the soltary wave, although this wave cannot itself occur under the circumstances 

 considered. 



To find the shape of the free surface very close to the separation point, other 

 methods have to be used. In the example of a vertical sluice-gate, I was able to apply 

 a fairly straight-forward method based on conformal transformation whereby the free- 

 surface profile in the hodograph plane is approximated by an arc of an ellipse. A sim- 

 ilar method is also adequate when the obstacle is an inclined plane. It appears that 

 these methods will provide estimates of the drag with an accuracy considerably better 

 than 1%. The results of these calculations compare favourably with the solutions 

 which Southwell and Vaisey- obtained by relaxation methods. 



Although this work has no direct bearing on the problem of the resistance of 

 ships in deep water, it may possibly have applications to other problems within the 

 sphere of this symposium. For instance, it seems to be relevant to the performance of 

 hydroplanes and seaplane floats in shallow water. 



/. J. Stoker 



Both of the lecturers this afternoon have made reference to the work of Arthur 

 Peters and myself about the motion of ships in a seaway. I thought I might try to tell 

 you what we had in mind to do, although it is impossible to give any details. 



For this audience the best way to put it would probably be to say that what we 

 did was to formulate a theory which is a thoroughgoing generalization of the classical 

 theory of Michell. We make the same basic assumptions as in that theory. Those are 

 principally that we have a perfect fluid, so that the turbulent wake is ignored, for ex- 

 ample; and then on top of that we make an assumption that is necessary to linearize 

 the problem, namely that the ship is slender in some sense so that it is possible for it to 

 have a forward motion with finite velocity, and still create waves of small amplitude. 

 The result is a theory which is a generalization of the Michell-Havelock theory, in the 

 sense that the latter results when we make the restrictive assumptions that are necessary 

 for that case. What we have done, in effect, is to give up the strong assumption that 

 the ship is held fixed in space, while the water streams past. That is what Michell as- 

 sumed, and the one quantity he computed is the wave resistance, the force necessary to 

 maintain the body in that position. 



We assume that the body floats freely, calculating its motion from the pressure 

 force of the water, treating the combined system of ship and sea without making any 

 a priori assumptions with respect to the coupling of the various degrees of freedom of 

 the ship or the coupling of the motion of the ship and the motion of the water. 



In developing the theory our procedure was to carry out a formal development 

 of all quantities with respect to a small parameter, which was a thickness-length ratio 

 of the ship's hull. Depending upon the manner in which this parameter is chosen, 

 various theories arise. If, lor example, the ship is regarded as small in the beam-length 

 ratio, our generalization of Michell's theory arises. If it is the draft-length ratio that is 

 assumed small, still a different theory arises which might be called a theory of planing 

 ships. For a yacht-sbaped hull, which is a combination of the previous two cases, still 

 a third theory is derived. 



In the first case, i.e. the case of a ship with a thin vertical section — a ship of 

 Michell's type — it turns out that a more general problem than that of Michell can be 

 solved explicitly. That is the problem of motion of the ship when it is assumed that it 

 moves only in the vertical longitudinal plane through waves which are symmetrical 

 with respect to that plane. The amplitude of the oscillations, the trim, and the wave 

 resistance can be calculated by explicitly given integrals. 



However, it is an inescapable consequence of this generalization of Michell's 

 theory that the pitching and heaving modes of oscillation are not damped. The reason 

 for that is simple: the ship is like a knife-blade placed vertically in the water, and small 



2 Phil. Trans. A, 240, 117. 



135 



