Inertia forces acting on a horizontal circular cylinder in regular and irregular waves significantly decrease at low Keulegan-Carpenter numbers ( Kc). The rapid reduction of inertia force is caused by a circulating flow around the cylinder. In order to clarify the mechanism of the occurrence of the circulation, flow visualizations around a horizontal circular cylinder in regular waves and a circular cylinder moving along a circular orbit were carried out. Simple formulae to estimate the circulation strength and the inertia force reduction in regular waves were deduced from a starting vortex model with reference to the results of the flow visualizations. The prediction method was applied to the cylinder inclined to wave crests. The estimated inertia coefficients in regular waves are in good agreement with the experimental results at Kc less than 2.5. The inertia force reduction in irregular waves was also estimated by considering the circulation effect for each half wave period which was calculated by the prediction method in regular waves. The estimated results are in good agreement with the experimental results.