Theory of Long Waves. 259 



and this gives by (1) and (14) 



{l-8(*-A)M*=/i?(«-W+/i ; (*+V0, • (19) 

 and by (2) 



u/V=-/,'(a-V0+/ 2 'O+V<). • • • (20) 



From (17) we see that the most general motion in a channel 

 of such a cross section as is given by (15) may be regarded as 

 compounded of two series of waves of horizontal displacement 

 moving in opposite directions with equal constant velocities. 

 The constant velocity Y given by (18), and here appearing 

 as an exact result, is of the same form as Kelland's well- 

 known approximation for the mean rate of propagation in a 

 channel of any section, which latter result may of course be 

 derived immediately from (5). From (16) it may be noted 

 that the constant c is the average depth over the cross 

 section. If we consider the solution obtained by taking only 

 one of the arbitrary functions in (17) and (18), we see that it 

 represents a propagation of waves in one direction only, with 

 a constant velocity and without change of form ; hence a 

 channel with such a cross section as we are considering may 

 be called a channel of uniform propagation. It is obvious, 

 moreover, that it is only that part of the section which lies 

 between the highest crest and lowest trough that need have 

 the form given by (15), provided the section below this level 

 is still such as to make the area equal to A. Thus, any 

 channel may be made a channel of uniform propagation by 

 suitably sloping the sides between the levels of crest and 

 hollow of such waves as traverse it, provided always of course 

 that the slope required be not such as to imply a greater 

 transverse motion than is contemplated in the formation of the 

 fundamental equations (1) to (5). If denote the required 

 inclination of each of the two sides to the vertical, corre- 

 sponding to the ordinate z, taking for simplicity the case of 

 equal slope on each side, then by (15) 



2 tan 6 = <j>"(z) = 3A/c 2 {l-2(z-h)}\ . . (21) 



and, if O be the inclination at the mean level A, 



2 tan 6 = SA/c 2 = 3b 2 / A {22) 



The equation to the form of the disturbed surface at any 

 time t will be obtained explicitly by eliminating x between 

 (17) and (19), but generally it will be more convenient to 

 leave x uneliminated, regarding it rather as a variable para- 

 meter. It is important to notice that, though waves advancing 

 in one direction only are propagated unchanged, in the general 

 case the form of the surface cannot be obtained by simply 



