Water Waves Generated by Thin Ships

The problem of interest is that of the water waves in a body of water of infinite depth generaied by a thin ship of given hull form, moving with constant velocity U along a straight course on the otherwise undisturbed water surface. A particular method is evaluated for computing the velocity field at an arbitrary distance (not too near the ship) fixed in the fluid. A new proposal is made here that the hull profile be represented by a double Fourier series with its half-periods spanning over the region occupied by the longitudinal mid-section of the ship. The convergence of this series representation is found to be satisfactorily rapid, especially when the tangent plane of the hull is everywhere continuous. In the latter case the longitudinal slope of the hull, which is the only partial derivative appearing in the analysis, is found in a specific case to be well represented by the partial derivative of the series. With this series representation of the hull, the analysis of the velocity-field calculation is greatly reduced so that the final result can be expressed in terms of a combination of several single and double Fourier integrals which are susceptible to numerical methods. However, for large values of or, where r is the distance from the ship, a = gL/U2, with g being the acceleration of gravity and L the ship length, these integrals can be evaluated with good approximation by asymptotic methods. The method of stationary phase and other asymptotic methods are employed in different regions in the water and the final expression for the velocity field is given explicitly. The numerical result for a specific ship will be given elsewhere.