274: Lord Hayleigh on Waves. 



by Green's theorem, c/S denoting an element of the surface 

 bounding the mass, and .--.the rate of variation of (j> in a nor- 

 mal direction outwards. The surface S consists of three parts — 

 the bottom of the vessel, the cylindrical side of the vessel, and 

 the upper surface of the fluid. Over the first two of these, 



- 7 — = 0, and thus 



da ' 



Now when z=0, 



v «* + €-«' «« 



(b = 2t- ,- 7 — u K . 



# v. 



so that 



the product of any two functions u K , v K > vanishing when inte- 

 grated over the area. 



We have now to calculate the work done by impressed forces 

 corresponding to the displacement represented by 8* K . It must 

 be remembered that these forces are limited to such as have a 

 potential. Let Bp denote the variable part of the pressure at 

 the surface, supposed to remain in its position of rest, whether 

 applied directly or due to impressed body-forces, then 



work done on system = — j j Bp Bh dx dy. 



If Bp be expanded in the series, 



Bp = ^/3 K n K (xy), 



work = — (TEA u K . XScc k u K . dx dy 



= — 2& Bci K JJ u\ dx dy. 



We can now form the equations of motion in terms of the 

 generalized coordinates ot K . By Lagrange's method, 



is the equation determining the variation of the coordinate a K , 

 where 



fi* = IS & Uk dx d y~*~ JT w * dx d y- 



When the oscillations are free, /S K = 0. If the period be r K) and 



