FIG 8 



From the naval point of view there are three main problems whose exact solu- 

 tion still eludes us, namely the problems of periodic surface waves on deep water, 

 surface waves on water of finite depth, and the solitary wave. The combination of 

 aerial photographs with a knowledge of the relation between wave length and wave 

 speed played an important part in determining depth and slope on the Normandy 

 beaches. 



Let us look at what is involved in the problem of the two-dimensional progres- 

 sive wave of permanent type on water of finite depth. As long ago as 1925 Levi-Civita 

 [18] stated the boundary value problem in the form 



m g 



— = e~ ?,T sin d, \f/ = 0, q = ce T . 



dip c 3 



(32) 



Here 6 is the inclination of the velocity vector to the horizontal. 



By mapping the strip defined by a period on the unit circle and then obtaining 

 a Taylor expansion, Levi-Civita established the existence of this type of wave. 



One might inquire why such an existence theorem should be necessary when 

 waves are to be seen any day. 



But it is only fair to remember that the wave here considered is perfectly 

 regular and is propagated in an inviscid fluid, conditions to which observed waves only 

 approximate. 



The intrinsic difficulty of the problem here envisaged is the non-linearity of 

 the boundary condition, quite apart from the fact that the form of the boundary is 

 part of the solution. 



Putting + h =■ o>, the linearized approximation is obtained by assuming 



|o,| = (02 _|_ T 2)l/2 (33) 



to be small of the first order. From this assumption follows the usual theory of waves 

 of small amplitude and slope. 



There is, however, a serious limitation to the use of the linearized approxima- 

 tion. A wave will break at the crest when the fluid velocity there exceeds the velocity 

 of the wave. The critical case is when the fluid velocity at the crest is equal to the 

 velocity of the wave, that is to say q = ce T = 0, so that T — — co . 



It follows that no approximation based upon the assumption that r is small 

 can throw any light on the case of breaking. 



A way to avoid this difficulty has been proposed by T. V. Davies [19] namely 

 in Levi-Civita's boundary condition to replace sin d by sin 3$. 



This substitution replaces one non-linear boundary condition by another. It 

 still preserves the essential feature but allows r to be large. 



10 



