One method of making waves in a ship tank is by means of the periodic heaving 

 motion of a cylinder. The result in general is to generate straight crested longitudinal 

 waves. 



If, however, the frequency of the oscillations of the cylinder exceeds a certain 

 well defined critical value, transverse waves are also set up and a choppy sea results. 

 You would hardly credit this without seeing it. I believe that no theoretical explana- 

 tion of this phenomenon has so far been found. The suggestion that it is connected 

 with the presence of the vertical walls of the tank is apparently without foundation. 

 Here then is a matter to which research workers might usefully direct their attention. 



What I have been describing is a method of approximation, very much superior 

 to linearization but still an approximation. 



It is therefore with gratification that we can turn to another method of approach 

 to the free surface problem which reduces the attack on the two-dimensional case to 

 the solution of a linear partial differential equation of the second order of parabolic 

 type. 



This method was originated by F. John [21] and reposes on the fact that on 

 a surface of constant pressure the acceleration of the liquid relative to the acceleration 

 due to gravity is a vector whose direction is normal to the surface of constant pressure. 

 This statement is a simple consequence [22] of the equation of motion in the form 



1 

 a- g = - - Vp. (38) 



P 

 If then the surface has the equation 



z = f(a, 0), z = x + iy, (39) 



where a is a real Lagrangian parameter, we have 



dH df 



— +ig = ir(a, t) — , (40) 



dP da 



where r( a , t) is an arbitrary function. This is the equation just mentioned. Every 

 two-dimensional continuous free surface must satisfy this equation, whether the motion 

 is rotational or irrotational, steady or unsteady. 



When the motion is irrotational it is not difficult to deduce the equation satisfied 

 by the velocity potential. 



Further John has shown that the form y rr k(x, t) of the free surface can be 

 prescribed. For a particle, x, y are functions of t and the constant pressure condition 

 then implies that x as a function of t satisfied a non-linear ordinary differential equation 

 of the second order. 



In the case of steady motion the problem can be simplified still further to depend 

 on the solution of an ordinary second order linear equation of the form 



f"(0) + ig = iSWGS), (41) 



where S(/3) is an arbitrary function. 



It follows from these considerations that two-dimensional free surface problems 

 can be reduced to the study of a limited class of differential equations. Nevertheless 

 progress will necessarily depend on divining the proper form of the arbitrary function 

 involved. 



On the other hand we have here a means of generating an unlimited number 

 of free surfaces by assigning the arbitrary function. In particular putting S(fi) =. co, 

 a constant, leads to a trochoidal free surface, from which we can proceed to a trochoidal 

 progressive wave. 



Unfortunately this wave has to be associated with a moving ocean bed since 

 the singularities of the progressive wave are no longer fixed. 



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