The resistance of a ship among waves

1—The wave resistance of a ship advancing in still water may be calculated under certain assumptions, which amount to supposing the forced wave motion to be small so that squares of the fluid velocity may be neglected; moreover, the ship is supposed to advance with constant velocity in a horizontal line. It does not appear to have been noticed that we may super pose on the solution so obtained free surface waves of small amplitude, and that the addition to the resistance may be calculated, to a similar degree of approximation, as the horizontal resultant of the additional fluid pressures due to the free surface waves; this additional resistance, which may be negative, depends upon the position of the ship among the free waves. Various calculations are now made from this point of view. We consider first transverse following waves moving at the same speed as the ship, and then a ship moving in the waves left by another ship in advance moving at the same speed; finally, we examine the more general case of a ship moving through free transverse waves of any wave-length. All the cases are discussed with reference to such experimental results as are available. 2—We treat the problem first as one of steady motion with the ship at rest in a uniform stream of velocity c in the negative direction ofOx ; we take the originO in the undisturbed water surface, andOz vertically upwards. The velocity potential is given byϕ =cx +ϕ 1 , (1)