258 Mr. J. McCowan on the 



of steep waves, it seems more probable that the ultimate fate 

 of the waves will depend entirely on the initial circumstances. 



3. The Channel of Uniform Propagation. 



The solution of the problem of the propagation of long 

 waves in one direction only which we have obtained affords 

 little or no clue to the treatment required for the general case 

 of propagation in both directions. The complete solution of 

 this latter problem in finite terms is in general unattainable ; 

 bat, as we shall see immediately, there is an infinite variety of 

 forms of cross section for which such solutions may be obtained. 

 Of these forms of section, however, there is one which, from 

 its special interest and simplicity, deserves a separate dis- 

 cussion. 



From the form of equation (5) we see that it will reduce to 



d^QJf) _ d 2 <j>(h) 

 dt 2 cj>(z) " gC dx l $(?)' 



or by (1) to 



3-2 <"> 



if we take </>(,?) such that 



l* (*) J 4>{z) " c' 



which gives on integrating 



Wtf= l -w < 18 > 



Let y denote the breadth of the cross section corresponding 

 to the ordinate^, let b denote the breadth at the mean level A, 

 and let A be written for (f>{h), the area of the mean cross 

 section. Then, by (13), 



<M*)=A/U-2(*--A)/4*; • • • (U) 



.-. y = <j>'(z)=A/c\l-2(z-h)c}*; . . (15) 



.-. b = 4>'(h) = A/c, 



or c = A/b. . .' (16) 



The complete solution of (12) is 



f =7i'(*-V0 +/i(* + Vl), .... (17) 

 where _ 



V = V</ C = •GrA/ft) ; (18) 



