A. J. Hermans

The influence of water depth on the hydroelastic response of a very large floating platform

The hydroelastic response of a two-dimensional very large floating platform to plane incident wave is investigated for three different cases: infinite, finite and shallow water depth. An integro-differential equation is presented to describe the deflection of the platform due to incident waves. Reflection and transmission coefficients are obtained as well. We consider the case of a strip and a half-plane. Numerical results are obtained for various values of the parameters. The results for the strip and for the semi-infinite platform are compared for different values of depth.

A geometrical-optics approach for the deflection of a floating flexible platform

In an earlier paper, see [J Fluids Struct 14 (2000) 943], a derivation of an integro-differential equation for the determination of the deflection of a large floating flexible platform, excited by waves, is given. In this paper, we investigate the possibility to derive a short wave formulation for this problem. It will be shown that for a flexible half-plane and a strip in deep water, the integral equation can be used to find such a formulation. The asymptotic results shall be compared with the numerical results obtained earlier.

Interaction of free-surface waves with a floating dock

A new method to describe the interaction of waves with a rigid or flexible dock, with zero draft, is derived. By means of Greens theorem an integral equation along the platform for either the velocity potential or the deflection is obtained. In the two-dimensional case this equation is solved by means of a superposition of exponential functions. With a specific choice of the Green function the integration with respect to the space coordinate can be carried out analytically. The integration left is the integration in the k-plane that occurs in the chosen Green function. Subsequently the contour of this integral is modified in the complex plane. This results at first in a dispersion relation for the phase functions in the expansion. Then the set of algebraic equations for the amplitude coefficients follows from the same singularity analysis in the complex plane. These equations are very simple and easy to solve. In contrast to the classical approach of eigen-mode expansions, there is no need to split the problem in a symmetric and antisymmetric one. An other advantage is that the transmission and reflection coefficients are determined seperately by means of Greens theorem, applied at the free surface in the far field. The method is first explained for the semi-infinite rigid dock, followed by the rigid strip, the moving strip and the flexible moving platform. In the appendix it is explained how to derive a set of algebraic equations in the case when the incident wave is not perpendicular to the strip.

A DEVICE TO EXTRACT ENERGY FROM WATER WAVES

In recent years attention has been focused on means of extracting energy from ocean waves. As a result, a number of systems have been devised by means of which wave energy, which consists of potential and kinetic fluid energy, can be captured and transformed into either mechanical, hydraulic or electrical energy. The majority of the systems transform the wave motion into a reciprocating motion of a mechanical system which in turn drives some other form of energy transformation system finally yielding electric or hydraulic energy. The selection of a reciprocating system to transform wave motion into mechanical motion is an obvious choice when the waves are interpreted in terms of being a free-surface phenomenon. It can be shown however, that it is possible to transform wave energy directly into rotational mechanical energy based on a different viewpoint of wave motion. In this paper the fundamental characteristics of such device are discussed and a mathematical model developed at the Delft University of Technology describing the properties of the device will be verified by means of results of model tests carried out at the Maritime Research Institute Netherlands.

Low-Frequency Second-Order Wave-Drift Forces and Damping

The effects of free-surface waves on floating structures is of great importance in the offshore industry. Besides the first-order responses to the waves, second-order effects play a role, in particular, in the situation where an anchored object may be excited in its frequency of resonance. This paper concerns the study of this phenomenon. Two different approaches are developed, each with its own advantage. Special attention is paid to excitation forces generating the motions; also a theory to determine the inviscid damping that limits the extreme excursions is treated. Numerical results are presented for a sphere, while numerical results for two classes of tankers, namely for a VLCC and a LNG-carrier, and a semi-submersible are compared with experimental data obtained at the Maritime Research Institute in the Netherlands (MARIN).

A quantum mechanical treatment of the mirror electron microscope

SummaryA wave mechanical description of the electron beam in a mirror electron microscope is given. First the Schrödinger equation of the electron beam is set up, from which the wave function is obtained. Then the probability current density is derived, which turns out to be a useful expression to clarify the formation of the image on the screen.

Slowly moving hull forms in short waves

The purpose of this paper is to provide a mathematical tool to improve the optimal design of ship forms. It is common practice that hull forms are designed such that they have minimum wave resistance in calm water. In this paper a theory is described by which the effect of short waves may be incorporated.The basic tool we use is the ray theory. First, the appropriate free-surface condition is shown. Then, the standard ray method, well-known in geometric optics, is formulated in the fluid region and at the free surface. After an elimination process the eiconal equation and the transport equation are obtained. The characteristic equation for the nonlinear eiconal equation is derived, keeping in mind that the characteristics are not perpendicular to the wave fronts, due to the effect of the double-body potential due to the forward speed of the ship, which is assumed to be a good approximation for the steady potential.Numerical computations are carried out by means of the RK4 method to obtain the ray pattern. After some manipulations the amplitude may be computed just as well. Finally, the nonlinear added-resistance force is calculated. Pictures of ray patterns for several angles of incidence are shown. Also the forces are shown.

A perturbation method for the radiation of surface waves

SummaryIn this paper we derive a straight forward asymptotic method to find the wave solution for the case that a circular cylinder is heaving in a free surface. The wave period is supposed to be small. The methods used are similar to methods used in the theory of geometrical optics and the theory of boundary layer expansions. It turns out that not only the lowest order approximation can be easily calculated, higher order approximations follow as well.

Diffraction of a wave at high frequencies by an arbitrary smooth convex object

SummaryIn this paper the three dimensional scattering of a plane and a spherical wave by an arbitrary smooth convex object will be considered.These problems are solved for large values of frequency by means of ray theory and the theory of boundary layer expansions.

A discussion on the second-order wave forces and wave drift damping

Abstract In this paper we derive a formulation for the second-order wave drift force on an object sailing at low forward speed. The method used consists of the application of the law of conservation of impulse over the fluid domain outside the body. We are interested in the large second order resonant motions where the damping coefficients in the equation of motion are mainly determined by the influence of the motion on the mean drift force at the frequency of excitation. The results are compared with results obtained by the direct pressure integration method. Due to the small first order damping at small frequencies the second order wave drift damping becomes significant. We present a formulation for this coefficient based on the speed dependency of the wave drift force. It is shown that an approximate formulation of Aranha [Aranha, J. A. P., A formula for “wave damping” in the drift for a floating body. J. Fluid Mech. , 275 (1994) 147–155.] gives reasonable results in some cases. However, it leads to large errors in the case that the first order motions are near resonance.