202 Mr Cookson, The Oscillations of a Fluid 



in which tc must satisfy (9) and m is given by (8). There are an 

 infinite number of roots (9) for all values of n from n = to n=oo : 

 so that the complete expression for <£ is a doubly infinite series of 

 terms of the form (10). 



For shortness put 



B n (/cr) = J n (/cr) - y >( K0 \ Yn ( Kr )> 



M = oo 



thus <f> = 2 2 K A n>K B n (/cr) . sin n^. cosh k (z + h) cos mt. 



n = 



Instead of the single trigonometrical factor A cos mt sin n6 in 

 the typical term, we might put 



(J.. cos mt+ B sin mt) sin n6 + (G cos mt + D sin mt) cos n6. 



Let now rj denote the elevation of the free surface at any 

 moment above the mean level : then 



V = \ 7T-) = ( ^- ) to the order required, 



and since the liquid was originally at rest, 17 must be zero when 

 t — 0, so that we require sin mt and not cos mt in the expression 

 for </>. The typical term is therefore 



(j) = A n B n (/cr) n6 . cosh k {z + h) . sin mt. 



sm 

 Hence 97 = icA n B n (kv) n0 . sinh kU . sin mt, 

 ' v y cos 



/c sm 



and 77 = — A n B n (/cr) n0 . sinh ich . cos mtf (10), 



7 m v 7 cos v 



a possible form of the initial free surface is defined by putting 

 t = in this expression for rj. 



By superposition of two fundamental modes of the same period 

 but in different phases, we obtain a solution 



7] = A n B n (kv) . sinh kJi . cos (n6 ±mt+ e), 



which represents a system of waves travelling unchanged round 

 the origin with angular velocity m/n in the negative or positive 

 direction of 6. 



We may write (10) in the form 



sin 

 7] = A n B n (/cr) nd . cos mt, 

 cos 



