304 TIDAL WAVES. [CHAP. VIII 



Thus, in the case of a rectangular boundary, if we take the 

 origin at one corner, and the axes of x } y along two of the sides, 

 the boundary conditions are that d^jdx = for x = and x = a, 

 and dQdy = for y = and y = 6, where a, b are the lengths of the 

 edges parallel to x, y respectively. The general value of f subject 

 to these conditions is given by the double Fourier s series 



^ v A mirx ^ 



?= 22.4m, COS- COS-y^ .................. (3), 



where the summations include all integral values of m, n from 

 to oo . Substituting in (1) we find 



& = ir*(m*/a* + n*/li ) ..................... (4). 



If a &amp;gt; b, the component oscillation of longest period is got by 

 making m = l, n = 0, whence ka = 7r. The motion is then every 

 where parallel to the longer side of the rectangle. Cf. Art. 175. 



187. In the case of a circular sheet of water, it is convenient 

 to take the origin at the centre, and to transform to polar 

 coordinates, writing 



x = r cos 6, y = r sin 6. 



The equation (1) of the preceding Art. becomes 



.................. . 



dr* r dr r* dd* 

 This might of course have been established independently. 



As regards its dependance on 6, the value of f may, by 

 Fourier s Theorem, be supposed expanded in a series of cosines and 

 sines of multiples of 6 ; we thus obtain a series of terms of the 

 form 



siri 



(2). 



It is found on substitution in (1) that each of these terms must 

 satisfy the equation independently, and that 



This is the differential equation of Bessel s Functions*. Its 



* Forsyth, Differential Equations, Art. 100. 



