Green Function of Steady Motion in Finite Water Depth

Abstract

The Green function associated with a steady translating source on a straight horizontal course in water with finite constant depth and infinite horizontal extent satisfying the classical free surface condition is studied by decomposing it into three parts: an array of Rankine singularities A, local disturbance D, and downstream wave part S. Each of the three parts is studied by several methods. This is used to verify the numerical scheme and find the most time-efficient procedure. The method of repeated averaging of partial sums for oscillating series is efficiently used to evaluate the infinite sum of Rankine singularities and the downstream wave part. The local disturbance needed in vertical force and pitch moment calculations is most demanding. The Green function is used in combination with thin ship theory to calculate wave resistance, vertical force, pitch moment, and far-field wash for a Wigley hull. The results are compared with Tuck's (1966) slender body theory for shallow water and experimental and theoretical results of wave resistance by Everest and Hogben (1970). The agreement is satisfactory. A shallow water wave resistance ratio r expressing the ratio between wave resistance in finite depth and infinite depth is introduced as an indirect way to minimize wash. It is demonstrated that a large influence of critical depth Froude number requires the ratio between fluid depth and ship length to be small.