to the wave excitation loads, added masses, and radiation dampings. Analytical, semi-analytical, and numerical methods of resolution are given, the first one for the case of one or several bottom-mounted vertical cylinders. Comparisons are offered with experimental results, where the merits and short-comings of the linearized theory are emphasized. Separate sections are then devoted to specific problems such as barge roll resonance, recovery of wave energy, coupling between seakeeping and sloshing in tanks, and resonances in moonpools and gaps.