94 Mb GREEN, ON THE MOTION OF WAVES IN CANALS. 



The theory of the motion of waves in a deep sea, taking the most 

 simple case, in which the oscillations follow the law of the cycloidal 

 pendulum, and considering the depth as infinite, is extremely easy, and 

 may be thus exhibited. 



Take the plane (a? as) perpendicular to the ridge of one of the waves 

 supposed to extend indefinitely in the direction of the axis y, and let 

 the velocities of the fluid particles be independent of the co-ordinate y. 

 Then if we conceive the axis % to be directed vertically downwards, 

 and the plane (xy) to coincide with the surface of the sea in equilibrium, 

 we have generally, 



dx 2 + d%* ' 



The condition due to the upper surface, found as before, is 



dd> d z d> 



0= edt-w- 



From what precedes, it will be clear that we have now only to 

 satisfy the second of the general equations in conjunction with the 

 condition just given. This may be effected most conveniently by 



taking 



<j> = He'T' sin -^ (v't - x), 



by which the general equation is immediately satisfied, and the condition 

 due to the surface gives 



*- T «^. «- , ., 



' - V£. 



where \ is evidently the length of a wave. Hence, the velocity of 

 these waves vary as \A, agreeably to what Newton asserts. But the 

 velocity assigned by the correct theory exceeds Newton's value in the 

 ratio *s/Hr to \/2, or of 5 to 4 nearly. 



