1 8 G\\YTll¥.K,Ra)ige of Progressive Waves in Deep Wafer. 



y= -—(h + ~h^ + — /^= COS-- (ct + a) 

 ■^ 27r\ 2 4 y A/ ' 



27rV6 36 / Xo ^ ' 



X T „ 67r, . 



- — • — /r COS ^ let + a) 



27r 24 A,/ ' 



- — • • — M^cos-r-ici + a). . . (27) 



27r 120 A/ ' ^ ' 



Hence the horizontal distance apart of two particles 

 indicated by (a,o) and (a + \,o) at any instant is A, so that 

 in the motion the wave-length (not only at the surface 

 but throughout the motion) is A. At a depth d the particles 

 will be indicated by (a,l?) and (a + X/^,d'), where the value of 

 Aj is readily expressed from (8) 



The motion of the particles is given in the next 

 section, but the general character of the motion is 

 illustrated by replacing A„ by A^ in the equations just 

 written and remembering that the coefficients will die 

 away exponentially. It will be noticed that A = A^ with 

 this notation. 



We have for the height of the wave 



27rV 2 24 y 



which on reversal becomes 



/l=z!!^- 3/^' Y _ I / 2 Tray 

 A 2V A / 24V A y ' 



and this might appear a convenient form for substitution 

 in the results just obtained. It can be shewn, however, 

 that this reversal is only justifiable for small values of /i, 

 and in fact for values of /i less than that value which will 

 be determined later as the limit of the range of /^. 



The general solution of Lagrange's equations for the 

 motion of a particle within the fluid mass is not of great 

 importance, but, as the necessary calculations are pre- 



