196 WAVES IN LIQUIDS. [CHAP. VII. 



where the summations include all integral values of ra, n from 1 

 to oo . Substituting in (40), we find 



If be the greater of the two quantities a, b, the component oscil 

 lation of longest period is got by making m = 1, n = 0, whence 



a 



the motion is then parallel to the longer side of the tank, and 

 consists of two systems of waves of the kind considered in Art. 156, 

 travelling in opposite directions, the wave-length being 2a. 



Circular Tank. 



163. In the case of a circular tank, whose axis is vertical, it 

 is convenient to take the origin in the axis, and to transform to 

 polar co-ordinates by writing 



x = r cos 0, y r sin 0. 

 The equation (40) then becomes 



....... 



2 * 



r ar 



Now whatever be the value of &amp;lt; it can be expanded by Fourier s 

 theorem in the form 



&amp;lt; = 2 (ty n cos n6 + ^ n sin nfy, 



where the summation embraces all integral values of n, and ty n , 

 % n are functions of r only. Substituting in (41), we have, to deter 

 mine T/r n , 



,J2__ 1 J^t- / M 2\ 



(42), 



with an equation of the same form for % n . 



The solution of (42), subject to the condition that ty n is finite 

 when r = 0, is 



^. = 24/.(fe-) (43), 



