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.

Morgan and Moulden have shown that the café wall illusion can be explained by the effects of processing with a difference-of-Gaussians filter which reveals Fraser twisted cords in the figure. It is deduced that their account leads to the novel prediction of the reverse tilt illusion if the width of the mortar lines is suitably increased. This prediction is confirmed by demonstration.

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Upwind and central difference schemes for laminar and turbulent flows over a step in a two-dimensional channel are compared with each other and with experiment. Vorticity u and stream function (I are used as dependent variables and it is shown that an upwind difference method can give predictions which agree with experiment for high Reynolds number flows. The numerical implementation of the boundary conditions is found critically to determine the solutions obtained. Explicit prescription of $ and w at the inlet leads to incorrect values of the inlet velocity component perpendicular to the flow and a solution that does not agree with experiment. Experimental evidence is not available at low Reynolds numbers, but it is found that upwind differences give recirculation zones 8 per cent shorter and 8 per cent less intense than the more conventional conditionally stable central difference method.The distribution of the false diffusion effect in the upwind scheme is considered. and it is shown by use of a simple example that previous statements as to possible minimization of this effect are not generally true.The difficulty of determining the pressure distribution from the vorticity and stream function model is analysed and illustrated.Further, turbulent separated flows are seen to contain regions where two-dimensional time-dependent flow does not exist, and conventional theories do not give good results. + Now at Admiralty Marine Technology Establishment, Teddington, Middx., U.K.

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