CHAPTER IX. 



SURFACE WAVES. 



216. WE have now to investigate, as far as possible, the laws 

 of wave-motion in liquids when the restriction that the vertical 

 acceleration may be neglected is no longer imposed. The most 

 important case not covered by the preceding theory is that of 

 waves on relatively deep water, where, as will be seen, the agita- 

 tion rapidly diminishes in amplitude as we pass downwards from 

 the surface ; but it will be understood that there is a continuous 

 transition to the state of things investigated in the preceding 

 chapter, where the horizontal motion of the fluid was sensibly the 

 same from top to bottom. 



We begin with the oscillations of a horizontal sheet of water, 

 and we will confine ourselves in the first instance to cases where 

 the motion is in two dimensions, of which one (x) is horizontal, 

 and the other (y) vertical. The elevations and depressions of the 

 free surface will then present the appearance of a series of parallel 

 straight ridges and furrows, perpendicular to the plane xy. 



The motion, being assumed to have been generated originally 

 from rest by the action of ordinary forces, will be necessarily 

 irrotational, and the velocity-potential </> will satisfy the equation 



*=0 (1) 



* 



dx* 



with the condition - = . . (2) 



dn 



at a fixed boundary. 



