292 TIDAL WAVES. [CHAP. VIII 



Between (2) and (3) we may eliminate either 77 or f ; the 

 equation in t) is 



* ( 



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 = (gh$ .................................... (i), 



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



dt* 



where the accents denote differentiations with respect to 6. If b and h were 

 constants, the equation would be satisfied by r) = F(6 t), as in Art. 167; in 

 the present case we assume, for trial, 



where is a function of 6 only. Substituting in (ii), we find 



S-?+S+(WJ) (?+?)- 



The terms of this which involve F will cancel provided 



or Q = Cb~h~ ....................................... (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"/0' and 

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

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

 parison with if~ l .dq/dx. If, now, A denote a wave-length, in the general 

 sense of Art. 169, drjjdx is of the order Tj/A, so that the assumption in 

 question is that \dbjdx and \dhfdx 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"/0' in 

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

 dbjdx and dhjdx. 



* "On the Motion of Waves in a Variable Canal of small depth and width." 

 Camb. Trans., t. vi. (1837), Math. Papers, p. 225; see also Airy, " Tides and Waves," 

 Art. 260. 



