356 Mr. W. Gr. Fraser on 



operator of Laplace, and « has the meaning which we have 

 assigned to it in § 1; similarly two analogous equations are 

 found. These three equations are obviously the equations of 

 motion of the fluid, which it was our object to study. 



In a future contribution we hope to be able to give some 

 applications of the theory which we have developed. 



XXXV. On the Breaking of Waves. 

 By W. G. Fraser, 31. A., Queen's College, Cambridge*. 



IX the ordinary theory of the reflexion of waves at a fixed 

 barrier, it is shown that an incident train of waves gives 

 rise to a reflected train of the same type and amplitude, the 

 two trains combining to form a system of standing waves. 

 Xow the theory claims to be no more than a first approxima- 

 tion, applicable only to small disturbances ; but it is a matter 

 of everyday knowledge that waves which, if left to themselves, 

 would proceed for a considerable distance without sensible 

 change of type, may nevertheless be too high to be reflected 

 at a wall according to the ordinary theory ; instead of this, 

 they break into spray against the wall. Thus it appears that 

 the theory of progressive waves is a better approximation to 

 fact than the theory of reflected waves. 



An attempt is here made to find part, at least, of the reason 

 for this in the friction of the wall. It appears that, in deep 

 water, if the ratio of the amplitude to the wave-length exceeds 

 a certain small amount, the wave will break ; in shallow water 

 the breaking amplitude is somewhat less than is indicated by 

 this ratio. 



In the ordinary theory, if the incident train of waves have 

 a velocity potential <£ we ascribe to the reflected train a 

 potential <//, so adjusted as to bring the water in contact with 

 the barrier into a state of motion that is purely vertical ; so 

 that, if u, v be the horizontal and vertical components of the 

 velocity of an element of liquid close to the barrier due to 

 the incident wave, the components clue to the reflected wave 

 are — u, v. The two velocity systems therefore bear the 

 same relation to one another as the velocity of a particle 

 before and after impact at a smooth vertical elastic wall. 

 Xow, while the horizontal motion of the liquid in contact 

 with the wall must be annihilated, it is only natural to suppose 

 that the wall exerts an appreciable drag on the vertical 

 motion. It is proposed, therefore, to take, as the velocities 

 in the reflected wave —it, mv, where m is slightly less than 



* Communicated bv the Author. 



