56 Mr. J. M c Cowan on the Solitary Wave. 



Hence, substituting for U, we have to the same order 



T=y P 7na 2 h 2 (l+^ma); (49) 



so that the kinetic and potential energies are equal to this 

 order of approximation at least. 



10. The Limiting Height of the Wave. 



It is found by experiment that there is a limit to the 

 height of the solitary wave depending on the mean depth of 

 the liquid : when an attempt is made to form a higher wave, 

 it breaks at the crest. 



Since q 2 cannot be negative, the limiting form will be that 

 for which ^ = at the crest, and therefore by (14) and (16), 

 the greatest elevation of crest will be given by 



2?w97 = tanm/* (50) 



Now by (6) we see that when u=0 at the crest 



ma = 2 cos 2 ^m(h + %), 



and therefore by (18) 



m?7 = sin m(h + 7j ) (51) 



Solving (51) and (52) for mh and m% we find 



m/i= l'l, mr} ='9 (52) 



so that the wave will break for an elevation rather less than 

 the mean depth. It is needless to seek to specify the breaking 

 elevation more exactly, for the approximation is here pushed 

 to an extreme limit. In fact, by (4), the crest when q = Q 

 becomes a double point on ^jr= —TJh, and the branches cut at 

 right angles, whereas Stokes has shown that for a free surface 

 the crest angle must be 27r/3 at the breaking-point. Our 

 approximation, however, considering the extreme circum- 

 stances, is sufficiently fair to indicate that the conditions (52) 

 for breaking should not be far wrong. Scott Russell's 

 experiments confirm this : he found that the wave broke when 

 the elevation was about equal to the depth ; but from some 

 experiments of my own I am inclined to think that ^ =3/4/i 

 is a closer approximation for the elevation at the breaking- 

 point. 



11. Approximate Theory of Breakers. 



Some account can be given of the gradual increase in height 

 and ultimate breaking of waves rolling in on a gently sloping 

 beacli. 



