This section lays the foundation for the analysis of random marine dynamics. A platform’s dynamics, which result from excitation due to irregular waves, can generally by expressed in a Fourier series - a consequence of linearity and the principal of linear superposition. Fourier representation, either through Fourier series or Fourier transforms, allows for frequency or time domain analysis, both of which are developed in this chapter. The frequency domain representation implies a harmonic solution in time. Consequently, the system of second order ordinary differential equations with constant coefficients become a set of simultaneous linear algebraic equations whose solutions are the complex motion amplitudes. This system of equations represents the response to harmonic forcing and does not include transient behavior associated with initial conditions. A time domain representation of floating bodies requires a means to include system memory effects. These memory effects are modeled by convolution integrals in the equations of motion where the kernel function in the convolution integral is related to the Fourier cosine transform of the damping coefficient of the floating body.

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 primary source of excitation in marine dynamics is the ocean environment, which is often characterized as a random process. Therefore, objective analysis of resulting dynamics is presented in terms of averages, or probabilities. For example, it is possible to determine, within the limits of the modeling assumptions, the average of the l/3 largest waves, or the average of the 1/1000 bow accelerations. The basis for these averages is linear theory and the Fourier transform. This chapter shows how the frequency decomposition of a time series can be achieved by Fourier analysis, resulting in a “mean square density” spectral density function. The assumption that the process is stationary and ergodic results in temporal statistics, e.g. process mean and mean square, are equal to ensemble statistics. Therefore a single time series record may be used to estimate probability density functions and statistical properties. Probability density functions (PDF) for the elevations (Gaussian) and amplitudes (Rayleigh, if the process is narrow banded) are given. Extreme value PDF’s and most probable maxima relations are derived allowing for the estimate of the largest response in N encounters.

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 uses the ideas of hydrodynamics introduced in the last chapter to formulate the hydrodynamic theory of the flocking problem (i.e., the “Toner–Tu” equations).

This chapter “derives” the scaling laws found in the previous chapter for incompressible flocks, using a simple heuristic argument which gives some physical insight into the mechanism, and its essentially nonequilibrium nature.

I present a purely dynamical derivation of the Mermin–Wagner–Hohenberg theorem, and compare it with the standard equilibrium derivation. This also provides an opportunity to introduce diffusion equations and gradient expansions, both of which play a large role in what follows.

This chapter deals with incompressible flocks in spatial dimensions d > 2, for which the dynamical renormalization group yields exact scaling laws.

I introduce, and describe in detail, the dynamical renormalization group, using the KPZ equation as an example. In addition to spelling out the mechanics of the technique in great detail, I also emphasize its philosophical importance, as the answer to Einstein’s famous question “Why is the universe intelligible?” and its role as a guide to the formulation of hydrodynamic theories.

This chapter treats incompressible flocks in two dimensions, and shows that they map onto both equilibrium two-dimensional smectics, and our old friend the KPZ equation (albeit in one dimension), as well as a peculiar type of constrained magnet. Exact scaling laws are again found, this time by exploiting these mappings.