292 TIDAL WAVES. [CHAP. VIII 



Between (2) and (3) we may eliminate either rj or f ; the 

 equation in 77 is 



d dr\ 



l 



The laws of propagation of waves in a rectangular canal of 

 gradually varying section were investigated by Green*. His 

 results, freed from the restriction to a special form of section, 

 may be obtained as follows. 



If we introduce a variable 6 defined by 



dxld6 = (gKp .................................... (i), 



in place of #, the equation (4) transforms into 



I/ 



where the accents denote differentiations with respect to 0. If b and h were 

 constants, the equation would be satisfied by rj F(B-t), as in Art. 167; in 

 the present case we assume, for trial, 



1 = B.F(0-t) .................................... (iii), 



where is a function of B only. Substituting in (ii), we find 



The terms of this which involve F will cancel provided 



or e = Cb~*k- ....................................... (v), 



C being a constant. Hence, provided the remaining terms in (iv) may be 

 neglected, the equation (i) will be satisfied by (iii) and (v). 



The above approximation is justified, provided we can neglect 0&quot;/0 and 

 /0 in comparison with F /F. As regards Q /Q, it appears from (v) and 

 (iii) that this is equivalent to neglecting b~ l . dbjdx and hr l . dhjdx in com 

 parison with r)~ l .drj/dx. If, now, A denote a wave-length, in the general 

 sense of Art. 169, drj/dx is of the order rj/X, so that the assumption in 

 question is that \dbjdx and \dhjdx are small compared with b and h, re 

 spectively. In other words, it is assumed that the transverse dimensions of 

 the canal vary only by small fractions of themselves within the limits of a 

 wave-length. It is easily seen, in like manner, that the neglect of 0&quot;/0 in 

 comparison with F /F implies a similar limitation to the rates of change of 

 db\dx and dhjdx. 



* &quot;On the Motion of Waves in a Variable Canal of small depth and width.&quot; 

 Camb. Trans., i. vi. (1837), Math. Papers, p. 225; see also Airy, &quot; Tides and Waves,&quot; 

 Art. 260. 



