147, 148.] WAVES IX CANALS. 177 



So far the only assumption made is that the vertical motion is 

 small compared with the horizontal. If we now assume in addi 



tion that -j- is also a small quantity, in other words that the rela 



tive displacement of the two neighbouring elements of fluid never 

 amounts to more than a small fraction of the distance between 

 them, (2) may be written 



-_ 



h~ dx 



and (3) becomes 



The elevation 77 then satisfies an equation of exactly the same 

 form. 



The above investigations apply to a straight canal of rectangular 

 section. We can however easily extend them to the case of a 

 canal whose section is any whatever, provided it be uniform. Thus, 

 if A be the area of the section, b the breadth of the canal at the 

 undisturbed level, we have instead of (2) 



whence rj = -^ ^, 



6 dx 



and ^f = ^^ (6) . 



148. If for shortness we write 



c 2 =gk* (7), 



x ct = ff l3 cT + ct = .r 2 , 



and transform (5) by making x iy x z the independent variables 

 instead of x } f, the equation becomes 



_**-- o 



j j v i 



* Or c 2 =^-,in(6). 



12 



