The transient motion of a floating body

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Abstract
An analytical method of calculating the body motion was given in an earlier paper. Viscosity and surface tension were neglected, and the equations of motion were linearized. It was found that, for a half-immersed horizontal circular cylinder of radius a, the vertical motion at time τ(a/g)½ is described by the functions h1(τ) (for an initial velocity) and h2(τ) (for an initial displacement) where \begin{eqnarray*} h_1(\tau) &=& \frac{1}{2\pi}\int_{-{\infty}}^{\infty}\frac{e^{-iu\tau}du}{1-\frac{1}{4} \pi u^2(1+\Lambda(u))}\\ {\rm and}\qquad\qquad\qquad h_2(\tau) &=& -\frac{1}{8}i\int_{-\infty}^{\infty}\frac{u(1+\Lambda(u))e^{-iu\tau}du}{1-\frac{1}{4}\pi u^2(1+\Lambda (u))}. \end{eqnarray*} The function ∧(u) in these integrals is the force coefficient which describes the action of the fluid on the body in a forced periodic motion of angular frequency u(g/a)½. To determine ∧(u) for any one value of u an infinite system of linear equations must be solved.In the present paper a numerical study is made of the functions h1(τ) and h2(τ). The integrals defining h1(τ) and h2(τ) are not immediately suitable for numerical integration, for small τ because the integrands decrease slowly as u increases, for large τ because of the oscillatory factor e−iur. It is shown how these difficulties can be overcome by using the properties of ∧(u) in the complex u-plane. It is found that after an initial stage the motion of the body is closely approximated by a damped harmonic oscillatory motion, except during a final stage of decay when the motion is non-oscillatory and the amplitude is very small. It is noteworthy that the motion of the body can be found accurately, although little can be said about the wave motion in the fluid.

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