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



impressed velocity, say — U, may be directly obtained from 

 tbe current function, say ty. For 



r- - d *- - 1 *¥. - k^t. - i/m 



4 dg ~ U dt ~ U B«~ 



d/tft being used to denote partial, and "dfdt complete differentia- 

 tion with respect to t. 



From (39) we see that a particle starting from the level z 

 returns to the same level after attaining a maximum elevation 

 go given by 



fo = atan4m(,: + f ), (40) 



which includes the special case of a surface particle given bv 



( 18 )- 



To obtain (f it is necessary to proceed by successive approxi- 

 mations. We find at once 



v sinh mx' ,... 



t—a — a — i i t+t, . . . (41) 



cos mJ -f cosn mx v 



where the a is added to make the first part vanish when 



x ! = oo , or t= — co , and t, which is of the order ma'\ is given 



by 



B^ U (cos ?/^ / + cosh mx') 2 ' ' \ ) 



If we integrate this, using the transformation of which (29) 

 is a case with special limits, and for brevity take advantage of 

 the expressions (36) and (37], we get 



9 f sin mz—mz cos mz £l<r — sin - l £l<r. cos m^' ") , . ON 



T=)M 2 < =-s W =-?* / [• +Ti, (43) 



t sin mz sin mz J v ' 



where again the first term is added to make the first part 

 vanish when #'=oo , and t 1? which is of the order m 2 a 3 , may 

 be similarly approximated to when wanted. 



If we neglect r, we find from (41) and (43) for the total 

 displacement 6* of any particle 



<s a f ., sin mz— mz cos mz~) 



o = za < 1 + ma .— z — > , 



I sm mz J 



or, neglecting terms of the order m s a 2 z 2 , 



S==2a{,l+lma},. ..... (44) 



which shows that to this order all particles are equally dis- 

 placed by the wave. The agreement of (44) with (32), which 

 is obtained very differently, may be noted in passing. 



