Slender oscillating ships at zero forward speed

Abstract

The ship is assumed to be a slender body of revolution with its axis in the mean free surface and making periodic oscillations of small amplitude. The theory presented here is a generalization of the well-known slender-body theory of incompressible aerodynamics in which the fluid is externally unbounded. One version of that theory goes as follows: approximate the body by axial line-distributions of known point singularities (sources and multipoles), whose strength is to be determined; by means of the Fourier convolution theorem express the velocity potentials of these line distributions in terms of the Fourier transforms (in the axial direction) of the point-singularity potentials; expand these Fourier transforms in powers of the radius and retain only the leading terms (it is here that the slender-body assumption is introduced); by means of the Fourier convolution theorem interpret the resulting expressions. By this procedure it is not only shown that near the body the potential is two-dimensional harmonic in every plane normal to the axis; but also the interaction between sections is shown to be involved in the ‘constant’ term and to depend in an explicit manner on the coefficient functions, which can be found without difficulty by applying the prescribed boundary conditions. This foregoing procedure can be justified when the body is slender and sharply pointed.In the present paper the same procedure is adapted to an oscillating surface ship a t zero speed. The fluid is now bounded by the ship, and also by a horizontal plane (the mean free surface) on which a wave boundary condition must be applied. The point singularities are now wave sources and wave-free potentials, each satisfying the free-surface condition. The Fourier transforms of these singularities are found and are expanded near the axis; the expansions near the axis are the only parts of the argument that present any serious difficulty. When only the leading terms are retained and the results are interpreted by the convolution theorem, explicit two-dimensional potentials are again obtained.It is assumed that the ratios (ship-radius/ship-length) and (ship-radius/wavelength) are small whereas the ratio (ship-length/wave-length) may have any value. Expressions are given which are valid according as this last ratio is not large or not small. The potential on the body is found, and forces, moments and wave damping are calculated. It is believed that the expansions can be extended with little trouble to certain other ranges of those ratios, to other cross-sections and to the boundary condition for ships moving a t non-zero speeds.