The Exciting Forces on Fixed Bodies in Waves

Abstract

General expressions, originally given by Haskind, are derived for the exciting forces on an arbitrary fixed body in waves. These give the exciting forces and moments in terms of the far-field velocity potentials for forced oscillations in calm water and do not depend on the diffraction potential, or the disturbance of the incident wave by the body. These expressions are then used to compute the exciting forces on a submerged ellipsoid, and on floating two-dimensional ellipses. For the ellipsoid, the problem is solved using the far-field potentials, and detailed results and calculations are given for the roll moment. The other forces agree, for the special case of a spheroid, with earlier results obtained by Havelock. In the case of two-dimensional motion the exciting forces are related to the wave amplitude ratio A for forced oscillations in calm water, and this relation is used to compute the heave exciting force for several elliptic cylinders. Expressions are also given relating the damping coefficients and the exciting forces. A = wave amplitude A = wave-height ratio for forced oscillations(a1 a2 a3) = semi-axis of ellipsoidBij = damping coefficientsC4 = nondimensional roll exciting-force coefficientDj = virtual-mass coefficients, defined by equations (18) and (19)g = gravitational accelerationh = depth of submergencei = √ — 1j = index referring to direction of force or motionn(z) = spherical Bessel function, K = wave number, K = ω2/gPj = functions defined following equation (17)R = polar coordinateV, = velocity components (x, y, z) = Cartesian coordinatesαi = Green's integrals, defined by equation (20)β = angle of incidence of wave systemθ = polar coordinateρ= fluid densityφj = velocity potentialsω = circular frequency of encounter