100 WAVES IN LIQUIDS. [CHAP. VII. 



potentials of distinct systems of waves of the particular kind above 

 considered, then 



(33) 



will be the velocity-potential of a possible form of wave-motion, 

 with a free surface. Since when &amp;lt;f&amp;gt; is determined the equation of 

 the free surface is given by 



the elevation above the mean level at any point of the surface, in 

 the motion given by (33), will be equal to the algebraic sum of 

 the elevations due to the separate systems of waves ^, &amp;lt; 2 , &c. 

 Hence each of the latter systems is propagated exactly as if the 

 others were absent, and produces its own elevation or depression 

 at each point of the surface. 



We may in this way by adding together terms of the form (29), 

 with properly chosen values of a, build up the solution of the 

 general problem of Art. 156 in the case where the initial conditions 

 are any whatever. Thus, let us suppose that, when = 0, the 

 equation of the free surface is 



- - y =/(*), 



and that the normal velocity at that surface is then F (x), or, to 

 our order of approximation, 



The value of &amp;lt;/&amp;gt; is found to be 

 dk &amp;lt;**&amp;lt;+*&amp;gt; +-*&amp;lt;+*) 



rdk g &amp;lt;*+ +- fl f p 



&amp;lt;f&amp;gt; = - . - =^~- ^ \\ d\F(\) cos k(\ - x] \ cos to 

 r Jo TT kc e kh + e-* h |jbcy_ ) 



( r )l 



- \ d\f(\) cos k (\ - x) \ sin kct , 



(7-00 J J 



and the equation of the free surface is 



r 00 fiif r ( r &quot;i 



y = \\ d\f(\) cos A^(X - a?) I cos kct 



JO 7T [_[J _oo J 



-f j- \ f d\F(\) cos k(\ - x)\ sin kct] . 



kc [ J _ oo J 



