A body floting on the free surface of water is given a small vertical displacement from its equilibrium position and is then held fixed. When the fluid has again come to rest the body is released. The subsequent damped motion is investigated when viscosity and surface tension are neglected and the equations of motion are linearized. The method applies to bodies of arbitrary shape in two or three dimensions, and is described in detail here for the heaving motion of a horizontal half-immersed circular cylinder of radius a.

The forced periodic motion of such a cylinder has been studied in earlier papers. In particular, the hydrodynamic forces exerted by the fluid on the body can be described by a dimensionless coefficient, $\Lambda(\omega (a|g)^{\frac{1}{2}})$, where ω is the (real) angular frequency. The function ∧ can be found by convergent infinite processes, but not explicitly, and the difficulties of the problem are due to this. The free motion of the cylinder is solved in the present paper by Fourier methods. The motion is regarded as the superposition of simple harmonic motions, and the displacement y(t) is thus obtained in the form of a Fourier integral $y_0(\tau (a|g)^{\frac{1}{2}})= -{\frac {1}{8}}iy_0(0) \int ^\infty_{-\infty}{\frac {u(1+ \Lambda (u))\; e^{-iu \tau }du} {1- \frac{1}{4}\pi u^2(1+ \Lambda(u))}}.$

It is seen that the integrand involves the force coefficient ∧ and is thus not strictly an explicit expression. The asymptotic behaviour for large times can be found explicitly when the depth is infinite: $y_0(t) \sim -\frac{4}{\pi}y_0(0) {\frac {a}{gt^2}}.$

A damped harmonic behaviour had been expected. The slow monotonic decay occurs because the function $\Lambda (\omega(a|g)^{\frac {1}{2}})$, when continued into the complex ω-plane, can be shown to be many-valued near ω = 0. No physical interpretation has yet been found for this property. The free motion of a cylinder set in motion by an applied force is also treated, with similar results.

Reasons are given why there are no rapidly oscillatory terms in the asymptotic expression. For finite constant depth the function $\Lambda (\omega(a|g)^{\frac {1}{2}})$ is single-valued near ω = 0, and the asymptotic expression for this case is not yet known.