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 dQdx = 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 



u vv A m7rx njry . 



5- wi.nCC ^ ^ ..('), 



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

 to oo . Substituting in (1) we find 



If a>b, the component oscillation of longest period is got by 

 making m = 1, n = 0, whence ka = TT. 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 2 dd 2 

 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 



COS] S ...(2). 



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

 satisfy the equation independently, and that 



(r) = ............ (3). 



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

 * Forsyth, Differential Equations, Art. 100. 



