Applying the velocity potential of a two-source system to the problem of a thin ship moving in a rectangular channel with time-varying speed

The linear non-stationary problem of determining the amplitude picture of the surface disturbances generated by a thin ship with symmetric outlines moving with time-dependent velocity and attached flow in the channel with rectangular cross-section along its longitudinal axis is solved. A methodology considering the sources distributed over a wetted surface is applied to obtain the solution. It is based on the use of the velocity potential of two laterally symmetric sources moving under the free surface with time-dependent velocity and intensity. This potential satisfies boundary conditions on the free surface, bottom, and channel walls. It is represented in analytical-and-quadrature form and is expanded into a cosine Fourier series with known coefficients. In the case of the ship of arbitrary geometry, the use of this method decreases the number of equations required for defining a distribution of sources over the wetted ship surface at least by an order. Its important feature is the possibility to solve the non-stationary problem at a predetermined moment. This is due to the lack of a procedure for finding the intermediate solution at every time step. In the case of a thin ship, source distribution intensity is written as a simple function of its geometry. As a result, the problem solution is reduced to calculating the double integrals over a wetted surface and integrals over a wave number and time. The obtained solution includes a system of ship waves with known wave numbers, the disturbances that attenuate as they move away from a ship, and disturbances determined by an added mass that is typical for non-stationary body motion. A sufficiently simple geometry of a wetted surface is used in the numerical studies. Variation of the ship's velocity motion corresponds to its two typical stages. In the first stage, the vessel starts motion from the state of rest and reaches a determined velocity. In the second stage, the ship moves in the deceleration mode till it stops. For such a motion pattern, the surface disturbance structure is determined only by decaying components from the ship body and added mass. The obtained relationships between free surface disturbances, Froude number for each motion stage, ship geometry, vessel draft, and channel geometry are analyzed.